Theoretical Computer Science, Volume 281, Issues 1-2, Pages 1-630 (3 June 2002), Selected Papers in Honour of Maurice Nivat

Theoretical Computer Science 281 (2002) 1

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Preface With the publication of this Special Volume, the Theoretical Computer Science community wishes to celebrate the career and achievements of Maurice Nivat on the occasion of his retirement and, at the same time, wishes to express to him its deep gratitude for the contributions that he has given us which have done so much to foster the growth of the )eld. Throughout his long career, Maurice Nivat has been one of the most in*uential )gures in the history of Theoretical Computer Science. His prominence stems in equal measure from the many important results he has produced in a diverse variety of research )elds, from the generations of scholars he has educated, and from the role that he has played in helping our discipline reach scienti)c maturity and respect among older disciplines. A great deal of this has been achieved through the energy and intellectual vision that Maurice Nivat has provided as founder and editor-in-chief of Theoretical Computer Science, the journal that he directed over the last twenty-)ve years and that, since its inception in 1975, has progressively risen to its current stature as one of the leading international computer science journals. This volume consists of three parts. The )rst part contains two contributions that retrace some of Maurice’s remarkable scienti)c achievements and his role in shaping Theoretical Computer Science as a new mathematical discipline. The second part, and by far the largest, is dedicated to new scienti)c contributions, ranging over many aspects of theoretical computer science. Indeed, the guest editor, Pierre-Louis Curien, intended that the scienti)c contributions to this volume should mirror the diversity of )elds to which Maurice Nivat has contributed, from formal languages to discrete geometry all the way through to semantics and concurrency. Some contributors are former students of Maurice, while others are fellow scholars who have shared great experiences collaborating with Maurice, some are colleagues of his generation, while others are in their thirties; this also is part of the diversity, of the breadth, and of the liveliness of the whole area covered by the journal Theoretical Computer Science. Finally, the third part is a “clin d’oeil” to the many other passions of Maurice, that will lead the reader from the mysteries of Indian esoterism to the more terrestrian hidden beauties of the Postscript language. We thank all the authors for their scienti)c contributions to this Special Volume; the Guest Editor, Pierre-Louis Curien, for coordinating this impressive volume; and our publisher, Elsevier, in the person of Arjen Sevenster, for having provided all the needed support for such an exceptional e
c 2002 Published by Elsevier Science B.V. 0304-3975/02/$ - see front matter
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Une breve biographie scienti"que de Maurice Nivat Pierre-Louis Curien PPS-Case 7014, Univ. Paris 7, 2 pl. Jussieu, F-75251 Paris Cedex 05, France

1. An even shorter biography Maurice Nivat was born on December 21, 1937. His parents were both high-school teachers—his father in French, Latin, and Greek, his mother in Mathematics. Maurice grew up in the city of Clermont-Ferrand. His elder brother, who has recently retired from the University of Geneva, became a scholar in Slavic studies. His younger sister, Aline Bonami, is a mathematician whose speciality is harmonic analysis. Maurice is married to Paule Nivat, who has just retired from a position of professor of statistics at University Paris 13. He has three children, Dominique, Jean-Luc, and Catherine, and 10 grand-children (so far). Since 1983, he has been a Corresponding Member of the Acad9emie des Sciences. In 1997 he was made doctor honoris causa of the University of Bologna. Maurice and his family moved to Paris in 1954, where Maurice prepared for the Ecole Normale Sup9erieure’s entrance examination, where he was admitted in 1956. In 1959, he began to work at the Institut Blaise Pascal, which was the newly set up computing centre of the CNRS (National Centre of Scienti"c Research). There he met Louis Nolin, and started to learn about programming languages (Lisp, Algol) and compilers. Maurice would then meet Marcel-Paul SchButzenberger and begin research work under his supervision. From 1965 to 1969, Maurice held positions in the Universities of Grenoble and then of Rennes. Maurice gained his “These d’Etat” in 1967 [28]. In 1969, he was appointed as professor at the University of Paris, a position that he has held until now. During the early 1970s, in parallel with his scienti"c work, Maurice began to devote much time and energy to building research teams, both at IRIA and at the university. IRIA (Institut de Recherche en Informatique et Automatique) had been created by the French government in 1967. In 1970, Nivat and SchButzenberger gave an inHuential talk at IRIA where they proposed the name “Informatique Th9eorique” to designate all the research on automata and formal languages, on programming languages, and on E-mail address: [email protected] (P.-L. Curien). 0304-3975/02/$ - see front matter PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 0 4 - X

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algorithms. Shortly after, Maurice created a research project at IRIA, which would be joined by Philippe Flajolet, Jean-Marc Steyaert, Bruno Courcelle, G9erard Huet, JeanJacques L9evy, and G9erard Berry. During this period, research at IRIA was very lively, and bene"ted from a fruitful interaction with young researchers like Gilles Kahn, Jean Vuillemin, and G9erard Huet who had learned advanced programming practice in the USA. Maurice’s project lasted for about 10 years. In the 1980s, new closely cooperating INRIA (the “N”—for National—was added in 1980) projects, such as Formel (Huet), VLSI (Vuillemin), and Para (L9evy), have arisen out of this research community. At the University of Paris (and then at the Universities Paris 7 and Paris 6), with SchButzenberger and Nolin, Maurice built a research team, and a few years later he was at the centre of the creation of LITP (Laboratoire d’Informatique Th9eorique et Programmation) in 1975, of which he would be co-director until 1985, "rst with Jacques Arsac, then with Bernard Robinet. Maurice also launched the “Ecoles de Printemps d’Informatique Th9eorique”, which have been held each year since 1973. These convivial spring schools played an important role in the creation of a national community in theoretical computer science. During this decade, Maurice was no less active internationally. He created essential vehicles for the European computer science research community. He founded the EATCS (European Association for Theoretical Computer Science), with the help of Jaco de Bakker, Corrado BBohm, and Mike Paterson. He organized the "rst ICALP (International Colloquium on Automata, Languages and Programming) at IRIA in 1972 (see the text of Arto Salomaa and Grzegorz Rozenberg in this volume). Finally, he created the journal TCS which started to appear in 1976 (we refer to the editorials of Giorgio Ausiello, Don Sannella, and Mike Misolve, and of Maurice Nivat in the TCS Vol. 250, for some recollections of the beginnings of the journal and of its expansion over the years). The inHuence of Maurice on international scienti"c publishing extends much beyond this journal (and its more practice-oriented cousin Science of Computer Programming, and the more recent Electronic Notes in Theoretical Computer Science). For instance, together with Engeler, Huet, Jouannaud, Milner, and Wirsing, Maurice has been a member of the editorial board of the series Progress in Theoretical Computer Science (Pitman, and then BirkhaBuser), directed by the late Ron Book, a great friend of Maurice. During the early 1980s, Maurice was intensively involved in the promotion and the organization of research in computer science at the national level. He launched the PRC programmes (Programmes de Recherche Concert9es) that gave a decisive impulse to the research community in this country.. 1.1. Scienti,c works (I) Formal language theory: In his thesis, Maurice gave an algebraic characterization of rational transducers, on which many later works of the “French school” of formal languages are based. The notions of rational cone and of hierarchy are already present [29]. Then, Maurice initiated a study of languages de"ned by congruences [17], which later led to the works of G9eraud S9enizergues (see his contribution to the volume). Maurice and his students Luc Boasson, JoProy Beauquier, and Jean-Michel Autebert

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worked extensively on understanding the structure (and the generators [7]) of contextfree languages. A remarkable complexity measure, which has been much used since, was proposed in [12]. A synthetic account of many of these results can be found in Jean Berstel’s book [11]. In the late 1970s, in connection with his work on semantics, Maurice began to study in"nite words [31]. He introduced the notion of the centre of a language and uncovered some of the underlying topological properties. During these years, the research carried out in France in this area became quite well-known internationally, and this is certainly largely due to ideas whose origin can be found in Maurice’s works. (II) Semantics: Maurice’s interest in semantics goes back to a meeting of the IFIP WG 2.2 in Colchester in 1969, on the semantics of programming languages—a subject that was just starting at the time. Maurice learned from David Park and Mike Paterson about program schemes. This led to a large body of work for Maurice and a number of students (Laurent Kott, G9erard Boudol, Irene Guessarian, Bruno Courcelle, Guy Cousineau, Jean-Marie RiQet). First, Maurice extended a well-known theorem of SchButzenberger by characterizing the set of terms generated by a context-free tree grammar as the smallest solution of this grammar considered as a system of equations on sets of terms. Then he showed that when the grammar represents a (recursive) program scheme, this algebraic solution in turn characterizes its semantics: the interpretation of the solution coincides with the operational semantics of the given program scheme [30]. The subject of “algebraic semantics” [18] developed progressively (see the book of Irene Guessarian [24]), based on the construction of an in"nite algebraic object—similar to the BBohm tree of a
-term—that factorizes the semantics of a program scheme. This algebraic object characterizes the semantic equivalence class of the associated program scheme, and in some cases (such as for the “rational” program schemes), it allows us to design algorithms for deciding this equivalence. The investigations of in"nite trees—for themselves—culminated in Bruno Courcelle’s thesis (see [19]). Subsequently Maurice set out to generalize his results to the non-deterministic case. Serious diRculties arose here, as denotational semantics did not oPer a satisfying notion of power-domain. Nivat and Arnold introduced the use of metric spaces [1]: in this framework, they de"ned a "xpoint semantics of non-deterministic program schemes. This metric space approach to semantics (a synthetic exposition of which can be found in [33]) has since been followed by various authors, and in particular by the group of Jaco de Bakker in Holland. However it is fair to say that this was not a complete success, as the coincidence with operational semantics (as proposed by G9erard Boudol [13] following works by G9erard Berry and Jean-Jacques L9evy [16]) does not hold any more, except in some special cases. This “chapter” of Maurice’s scienti"c activity ends with the publication of [34], based on a joint French-US meeting held in Fontainebleau in 1982. Next, Maurice addressed the semantics of parallelism. The initial insight was in his study of in"nite words viewed as in"nite behaviours of processes, which was to generate numerous works under the supervision of Maurice: those of FranScoise Gire, Daniele Beauquier, Hugues Fauconnier, Paul Gastin, and Andreas Podelski. For more details about this period of his scienti"c activity, we refer to the contribution of Andr9e

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Arnold in this volume. Certainly, Maurice was a pioneer of model-checking when he worked on the synchronization of automata [32]. (III) Tilings and discrete tomography: In the mid-1980s, after his national responsibilities were over, Maurice changed his research direction once more. He turned his attention to the study of discrete geometrical objects, while keeping an active interest in concurrency and tree languages. Some young researchers advised by Maurice in this area are Alain Daurat, Yann G9erard, and Laurent Vuillon. One of his former students, Ahmed Saoudi, who unfortunately died since, has pushed much Maurice in this direction. With Daniele Beauquier, Maurice "rst worked on the decidability of the problem of whether a given "nite set of polyominoes forms a plane code. The problem turned out to be diRcult, but they "nally obtained a negative answer in 1993–1994 (yet to be published). In [8], the two authors tackled the problem of tiling the plane by translated instances of a single tile. They showed that such tilings are possible if and only if this tile is a pseudo-hexagon, and moreover that the tiling is then half-periodic (i.e., invariant under some translations). Beauquier and Nivat have thus found an interesting solvable family of tiling problems. In [9], Maurice and his co-authors showed that for general regions of the square lattice the tiling problem is NP-complete for horizontal dominoes and vertical trominoes, and for any other pair of bars where either has length ¿2. Recently, by using a similar approach, Moore and Robson [27] proved some additional results of this kind. Maurice is also among the pioneers of discrete tomography, which deals with the reconstruction of discrete geometrical objects from multiple X-ray projections [25]. Maurice tackled the problem of reconstructing binary matrices that are only accessible through a "nite number of projections counted with multiplicities along some lines. For the much studied case of two projections (horizontal and vertical), existence and uniqueness results are known [35]. But as the number of solutions can be rather large in general, the researchers interested in this problem have looked at particular cases. In [5,6] (see also [20] for a synthetic presentation), Maurice, in joint work with Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, provided some polynomial-time algorithms to reconstruct binary matrices having some convexity and=or connectivity properties. These works have been extended by several authors [3,15,26]. The problem of reconstructing binary matrices from three or more X-rays is NP-complete [21]. Maurice and his co-authors have found an interesting class of polyominoes (having convexity and connectivity properties, again) for which the problem is solvable in polynomial time [4]. An extension of this result to the reconstruction of convex lattice sets has been given in [14]. Discrete Tomography is a relative young and actively studied "eld, and Maurice is one of its most active promoters. During recent years, Maurice and Peter Gritzmann have organized two international meetings devoted to this "eld. Thus, the spectrum of Maurice’s scienti"c works has covered (so far!) all the three areas that he had cited as the cornerstones of Theoretical Computer Science: the theory of formal languages and automata, programming languages, and algorithms. More widely, the contributions of Maurice to the whole computer science community, nationally and internationally, are immense.

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2. Quelques dates En d9ecembre 1997 se tenait a l’Ecole Normale Sup9erieure un Jubil9e en l’honneur de Maurice Nivat. Cette r9eunion amicale, dont l’initiative revient a Jean-Eric Pin, fut organis9ee par Andr9e Arnold et Jean-Marc Steyaert, avec l’aide de Paule Nivat. Quatre ans plus tard, Maurice a quitt9e ses fonctions de r9edacteur en chef de la revue Theoretical Computer Science, et s’apprˆete a prendre sa retraite de professeur a la "n de l’ann9ee universitaire en cours. Ce volume lui est d9edi9e, et les lignes qui suivent tentent de retracer quelques faits marquants de sa carriere. Maurice Nivat est n9e le 21 d9ecembre 1937. Ses parents sont tous deux professeurs de lyc9ee: le pere, normalien, enseigne le franScais-latin-grec, la mere les math9ematiques. Maurice grandit a Clermont-Ferrand. La r9eussite scolaire est un objet d’attention tout particulier dans la famille, qui sera combl9ee de ce point de vue: les trois enfants seront normaliens. Le frere aˆYn9e Georges est slavisant. Il vient de prendre sa retraite de professeur d’universit9e a Geneve. Sa sZur cadette, Aline Bonami, est math9ematicienne, sp9ecialiste d’analyse harmonique. Au grenier, le jeune Maurice d9evore les livres qui avaient appartenu a son grand-pere architecte: ce goˆut pour l’architecture, et singulierement les styles architecturaux de chaque in"me r9egion de France, ne le quittera jamais. Maurice est mari9e a Paule Nivat, professeur de statistiques a l’universit9e Paris 13, qui vient de prendre sa retraite. Il a trois enfants: Dominique (1960, e9 conomiste), Jean-Luc (1963, informaticien), et Catherine (1967, m9etallurgiste). Il a aujourd’hui dix petits-enfants. Il est chevalier dans l’ordre de la L9egion d’Honneur et oRcier dans celui du M9erite. Il est membre correspondant de l’Acad9emie des Sciences depuis 1983, et docteur honoris causa de l’Universit9e de Bologne depuis 1997. 3. Annees d’apprentissage Maurice obtient brillamment son baccalaur9eat en 1954, avec plus de 18 de moyenne: un exploit r9egional! Son professeur de math9ematiques en Math Elem, Monsieur Dumas, l’a initi9e a la g9eom9etrie des coniques. La famille se d9eplace alors vers Paris; Maurice rentre en Classes Pr9eparatoires au lyc9ee Jeanson de Sailly. Il est admis au concours d’entr9ee a l’Ecole Normale Sup9erieure en 1956. Il se laisse d’abord s9eduire par la physique: il fr9equente un an le laboratoire de Pierre Aigrain, mais n’arrivant pas a faire e9 mettre d’ondes a un klystron capricieux, il pr9efere en rester la. En ce temps la, il n’y avait pas de maˆYtrise. Il fallait passer pour la licence trois certi"cats, auxquels s’ajoutaient des cours d’option. En premiere ann9ee, Maurice passe Calcul DiP9erentiel et Physique G9en9erale, en fournissant un travail tres mod9er9e. Car il rattrapait le temps: il allait voir jusqu’a trois "lms par jour! En option, en deuxieme ann9ee, il passe (et rate) Topologie G9en9erale de Gustave Choquet. Comme il e9 tait d’usage, les normaliens n’allaient pas aux cours, d9el9eguant l’un des leurs pour tenir les autres au courant. En lieu de quoi ils travaillaient ensemble assez s9erieusement la Topologie G9en9erale de Bourbaki, tres a l’oppos9e du style de Choquet, lequel pr9ef9erait l’intuition g9eom9etrique, les contre-exemples... Maurice remet copie blanche a l’examen. Ses condisciples ne font

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pas d’9etincelles non plus. Cet e9 chec collectif sert de leScon, et a certainement contribu9e a la diversi"cation des vocations! Pierre L9ena se tournera vers l’astrophysique, Jean Bretagnolle vers les probabilit9es. Il restait a passer M9ecanique G9en9erale. Ici l’9echec est plus p9enalisant. Maurice est suspendu un an sans traitement. Pour gagner un peu d’argent, il donnera des colles a Jeanson. Re-9echec a M9ecanique G9en9erale. Il travaille peu... La direction de l’Ecole Normale s’en e9 meut. Il est reScu et sermonn9e par le sous-directeur pour les sciences, Albert Kirrmann. Henri Cartan, directeur des math9ematiques d’alors pour les normaliens, lui fait trois propositions: 1. Faire de la th9eorie des carapaces avec lui (th9eorie qu’il d9eveloppait pour les normaliens, et qui n’a pas surv9ecu). 2. Faire de la m9ecanique c9eleste (nonobstant ses d9emˆel9es avec la “terrestre”). 3. Travailler au nouveau Centre de Calcul du CNRS, l’Institut Blaise Pascal, dirig9e par Ren9e de Possel. Maurice choisit la solution 3. De Possel est un personnage extraverti, jovial mais distant en mˆeme temps. Maurice est calculateur adjoint (il est toujours suspendu de l’Ecole). L’Institut Blaise Pascal est dans les caves de l’IHP. Plusieurs dames y travaillent, ex9ecutant des calculs sur les machines a manivelles. Au Centre de Calcul, Maurice rencontre Louis Nolin, avec lequel il nouera une amiti9e ind9efectible jusqu’a la mort de ce dernier, survenue en janvier 1997. A parte: D’origine modeste, Louis Nolin a commenc9e sa formation par des e9 tudes de philosophie avant de rencontrer l’informatique. Il fallait dans les ann9ee 60 (et a vrai dire jusqu’a l’avenement des PRC, voir plus bas) beaucoup de courage pour entreprendre de concevoir et mettre en Zuvre des langages de programmation a l’universit9e: ni moyens, ni reconnaissance de la discipline. Nolin a donc d’abord conScu des langages de programmation sur le papier. Son langage ATF (“a tout faire”!) a inspir9e Excel, un langage d9evelopp9e plus tard au laboratoire LCR de la Thomson (voir plus bas). Louis Nolin avait une passion pour les machines-outils et pour les vins en g9en9eral et ceux de sa r9egion en particulier (Bergerac). Sa "gure discrete mais chaleureuse reste pr9esente a l’esprit de tous ceux qui l’ont cˆotoy9e au LITP. Maurice rencontre aussi Andr9e Lentin. Il fait ses premiers expos9es sur Lisp, Algol, au s9eminaire anim9e par FranScois G9enuys (IBM) qui se tient a l’Institut d’Astrophysique. Maurice sera d’ailleurs plus tard coauteur d’un livre sur Algol [2]. Le s9eminaire est fr9equent9e par des physiciens (dont Jacques Arsac, qui a commenc9e par l’astrophysique, et Bernard Vauquois qui deviendra sp9ecialiste du traitement automatique des langues a Grenoble), et aussi des logiciens: Lacombe, Porte. Maurice fr9equente e9 galement le s9eminaire de Lacombe, qui s’int9eresse a la d9emonstration automatique. Il est devenu informaticien (mˆeme si le mot n’existe pas encore—il sera cr9ee9 au milieu des ann9ees 60). Il passe en"n M9ecanique G9en9erale, malgr9e un 2,5 a l’9ecrit, rattrap9e a l’oral. Ceci permet a Maurice d’ˆetre r9eint9egr9e a l’ENS, ou il pr9epare l’agr9egation de math9ematiques, qu’il obtient en 1961. La pr9eparation est assur9ee par Henri Cartan, dont les conseils et critiques sont tres formateurs. Il fr9equente alors les e9 leves d’une autre promotion, mais les connaˆYt peu: il est d9eja mari9e. De ces ann9ees normaliennes, Maurice gardera un attachement constant pour Choquet et Cartan.

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4. A la rencontre de Schutzenberger A la rentr9ee universitaire de 1961, De Possel propose a Maurice un poste d’assistant. En le comptant, il y a alors en tout et pour tout sept assistants a la Facult9e des Sciences de Paris! Maurice aurait pu aussi rentrer au CNRS, mais il n’a pas encore d’id9ee de ce que peut eˆ tre la recherche, tandis qu’il peut ais9ement imaginer a quoi peut ressembler le m9etier d’enseignant. Il assure les Travaux Dirig9es de De Possel (polynomes de Tchebytchev), ainsi que des Travaux Pratiques (calcul sur les machines du Centre de Calcul de racines r9eelles d’9equation du troisieme degr9e). Il est un assistant heureux: peu de travail, une paix royale, un peu d’administration. A parte: De Possel poursuivait une carriere universitaire a Alger avant d’ˆetre rapatri9e et de prendre la responsabilit9e du Centre de Calcul. Il tient tous les lundis une sorte de salon, ou l’on parle de tout: il mange d’9enormes sandwichs, propose des problemes ou des d9e"s que les participants a ces r9eunions sont invit9es a r9esoudre sur le champ. Maurice fait son service militaire d’octobre 1962 a mars 1964. Un mois a Brest, deux mois a Toulon, et le reste dans un laboratoire du Service Technique des Constructions et Armes Navales, a Paris. Souvenir d’une machine Bull qui ne fonctionne pas... Maurice fait aussi l’exp9erience de la manie du secret. Il ne connaˆYtra jamais l’id9ee qu’un ing9enieur de ce centre venu le voir rue du Maroc (ou s’9etait transport9e l’Institut Blaise Pascal) renoncera a lui communiquer par peur d’oreilles qui traˆYneraient. Par le jeu des relations normaliennes, Maurice rentre aussi comme Conseiller a la CSF (qui deviendra Thomson-CSF), y fr9equente Jean-Pierre Vasseur, puis plus tard Gilles Ruggiu. Plusieurs jeunes chercheurs aiguill9es par Maurice y feront tout ou partie de leur these de troisieme cycle ou d’Etat: Patrice Enjalbert, Guy Cousineau, Fr9ed9eric Boussinot. Maurice souhaite rencontrer Marcel-Paul SchButzenberger. C’est Lentin, dont la these, fort e9 l9egamment calligraphi9ee, portera sur les e9 quations dans les monoBYdes, qui le connaˆYt le mieux. Maurice attendra un an avant de le voir. Il est alors en poste a Poitiers, mais passe beaucoup de temps aux USA. Nolin et Nivat lui pr9esentent un compilateur pour ALGOL bas9e sur une notion d’automate a taquets. SchButzenberger e9 coute, puis parle, et ne cessera plus de fasciner Maurice... Il lui propose de travailler sur la th9eorie des codes. Il a une conjecture sur la g9en9eration des codes "nis par surcodages, conjecture que Yves C9esari prouvera fausse plus tard. Maurice va commencer la recherche. Pour aborder ces rivages ardus, en dehors des sentiers battus, il faut beaucoup d’9energie, de rage mˆeme, et Maurice n’en manque pas, endurci par de dures e9 preuves familiales (suicide de sa premiere femme en 1965). A parte: SchButzenberger est un “self-made man”. Il a vingt ans au d9ebut de la seconde guerre mondiale. Il est communiste, tendance Tillon, se fait pour la bonne cause faussaire. Membre du cabinet de Tillon jusqu’a la mise a l’9ecart de ce dernier par le Parti Communiste, il b9en9e"cie de la loi autorisant des e9 tudes acc9el9er9ees pour les r9esistants. Il fait sa m9edecine, et part travailler pour l’OMS (Asie du Sud-Est, Pakistan...). La-bas il fait aussi des math9ematiques. Il invente les groupes qui portent son nom. Il fait de la th9eorie des treillis, trouve ind9ependamment le th9eoreme des vari9et9es de BirkhoP. Ses travaux sont publi9es aux CRAS. Puis SchButzenberger rentre en France, passe une these en th9eorie des probabilit9es, et est nomm9e professeur a Poitiers. En

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1957–1958, il part aux USA: il voulait faire de l’9epid9emiologie au Harvard Medical School. La, il rencontre Zellig Harris (linguiste formel), et son e9 leve Noam Chomsky. Mais il s’int9eresse aussi a la th9eorie des codes (Shannon, HuPman). Bref, il est curieux de tout! Il y a d9eja quelques ann9ees que la th9eorie des automates est un sujet tres actif, depuis le th9eoreme de Kleene de 1953 (reliant automates et expressions r9egulieres) [26]. Apres Rabin, Scott et d’autres, SchButzenberger rattrape le temps perdu, publie beaucoup avec Chomsky (transducteurs). Il est nomm9e a Paris au milieu des ann9ees 60. Il prend des e9 leves: Dominique Foata, Maurice Nivat, Jean Berstel, Jean-FranScois Perrot, Michel Fliess, Robert Cori, G9erard Viennot, Dominique Perrin, Christian ChoPrut. Et aussi Maurice Gross, qui s’orientera vers la linguistique. En mai 68, SchButzenberger retrouve des e9 lans de communiste tendance libertaire. Mais il est en mˆeme temps farouchement e9 litiste. De cette contradiction r9esultera une r9esolution: il ne prendra plus d’9eleves. Mais l’9ecole de pens9ee se poursuit: Perrot sera le directeur de these de Jacques Sakarovitch et de Jean-Eric Pin; Berstel de Christophe Reutenauer, ... SchButzenberger reste naturellement scienti"quement tres actif et continue d’exercer fortement son inHuence scienti"que.. A la “g9en9eration” suivante, Daniel Krob par exemple a ainsi eu la chance de communiquer r9egulierement avec lui. Marcel-Paul SchButzenberger est mort en 1996. En 1964, Maurice va a une Ecole de Printemps a Ravello (sur la th9eorie des codes). Il partage son logement avec Corrado BBohm. Site superbe. A Paris, il apprend de la th9eorie des langages, et tel un grand frere donne des petits cours a ses cadets Jean Berstel, Jean-FranScois Perrot, Michel Fliess. Il y avait alors un chapitre important: celui des s9eries formelles en variables non commutatives et de leurs applications a la combinatoire. SchButzenberger tenait un s9eminaire hebdomadaire dans les caves de l’IHP, diRcile a suivre, mˆeme pour les quelques initi9es. De 1965 a 1967, Maurice est Charg9e de Conf9erences a Grenoble; puis, de 1967 a 1969, a Rennes. Il a deux e9 leves a Grenoble: Philippe Butzbach (qui vient malheureusement de mourir) et Michele Benois, et un a Rennes, Yves Cochet (aujourd’hui ministre de l’Environnement, devenu e9 cologiste dans son mode de vie avant mˆeme la constitution de mouvements e9 cologistes). Avec ses e9 leves grenoblois, Maurice s’int9eresse a la r9ee9 criture de mots. Les problemes sont diRciles, et fort peu e9 tudi9es alors. Certaines questions n’ont trouv9e de r9eponses que beaucoup plus tard. Par exemple, c’est seulement au milieu des ann9ee 80 que Squier [36] montrera en utilisant l’homologie que certaines th9eories e9 quationnelles n’admettent aucune pr9esentation orient9ee qui soit conHuente et nZth9erienne. Peu de gens s’int9eressaient a la r9ee9 criture: l’engouement autour de Knuth-Bendix, c’est une dizaine d’ann9ees plus tard. Maurice identi"ait ainsi d9eja la notion de systeme conHuent (quasi-parfait dans sa terminologie). En 1967, Maurice soutient sa these d’Etat [28]. SchButzenberger e9 tait un peu inquiet. Le jury ne manquait pas de math9ematiciens prestigieux parmi lesquels Lelong, conseiller de De Gaulle, et pr9esident d’un Comit9e de Sages qui avait recommand9e la cr9eation de l’IRIA, et Ehresmann, le pere franScais de la th9eorie des cat9egories. Qu’allaient-ils penser de ces travaux aux marges de la math9ematique? La soutenance est un peu perturb9ee par une pol9emique qui surgit a propos de la deuxieme these. Maurice devait commenter une note e9 nonScant un r9esultat r9ecent en th9eorie de la mesure. Cartan, qui e9 tait dans la salle, dit que le r9esultat est faux... Faute de temps,

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il n’y aura pas de questions. Mais le jury fait des compliments, SchButzenberger est rassur9e. En 1969, Maurice se pr9epare a devenir professeur a Strasbourg, quand un poste suppl9ementaire en informatique se d9egage a la Facult9e des Sciences de Paris. Il y suit de pres Louis Nolin. Ce dernier avait e9 t9e e9 vinc9e brutalement en 1967 de l’Institut Blaise Pascal: il e9 tait le bouc e9 missaire d’un rapport d’9evaluation s9evere. Louis Nolin se r9efugie a l’IRIA pour y pr9eparer sa these, qu’il r9edigera a Montr9eal, et qu’il passera discretement a l’IHP, en pleine d9ebˆacle du mouvement e9 tudiant de mai 68. Il est recrut9e professeur la mˆeme ann9ee. Maurice d9ecouvre la s9emantique des langages de programmation a une r9eunion du WG 2.2 de l’IFIP a Colchester, en septembre 1969: Dana Scott, Robin Milner, Christopher Strachey, Peter Landin, David Park y participent. C’est avec ce dernier, et avec Mike Paterson, tous deux a Warwick, que Maurice aura le plus d’atomes crochus: Paterson lui parle des sch9emas de programmes, sur lesquels Maurice travaillera dans les ann9ees 70 avec ses e9 leves (Laurent Kott, G9erard Boudol, Irene Guessarian, Bruno Courcelle, Guy Cousineau, Jean-Marie RiQet). Landin est fort brillant et impressionne Scott.

5. L’“ecole humaine” de Maurice C’est au d9ebut des ann9ees 70 que, parallelement a ses travaux, Maurice commence a consacrer en pionnier beaucoup de temps et d’9energie a construire des e9 quipes de recherche, a l’IRIA et a Paris 7, institutions toutes deux fraˆYchement cr9ee9 es. A parte: Quand les Chinois "rent exploser leur bombe H avant les FranScais, les militaires ont invoqu9e le manque d’ordinateurs. De Gaulle commande alors un CDC6600, et les Am9ericains font l’embargo. D’ou le Plan Calcul, ensemble de dispositions visant a doter la France d’une industrie informatique autonome. L’IRIA, cr9ee9 "n 1967 et install9e a Rocquencourt dans des casernements lib9er9es par l’OTAN, devait eˆ tre l’un des fers de lance de ce plan. Les premieres ann9ees de l’IRIA (1968–1971) sont un peu balbutiantes. Le pilotage scienti"que est assur9e par un directoire, qui comprend notamment, autour de l’Ing9enieur G9en9eral Boucher: Michel Laudet, Jacque-Louis Lions, Pierre Faurre, Marcel-Paul SchButzenberger, Andr9e Lichnerowiz. Lions prend les rˆenes en 1972. (disparition du Directoire). 5.1. IRIA En 1970, SchButzenberger introduit Maurice a l’IRIA. Tous deux donnent une conf9erence a l’IRIA, proposant la d9enomination d’Informatique Th9eorique pour d9esigner les recherches sur les automates et langages formels, sur les langages de programmation et sur les algorithmes. Des le d9ebut, l’IRIA est organis9e en projets. Celui de SchButzenberger, “Automates et Langages Formels” ne dure que quelques ann9ees. Maurice cr9ee en 1971 un projet “S9emantique Formalis9ee des Langages de Programmation”. Plusieurs facteurs se conjuguent ici. Maurice vient d’attraper le virus de la

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s9emantique a Colchester. Les travaux en th9eorie des langages commenScaient a buter sur des problemes diRciles pour lesquels il semblait raisonnable de laisser passer du temps et d’attendre que d’autres id9ees viennent d’ailleurs. Le contexte de l’IRIA l’engageait a proposer des themes dont les perspectives d’application seraient plus faciles a d9efendre. Il commenScait ainsi a militer pour sensibiliser les th9eoriciens aux applications, et r9eciproquement, combat qu’il devra mener maintes et maintes fois, et ce au sein mˆeme de son entourage scienti"que imm9ediat. Le projet de Maurice accueillera Philippe Flajolet et Jean-Marc Steyaert, Bruno Courcelle, G9erard Huet (qui avait fait un PhD aux Etats-Unis), G9erard Berry, et aussi pour une courte p9eriode Jean-Jacques L9evy (qui avait commenc9e par les “Operating Systems”, dans le projet de Krakowiak, Kayser et B9etourn9e). La th9ematique reScoit vite le renfort d’autres jeunes chercheurs revenus des USA: Jean-Marie Cadiou, Gilles Kahn, Jean Vuillemin. Ces derniers fondent en 1973 le projet “Analyse des Algorithmes, Validit9e des Programmes”. Cette confrontation entre une e9 cole franScaise tres th9eoricienne et aux allures parfois un peu sectaires, sans doute imprim9ees par la personnalit9e forte mais altiere de SchButzenberger, et des chercheurs ramenant des Etats-Unis une avance certaine dans le mariage entre une bonne th9eorie et une v9eritable activit9e de programmation ne pouvait se faire sans certains heurts. Fort heureusement, des passerelles se sont rapidement e9 tablies, et de fructueuses collaborations se sont nou9ees: Kahn et Huet, Vuillemin et Flajolet... De nouveaux projets naissent: “Conception et R9ealisation d’Outils d’Aide a la Programmation” (CROAP) (Kahn et Huet) et “Analyse et Optimisation des Algorithmes” (Vuillemin, rejoint par Flajolet). Les Ecoles de Printemps (cf. infra) ont notamment servi a croiser abondamment le “groupe de Paris 7” et le “groupe du bˆatiment 8” (lequel abritait— et abrite encore avec le bˆatiment 11—ces projets—et ceux qui en sont issus). Le projet de Maurice, rebaptis9e “Th9eorie de la Programmation”, durera jusqu’a 1979 (mais il restera conseiller de Lions, puis de son successeur, Alain Bensoussan, jusqu’en 1996). En 1980, l’IRIA prend un “N” (pour National), et, par essaimage, de nouveaux projets collaborant e9 troitement verront le jour, notamment: Formel (Huet), VLSI (Vuillemin), Para (L9evy). 5.2. La cr3eation du LITP Quand Maurice est nomm9e a Paris, c’est encore la Facult9e des Sciences. Il est int9egr9e dans l’Institut de Programmation, structure universitaire cr9ee9 e quelques ann9ees plus tˆot pour compl9ementer le dispositif du Centre de Calcul. Initialement, l’Institut de Programmation devait servir a former les cadres techniques dont le CNRS aurait besoin pour entretenir les parcs de machines naissants. Puis Arsac nomm9e professeur est appel9e a la direction de l’Institut de Programmation au milieu des ann9ees 60. En collaboration avec les deux autres centres informatiques universitaires qui se d9eveloppaient alors a Grenoble (Kuntzman) et Toulouse (Laudet), Arsac obtient la cr9eation d’un cursus national de licence et maˆYtrise en informatique. Les maˆYtrises venaient d’ailleurs d’ˆetre instaur9ees (r9eforme Fouchet). Mais la fondation de Paris 6 et de Paris 7 se pr9epare. Le

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cursus informatique complet se poursuivra a Paris 6, tandis qu’a Paris 7 l’enseignement de l’informatique commencera sous forme de cours a option pour les e9 tudiants de math9ematiques. Il faudra attendre le milieu des ann9ees 80 pour que Paris 7 se dote a son tour d’une licence et maˆYtrise en informatique. Le groupe form9e par SchButzenbergerNivat-Nolin ira a Paris 7. Une structure CNRS (ERA) est cr9ee9 e autour de ce petit noyau. Le premier budget annuel reScu du CNRS e9 tait de 2000F, ce qui mˆeme pour l’9epoque e9 tait fort peu: de quoi tenir des r9eunions a Fontainebleau... Ce laboratoire ne s’appelait pas encore LITP, mais on pourrait parler d’un “premier LITP”, car pour beaucoup, l’ˆage d’or de ce laboratoire s’identi"e bien avec la d9ecennie 1970. Tous les membres de l’9equipe se r9eunissaient une journ9ee par mois, salle Dussane a l’ENS, et mangeaient ensemble dans un restaurant du quartier. JoProy Beauquier se souvient de la premiere de ces r9eunions: “il y avait une atmosphere extraordinaire et un enthousiasme perceptible”. Dans le mˆeme esprit, Maurice lance les Ecoles de Printemps d’Informatique Th3eorique, qui ont jou9e un rˆole primordial dans la constitution d’un v9eritable r9eseau: ces r9eunions de l’ensemble des informaticiens th9eoriciens une semaine par an dans un lieu agr9eable ont largement contribu9e a souder la communaut9e. Telle e9 tudiante y a trouv9e sa vocation de chercheur, tel autre y a fait ses premieres armes. La convivialit9e, la tres faible moyenne d’ˆage des participants et les multiples libations et plantureux repas ont contribu9e a la cr9eation de la famille! Dans ces occasions, la presse r9egionale est de la partie e9 galement. L’organisation mat9erielle de ces e9 coles a longtemps e9 t9e assur9ee par un professeur de l’ENSTA, Jean-Pierre Crestin. Au moment de renouveler le jeune laboratoire, quatre ans plus tard, Maurice Zuvre a l’9elargir, a"n de r9eunir des th9eoriciens et des programmeurs, r9epartis sur les deux universit9es Paris 6 et 7. Son pari, qui sera gagn9e, e9 tait d’atteindre une meilleure visibilit9e, et aussi de favoriser des rencontres entre th9eoriciens et praticiens de l’informatique (Fig. 1). De fait, le laboratoire est tres rapidement connu en France et dans le monde, et les "nancements publics deviennent plus substantiels. Ainsi est cr9ee9 le LITP (Laboratoire d’Informatique Th9eorique et Programmation, UA 248), en 1975. Maurice en sera le co-directeur jusqu’en 1985, avec Jacques Arsac puis avec Bernard Robinet. La gestion administrative du LITP a e9 t9e assur9ee par Paulette Le Louet depuis le milieu des ann9ee 70 jusqu’a son d9epart a la retraite en 1996. Sa sZur cadette, Nicole Robinet, a un record de long9evit9e plus grand. encore, puisqu’aujourd’hui encore en activit9e a Paris 6, elle est arriv9ee au d9ebut des ann9ees 60 pour assurer le secr9etariat de l’Institut de Programmation naissant. A partir des ann9ees 80, l’histoire du LITP est malheureusement travers9ee de conHits, complexe alchimie d’oppositions de styles, d’orientations, et de comp9etition entre les deux universit9es sur lequel le laboratoire est implant9e. Le LITP a disparu en 1997. Mais le spectre de l’informatique fondamentale, appuy9e sur les deux pˆoles de l’algorithmique et de la programmation, est aujourd’hui en quelque sorte r9eincarn9e a Paris 7, ou ont e9 t9e cr9ee9 s en 1997 le LIAFA (Laboratoire d’Informatique Algorithmique Fondamentale et Appliqu9ee) et en 1999 l’Equipe PPS (Preuves, Programmes et Systemes). Inlassable p9elerin, Maurice oPrira un actif soutien aux partisans de ce renouveau, r9edigeant notamment de sa plume un beau texte a l’appui de la demande de cr9eation du LIAFA.

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Langages alg3ebriques, Bonascre (J.-P. Crestin et M. Nivat), 1973 Complexit3e des algorithmes, Ile de Berder (Ph. Flajolet), 1974 Mono9:des syntactiques, Vic-sur-Cere (J.-F. Perrot), 1975 S3emantique des langages de programmation, Molines-en-Queyrac (M. Nivat), 1976 S3eries formelles, Vieux-Boucau-les-Bains (J. Berstel), 1977
-calcul, La Chˆatre (B. Robinet), 1978 Th3eorie des codes, Jougne (D. Perrin), 1979 Parall3elisme, Colleville (G. Roucairol), 1980 Langages alg3ebriques, Murol (L. Boasson), 1981 Compilation, Bareges (L. Nolin), 1982 Algorithmique, Ile de R9e (M. Fontet), 1983 Automates et mots in,nis, Le Mont Dore (D. Perrin) 1984 Logique combinatoire et
-calcul, Val d’Ajol, (G. Cousineau et P.-L. Curien), 1985 R3eseaux d’automates, Argeles-sur-Mer (C. ChoPrut), 1986 Linguistique et informatique, Ile d’Ol9eron (M. Gross et D. Perrin), 1987 Automates ,nis et applications, Ramatuelle (J.-E. Pin), 1988 Logique et informatique, Albi (B. Courcelle), 1989 S3emantique des syst;emes de processus concurrents, La-Roche-Posay (I. Guessarian), 1990 Math3ematiques et informatique th3eorique, M9ejanes-Le-Cap (S. GrigorieP), 1991 Parall3elisme, Les Sables d’Or les Pins (M. Cosnard et Y. Robert), 1992 R3ee3 criture de termes, Font-Romeu (H. Comon et J.-P. Jouannaud), 1993 Programmation logique avec contraintes, Chatillon-sur-Seine (A. Podelski), 1994 G3eom3etrie et topologie discr;ete, Super Lioran (J.-P. R9eveilles et D. Richard), 1995 Automates cellulaires, Saissac (J. Mazoyer), 1996 Algorithmique, Longefoy (D. Krob et M. Morvan), 1997 Alg;ebres Max-Plus et applications en informatique et automatique, Ile de Noirmoutier (S. Gaubert, J.-J. Loiseau, J. Mairesse, J.-E. Pin), 1998 Codage et cryptographie, Batz-sur-mer (A. Canteaut, C. Carlet, P. Charpin, M. Girault, B. Vall9ee), 1999 Pavages du plan, Branville (B. Durand, M. Nivat, L. Vuillon), 2000 Arithm3etique des ordinateurs, Prapoutel-Les-Sept-Laux (C. Frougny et J.-M. MBuller), 2001 S3emantique des langages de programmation, Agay (P.-L. Curien, V. Padovani, J.-M. RiQet), 2002 Fig. 1. Liste des ecoles de printemps d’informatique th9eorique.

5.3. L’informatique th3eorique europ3eenne Mais revenons au d9ebut des ann9ees 70. Maurice ne d9eborde pas seulement d’activit9e sur la scene nationale. Il va forger des outils essentiels pour la communaut9e de recherche europ9eenne en informatique. Il rencontre le prince Caracciolo di Fiorino, qui Zuvre a l’OCDE et a la Commission Europ9eenne. Il se voit con"er la r9edaction d’un rapport

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OCDE sur l’Informatique, qui aura la plus grande inHuence ... sur le gouvernement polonais (l’un des premiers pays a adopter le mot informatique, comme dans la revue Fundamentae Informaticae). A Bruxelles, le prince plaidait en vain la cr9eation d’un Institut du Software, mais une id9ee moins ambitieuse eut plus de succes: il suggere a Maurice la cr9eation d’une Association Europ9eeenne d’Informatique Th9eorique, qui serait soutenue par Bruxelles. Il n’y aura pas de soutien de la Commission Europ9eenne, mais l’EATCS (European Association for Theoretical Computer Science) est fond9ee comme association de droit belge. Le groupe fondateur comprend autour de Maurice: Jaco de Bakker, Corrado BBohm, Mike Paterson. Le premier “International Colloquium on Automata, Languages and Programming” (ICALP) est organis9e en 1972 a l’IRIA. Beaucoup de vedettes sont la: Rabin, Paterson, Scott, Meyer, Eilenberg,... . On trouvera plus loin davantage d’informations sur l’EATCS et l’ICALP dans le texte d’Arto Salomaa et Grzegorz Rozenberg. Plus tard, Maurice cr9eera aussi l’Association FranScaise d’Informatique Th9eorique (AFIT), avec l’aide de Paul Gastin et Brigitte Rozoy. 5.4. Naissance de TCS Fruit de conversations avec Inar Fredrikson, e9 diteur chez North Holland, le journal paraˆYt depuis 1976. Dans toute son histoire, il n’a subi que deux attaques, auxquelles il a r9esist9e sans diRcult9e, l’une e9 manant de collegues d’outre-atlantique qui ont d9emarch9e aupres de North Holland pour tenter d’empˆecher la cr9eation d’un tel titre en Europe, et dix ans plus tard de la part de collegues hostiles a l’informatique th9eorique. Nous renvoyons aux e9 ditoriaux de Giorgio Ausiello, Don Sannella et Mike Misolve, et de Maurice Nivat, dans le num9ero 250 de TCS, pour plus d’9el9ements sur la revue et sur sa prodigieuse expansion depuis un quart de siecle. L’inHuence de Maurice sur le milieu international de l’9edition scienti"que d9epasse de beaucoup ce journal (auquel il faut ajouter Science of Computer Programming, un cousin plus orient9e vers la pratique de la programmation, et plus r9ecemment les Electronic Notes in Theoretical Computer Science). Maurice faisait par exemple partie, avec Engeler, Huet, Jouannaud, Milner et Wirsing du Comit9e Editorial de la s9erie Progress in Theoretical Computer Science (Pitman, repris par BirkhaBuser) dirig9ee par le regrett9e Ron Book, grand ami de Maurice. 6. Des langages algebriques a( la semantique Je reprendrai plus loin l’expos9e des r9ealisations de Maurice pour l’informatique franScaise et europ9enne, mais je souhaite ne pas perdre le "l du temps. Parlons donc un peu des travaux scienti"ques de Maurice depuis la "n des ann9ee 60 jusqu’au d9ebut des ann9ees 80. 6.1. Th3eorie des langages Dans sa these, Maurice avait e9 tabli une caract9erisation alg9ebrique des transductions rationnelles. Celle-ci est a la base de nombreux d9eveloppements ult9erieurs de l’“9ecole

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franScaise” de th9eorie des langages. Les notions de cˆone rationnel et de hi9erarchie sont d9eja la [29]. Peu de temps apres, Maurice entame l’9etude des langages d9e"nis par congruence [17]; cette e9 tude conduira beaucoup plus tard aux travaux de G9eraud S9enizergues (voir la contribution de ce dernier dans ce volume). Un ePort continu est poursuivi par Maurice et ses e9 leves (Luc Boasson, JoProy Beauquier, Jean-Michel Autebert) en vue de comprendre la structure (et notamment les g9en9erateurs [7]) des langages alg9ebriques, ou “context-free”, et menera en particulier a la d9ecouverte d’une mesure de complexit9e remarquable [12] qui a e9 t9e beaucoup utilis9ee. Un expos9e synth9etique d’un certain nombre de ces r9esultats peut eˆ tre trouv9e dans le livre de Jean Berstel [11]. En"n, a la "n des ann9ees 70, et en relation avec le d9eveloppement de ses travaux en s9emantique (cf. infra), Maurice entamera l’9etude des mots in"nis engendr9es par une grammaire alg9ebrique [31]. Il introduit la notion de centre d’un langage et met en e9 vidence les propri9et9es topologiques sous-jacentes. Grˆace a ce foisonnement d’id9ees neuves, la France s’est taill9ee une r9eputation internationale sur ces sujets et bon nombre de chercheurs du domaine lui sont redevables d’id9ees fondatrices dont il est l’auteur. 6.2. De la s3emantique alg3ebrique a; celle du parall3elisme Rappelons que Maurice d9ecouvre les sch9emas de programmes a Colchester. Il e9 tend d’abord un fameux th9eoreme de SchButzenberger, caract9erisant l’ensemble des termes engendr9es par une grammaire alg9ebrique (d’arbres) comme la plus petite solution de cette grammaire, consid9er9ee comme un systeme d’9equations sur les ensembles de termes. Puis il montre que, dans le cas ou la grammaire repr9esente un sch9ema de programme (r9ecursif), cette solution alg9ebrique caract9erise a son tour la s9emantique du sch9ema de programme, au sens ou l’interpr9etation de cette solution coBYncide avec la s9emantique op9erationnelle du sch9ema de programme consid9er9e [30]. Peu a peu se d9egage ainsi la “s9emantique alg9ebrique” [18] (voir aussi l’ouvrage [24] d’Irene Guessarian), qui repose sur la construction d’un objet alg9ebrique in"ni— similaire a l’arbre de BBohm d’un lambda-terme—qui factorise la s9emantique d’un sch9ema de programme. Cet objet alg9ebrique caract9erise ainsi la classe d’9equivalence s9emantique du sch9ema de programme auquel il est associ9e. Ceci permet dans certains cas (sch9emas de programmes “rationnels” par exemple) de donner des algorithmes pour d9ecider de cette e9 quivalence. L’9etude des arbres in"nis—pour eux-mˆemes—culmine avec la these de Bruno Courcelle (voir [19]). Maurice entreprend ensuite de g9en9eraliser ses r9esultats au cas non-d9eterministe. Les diRcult9es s’averent s9erieuses, notamment parce que la s9emantique d9enotationnelle est impuissante a d9e"nir une notion satisfaisante de “power-domain”. En collaboration avec Andr9e Arnold, il introduit alors l’utilisation des espaces topologiques, et plus particulierement des espaces m9etriques [1], a"n de d9e"nir, toujours dans l’esprit du th9eoreme de SchButzenberger—c’est-a-dire par point "xe—, l’objet s9emantique associ9e a un sch9ema de programme non-d9eterministe. Cette faScon de voir la s9emantique, via les espaces m9etriques, dont l’expos9e le plus synth9etique est [33], sera reprise par de nombreux auteurs par la suite, notamment par le groupe de Jaco de Bakker en Hollande.

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Mais il est juste de dire que le succes ici n’est pas complet, car la coBYncidence avec la s9emantique op9erationnelle (telle que d9e"nie par G9erard Boudol [13] a la suite des travaux de G9erard Berry et Jean-Jacques L9evy [10]) ne subsiste que dans certains cas particuliers. Ce “chapitre” de l’activit9e scienti"que de Maurice se clˆot avec la publication de [34], qui collecte les communications faites a un s9eminaire francoam9ericain tenu a Fontainebleau en 1982. Il va ensuite s’int9eresser a la s9emantique du parall9elisme. L’articulation technique est ici fournie par l’9etude des mots in"nis vus comme des comportements in"nis de processus, qui engendrera de nombreux travaux men9es sous la direction de Maurice: ceux de FranScoise Gire, Daniele Beauquier, Hugues Fauconnier, Paul Gastin, Andreas Podelski. L’activit9e de consultant de Maurice a la Thomson joue aussi un rˆole d9eterminant: au tournant des ann9ees 70=80, Ruggiu organise a Corbeville des r9eunions r9egulieres avec quelques ing9enieurs travaillant sur la programmation des centraux t9el9ephoniques. Ces rencontres ont favoris9e la maturation des travaux de Maurice [32], et—a la suite— d’Andr9e Arnold, sur la synchronisation d’automates. Pour plus de d9etails sur cette p9eriode de l’activit9e de Maurice, nous renvoyons a la contribution d’Andr9e Arnold dans le pr9esent volume. Disons seulement que dans ce domaine, il a e9 t9e en bonne part un pr9ecurseur, faisant du “model-checking” avant l’heure. Il a aussi a cette e9 poque beaucoup discut9e de concurrence avec des chercheurs venus d’horizons pratiques, tels que G9erard Roucairol, que Maurice avait invit9e a fonder une e9 quipe sur le parall9elisme au LITP, ou Jean-Pierre Verjus (Rennes). 7. Travaux d’Hercule En 1982, par l’interm9ediaire de Lions, Maurice est charg9e par Chevenement et Savary, respectivement ministres de la Recherche et de l’Education, de la r9edaction d’un rapport sur la "liere e9 lectronique. Cette tˆache prendra pres d’un an. Maurice consulte beaucoup. La rapport s’intitule “Savoir et Savoir-faire en Informatique”. Lorsqu’il est termin9e, Fabius a succ9ed9e a Chevenement, a la tˆete d’un ministere de la Recherche et de l’Industrie. Maurice ira tout de mˆeme pr9esenter son rapport a Chevenement, lequel, tout a sa libert9e politique retrouv9ee, lui oPrira en tˆete-a-tˆete un long et impressionnant discours de g9eopolitique. Fabius propose a Maurice de pr9esider le Conseil Scienti"que du Programme Mobilisateur de la Filiere Electronique. Maurice met en place les Programmes de Recherche Coordonn9es (PRC), qui seront pendant quelques ann9ees richement dot9es et donneront un fantastique coup de fouet a la recherche franScaise. Dans les domaines qui nous sont proches, les projets suivants se d9eploient: • Programmation (Robert Cori), • Math9ematiques pour l’Informatique (Dominique Perrin, puis Philippe Flajolet). • C3 (Coop9eration, Concurrence, Communication) (Jean-Pierre Verjus). • BD3 (Bases de donn9ees) (Delobel). • Informatique linguistique (Maurice Gross). C’est aussi a ce moment qu’intervient la "n du monopole de Bull, qui empˆechait les laboratoires de s’acheter des machines Unix. Les PRC ont permis l’achat des premiers VAX, qui e9 taient des machines coˆuteuses. Toutes les e9 quipes utilisaient la mˆeme ma-

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chine a travers le r9eseau TRANSPAC (qui venait d’ˆetre mis en service), ce qui a contribu9e a faire nouer des collaborations et a souder des communaut9es. Ce travail important et pro"table se d9eroulait ind9ependamment d’une p9erip9etie grandiloquente, le Centre Mondial de l’Informatique, lubie d’un journaliste politicien, JeanJacques Servan-Schreiber, qui prenait son essor sous de hautes protections politiques (DePerre, Mitterrand). Appel9es en gourous, Seymour Papert (ancien th9eoricien des langages formels devenu apˆotre de l’IA au MIT) et Raj Reddy (directeur du Centre de Robotique de CMU) promettaient la suppression des universit9es, ou ce qui revient au mˆeme leur distribution sur les plus in"mes coins du territoire ou les villages de la brousse africaine, tout le savoir pouvant eˆ tre enferm9e sur un disque dur, et pouvant eˆ tre interrog9e interactivement. Henri Emmanuelli se porte candidat pour les Landes! 8. Pavages et tomographie discr(ete D9echarg9e en 1985 de ses responsabilit9es nationales, Maurice se remet a la recherche. Il garde un int9erˆet pour les langages d’arbres et pour la concurrence. Mais surtout, il d9ecide une nouvelle fois d’inH9echir—plus radicalement sans doute cette fois—son domaine de recherche, et va s’int9eresser aux objets g9eom9etriques discrets, et notamment aux problemes de pavage qui le fascinent [23,22]. Il suit les travaux de plusieurs jeunes chercheurs sur ces sujets: Alain Daurat, Yann G9erard, Laurent Vuillon. Un de ses anciens e9 tudiants, Ahmed Saoudi, malheureusement d9ec9ed9e depuis, l’a beaucoup pouss9e dans cette direction. 8.1. Pavages Avec Daniele Beauquier, Maurice se pose la question suivante: est-il d9ecidable de savoir si un nombre "ni de polyominos forme un code plan? La r9eponse, n9egative, qu’ils obtiennent vers 1993–1994, sera longue a trouver, et doit encore eˆ tre publi9ee. Dans [8], ces deux auteurs s’attaquent au probleme du pavage du plan par translation d’une seule piece. Ils montrent que de tels pavages sont possibles si et seulement si la piece est un pseudo-hexagone, et que tout tel pavage est semi-p9eriodique (c’est-a-dire invariant par certaines translations). Beauquier et Nivat ont ainsi trouv9e une famille int9eressante et d9ecidable de problemes de pavage. Dans [9], Maurice et ses coauteurs montrent la NP-completude des pavages par des dominos horizontaux et des trominos verticaux, et plus generalement pour toute paire de barres dont l’une a une longueur sup9erieure a deux 2. D’autres r9esultats dans la mˆeme veine ont e9 t9e obtenus r9ecemment par Moore et Robson [27]. 8.2. Tomographie discr;ete Maurice est aussi l’un des pionniers de la tomographie discrete, dont l’objet est de reconstruire un objet discret a partir de projections [25]. Il s’est int9eress9e au probleme de la reconstruction de matrices binaires a partir d’un nombre "ni de “radiographies discretes”, c’est-a-dire de projections sur certaines lignes compt9ees avec des

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multiplicit9es. Le cas de deux projections (horizontale et verticale) a e9 t9e le plus e9 tudi9e. Des r9esultats d’existence et d’unicit9e sont connus [35]. Mais comme le nombre de solutions peut eˆ tre tres e9 lev9e en g9en9eral, les chercheurs du domaine se sont int9eress9es a des cas particuliers. Dans [5,6], Maurice, en collaboration avec Elena Barcucci, Alberto Del Lungo et Renzo Pinzani, donne des algorithmes polynomiaux pour la reconstruction de matrices binaires ayant des propri9et9es de convexit9e et/ou connexit9e. Ces travaux, dont une synthese est pr9esent9ee dans [20], ont e9 t9e prolong9es par plusieurs auteurs [3,15,16]. Le probleme de la reconstruction de matrices binaires a partir de trois projections ou plus est NP-complet [21]. Maurice et ses coauteurs ont trouv9e une classe de polyominos (convexes et connexes, toujours) pour lesquels la reconstruction peut se faire en temps polynomial [4]. Une extension de ces r9esultats aux treillis convexes est donn9ee dans [14]. La tomographie discrete est un sujet relativement r9ecent, riche d’applications et activement e9 tudi9e, que Maurice contribue a promouvoir. Au cours des dernieres ann9ees, il a organis9e deux colloques avec Peter Gritzmann: Workshop on Discrete Tomography: Algorithms and Complexity (Dagstuhl, 1997) et Workshop on Discrete Tomography and Related Problems (Nancy, 1999). 9. Un homme de conviction Ainsi arriv9e au terme de ce bref survol d’une carriere qui n’est pas "nie—car je forme le vZu que Maurice ait une retraite des plus actives—, je voudrais laisser la parole aux collegues et amis—les deux se confondent souvent, tant Maurice a su chaleureusement accueillir et admettre dans sa famille les nombreux collegues avec lesquels il a e9 tabli des relations de con"ance et d’estime—qui m’ont fait parvenir leurs t9emoignages sur leurs rencontres avec Maurice. Ce qui suit est principalement le fruit d’une synthese anonyme de leurs propos, dont j’espere ne pas avoir trahi l’esprit. D9ebarrassons-nous d’abord des ombres au tableau. Les relations de Maurice avec quelques collegues et amis, tout comme sa vie personnelle, ont e9 t9e travers9ees d’9episodes diRciles, ins9eparables d’une personnalit9e complexe et attachante, que rien n’9evoque sans doute mieux que de longs silences de Maurice au milieu d’une conversation t9el9ephonique, nocture de pr9ef9erence! Au total, il faut surtout saluer une activit9e d9ebordante, une vaste imagination, une inlassable curiosit9e, et ... un solide sens de l’humour. Maurice a d9eploy9e des ePorts constants pour diPuser sa vision de l’informatique: discipline scienti"que a part entiere et devant construire ses fondations. Son passage a Grenoble et a Rennes lui a montr9e l’importance de cette “mission”. Il faut se souvenir qu’au d9ebut des ann9ees 70, en France, certains universitaires informaticiens des gros centres de province d9efendaient une informatique “pratique” qui leur servait a justi"er une absence de d9emarche scienti"que. La cr9eation du LITP n’est pas all9ee tout-a-fait de soi au CNRS. En 1983, il arrive encore qu’un th9esard du LITP puisse eˆ tre mal reScu dans un laboratoire d’informatique, sur la seule base de son appartenance au “laboratoire de Nivat”! Le m9erite de Maurice est d’avoir compris qu’il fallait non seulement “produire” des gens qui auraient cette vision de l’informatique et propageraient la “bonne parole”, mais aussi trouver au sein des groupes existants les gens qui comprendraient les enjeux

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scienti"ques et la n9ecessit9e de sortir de cette d9efense et illustration d’une “informatique pratique a la franScaise” a"n d’ˆetre pr9esent sur la scene scienti"que internationale. Maurice ne s’est pas content9e de d9efendre et diPuser ses convictions aupres des praticiens. Il a aussi constamment plaid9e parmi les th9eoriciens pour l’ouverture vers les applications ou vers de nouveaux champs a explorer. Si le lecteur n’en e9 tait pas encore convaincu, on pourrait encore noter l’int9erˆet de Maurice pour les rapports entre informatique et culture (habilitation de Philippe Aigrain), qui a sans doute pour racine sa passion pour le patrimoine rural ou l’arch9eologie industrielle. Les liens profonds que Maurice a nou9es avec Louis Nolin des le d9ebut de sa carriere ont certainement contribu9e a cr9eer les conditions pour la maturation d’un aussi remarquable spectre scienti"que. Maurice aura ainsi e9 tendu son “9ecole humaine”, pour reprendre le titre d’une section de ce texte (que je dois a Jean Vuillemin), jusqu’a: • Toulouse. Des chercheurs du LITP y ont longtemps donn9e des cours de DEA, qui ont engendr9e des theses (FranScois Rodriguez, Patrick Sall9e, Jean-Louis Durieux,...). Jean Vignolle—qui a travaill9e sur la premiere machine franScaise: le GAP 500—a encourag9e et supervis9e cet apport. • Lille (Dauchet, Arnold, Latteux, Comyn, Jacob). Ici, l’initiative revient a Andr9e Arnold et Max Dauchet, qui forment en 1974 un groupe de travail sur les langages d’arbres, et r9ealisent bientˆot que Maurice travaille d9eja dans ce domaine. Il devient leur directeur de these. Il leur suggere d’organiser un colloque a Lille sur les arbres en algebre et en programmation, qui a ePectivement eu lieu en f9evrier 1976. C’9etait le premier CAAP, qui s’est tenu a Lille pendant une demi-douzaine d’ann9ees, puis dans toute l’Europe. Le CAAP a v9ecu 20 ans, son h9eritier est FOSSACS. • Nancy: ici, il s’agit surtout de nombreux contacts et e9 changes (Jean-Pierre Jouannaud, Claude et H9elene Kirchner, Pierre Lescanne). • Et, via des chercheurs de l’9ecole parisienne nomm9es en province: Amiens (Boasson, aujourd’hui professeur a Paris 7), Bordeaux (Cori, puis Courcelle, puis Arnold), Poitiers (Arnold), Rennes (Kott), Rouen (Perrin, aujourd’hui professeur a Marne-laVall9ee). Notons que le contexte e9 conomique ou reglementaire peut donner un coup de pouce a d’exemplaires mouvements. Ainsi, cet essaimage vers la province doit un peu au premier choc p9etrolier, qui a engendr9e une grande p9enurie de postes, poussant des gens qui ne l’auraient peut-ˆetre pas souhait9e a s’9eloigner de la capitale. Dans un autre ordre d’id9ee, j’ai pu constater le rˆole de l’anti-militarisme dans les vocations vers la "n des ann9ees 60 et le d9ebut des ann9ees 70. Quelques polytechniciens sont arriv9es vers l’informatique en b9en9e"ciant de la “botte recherche”, une disposition qui a l’X permettait entre autres d’“9echapper” au service militaire en allant faire de la recherche! A cette liste, on peut ajouter l’Ecole Normale Sup9erieure de Lyon, ou Maurice fait de multiples s9ejours au d9ebut des ann9ees 90, et noue de nombreux contacts (Mazoyer, Durand, Remila). Maurice a depuis longtemps une vision a long terme de la recherche en informatique, et une compr9ehension remarquable de l’importance relative des r9esultats scienti"ques. Il sait d9etecter les talents et a toujours cherch9e a permettre a de jeunes chercheurs de d9evelopper leurs propres travaux. Il a su faire se rencontrer beaucoup de gens a

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l’9echelle franScaise et leur faire rencontrer beaucoup d’9etrangers, et construire ainsi un tissu de relations qui a aid9e l’informatique franScaise a rattraper les centres d’excellence comme Edimbourg. Il a constamment cherch9e a e9 largir les domaines de recherche de son entourage, en commenScant par lui-mˆeme. Il n’a eu de cesse de sensibiliser la communaut9e scienti"que autant que les d9ecideurs aux enjeux e9 conomiques ou sociaux de l’informatique. Mais il n’a pas toujours pu assez faire entendre les points de vue dont il e9 tait porteur. Ce n’est pas faute d’y avoir travaill9e, pendant des ann9ees! Appel9e de nouveau par le Ministere charg9e de la Recherche a pr9esider l’Observatoire de la Recherche en Informatique en France, il d9epensera beaucoup d’9energie en 1993 et 1994 a coordonner les travaux d’une vingtaine d’experts chercheurs et industriels, y d9eployant tout le talent de sa belle plume, tout cela pour un rapport si vite enterr9e... Aujourd’hui comme hier, Maurice regrette que les politiques et les grands industriels n’aient pas d9eploy9e les ePorts suRsants pour cr9eer une puissante industrie informatique nationale ou europ9eenne. On se souviendra par exemple de la SM90, dont quelques dizaines d’exemplaires furent produits au d9ebut des ann9ees 80. Une belle architecture, initialement conScue au CNET. La machine aurait pu rivaliser avec le SUN, mais il aurait fallu y mettre dix fois plus de personnes... Notons tout de mˆeme, sur une note plus optimiste, que l’9equipe gouvernementale suivante a reconnu au tournant du mill9enaire l’importance du secteur des STIC (Sciences et Technologies de l’Information), et s’emploie a compenser ce r9eveil tardif par une impulsion forte en termes de programmes et de moyens humains, cr9eant notamment des postes de chercheurs en nombre important dans ce secteur, a l’INRIA et au CNRS, dot9e d’un nouveau d9epartement des Sciences et Technologies de l’Information. Maurice est, en"n, un grand voyageur. Il est impossible de faire la liste de tous les pays qu’il a visit9es. Mentionnons ici seulement ses liens nombreux et "deles avec l’Italie, l’Inde, le Japon, et plus r9ecemment le Chili.. Ainsi s’acheve cette courte et donc bien partielle promenade dans la carriere de Maurice Nivat. Mes remerciements iront d’abord a lui pour ce que je lui dois en g9en9eral, et pour les e9 l9ements de son histoire qu’il a bien voulu me communiquer au cours de deux longues conversations, qui ont fourni la matiere d’une bonne partie de ce texte. Je remercie aussi tous les collegues qui m’ont aid9e dans cette modeste entreprise de m9emoire. Apres une trentaine d’ann9ees d’existence, il reste a souhaiter longue vie a l’informatique fondamentale, science du logiciel, ouverte sur les autres disciplines (biologie, linguistique,...) et les applications, et appuy9ee sur les math9ematiques (algebre, logique, g9eom9etrie), assez inventive cependant pour relever de nouveaux d9e"s comme celui de la programmation distribu9ee et de l’internet, pour lesquels les math9ematiques n’ont pas plus pr9epar9e d’outils tout faits que lorsque les pionniers de la th9eorie des automates, de la s9emantique d9enotationnelle, des calculs de processus ou de l’algorithmique ont commenc9e d’apporter leurs pierres a l’9edi"ce des fondements de l’informatique. References [1] A. Arnold, M. Nivat, Formal computations of non-deterministic recursive program schemes, Math. Systems Theory 13 (1980) 219–236. [2] J. Arsac, A. Lentin, L. Nolin, M. Nivat, Algol, th9eorie et pratique, Gauthier-Villars, 1965.

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[3] E. Balogh, A. Del Lungo, C. D9ev9enyi, A. Kuba, Comparison of algorithms for reconstructing hv-convex discrete sets, Linear Algebra Appl., to appear. [4] E. Barcucci, S. Brunetti, A. Del Lungo, M. Nivat, Reconstruction of lattice sets from their horizontal, vertical and diagonal X-rays, Disc. Math., to appear. [5] E. Barcucci, A. Del Lungo, M. Nivat, R. Pinzani, Reconstructing convex polyominoes from their vertical and horizontal projections, Theoret. Comput. Sci. 155 (1996) 321–347. [6] E. Barcucci, A. Del Lungo, M. Nivat, R. Pinzani, Medians of polyominoes: a property for the reconstruction, Internat. J. Imaging Systems Technol. 8 (1998) 69–77. [7] J. Beauquier, G9en9erateurs alg9ebriques et paires it9erantes, Theoret. Comput. Sci. 8 (1979). [8] D. Beauquier, M. Nivat, Tiling the plane with one tile, Symp. on Computational Geometry, 1990, pp. 128–138. [9] D. Beauquier, M. Nivat, E. Remila, M. Robson, Tiling "gures of the plane with two bars, Comput. Geometry Theory Appl. 5 (1) (1995) 1–25. [10] G. Berry, J.-J. L9evy, Minimal and optimal computations of recursive programs, JACM 26 (1979) 148–175. [11] J. Berstel, Transductions and context-free languages, Teubner (1979). [12] L. Boasson, B. Courcelle, M. Nivat, A new complexity measure for languages, Conf. on Theoretical Computer Science, 1977. [13] G. Boudol, Computational semantics of term rewriting systems, in: M. Nivat, J. Reynolds (Eds.), Algebraic Methods in Semantics, Cambridge University Press, Cambridge, 1985. [14] S. Brunetti, A. Daurat, Reconstruction of discrete sets from two or more projections in any direction, Proc. of the Seventh International Workshop on Combinatorial Image Analysis (IWCIA 2000), Caen, 2000, pp. 241–258. [15] S. Brunetti, A. Daurat, A. Del Lungo, An algorithm for reconstructing special lattice sets from their approximate X-rays, Lecture Notes in Computer Science 1953, 9th Conf. on Discrete Geometry for Computer Imagery (DGCI00), Uppsala, Sweden, 2000, pp. 113–125. [16] M. Chrobak, C. DBurr, Reconstructing hv-convex polyominoes from orthogonal projections, Inform. Process. Lett. 69 (6) (1999) 283–289. [17] Y. Cochet, M. Nivat, Une g9en9eralisation des ensembles de Dyck, Israel J. Math. 9 (1971). [18] B. Courcelle, M. Nivat, The algebraic semantics of recursive program schemes, Proc. 7th MFCS, Lecture Notes in Computer Science, Vol. 64, Springer, Berlin, 1978, pp. 16 –30. [19] B. Courcelle, Fundamental properties of in"nite trees, Theoret. Comput. Sci. 25 (1963) 95–169. [20] A. Del Lungo, N. Nivat, Reconstruction of connected sets from two projections, Chap. 7 (Foundations), in: G.T. Herman, A. Kuba (Eds.), Discrete Tomography: Foundations, Algorithms and Applications, Birkhauser, Boston, Cambridge, MA, 1999. [21] R.J. Gardner, P. Gritzmann, D. Prangenberg, On the computational complexity of reconstructing lattice sets from their X-rays, Discrete Math. 202 (1999) 45–71. [22] S.W. Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings, Academic Press, Princeton, 1996. [23] B. Grunbaum, G.C. Shepard, Tilings and patterns, W. H. Freeman and Company, New York, 1986. [24] I. Guessarian, Algebraic Semantics, Lecture Notes in Computer Science, vol. 99, Springer, Berlin, 1981. [25] G.T. Herman, A. Kuba (Eds.), Discrete tomography: Foundations, Algorithms and Applications, Birkhauser, Boston, Cambridge, MA, 1999. [26] S.C. Kleene, Representation of event nets and "nite automata, in: C.E. Shannon, J. MacCarthy (Eds.), Automata Studies, Princeton University Press, Princeton, NJ, 1956, pp. 3–42. [27] C. Moore, M. Robson, Hard tiling problems with simple tiles, Discrete Comput. Geometry, to appear. [28] M. Nivat, Transductions des langages de Chomsky, Ann. Inst. Fourier 18 (1968) 339–455. [29] M. Nivat, These d’Etat mim9eographi9ee, Chap. VI, 1966. [30] M. Nivat, On the interpretation of recursive polyadic program schemes, Symp. Mathematica, vol. XV, Istituto di Alta Mathematica, Bologna, 1975, pp. 255 –281. [31] M. Nivat, Sur les ensembles de mots in"nis engendr9es par une grammaire alg9ebrique, RAIRO Inform. Th9eor. 12 (1979). [32] M. Nivat, Sur la synchronisation des processus, Rev. Tech. Thomson-CSF 11 (1979) 899–919. [33] M. Nivat, In"nite words, in"nite trees, in"nite computations, Math. Cent. Tracts 109 (1979) 1–52.

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[34] M. Nivat, J. Reynolds (Eds.), Algebraic Methods in Semantics, Cambridge University Press, Cambridge, 1985. [35] H. Ryser, Combinatorial mathematics, The Carus Math. Monographs, Vol. 14, Mathematical Association of America, 1963. [36] C.G. Squier, Word problems and a homological "niteness condition for monoids, J. Pure Appl. Algebra 49 (1987) 201–217.

Theoretical Computer Science 281 (2002) 25 – 30

www.elsevier.com/locate/tcs

ICALP, EATCS and Maurice Nivat Grzegorz Rozenberga; b; ∗ , Arto Salomaac a Leiden

Institute of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands b Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309, USA c Turku Centre for Computer Science (TUCS), Lemmink* aisenkatu 14A, 20520 Turku, Finland

Abstract The purpose of this writing is to tell something about the quite unique and fundamental role of Maurice Nivat, both in launching ICALP and EATCS, as well as in guiding them through the di/cult early stages. Our account is based on personal recollections, as well as some original documents and letters. To preserve the personal character of our recollections, we have not tried to 2ll the gaps in our information through other channels such as interviewing some of the people involved. In other words, our personal account is not intended to be a comprehensive c 2002 Elsevier Science B.V. All rights reserved. history. It only tells how things seemed to us.


1. ICALP In our days it is hard to visualize any more the scienti2c landscape and academic surroundings in the early 1970s in 2elds now referred to as computer science. Everything was rather unorganized and scattered. Automata theory was already very advanced. On the other hand, what was labeled as theoretical computer science or theoretical informatics varied enormously and could be, for instance, numerical analysis or some parts of theoretical physics. Maurice Nivat apparently wanted to have some order and uniformity, and was also very successful in his e8orts to achieve this goal. The impact of his work was by no means restricted to Europe, since both ICALP and EATCS were quite international from the very beginning. Both writers of this account knew already in summer 1971 that a big International Colloquium on Automata, Languages and Programming was coming up in Paris the following summer 1972. Later on, Maurice Nivat wrote to Arto Salomaa in a letter dated November 7, 1971: ∗

Corresponding author. Tel.: +31-71-277-063; fax: +31-71-276-985. E-mail addresses: [email protected] (G. Rozenberg), [email protected] (A. Salomaa).

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 0 5 - 1

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Dear Arto, At last you will receive very soon a number of call for papers for the colloquium that SchCutzenberger and myself organize in IRIA next July (3–7). Will you be so kind as to spread these call for papers among your students, friends and colleagues whom you think may be interested by this colloquium. The program committee is composed, beside yourself, SchCutzenberger and myself of S. Eilenberg, S. Ginsburg, M. Rabin, A. van Wijngaarden, P. Fischer, D. Park, G. Hotz, C. Boehm and L. Nolin. Our work in choosing the papers to be presented at the conference will take place in April. I con2rm that your travel and stay expenses will get paid by IRIA when, which I hope, you come in July. . . . The letter continues with other matters. Clearly, both the schedule for submitting the papers and the schedule of the Program Committee were very tight. For most of the committee members, the work in April was by mail. Everybody in the committee received all the submitted papers. It is needless to emphasize that mail here refers to ordinary mail! Maurice Nivat apparently did all the heavy work. The scienti2c success of the conference is well known. It was a big success also socially and initiated the friendly and leisurely atmosphere characteristic of so many later ICALP’s. The program of the excursion was printed as follows. Chantilly: The castle and the gardens Senlis: 10th century old little town Royaumont: Vestiges of a beautiful mediaeval abbey Dinner: Refectory of the Abbey There was also a formal reception and dinner for the dignitaries, with the following invitation: A l’occasion du Colloque sur la ThKeorie des Automates, des Langages, et de la Programmation Messieurs M. Nivat, L. Nolin et M.P. Sch*utzenberger recevront Monsieur et Madame . . . le Mardi 4 Juillet. Time and place were indicated separately. Still two important facts should be mentioned. • As far as we know, the Paris ICALP in 1972 was not intended to start a series of conferences. A somewhat similar meeting took place in Haifa in 1971, with no direct continuations. • EATCS did not yet exist. Only later ICALP became the annual conference of EATCS. In the 2rst ICALP’s the procedures (program committee methods, etc.) and time schedules worked on an ad hoc basis. Only gradually things started to become more systematic. As far as we know, Maurice Nivat carried alone the responsibility of keeping ICALP going during the early years. He came to talk to Arto Salomaa during the Oberwolfach meeting in November 1975, asking whether it was possible to organize ICALP in Turku. This was the beginning of ICALP-77. It is well known that ICALP took place only every second year until 1976; after that it has been every year.

G. Rozenberg, A. Salomaa / Theoretical Computer Science 281 (2002) 25 – 30

27

2. EATCS Maurice Nivat was the driving force in creating the European Association for Theoretical Computer Science, as well as getting new members and keeping the association alive in the early days. (Also the French name Association Europ;eenne d’Informatique Th;eorique appeared in the original stationery, and was the main name in the 2rst Bulletin.) Since we have quite much original material, we will rely on many quotations in the sequel. The whole story is actually told in the following letter by Maurice Nivat. It has no date but was obviously written some time in 1972. Dear Colleague, This letter is an invitation to join the young European Association for Theoretical Computer science. Let me begin with some history: in January 1972 a small meeting was held in the building of the European Communities Commission; M. Caracciolo chaired a discussion between a few computer scientists from six countries known for their theoretical work. First, a text written by the members of the University of Paris VII who were present at the meeting was presented and discussed; this text aimed at the de2nition of what can be called Theoretical Computer Science, i.e. the chapter of Computer Sciences which uses mathematical and logical tools to clarify and study the notion of computation. The principal subchapters would be: • Theory of algorithms and complexity • Theory of automata and formal languages • Theory of programming (formal semantics) but this list is certainly not complete. At the time the creation of a European Association was proposed, to promote the development of this 2eld by organizing meetings, exchanges of researchers, professors and students, and by increasing the communication of all sorts of information (it is rather di/cult to know what happens in a 2eld which changes completely in one year if one considers that two years are necessary for the publication of a paper in a journal). An original feature of this association would be to accept both individual members, known scientists working in the 2eld, and institutional members, University departments, research laboratories or any other organization interested in the development of Theoretical Computer Science. It appeared that the Belgian law on scienti2c association allows such a mixture of two kinds of members and more generally o8ers very liberal and Mexible possibilities. This is why, after some months of work, essentially done by our 2rst temporary President M. Sintzo8, statutes were written according to this law and the Belgian authorities were asked to create such an association. It was necessary that some people should act as founding members and share the 2rst responsibilities. These people are: L. Verbeek (President), M. Paterson and M. Paul (Vice-Presidents), M. Sintzo8 (Treasurer), M. Nivat (Secretary), J. de Bakker and G. Ausiello.

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Obviously these people were designated only by chance and each of them represents only himself. And no less obviously, this association will play the role we wish it to play, in the development of the 2eld, only if a large part of the scientists working in this 2eld become members. It is therefore very important that you agree to increase the membership of this association and that, together with the founding members, you not only work to make the association viable but also help in the de2nition of its scope, aims, composition and means of action. The temporary o/cers of the association look forward to your active participation. I cannot indeed tell you much about future activities, which will depend on you and all the people who receive this letter. M. Paterson has generously o8ered the University of Warwick as the location for a necessary General Assembly, and at the same time an informal colloquium which will be the 2rst scienti2c activity of the association. This will take place in March 1973. We have also discussed the idea of a journal or periodic letter to convey information among members (and non-members!). Suggestions concerning this and other actions will be most welcome, in particular an early priority is to assure the 2nancing of our activities. The enclosed copy of the statutes will help you to get a better idea what we are looking for and, I hope, will give you strong reasons to join our common e8ort. Yours sincerely, M. Nivat Together with the letter also a French copy of it was sent, as well as a copy of the Statutes, only in French. The letter tells the early history in quite a detailed manner. Between the lines one can read how Maurice Nivat was involved in everything. The Statutes, very formal law text, do not give much information that would interest us today. The Council was supposed to have between seven and 2fteen members, and the Board (President, two Vice-Presidents, Treasurer, Secretary) was supposed to contain one Belgian member. There were two types of members: personnes morales and personnes physiques. We might easily go wrong if we try to translate these terms! The 2rst General Assembly of EATCS took place at the University of Warwick on March 24–26, 1973. We mention two points from the invitation letter by Mike Paterson to this meeting. • If you would like to talk (in any language you like), please send me a title and a time estimate immediately. I can guarantee you a room and a blackboard but not, of course, an audience. • If you let us know your time of arrival in advance we can meet you either at Birmingham Airport or Coventry Station. Wave a magnetic tape or carry a Turing machine for identi2cation. After Nivat was elected the 2rst (real) President of EATCS in Warwick, he was very active. Some of the matters are described in the following two letters. The 2rst one is written to Arto Salomaa on November 8, 1973.

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Dear Arto, Thank you both for your acceptance to become an editor of the Journal of Theoretical Computer Science and the text you sent me for the Bulletin of the European Association of the same name. You are the 2rst and only one to send some things about TCS to be published! I write you in connection of our Association precisely: in our last meeting in Hamburg we expressed the wish to have you, Z. Manna and E. Engeler 2ll the three empty seats left in the council. I do hope you can accept that too, since it will not be a big burden. We expect to hold the next meeting of the council in SaarbrCucken during the course on theory of programming in late February. . . . Best regards to your wife and you, M. Nivat The EATCS Council consisted by the end of 1973 of the following members: G. Ausiello, C. Boehm, W. Brauer, J. de Bakker, E. Engeler, Z. Manna, B. Mayoh, R. Milner, M. Nivat, M. Paterson, J.F. Perrot, A. Salomaa, D. Scott, M. Sintzo8, L. Verbeek. The second letter we would like to quote is dated December 20, 1973, and was sent to the Council members. Dear Colleague, This is to inform you about our Association EATCS. 1. The 2rst issue of the bulletin is being printed in IRIA and will be distributed in January. It contains about 20 pages mainly informations about Theoretical Computer Science in Holland (furnished by J. de Bakker), in Torino (furnished by C. Boehm) and in the University of Paris VI (furnished by J.F. Perrot). This is completed by various announcements of forthcoming events, an account written by A. Salomaa of the MFCS Conference in the High Tatras, the accounts of the two council meetings held in Warwick and Hamburg written by B. Mayoh and an editorial by me. I believe we should all think about the next issue and also about whom to send this 2rst issue to. People will, I think, be encouraged to become members when they see this bulletin, and for this reason we should spread widely this 2rst issue. 2. The next meeting of the council will be held in the Institut fCur Informatik of SaarbrCucken on Saturday February 23rd at 14.30. The main point on the agenda is the organization of a working conference in 1975: I expect de2nite proposals to be made by C. Boehm. Another point is the cooperation with SIGACT: I started talking about it with Michael Harrison and I hope that I’ll be able to make, on this point too, de2nite proposals. I wish you all a merry christmas and happy new year. Best regards, M. Nivat P.S. Though it no longer interests directly EATCS, I’ll give in SaabrCucken information about the journal Theoretical Computer Science created by North Holland. Let me say now that the negotiations progressed favorably and the creation of

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the journal will be announced very likely in January for a 2rst issue to appear in October. The suggestion about cooperation with SIGACT was responded very positively by Michael Harrison: We look forward to a collaboration with you which should be very bene>cial to both groups. Explicitly, Harrison proposed reduced registration fees and an exchange of Newsletters. It was also clear in early 1974 that EATCS will sponsor the SaarbrCucken ICALP in summer 1974. However, ICALP was not yet speci2ed as the conference of EATCS. Sometimes the going apparently seemed rough to Nivat. The tone of his letter dated January 17, 1974, is quite frustrated. The shipment of the 2rst issue of the Bulletin was delayed very much because of a major postal strike in France. Very few contributions arrived, at a very slow pace, for the second issue, and the printing took long. The whole letter begins with a remark: This is a suggestion to keep our association alive. (The actual suggestion was to combine the membership fee with a subscription of TCS.) But everything worked out 2ne. When we 2nally received the 2rst issue of the EATCS Bulletin, we could read in the editorial: A last word on the scope of Theoretical Computer Science: a general agreement was reached to consider as the major points of Theoretical Computer Science the following 2elds of interest. • Theory of Automata • Formal Languages • Algorithms and their complexity • Theory of programming, if such a thing exists: we may now consider as chapters of this theory formal semantics, proving properties of programs, and all applications of logical concepts and methods to the study and design of programming languages. The list is by no means limitative: de2ning the limits of Theoretical Computer Science is at least as di/cult as de2ning the limits of Computer Science itself. And we strongly believe that a science is what the scientists at work make it: certainly new areas of Computer Science will be open to theory in a very near future. Let us start small and grow: we are sure many of you will help to achieve this necessary growth up to the point where our Association will be a natural link between all European Theoretical Computer Scientists. M. Nivat PrKesident de l’EATCS We started small and grew. Besides ICALP and EATCS, numerous subsequent conferences and associations carry on the work. We are convinced that the entire Theoretical Computer Science community will join our acknowledgement: Thanks, Maurice, for what you gave us!

Theoretical Computer Science 281 (2002) 31 – 36

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Nivat’s processes and their synchronization Andr%e Arnold Universite de Bordeaux I, LABRI, CNRS UMR 5800, 351 cours de la Liberation, F-33405 Talence, France

Abstract This short paper retraces how the notion of synchronization of processes introduced by Maurice c 2002 Elsevier Science B.V. All rights Nivat in 1979 has evolved over more than 20 years.
reserved.

At the end of the 1970s, Maurice Nivat took interest in the semantics of parallel and concurrent systems. With G. Ruggiu of the Laboratoire central de recherches of the French company Thomson-CSF, he organized a joint seminar on this topic. That is why his seminal paper on the synchronization of processes [14] appeared in a technical journal of this company and, therefore, was unfortunately not widely known. An extended version of this paper [15] was published in a yet more con;dential place. However, the ideas expressed in these papers have made their way; for instance, they are implemented in the model-checker MEC [4] and are the basis of the AltaRica formalism [6]. In this paper, we wish to explain how these ideas have evolved along time and to compare them with some other similar ideas. In [14], a process is de;ned as a set of in;nite sequences of actions or events. Each action or event is denoted by a letter from an alphabet A, so that a process is de;ned as a set of in;nite words over A. It appeared very soon that this de;nition was not general enough, in particular, because it did not include the notion of a deadlocking process, which was nevertheless considered as a very important notion in the last part of the paper. Therefore, in [15], a process is de;ned as a triple of languages over A: the set of in;nite sequences, the set of ;nite terminated sequences, and the set of all ;nite sequences the process is able to perform. Clearly the latter set contains all the pre'xes (or initial subsequences) of the former two. Such a process is said to be deadlock-free (or nonblocking) if each ;nite sequence which is not terminated can be extended by at least one letter into another ;nite c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter
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sequence. It follows that every ;nite sequence which is not terminated can be extended into a terminated sequence or into an in;nite word over A all of whose pre;xes are in the process. But it is not guaranteed that such an in;nite word belongs to the process. If it is always the case, the process is said to be safe. The above de;nitions are intuitively quite natural and it is not a surprise that they appear elsewhere, for instance, in the fundamental papers by Ramadge and Wonham on the control of discrete event systems [16,17]. They de;ne a discrete event system as a Nivat’s process without in;nite sequences, with the restriction that both languages (of all ;nite sequences and of all terminated ones) are recognizable, and they say that it is nonblocking if every non terminating sequence can be extended into a terminated one, which is exactly the notion of safeness for a process which has no in;nite sequences. The restriction to recognizable processes is quite natural as soon as one is interested in decision problem (whether a process is safe or not) or eDective constructions (e.g., of the greatest safe subprocess of a given process) and was considered in [15]. The paper [8] investigates such problems for some kinds of algebraic (or context-free) processes. The main idea of [14] is to de;ne the interactions between processes (communications, synchronizations) on a very high level of abstraction, so that almost all mechanisms encountered so far in the literature become particular cases of a very general concept: the synchronization vectors. Actually, the eDect of a synchronization mechanism is to force some actions of some processes to be performed simultaneously, or, on the contrary, to forbid their simultaneous occurrences. Therefore, it can be described by giving the list of tuples of action that must or may occur simultaneously (or its complement, the set of tuples that can never occur simultaneously). Formally, for i = 1; : : : ; n, let Li be a process over the alphabet Ai . Let A0i be the set Ai ∪ {}, where  is a symbol to be interpreted as “no action”. A synchronization vector is any element of the product A = A10 × · · · × A0n . A synchronization constraint is a set I ⊆ A of synchronization vectors The synchronized product of L1 ; : : : ; Ln with respect to I is the process over I de;ned as follows. Its ;nite (terminated) sequences are those whose projections are ;nite (terminated) sequences of the corresponding component, its in;nite sequences are those whose projections are in;nite or terminated. It is not diIcult to convince oneself that communication mechanisms of CSP [10], CCS [13], COSY [12] can be described in this way. Even Petri nets can be seen as synchronized products where the processes are places and synchronization vectors are transitions. Indeed, the synchronization mechanism proposed in [15] is a little more general: the synchronization constraint is a process over I , acting as a controller or supervisor, and the synchronized product is de;ned by the additional condition that its sequences must also belong to the control process. The former de;nition is a particular case of the latter when the control process consists of all ;nite and in;nite sequences over I . The diDerence between them is quite similar to the diDerence between vector addition systems and vector addition systems with states.

A. Arnold / Theoretical Computer Science 281 (2002) 31 – 36

33

However, in a few lines argument, we explained in [15] that a speci;c control process is not really needed and that the ;rst de;nition is quite suIcient, because one can apply it to a vector of processes containing also this control process as an additional component and then hide the actions of this control process by taking a homomorphic image of the synchronized product. Actually, the notion of a control process no longer appeared in subsequent papers on synchronized products [2,3] and felt into oblivion. This forgetting was an unforgivable blunder, we completely missed the point that a controller for a system of interacting processes is not a process similar to others as it was established by Ramadge and Wonham [16,17] and, however strange it may appear, the notion of a speci;c control process was reintroduced in AltaRica [6] without any conscious reminding of this ;rst attempt. Recognizable processes are those each of the three languages of which is recognizable. Therefore, they can be represented by a ;nite automaton (or labeled transition system), and the synchronized product of recognizable processes is easily represented by the automaton obtained as a kind of product of the automata representing the component processes. We privately used this construction to build up examples, and even to prove some basic results, until it appeared that the synchronized product of labeled transition systems was indeed “the” basic notion. This is why, in [2], the synchronized product is de;ned on transition systems, and no longer on processes, although transition systems are just seen as a way of specifying Nivat’s processes. Actually, it is conceptually easier to work with transition systems than with languages. As an example, Bergeron [7] drastically simpli;ed some proofs of Ramadge and Wonham just by considering that the supervisor of a “plant” is a transition system rather than a language. The next step, that still took some time, was to make explicit the fact that a transition system allows us to de;ne not only a triple but a whole bunch of languages, according to the acceptance condition that is chosen for each of them. Then, the process is indeed the transition system, and the triples that actually started the story de;nitively went out of the picture. It only remained the notion of synchronized products of transition systems [3]. However, the fact that we arrived at this notion as it is explained had an important consequence: the states play no role at all in the synchronization mechanism (in contrast, for instance, with the s=r model of Kurshan [11]) and only the actions or events of each component can be used to synchronize them. Indeed, the states of a transition system are inside a black box, that the designer is allowed to open, but the other components of the system to be synchronized are not. It follows that the synchronization product is congruent with respect to the bisimulation equivalence. The emergence of the synchronized product of transition systems as a basic notion followed many experiments by hand. But it appeared very soon that it was impossible to manage by hand the “combinatorial explosion” and it was obviously necessary to implement this product in a software tool. Under the supervision of Maurice Nivat, Alaiwan realized such a tool, together with some algorithms to analyze the structure of transition systems [1]. It was the preliminary version of the tool christened MEC by Maurice Nivat (it is a shortening of “Meccano”, the french name of the game also known as “Erector set”).

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A. Arnold / Theoretical Computer Science 281 (2002) 31 – 36

Fig. 1. A synchronized product of transition systems.

Several versions of the tool have been developed, until the last one, which is described in [4] and which diDers from Alaiwan’s one by a much larger set of properties which can be checked on a transition system, so that MEC can be considered as a model-checker for a logic described by Dicky [9]. This tool was used to design and check the embedded software of a household electricity meter, as reported in [5] (Fig. 1 is a picture of this meter). Did Maurice

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35

Nivat imagine, when he published his paper [14], that, some years later, synchronized products of transition systems would be installed in thousands and thousands of homes? More recently, the synchronous product has been introduced as one of the conceptual bases of the AltaRica formalism for describing complex industrial systems [6]. This formalism is intended to serve as an input language for a software workbench devoted to risk analysis. In order to comply with industrial needs, the synchronization product has been extended in several ways. Some of these extensions are just notational short cuts, others consist in minor modi;cations of the basic notion of synchronization product, such as the introduction of priorities on the synchronization vectors. Two of them oDer a special interest in the perspective of this review. First, the complex systems are usually designed in a hierarchical way, and direct interactions between components at a diDerent level of the hierarchy are forbidden for modularity reasons (replacement of a subsystem by another one functionally equivalent). Only indirect interactions are allowed. To implement this constraint, we have reintroduced in the synchronization mechanism, the controller that was so early and unwisely given up, but with an additional role: to provide an interface between the lower and upper levels of the hierarchy. Second, we had to renounce the dogma that states play no role in the synchronization mechanism. In fact, in the systems we are interested in, in contrast with purely software systems, some components can communicate and exchange information on their respective states even in the absence of events or actions ( just think of a turbine and a steam generator connected by a pipe). The solution proposed in AltaRica is to split the state of a transition system into two parts, a control part, which remains hidden, and a visible part whose value depends on the value of the control part. For instance, let us imagine that there is a valve between the generator and the turbine. It can be in one of the two control states open or closed. The visible part of a state is the pair of steam pressures Pg and Pt at the generator and the turbine side of the valve, which can be high or low. If the valve is open then Pg = Pt . If it is closed then Pt = low. (This example explains why the visible parts of a state are named “Pows”.) In the generator–valve–turbine system, the information exchanged between the components are expressed by stating that Pg is equal to the pressure delivered by the generator, and that the pressure on the turbine is Pt . An action can modify only the control part of the state, while the visible part is modi;ed either as a consequence of the modi;cation of the control part of the state, or because of interactions with other components (for instance Pt can go from high to low if the valve is closed or if the generator fails). Although this synchronization mechanism can be expressed by synchronization vectors (but at the cost of dramatically increasing the alphabet of actions of each component), it is more natural and more usable to describe complex computerized systems whose components interact in two very diDerent ways, by physical mechanisms (pressure, voltage, etc.) as well as by electronic control devices (control program of a valve). The base of any concurrency theory is the de;nition of concurrent events. The synchronization product of Maurice Nivat is the way he formalized his view of this central question. Like many other notions introduced by him in computer science, it is both simple and powerful, because it directly addresses the core of the problem.

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Since this notion is not “marvellously intricate” it could not be the starting point of a large set of theorems, although it suggested or renewed many problems in language and automata theory. However, along the time, it revealed itself as a very fundamental notion on which many works about concurrent systems and their veri;cation are based. References [1] H. Alaiwan, Algorithmes d’analyse des automates ;nis et applications aux problQemes de synchronisation, Ph.D. Thesis, Universit%e Paris VII, 1983. [2] A. Arnold, Transition systems and concurrent processes, in: G. Mirkowska, H. Rasiowa (Eds.), Mathematical Problems in Computation Theory, Banach Center Publications, Vol. 21, Polish Scienti;c Publishers, 1988, pp. 9–20. [3] A. Arnold, Finite Transition Systems. Semantics of Communicating Sytems, Prentice-Hall, Englewood CliDs, NJ, 1994. [4] A. Arnold, D. B%egay, P. Crubill%e, Construction and Analysis of Transition Systems with MEC, AMAST Ser. Comput., Vol. 3, World Scienti;c, Singapore, 1994. [5] A. Arnold, D. B%egay, J.-P. Radoux, The embedded software of an electricity meter: an experience in using formal methods in an industrial project. Sci. Comput. Programming 28 (1997) 93–110. [6] A. Arnold, G. Point, A. GriDault, A. Rauzy, The AltaRica formalism for describing concurrent systems, Fund. Inform. 40 (2000) 109–124. [7] A. Bergeron, A uni;ed approach to control problems in discrete event processes, RAIRO-Inform. Th%eor. 27 (1993) 555–573. [8] L. Boasson, M. Nivat, Centers of languages, in: Proceedings of the Fifth GI Conference in Computer Science, Lecture Notes on Computer Science, Vol. 104, 1981, pp. 245 –251. [9] A. Dicky, An algebraic and algorithmic method of analyzing transition systems, Theoret. Comput. Sci. 46 (1986) 285–303. [10] C.A.R. Hoare, Communicating sequential processes, Commun. ACM 21 (1978) 666–677. [11] R.P. Kurshan, Modeling concurrent processes, in: Proceedings of Symposia on Applied Mathematics, vol. 31, American Mathematical Society, Providence, RI, 1985, pp. 45 –57. [12] P.E. Lauer, P.R. Torrigiani, M.W. Shields, COSY—a system speci;cation language based on paths and processes, Acta Inform. 12 (1979) 109–158. [13] R. Milner, Processes: a mathematical model of computing agents, in: H.E. Rose, J.C. Shepherdson (Eds.), Proceedings of Logic Collo., 1973, pp. 157–173. [14] M. Nivat, Sur la synchronisation des processus, Rev. Techn. Thomson-CSF 11 (1979) 899–919. [15] M. Nivat, A. Arnold, Comportements de processus, in: Collo. AFCET Les Math%ematiques de l’informatique, 1982, pp. 35 – 68. [16] P.J.G. Ramadge, W.M. Wonham, Supervisory control of a class of discrete events processes, SIAM J. Control Optim. 25 (1987) 206–230. [17] P.J.G. Ramadge, W.M. Wonham, The control of discrete event systems, in: Proceedings of the IEEE, Vol. 77, 1989, pp. 81–98.

Theoretical Computer Science 281 (2002) 37 – 80

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Basic analytic combinatorics of directed lattice paths Cyril Banderier ∗ , Philippe Flajolet Algorithms Project, INRIA, Rocquencourt, 78150 Le Chesnay, France

Abstract This paper develops a uni(ed enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci(ed by a (nite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially one-dimensional objects.) The theory relies on a speci(c “kernel method” that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated c 2002 Elsevier Science B.V. All rights reserved. to several basic parameters of paths.
Keywords: Lattice path; Analytic combinatorics; Kernel method; Singularity analysis; Generalized ballot problem; Catalan numbers

0. Introduction By a lattice path it is meant in all generality a polygonal line of the discrete Cartesian plane Z×Z. The lattice paths to be considered here are speci(ed by a (nite set of simple rules: typically, from each point, there is a (nite set of allowable moves that are both “time independent” and “space independent”. Throughout this study, we also assume the existence of some privileged direction of increase (the horizontal axis, say), so that paths become essentially similar to one-dimensional objects, namely, walks on the line. Such directed lattice paths intervene in many areas of mathematics and computer science. They play a rˆole, for instance, in probability theory (sums of discrete random ∗

Corresponding author. E-mail addresses: [email protected] (C. Banderier), philippe.;[email protected] (P. Flajolet). URLs: http://algo.inria.fr/banderier, http://algo.inria.fr/;ajolet

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variables), statistics (non-parametric tests), formal language theory, random generation of planar diagrams (animals and polyominoes), the analysis of dynamic data structures, and queueing theory models. In probability theory, lattice paths describe the evolution of sums of independent discrete random variables, for instance, the succession of your gains if a die is repeatedly cast and your capital is increased by j when face number j shows up. A typical question in this context is the following: Determine the probability of a “lucky game” in the sense that, at any time t, the partial gain is at least as large as the “mean gain”, 72 t. Such questions are indeed addressed by classical probability theory, with Brownian motion entering the game. However, by design, stochastic processes only provide a (rst-order asymptotic theory, while some purely discrete phenomena remain out of reach of this theory. Statistics, though not our primary motivation in this paper, is historically an other important source of problems regarding lattice paths. We may mention the Kolmogorov– Smirnov test in non-parametric statistics that aims at discerning as to whether two random variates have the same distribution (see, e.g., [47]). As a matter of fact, the early books on lattice path combinatorics and lattice path statistics by Mohanty and Narayana [57,59] speci(cally draw some of their motivations from such questions. In discrete mathematics, all sorts of constrained lattice paths serve to describe apparently complex objects. Two-sorted permutations are for instance equivalent to paths made of horizontal and vertical steps that connect the origin to a point lying on the main diagonal—such facts are directly relevant to the analysis of the mergesort and shellsort algorithms [48,69,74]. Dyck paths that are closely related to diagonal paths describe traversal sequences of general and binary trees; they belong to what Riordan has named the “Catalan domain”, that is, the orbit of structures counted by the Catalan numbers, 1=(n + 1)( 2n n ). The wealth of properties surrounding Dyck paths can be perceived when examining either Gould’s monograph [41] that lists 243 references or from Exercise 6.19 in Stanley’s book [72] whose statement alone spans more than 10 full pages. More generally, trees constrained by degrees—e.g., term trees in free magmas, of interest in formal semantics [60]—are known to be bijectively equivalent to Lukasiewicz words, themselves isomorphic to lattice paths of a special form; Lothaire’s book oNers a good description within the framework of combinatorics on words [52, Chapter 11]. Lattice paths also intervene in the analysis of dynamically evolving structures, and, as such, they surface in the continuous as well as discrete parts of the theory. On the discrete side, we have Flajolet’s combinatorial theory of continued fractions [29] motivated by FranPcon’s theory of “histories” of dynamic data structures [32,36] or Knuth’s dynamic storage allocation model (see [46, 2.2.2–13] for the statement of the problem and [30,75] for solutions). As regards continuous aspects, the Karlin–McGregor theory of birth–death processes (of which [33,58] oNer lattice-path perspectives), itself closely related to various queueing theory models, involves lattice paths that describe an interesting collection of events (the embedded Markov chain). The recent book by Fayolle et al. on random walks in the quarter-plane [26] is historically motivated by such queueing theory questions [25].

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

39

Word representations of lattice paths also provide many examples of context-free languages. This side of the coin is closely related to encodings of trees by words, so that Dyck paths (that are associated to general trees and binary trees) and Motzkin paths (that encode unary-binary trees) play an especially important rˆole. The theory of context-free languages and pushdown automata then combines nicely with the Chomsky–SchQutzenberger theorems [10,73], to the eNect that many types of paths can be a priori recognized as admitting generating functions that are algebraic. Examples are provided by Labelle and Yeh [49,50], Merlini et al. [56], and Duchon [22]. (In return, enumerative studies related to context-free languages can sometimes provide structural information on generation mechanisms and formal languages as is evidenced by the analytic theory of inherent ambiguity of [31].) Finally, because of the rich combinatorics surrounding them, lattice paths intervene at many places in the random generation of structured objects. The problem there is to draw a combinatorial object from some class C, and do so uniformly at random amongst all objects of size n in C. Strong decomposability properties of paths usually make random generation possible in low polynomial time (usually with a complexity between O(n) and O(n2 )). Consequently, any easily computable bijection between a class C and a class of simple enough lattice paths induces a random generation algorithm for C. Known examples include the random generation of two-dimensional diagrams like polyominoes and animals. For instance, the Delest–Viennot methodology of [18] allows us to generate parallelogram polyominoes in linear time; the rejection methods of the “Florence School” [8] make it possible to generate various types of directed lattice animals in a surprisingly eScient manner. The design of such algorithms is clearly dependent on the basic combinatorics of lattice paths while the corresponding performance analyses rely on (ne probabilistic estimates of characteristic properties of paths; see Louchard’s contribution [53] for a neat example and the paper [4] for algebraic techniques related to the present paper. In this introduction, we cannot do more than scratch the surface of such rich combinatorial, probabilistic, and algorithmic aspects of lattice paths. Accordingly we cut short our discussion of motivations at this point. Scope of the paper. This paper assembles combinatorics of words and paths, some algebra of formal power series, and complex analysis. Under this angle, we believe the enterprise to be original. Quite a lot is otherwise known regarding probabilistic properties of paths, as these represent sums of random variables. Accordingly, our treatment can be, to some extent, regarded as a parallel of probabilistic–analytic methods in the realm of enumerative combinatorics. In Section 2, we show that the counting generating functions of paths of various sorts are invariably algebraic functions. This algebraic character is predictable since the word encodings of the object considered are clearly recognizable by deterministic pushdown automata, hence are deterministic context-free languages. However, for directed lattice paths, we demonstrate that a strong algebraic decomposability prevails that is obtained by a speci(c technique, the “kernel method ” (historical remarks are given at the end of Section 2.2) and is not clearly visible on combinatorial and grammatical descriptions. Our purpose in this paper is to arrive eventually at a complete characterization of the singular structure of intervening generation functions (Section 3)—by virtue of

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

the method of singularity analysis, this leads to very precise asymptotic information on the counting quantities involved. At this level also, the decomposability granted by the kernel method is central as it enables us to determine the location and nature of dominant singularities. Then, once the singular structure of counting generating functions has been extracted, tight estimates on probability distributions of parameters follow easily: see Section 4 for a sample of what can be done. Section 5 sketches extensions to the enumeration of certain types of planar objects provided they satisfy a strong directedness condition.

1. Lattice paths and generating functions This section presents the varieties of lattice paths to be studied as well as their companion generating functions. Denition 1. Fix a (nite set of vectors of Z×Z, S= {(a1 ; b1 ); : : : ; (am ; bm )}. A lattice path or walk relative to S is a sequence v = (v1 ; : : : ; vn ) such that each vj is in S. The geometric realization of a lattice path v = (v1 ; : : : ; vn ) is the sequence of points −→

(P0 ; P1 ; : : : ; Pn ) such that P0 = (0; 0) and Pj−1 Pj = vj . The quantity n is referred to as the size of the path. In the sequel, we shall identify a lattice path with the polygonal line admitting P0 ; : : : ; Pn as vertices. The elements of S are called steps or jumps, and we also refer −→

to the vectors Pj−1 Pj = vj as the steps of a particular path. Various constraints will be imposed on paths. In particular, we restrict attention throughout this paper to directed paths de(ned by the fact that if (a; b) lies in S, then necessarily one should have a¿0. In other words, a step always entails progress along the horizontal axis and the geometric realization of the path naturally lives in the half plane Z¿0 × Z. (This constraint rules out paths like the ones occurring in PWolya’s “drunkard problem” as described in the attractive booklet of Doyle and Snell [19]; it also implies that the paths studied can be treated essentially as one-dimensional objects.) The following conditionings are to be considered (Fig. 1). Denition 2. A bridge is a path whose end-point Pn lies on the x-axis. A meander is a path that lies in the quarter plane Z¿0 × Z¿0 . An excursion is a path that is at the same time a meander and a bridge; it thus connects the origin to a point lying on the x-axis and involves no point with negative y-coordinate. A family of paths is said to be simple if each allowed step in S (De(nition 1) is of the form (1; b) with b ∈ Z. In this case, we also abbreviate S as S= {b1 ; : : : ; bm }. In the simple case the size of a path coincides with its span along the horizontal direction, that is, its length. The terminology of bridges, meanders, and excursions is chosen to be consistent with the standard one adopted in Brownian motion theory; see, e.g. [62].

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

41

Fig. 1. The four types of paths: walks, bridges, meanders, and excursions and the corresponding generating functions.

The main objective of this paper is to enumerate exactly as well as asymptotically paths, bridges, and meanders, this with special attention to simple families. Once the set of steps is (xed, we let W and B denote the set of paths and bridges, respectively (W being reminiscent of “walk”); we denote by M and E the set of meanders and excursions. Given a class C of paths, we let Cn denote the subclass of paths that have size n, and, whenever appropriate, Cn; k ⊂ Cn those that have (nal vertical abscissa (also known as “(nal altitude”) equal to k. With the convention of using standard fonts to denote cardinalities of the corresponding sets (themselves in calligraphic style), Cn = card(Cn ) and Cn = card(Cn; k ), the corresponding (ordinary) generating functions (GFs) are then

Cn z n ; C(z; u) = Cn;k uk z n : C(z) := n

n;k

This paper is entirely devoted to characterizing these generating functions: they are either rational functions (W ) or algebraic functions (B; E; M ). As we shall see, a strong algebraic decomposition prevails which, as opposed to other approaches, renders the calculation of the GFs eNective. Even more importantly, the decomposability of GFs makes it possible to extract their singular structure, and in turn solve the corresponding asymptotic enumeration problems in a wholly satisfactory fashion. Weighted paths. For several applications, it is useful to associate weights to single steps. In this case, the set of steps S is coupled with a system of weights = {w1 ; : : : ; wm }, with wj ¿0 the weight associated to (aj ; bj ) ∈ S; the weight of

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

a path is then de(ned as the product of the weights of its individual steps. Then the quantity Cn , still referred to as number of paths (of size n), represents the total weight of all paths of size n. Such weighted paths cover several situations of interest: (i) combinatorial paths in the standard sense above when each wj =1; (ii) paths with coloured steps, e.g., wj =2 means that the corresponding
step (aj ; bj ) has two possible coloured incarnations (say blue and yellow); (iii) wj = 1 corresponds to a probabilistic model of paths where, at each stage, step (aj ; bj ) is chosen with probability wj . 2. Algebraic structures and the kernel method In this section, we characterize the generating functions of the four types of directed paths (unconstrained, bridges, meanders, and excursions). For ease of exposition, we restrict attention to simple families of paths till Section 5, where we brie;y discuss the more general directed models. It will be seen that a speci(c algebraic curve, the “characteristic curve” plays a central rˆole. In this section, a modicum of analysis is introduced for convenience, but it is limited to the vicinity of z = 0, and consequently, it is largely equivalent to formal series manipulations. 1 Denition 3. Let S= {b1 ; : : : ; bm } be a simple set of jumps, with = {w1 ; : : : ; wj } the corresponding system of weights (wj ≡ 1 in the unweighted case). The characteristic polynomial of S is de(ned as the polynomial in u; u−1 (a Laurent polynomial) P(u) :=

m
j=1

wj u b j :

Let c = − minj bj and d= maxj bj be the two extreme vertical amplitudes of any jump, and assume throughout c; d¿0. The characteristic curve of the lattice paths determined by S is the plane algebraic curve de(ned by the equation 1 − zP(u) = 0;

or equivalently

uc − z(uc P(u)) = 0:

(1)

The quantity K(z; u) := u c − zu c P(u) is also referred to as the kernel and Eq. (1) as the kernel equation. As we shall see the characteristic equation plays a central rˆole, the second form being the entire version (that is, a form without negative powers). We also need to introduce technical conditions on periodicities. In a coin-tossing game (S= {−1; +1}) for instance, a bridge or an excursion only exists for even lengths; consequently, what is observed of a random path at time n depends on the residue class of n modulo 2 (Fig. 2). 1 Following a remark by a referee, we note that analyticity considerations in this section could be logically dispensed with; see Gessel’s paper [38] for a proper framework. However, the authors’ feeling is that purely algebraic proofs, though feasible, tend to be less transparent. More importantly, analyticity considerations developed here serve as a useful preparation for our “non-local” treatment of singularities in the next section.

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43

8 10 6

4 5 2

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

–2 –5 –4

–6 –10 –8

Fig. 2. Fragments of the sublattices accessible from the origin by the Dyck walk (S= {−1; +1}) and Duchon’s clubs (S= {−2; +3}). The periods are 2 and 5, respectively.


Denition 4. A Laurent series h(z) = n¿−a hn z n is said to admit period p if there exists a Laurent series H and an integer b such that h(z) = z b H (z p );

(2)

the largest p such that a decomposition (2) holds is called the period of h and is denoted by per(h). The series h is called aperiodic if per(h)=1. A simple walk de(ned by the set of jumps S is said have period p if the characteristic polynomial P(u) has period p. A simple walk is said to be reduced if the gcd of the jumps is equal to 1. In what follows, we systematically restrict attention to reduced walks since, up to a linear change of abscissa, any walk can be reduced. For instance, the walks corresponding to U= {−3; +3} are transformed (upon shrinking the vertical axis by a factor of 13 ) into the reduced form S= {−1; +1}. (Aperiodic walks are from their definition automatically reduced.) Periodic walks live on sublattices: the walks associated to S= {−1; +1} (Dyck walks) and T= {−1; 0; +1} (Motzkin walks) are naturally reduced, but Dyck walks are periodic with p = 2 (since uP(u) = 1 + u2 ), while Motzkin walks are aperiodic; “Duchon’s clubs” studied below and de(ned by S= {−2; +3} have period p = 5 (since u2 P(u) = 1 + u5 ), etc. Notice that, if we write P(u) =

m
j=1

wj u b j ;

wj = 0;

bj ∈ Z;

the period of P (and of the set of jumps S) is p = per(P) = gcd(b2 − b1 ; : : : ; bm − b1 ):

(3)

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

Also, by the strong form of the triangle inequality, for an aperiodic P(u), the strict inequality holds in |P(u)| ¡ P(|u|)

for all u ∈ C\R¿0 :

(4)

It proves convenient to rewrite P(u) =

d
k=−c

pk u k :

Examination of the asymptotic regimes consistent with the characteristic equation near z = 0 shows that the equation can only be satis(ed if one of the two relations pd zud ∼ 1 or p−c zu−c ∼ 1

(z → 0)

(5)

is satis(ed. The characteristic equation being of degree c + d in u is known to have generically c+d roots; these constitute the branches of a single algebraic curve de(ned by (1) and called the characteristic curve. Then, as suggested by (5), one expects, in the complex domain (for z near 0), c “small branches” that we write as u1 ; : : : ; uc and d “large branches” v1 ≡ uc+1 ; : : : ; vd ≡ uc+d satisfying (Fig. 3) uj (z) ∼ e2i(j−1) =c (p−c )1=c z 1=c ;

vk (z) ∼ e2i(1−k)

=d

(pd )−1=d z −1=d :

(6)

For determinacy, one restricts attention to the complex plane slit along the negative real axis, which allows us to talk freely of the individual branches in the sequel. The informal discussion summarized by (6) is vindicated by the classical theory of Newton–Puiseux expansions—the fundamental result in the elementary theory of algebraic curves that determines constructively all the possible behaviours of solutions of polynomial equations. For an exposition, we refer to one of the many excellent books on the basic theory of algebraic curves, e.g. [1,45]. Precisely, the general theory teaches us that the small branches are conjugate of each other at 0, and similarly for the large branches at ∞. This means that there exist functions A and B analytic at 0 and non-zero there, such that, in a neighbourhood of 0, one has uj (z) = !j−1 z 1=c A(!j−1 z 1=c ) = u1 (e2i(j−1) z); vk (z) = $1−k z −1=d B($k−1 z 1=d ) = v1 (e2i(k−1) z);

! = e2i

=c

$ = e2i

=d

:

(7)

In summary, the uj and v‘ organize themselves into two “cycles” of c and d elements, respectively; for analytic details, we refer to Hille’s crisp presentation based on monodromy and analytic continuation in [44]. The branch u1 de(ned near 0 by (6) is real positive and is called the principal (small) branch. The graph of branches is obtained by interchanging the axes in the graph of 1=P(u), with u1 appearing as the real positive branch near the origin; see Fig. 3 for an example. We shall prove in Section 3 that in a proper sense u1 “dominates” all the other small branches.

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80 40

1

30

0.8 0.6

20

0.4

10 –4

–3

–2

–1

0

45

0.2 1

–10

2 u

3

–4

4

–3

–2

–1

(τ,ρ)

0 –0.2

1

u

2

3

–0.4

–20

–0.6

–30

–0.8

4

–40

–1

u 2

v1 u1

–0.4

–0.2

u2

0.2 z

0.4

–2

–4

Fig. 3. Graphs associated to the set of jumps S= {−2; −1; 0; 1; 2; 3}, with characteristic polynomial P(u) = u−2 + u−1 + 1 + u + u2 + u3 : Top: the graphs of P(u) and 1=P(u) for real u. Bottom: the three real branches of the characteristic curve, one large of order z −1=3 , and two small of order ±z 1=2 (two complex branches of order e±2i =3 z −1=3 are not shown).

2.1. Walks and bridges We start with the easy case of unconstrained walks and bridges. This already makes use of the characteristic curve and some of its branches. Theorem 1. The bivariate generating function (BGF) of paths (with z marking size and u marking ;nal altitude) relative to a simple set of steps S with characteristic polynomial P(u) is a rational function. It is given by 1 W (z; u) = : (8) 1 − zP(u) The GF of bridges is an algebraic function given by B(z) = z

c u (z)
d j =z log(u1 (z) · · · uc (z)); u (z) dz j=1 j

(9)

where the expressions involve all the small branches u1 ; : : : ; uc of the characteristic curve (1). Generally, the GF Wk (z) of paths terminating at altitude k is, for −∞¡k¡c,   c c uj (z)
z d
−k (10) =− uj (z) Wk (z) = z k+1 k dz j=1 j=1 uj (z)

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

and for −d¡k¡ +∞, Wk (z) = −z

d
j=1

vj (z) z d = k+1 vj (z) k dz



d


j=1

 vj (z)

−k

;

(11)

where v1 ; : : : ; vd are the large branches. (For W0 (z), the second form is to be taken in the limit sense k → 0.) Proof. Set wn (u) = [z n ]W (z; u), the Laurent polynomial that describes the possible altitudes and the number of ways to reach them in n steps. We have w0 (u) = 1, w1 (u) = P(u), and wn+1 (u) = P(u)wn (u), so that wn (u) = P(u)n for all n. The determination of W (z; u) in (8) follows from
n¿0

P(u)n z n =

1 ; 1 − zP(u)

where the sum converges and represents an analytic function of both arguments for |z|¡1=P(|u|). Observe that the resulting series is entire in z but of the Laurent type in u (it involves arbitrary negative powers of u). For positive u, the radius of convergence of W (z; u) viewed as a function of z is exactly 1=P(u). Also, by dominance of coeScients (one has Bn 6P(1)n ), the radius of convergence of B(z) as a function of z is at least 1=P(1). Consider now |z|¡r, where r := 12 P(1)−1 . Then, since 1=P(u) is continuous and unimodal for u ∈ (0; +∞) (where P  (u)¿0, so that P is convex) and 1=P(0) = 1=P(∞) = 0, there exists an interval (&; ') such that for &6u6', one has 1=P(u)¿r. More generally, by positivity of the coeScients, the function W (z; u) is seen to be analytic in the product domain (z; u) ∈ {z | |z|¡r} × {u | &¡|u|¡'}: Thus, by Cauchy’s formula applied to the function W (z; u) (viewed now as a function of u analytic in a crown), one has 2  1 du B(z) = [u0 ]W (z; u) = W (z; u) : 2i u |u|=(&+')=2 Take z small enough, so that all the large branches that escape to in(nity lie outside of |u|6(& + ')=2 and the small branches are all distinct. Then, only the small branches remain inside, and, since there are only simple poles, one has   1 1 =− : (12) Res u=uj u(1 − zP(u)) zuj P  (uj ) 2

We make use of the conventional notation for coeScients of entire and Laurent series: [z n ]




n fn z

n

:= fn .

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

47

The integration contour is shrunk to 0, which is legitimate since W (z; u) remains O(1), and residues are taken into account. The residue theorem then gives B(z) as a sum of residues of form (12) over all small branches. The formula simpli(es to (9) since diNerentiation of the characteristic equation shows that P  (u)−1 = −z 2 u for any branch u. The same procedure is applicable to  1 du Wk (z) ≡ [uk ]W (z; u) = W (z; u) k+1 : 2i u |u|=(&+')=2 The integration contour can be shrunk to zero provided the integrand (which is of order u c−k−1 ) remains bounded as u → 0, which necessitates k6(c − 1). The result of (10) follows again from a residue calculation involving small branches. (The proof shows the formulX to be valid in a small enough neighbourhood of the origin. The identities are then a posteriori valid as identities between formal (fractional) power series.) When k¿ − d, which covers case (11) of an arbitrary positive k, the residue calculation is completed by extending the contour to a large circle at ∞; in this case, the large branches contribute. The algebraic character of B(z) and the Wk (z) (nally results from the well-known fact that algebraic functions are closed under sums, products, and multiplicative inverses. The quantity B(z) ≡ W0 (z) is equivalently given as the diagonal of a bivariate rational function,  
1 zn [z n ucn ] B(z) = 1 − zuc P(u) n and as such it must be algebraic: see PWolya’s paper [63] of 1921 and [37] for developments regarding diagonals of rational functions. Example 1 (Central binomial and trinomial numbers). These are perhaps the most famous examples, associated to the sets S= {−1; +1} and T= {−1; 0; +1}. The corresponding polynomials are P S (u) = u−1 + u and P T (u) = u−1 + 1 + u. In this case, the characteristic curve is of degree 2 and there is only one small branch, namely √ √ 1 − 1 − 4z 2 1 − z − 1 − 2z − 3z 2 u1S (z) = ; u1T (z) = : 2z 2z The algebraic generating functions of bridges are then BS (z) = √

1 = 1 + 2z 2 + 6 z 4 + 20 z 6 + 70 z 8 + 252 z 10 + · · · ; 1 − 4z 2

BT (z) = √

1 = 1 + z + 3z 2 + 7z 3 + 19z 4 + 51z 5 + · · · ; 2 1 − 2z − 3z

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

the coeScients being 3 EIS A000984 and EIS A002426   2n n S n 2 n [z ]B (z) = [t ](1 + t ) ≡ ; [z n ]BT (z) = [t n ](1 + t + t 2 )n : n The names of central binomial and trinomial numbers are suggested by the usual expansions of (1 + t 2 )n and (1 + t + t 2 )n : 1 1 1 +1t +t 2 1 + t2 1 +2t +3t2 +2t 3 +t 4 1 + 2t2 + t 4 2 4 6 1 + 3t + 3t + t 1 +3t +6t 2 +7t3 +6t 4 +3t 5 +t 6 4 2 6 8 1 + 4t + 6t + 4t + t 1 +4t +10t 2 +16t 3 +19t4 +16t 5 10t 6 +4t 7 +t 8 It is notable that these cases were already considered by Euler [24], who also gave linear recurrences (with polynomial coeScients) satis(ed by BnT . 2.2. Meanders and excursions In this section, we consider meanders, that is paths that never go below the horizontal axis. The meanders whose (nal altitude is 0 are called excursions, in accordance with De(nition 2, and they turn out to be the objects with the richest combinatorial properties. We continue with a simple system of paths de(ned by the set of jumps S, possibly endowed with weights. The new generating functions will again involve the characteristic curve together with its small and large branches. Let now Fn; k be the number of meanders of size (i.e., length) n that end at altitude k. The corresponding BGF is
Fn;k uk z n ; F(z; u) := n;k

which is now an entire series in both z and u. By the combinatorial origin of the problem, F(z; u) is bivariate analytic for |u|61 and |z|¡1=P(1). We also make use of the polynomials fn (u) that describe the possible positions after n steps and write

F(z; u) = fn (u)z n = Fk (z)uk : (13) n¿0

k¿0

Combinatorially, the natural decomposition is the one based on the last step added. For the fn (u), “adding a slice” is translated by the recurrence, f0 (u) = 1;

fn+1 (u) = P(u)fn (u) − {u¡0 }P(u)fn (u):

(14)

There, the notation {u¡r }g(u) means the sum of all the monomials with exponent less than r that appear in the Laurent series g(u) as   +∞ r−1

{u¡r } gj uj := gj u j : (15) j=−a

j=−a

3 References to EIS point to Sloane’s Encyclopedia of Integer Sequences [70], of which a version also exists in print [71].

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

49

Then, multiplying the terms of the recurrence by z n and summing yields F(z; u) = 1 + zP(u)F(z; u) − z{u¡0 }(P(u)F(z; u));

(16)

where {u¡0 } is to be understood as applied to the u-expansion of F(z; u) in (13). Relation (16) is the fundamental functional equation de(ning meanders. It reads as follows: “A path is either the empty path or it consists of a step (zP(u) describes the possibilities) added to a path except that the steps that would take the walk below level 0 (the operator {u¡0 }) are to be taken out”. Now, P involves only a (nite number of negative powers, so that F(z; u)(1 − zP(u)) = 1 − z

c−1
k=0

rk (u)Fk (z)

(17)

for some Laurent polynomials rk (u) that are immediately computable from P via (16): rk (u) := {u¡0 }(P(u)uk ) ≡

−k−1
j=−c

pj uj+k :

(18)

Theorem 2. For a simple set of steps, the BGF of meanders (with z marking size and u marking ;nal altitude) relative to a simple set of path S is algebraic. It is given in terms of the small and large branches of the characteristic curve of S by c d 1 1  j=1 (u − uj (z)) F(z; u) = c =− : (19) u (1 − zP(u)) pd z ‘=1 (u − v‘ (z)) In particular the GF of excursions, E(z) = F(z; 0); satis;es E(z) =

c d 1 (−1)d−1  (−1)c−1  uj (z) = : p−c z j=1 pd z ‘=1 v‘ (z)

(20)

Proof. The point is that the fundamental equation in its form (17) looks grossly underdetermined as it involves (c + 1) unknown functions; to wit, the bivariate F(z; u) and c−1 the univariate {Fk (z)}k=0 . The main idea of a method known as the “kernel method” (see also historical notes below) consists in binding z and u in such a way that the left-hand side vanishes. Indeed, substitute in (17) any small branch of the characteristic equation. Take |z|¡1=P(1) and restrict z to a small neighbourhood of the origin in such a way that: (i) all the small branches are distinct; (ii) all the small branches satisfy |uj (z)|¡1. Then the substitution is analytically legitimate and, taking all small branches into account, it provides a system of c equations in the unknown functions F0 ; : : : ; Fc−1 : u1c − z

c−1
k=0

u1c rk (u1 )Fk = 0

.. . ucc − z

c−1
k=0

ucc rk (uc )Fk = 0:

(21)

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

This system is non-singular for the reason that its determinant is a variant of the Vandermonde determinant and the small branches are clearly all distinct. This observation is enough to justify that each of the Fk is an algebraic function expressible rationally in terms of the algebraic branches uj . Instead of pursuing in the direction of determinantal calculations, here we make use of a cute observation of Bousquet-MWelou (introduced in [13] and employed in the parallel paper [4]). The quantity N (z; u) := uc − z

c−1
k=0

uc rk (u)Fk

(22)

is by (21), a polynomial in u whose roots are precisely all the uj . The leading monomial of this polynomial is u c , so that the polynomial factorizes as N (z; u) =

c 

(u − uj (z)):

j=1

(23)

Then, the constant term is at the same time the product (−1)c u1 · · · uc and the quantity −zp−c F0 , as is apparent from de(nition (22) and form (18) of the coeScients. The form of F0 follows. Finally, the result for the BGF F(z; u) derives from (17) made entire, F(z; u) =

N (z; u) uc (1 − zP(u))

and from the factorization (23). An immediate corollary of Theorems 1 and 2 is the generating function of all paths and meanders irrespective of their (nal altitude. Corollary 1. The generating functions of all paths and all meanders are W (z) ≡ W (z; 1) = M (z) ≡ F(z; 1) =

1 ; 1 − zP(1) c d  1 1 1  (1 − uj (z)) = − : 1 − zP(1) j=1 pd z ‘=1 1 − v‘ (z)

A somewhat deeper consequence is a direct relation between the GFs of excursions and bridges that obtains by comparing Eqs. (9) and (20). Corollary 2. The generating functions of bridges (B) and excursions (E) are related by E  (z) d (log E(z)) = 1 + z ; E(z) dz   z dt : (B(t) − 1) E(z) = exp t 0 B(z) = 1 + z

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

51

In the same vein, consider paths whose intermediate steps may be negative, but with a (nal altitude that is ¿0. Their BGF is W + (z; u) :=

∞
k=0

Wk (z)uk :

Then, comparison of the forms involving large branches for Wk (z) and F(z; u) and a trite calculation shows that W + (z; u) = 1 + z  F(z; u) = exp

d (log F(z; u)); dz z

dt (W (t; u) − 1) t 0 +

 :

Finally, with Fk (z) being the generating function of meanders that end at altitude k, one has Fk (z) = [uk ]F(z; u). Since F(z; u) is a rational function of u with a simple product expression in terms of the large branches, its expansion with respect to u is easily accessible via a partial fraction decomposition, and one (nds: Corollary 3. The generating function of meanders terminating at altitude k is Fk (z) =

d 1
.‘ v‘−k−1 ; pd z ‘=1

.‘ :=

 j=‘

1 : vj − v‘

Some of these relations admit combinatorial interpretations succinctly discussed in Section 4.1. Example 2. Ballot problem, Dyck paths, and Motzkin paths. These are the most famous problems in the area, and they are closely related to Example 1. The ballot problem asks for the probability, in a two candidate election between A and B that eventually results in a tie, of A dominating B throughout the poll. Recording the diNerence between the scores of A and B as time evolves, we model the problem as the counting of excursions associated with S= {−1; +1}. The characteristic curve is the one examined in Example 1 in connection with central binomial coeScients and the GF of excursions is   √ 2n
1 1 − 1 − 4z 2 S z 2n ; E (z) = = 2 2z n n¿0 n + 1 where the coeScients 1=(n + 1)( 2n n ) are the Catalan numbers (EIS A000108). For T= {−1; 0; +1}, one (nds similarly √
T n 1 − z − 1 − 2z − 3z 2 T E (z) = = En z ; 2 2z n¿0 where the coeScients are the Motzkin numbers (EIS A001006).

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Example 3. Lukasiewicz paths and tree codes. Consider generally a (nite set / that contains −1 as single negative value. The corresponding paths are known as Lukasiewicz paths. Set 0(u) := uP(u), which is a polynomial. There is only one small branch satisfying u1 (z) = z0(u1 (z))

(24)

and the GF of excursions is (1=zp−1 )u1 (z). vukasiewicz paths of type / encode trees whose node degrees are constrained to lie in 1 + /, this by virtue of a well-known correspondence [52, Chapter 11]. (Traverse the tree in preorder and output a step of d − 1 when a node of outdegree d is encountered.) In this way, it is seen that Eq. (24) gives the GF of trees counted according to the number of their nodes, an otherwise classical result [55]. By Lagrange inversion, the number of trees comprised of n nodes is Tn =

1 n−1 [w ]0(w)n ; n

where 0 can be directly interpreted as the characteristic polynomial of the allowed node (out)degrees. Example 4. Walks with steps in {−2; −1; 0; +1; +2}. This is our (rst example involving inherently more than one branch. The characteristic equation is u2 − z(1 + u + u2 + u3 + u4 ) = 0: The two small branches are conjugate and given by u1 (z) = +z 1=2 + 12 z + 58 z 3=2 + z 2 +

231 5=2 128 z

+ 3z 6 + · · · ;

u2 (z) = −z 1=2 + 12 z − 58 z 3=2 + z 2 −

231 5=2 128 z

+ 3z 6 + · · · :

Then, by (20), the (rst few terms of E(z) are easily determined as E(z) = −

u1 (z)u2 (z) = 1 + z + 3z 2 + 9z 3 + 32z 4 + 120z 5 z

+473z 6 + 1925z 7 + · · · : Similarly, for meanders, one has M (z) =

(1 − u1 (z))(1 − u2 (z)) = 1 + 3z+12z 2 +51z 3 +226z 4 + 1025z 5 + · · · : 1 − 5z

It is then a natural question to ask for an equation satis(ed directly by E(z) or F(z; 1). Regarding excursions, an equation may be obtained by elimination of u1 ; u2 from the system zE + u1 u2 = 0;

u12 − z(1 + u12 + u13 + u14 ) = 0;

u22 − z(1 + u2 u22 + u23 + u24 ) = 0:

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

53

Either resultants or GrQobner bases do the job. For instance, resultants give a polynomial equation of degree 12 satis(ed by E(z). The polynomial factorizes (this is expected as we did not impose conditions like u1 = u2 in the process). Eventually, it is found that E(z) satis(es a polynomial equation of degree 4: z 4 y4 − z 2 (1 + z)y3 + z(2 + z)y2 − (1 + z)y + 1 = 0:

(25)

We shall examine shortly a much better way to perform such computations. Example 5. Duchon’s clubs and underdiagonal paths. The following problem 4 was considered by Duchon [22] (under a diNerent formulation): A club opens in the evening and closes in the morning. People arrive by pairs and leave in threesomes. What is the possible number of scenarios from dusk to dawn as seen from the club’s entry? For instance, an event may be +2 (two enter), +2 (two more enter), −3 (three leave), +2 (two, again arrive), −3 (and the club closes). Naturally, the population inside the club is never negative and a business night starts with the empty club and ends with the empty club. The generalized problem then calls for the number of excursions with step ˆ {−3; +2} or, equivalently by time reversal, set {−c; d} (where Duchon’s case is S= S= {−2; +3}). We assume here without loss of generality that c and d are coprime integers, so that the system of paths is reduced. The characteristic polynomial is P(u) = u−c + ud and the kernel equation is equivalent to uc = z(1 + ue )

with e = c + d:

Thus, the period is e = c + d and the horizontal axis is only touched at places that are a multiple of e. Set z = t c , where t is a local uniformizing parameter at 0. Then, the quantity y(t) := u1 (t c ) satis(es the equation y = t(1 + ye )1=c ; which is Lagrangean. By Lagrange inversion [42], one (nds   n=c
1 tn: (26) y(t) = (n − 1)=e n¿1 n (By convention, ( ab ) = 0 if b is non-integral.) Let ! be a primitive cth root of unity; then all the branches admit an expansion similar to y(z). Indeed, by conjugacy, one has
yn !nj t n ; uj+1 (t c ) = y(!j t) = n¿1

where yn = [t n ]y(t) is given by (26). Then, the number of excursions is a convolution:
yn1 yn2 · · · ync !0n1 +1n2 +···+(c−1)nc : (−1)c−1 En = n1 +···+nc =c(n+1)

4

After this paper had been submitted, Christian Krattenthaler pointed us to Ref. [68] by Sato, dating from 1989. In that paper, Sato derives directly our Eq. (27) by matrix generating function methods and provides valuable additional results regarding underdiagonal paths in a strip.

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

It can be checked that En is automatically zero unless n ≡ 0 (mod e) (see also the discussion on periodicities in Section 3.3 below). In summary, taking ! any primitive cth root of unity, and setting nj = 1 + e3j , n = e3, we (nd Ee3 =




1 1 + 31 e

31 +···+3c =c3



1 ··· 1 + 3c e



(1 + 31 e)=c



31

(1 + 3c e)=c

 !031 +132 +(c−1)3c :

3c

(27)

In particular, for c = 1, no summation is needed and 1 1 + ne



1 + ne n



gives the number of excursions of length n and type {−1; e − 1}, which is also the number of e-ary trees having n internal nodes (Example 3). If c = 2 formula (27) yields a single convolution. For S= {−2; 3}, the result is E5n

2n (−1)3
= 3=0 1 + 53



(1 + 53)=2 3



1 1 + 5(2n − 3)



(1 + 5(2n − 3))=2 2n − 3



to be compared to E5n =

n
i=0

1 5n + i + 1



5n + 1 n−i



5n + 2i i

 ;

(28)

which Duchon obtained from quite speci(c series manipulations. In general, if the jump in the negative direction is −c, formula (27) is a (c − 1)-fold convolution of binomial coeScients. Duchon’s clubs can also be interpreted as underdiagonal paths. Consider paths in the Z¿0 × Z¿0 lattice whose allowed steps are of type either East (horizontal) or North (vertical), with a straight line barrier 5. It is assumed that 5 passes through the origin and has a rational slope, p=q61. The number of ways Nm; n of reaching point (m; n) by North and East steps then satis(es a recurrence of the same type as Pascal’s triangle but with boundary conditions. For instance, the case of slope 1 gives rise to the original formulation [54] of the ballot problem (Example 2). If one measures at each step of a path the vertical distance to 5, then, this distance can only evolve by +(p=q) for a horizontal step and −1 for a vertical step. Thus, up to rescaling, such an underdiagonal path is equivalent to a Duchon path of type {−q; +p}. The numbers Nm; n are then amenable to the analysis of the paper since their

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

55

determination is equivalent to counting meanders and excursions. For instance, here is a table of values for slope 23 :

1

1

1 1

2 2 1

5 3 1

9 9 4 1

23 23 14 5 1

66 43 20 6 1

136 136 70 27 7 1

377 377 241 105 35 8 1

1144 767 390 149 44 9 1

The sequence of numbers in this array that correspond to the number of ways of touching the boundary line is (EIS A060941) 1; 2; 23; 377; 7229; 151491; 3361598; 77635093; 1846620581; : : : ; which precisely coincides with the sequence of Duchon numbers, {E5n }n¿0 , in (28). Related enumerative results have been obtained by Durand [23] in the context of the “klam” recurrence that arises in complexity theory. Mohanty [57, p. 22] even quotes results of TakWacs relative to underdiagonal paths under a line of arbitrary slope. As the last example shows, the decomposability aNorded by the kernel method provides a grasp on the structural complexity of summatory formulX expressing the number of walks, excursions, etc. Following Comtet [15, p. 216], we observe that the “rank” (de(ned as the minimal number of summations) of the excursion formula in the general case is at most c(q − 1) − 1 if P(u) comprises q terms. For instance, Catalan numbers ((c; q) = (1; 2)) are of rank 0, Motzkin numbers ((c; q) = (1; 3)) and the Duchon numbers En of (28) (having (c; q) = (2; 2)) are of rank 1, etc. Some origins of the kernel method. What we named here the “kernel method” has been part of the folklore of combinatorialists for some time. Earlier references usually deal with the case of a functional equation of the form K(z; u)F(z; u) = A(z; u) + B(z; u)G(z) (with F; G the unknown functions), when there is only one small branch, u1 , such that K(z; u1 (z)) = 0. In that case, a single substitution does the job, and G(z) = −A(z; u1 )= B(z; u1 ). One clear source of this is the exercise section of the (rst edition (in 1968) of Knuth’s book [46]: the detailed solution to Exercise 2.2.1– 4 (see [46, pp. 536–537] and also Example 2.2.1.11) presents a “new method for solving the ballot problem”, for which the characteristic equation is quadratic. See also Odlyzko’s splendid survey [61, Section 15.4] for a discussion of a pebbling game and Prodinger’s recent note [64] for an original application to a quadratic problem arising from queueing theory. The kernel method in its more general version was used recently in a few unpublished works by the authors, including a systematization to directed lattice paths by Banderier in his memoir [2]. Independent combinatorial developments at the end of the last century are due to Bousquet-MWelou and PetkoYsek whose recent paper oNers

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

a penetrating perspective on the subject of multidimensional walks, recurrences, and kernels [13]. In fact, as indicated earlier, a remark of Bousquet-MWelou has been used to simplify our proof of Theorem 2 (see also [4] for another application). That probabilists had known a lot since the early 1950s regarding related questions is manifest upon reading Chapter XII of Fellers’ book [28]. It appears that our presentation parallels in some ways what is obtained by the famous Wiener– Hopf approach: refer in particular to the example on bounded arithmetic distributions in [28, pp. 407–408]. Such techniques prove in turn valuable in the theory of queueing systems: see, e.g., Robert’s book [66] for an account. The synthesis by Fayolle et al. [26] exposes the deep rami(cations of the theory in the harder case of walks in a quarter plane not satisfying directedness restrictiction (thus, a “pure” two-dimensional problem), but their methods only apply to nearest-neighbour moves. The book [26] itself draws some of its inspiration from the early paper [25] where a sophisticated use of the kernel method already plays a central rˆole (amongst other techniques like conjugacy and Riemann–Hilbert problems); see also the references to Flatto and Malyshev’s works in [61, p. 1208] and the historical comments in [26, pp. VII–XI]. 2.3. Computational aspects We discuss now a way to determine directly the equations satis(ed by the algebraic functions encountered so far. Because of Corollary 2, we know that bridges and excursions are tightly coupled, and the case of excursions will be detailed here. It is assumed that the characteristic polynomial P(u) is (xed. Then, what is needed in view of Theorem 2 is the equation satis(ed by the product Y = u1 · · · uc of c distinct roots of a polynomial of degree c + d. As roots are in general “indistinguishable”, we expect a polynomial of degree ( c+d c ) to cancel Y . Take a polynomial Q(u) of degree e in C(z)[u] normalized by Q(0) = 1 and assume it has distinct roots u1 ; : : : ; ue . For us, e = c + d, and Q(u) = −

1 (uc − zuc P(u)); zp−c

yet another reformulation of the kernel. We (rst develop the computational process when c = 2, so that the equation for Y = u1 u2 with u1 ; u2 two distinct roots of Q is sought. Write &; & for generic roots of Q. Then, since Q(0) = 1, one has Q(u) =

 &

1−

u ; &

while what we need to determine is   u R(u) = 1−  : && {&;&
}

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

57

(A sum or product over {&; & } means a sum or product over all unordered pairs of distinct elements.) Now, take logarithms. One has  

1 un 1 = Sn ; log with Sn := n Q(u) n & & n¿1  
(2) un
1 1 = Sn : log with Sn(2) := n & n R(u) n &
n¿1 {&;& } Then, a simple combinatorial reasoning shows that
{&;&
}

1

&n & n

=

1 1
1
1 ; n − n  2 (&;&
) & & 2 & &2n

so that Sn(2) = 12 Sn2 − 12 S2n :

(29)

The degree of R is ; := ( 2e ) a priori, and R can be recovered from the formula (“I am always the exponential of my logarithm!”) 

 ;
1 2 un R(u) := {u6; } exp − ; (30) (Sn − S2n ) n n=1 2 where {u6; } f means the truncation of the series expansion of f with all terms of degree 6; included (see the analogous notation (15)). The general formulX for c¿2 are easily found from the usual relations between elementary and power sum symmetric functions. Set xj = &j−n . What is sought is plainly a formula expressing the sum
the sum being on all subsets of c elements. Then we have Sn(c) =
1 4 24 Sn

2 − 14 Sn2 S2n + 13 Sn S3n + 18 S2n − 14 S4n :

(32)

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C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

These considerations give rise to a simple algorithm for computing the polynomial cancelled by the product of all small branches. Platypus Algorithm. Computes the polynomial R(u) ∈ C(z)[u] of degree ; = ( ec ) such that R(Y ) = 0, where Y = u1 · · · uc = (−1)c−1 zpc E(z) is the product of all small branches of the characteristic curve. The input is the characteristic polynomial of steps, P(u). 1. Set up the symbolic formulX of type (29) and (32) appropriate for the given value of c. To this eNect, perform the symbolic expansion of (31) with
 ;
un 6; : R(u) := {u } exp −
2(n + 7)(n + 8)(2n + 13)En+6 + · · · + 625(n + 1)(n + 2)(n + 3)En = 0:

(34)

3. Singular structures We now examine paths, bridges, meanders and excursions under the angle of asymptotics. As is well known, the asymptotic behaviour of counts is closely related to the

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

59

2.5

|v2 |, |v3 |

2

v1 u

|u 1 |, |v1 |

1 0.5

u1

0.1

|u 2 | z

0.2

0.3

Fig. 4. A rendering of the modulus of the (ve branches of the characteristic curve in the example of Fig. 3 illustrates the domination properties of the principal small and large branches.

singular structure of the corresponding generating functions [34,61]. Thanks to the factorizations aNorded by the kernel method, the singular forms of intervening generating functions become manageable. This part of the analysis makes use of global properties of branches followed by local analysis in the vicinity of a quantity called the “structural radius” ?. Lemma 1. Let P(u) be the polynomial associated to the steps of a simple walk. Then, there exists a unique number @, called the structural constant, such that P  (@) = 0;

@ ¿ 0:

The structural radius is de;ned by the quantity ? :=

1 : P(@)

Proof. DiNerentiating P twice as given in (3), we see that P  (x)¿0 for all x¿0. Thus, the real function x → P(x) is strictly convex. Since it satis(es P(0) = P(+∞) = +∞, it must have a unique positive minimum attained at some @, and P  (@) = 0. Structural constants a priori live in a (eld of degree e := c + d over the base (eld of weights. However, for symmetric walks (P(u) = P(u−1 )), they automatically reduce to the value @ = 1 and ? becomes automatically a member of the (eld of coeScients of P. In Section 2, we have de(ned the principal branch u1 (z) near the origin by means of its expansion at 0. We show here that this branch satis(es a useful domination property for 06z6?. Cf. Fig. 4 for an illustration. Lemma 2. For an aperiodic walk, the principal small branch u1 (z) is analytic on the open interval z ∈ (0; ?). It dominates strictly in modulus all the other small branches, u2 (z); : : : ; uc (z), throughout the half-closed interval z ∈ (0; ?].

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Proof. By the discussion of Lemma 1, the function 1=P(z) is continuously increasing for z ∈ [0; @]. Hence the equation (in u) z = 1=P(u) admits a unique positive solution, say u+ (z), that is less than @ when z ∈ [0; ?]. This positive solution u+ (z) must coincide with the branch u1 at 0+ (since the expansions at 0+ are the same). Also, the analytic version of the implicit function theorem guarantees that the positive solution u+ (z) remains analytic all along z ∈ (0; ?), so that the principal small branch u1 and the positive solution u+ must coincide throughout this interval. Consequently, u1 (originally only de(ned near 0+ ) increases from 0 to @ as ? increases from 0 to ?. Next, a general fact about polynomials with positive coeScients enters the game: if P(u) is aperiodic, then one has for positive r |P(reiA )| ¡ P(r)

for all A ≡ 0

(mod 2 )

(35)

as seen from the strong form of the triangle inequality. Fix z = x, with x real positive and x¡?, and let w be an arbitrary solution of the kernel equation 1 − xP(w) = 0 that is at most @ in modulus and not equal to u1 (x) (i.e., not real and positive). Then, one has by the strict inequality in (35) that x=

1 1 1 ¿ ; = P(u1 (x)) P(w) P(|w|)

which implies |w|¡u1 (x) since 1=P is increasing in the region considered, [0; @]. Thus, near 0+ and since the non-principal small branches u2 ; : : : ; uc are majorized by @ in modulus (they tend to 0), they must satisfy |uj (x)|¡u1 (x). Additionally, the domination property cannot cease to hold on (0; ?): by continuity of the modulus of any branch, this would imply that u1 (x) itself reaches the value @ for some x¡?, yielding a clear contradiction. Domination must (nally continue to hold at ?, since otherwise, there would be a contradiction with the strong triangle inequality (35). Stronger domination properties are in fact derivable from similar uses of the strong triangle inequality, under the aperiodicity condition (see also [3] for details). For |z|6?, one has: |uj (z)|¡u1 (|z|) for j = 2; : : : ; c; also, |u1 (z)|¡|v1 (z)| safe at z = ?. Simply put, the principal small branch u1 is the “largest” of all the small branches. In Section 4, it will also prove handy to have available the corresponding properties of large branches. For instance, the principal large branch, v1 , is in a similar sense the smallest of all large branches. Generally, the domination properties of large branches are counterparts of those of small branches, as can be seen by mimicking the arguments. Alternatively, one can introduce duality: If P(u) is a Laurent ˜ polynomial, then P(u) = P(u−1 ) is called its dual. It is then easy to see that the small and large branches, u˜j and v˜‘ of the dual are, respectively, the inverses of the large and small branches of the primal: u˜j vj = 1 and v˜‘ u‘ = 1. Duality thus exchanges small and large branches. (Combinatorially, duality may be realized either as a symmetry along the horizontal axis applied to steps, or by the time-reversal transformation that changes a path into another path obtained by reading steps backwards.)

C. Banderier, P. Flajolet / Theoretical Computer Science 281 (2002) 37 – 80

61

3.1. Bridges and excursions We (rst address the important problem of estimating the numbers of bridges and excursions. The discussion makes use of the assumption that the walk is reduced and aperiodic. Theorem 3. Consider a simple system of walks that is aperiodic. Let @ be the structural constant determined by P  (@) = 0; @¿0. The number of bridges of size n admits a complete asymptotic expansion a2 P(@)n  a1 1 P(@) Bn ∼ '0 √ + 2 + · · · ; '0 = 1+ : (36) n n @ P  (@) 2 n The number of excursions of size n satis;es   b2 b1 P(@)n √ + 2 + ··· ; 1+ En ∼ B0 n n 2 n3 where (the uj are the small branches, with u1 the principal branch) c  (−1)c−1 2P(@)3 1 B0 = Y1 (?); Y1 (z) := : uj (z); ? =  p−c P (@) P(@) j=2

(37)

(38)

By Lemma 2, the constant Y1 (?) is equivalently characterized as  Y1 (?) = C: |C|¡@;P(C)=?−1

Proof. The result for bridges is known as it is equivalent to the local limit theorem for sums of discrete random variables [40, Chapter 9], of which the (rst proof goes back to Laplace 5 in [51]. For completeness, we brie;y sketch the argument here. Start from the fact that the number of bridges of length n is [u0 ]P(u)n . By Cauchy’s coeScient formula, one has  du 1 Bn = P(u)n ; 2i u D where the contour D is any positively oriented loop about the origin. The positive real point @ is a simple saddle-point of P(u) (hence of P(u)n ), so that the choice of the circle |u| = @ as integration contour suggests itself by the saddle-point method [16]. By the aperiodicity condition, P(u) is uniquely maximal in modulus along the contour at 5

Quite remarkably, in his ThAeorie analytique des probabilitAes, in 1812 Laplace expresses the problem as a Cauchy coeScient formula presented by its Fourier series counterpart (analytic functions are not yet invented by Cauchy!) and proceeds with a saddle-point argument expressed as an application of the “Laplace method” that was speci(cally developed for that occasion (saddle-point integrals will only emerge half-acentury later!).

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u = @; see (4). Therefore, the following saddle-point approximations are justi(ed:  du 1 Bn = P(u)n 2i u |u|=@ 1 ∼ 2i ∼



P(@)n 2 @

   1 P  (@) du 2 3 exp n log P(@) + (u − @) + O((u − @) ) 2 P(@) u @e−iB @e+iB



+∞ −∞

e−nht

2

=2

P(@)n ; dt = √ @ 2 nh

h=

P  (@) : P(@)

By the usual √ process, the contribution is (rst localized near @, taking for instance B = (log n)= n, and local expansions are applied; then the contour is extended back to yield a complete Gaussian integral. This streamlined version of the method is then extended to a full asymptotic expansion in the usual way [43, p. 419], so that (36) results. The saddle-point method thus provides an easy access to the enumeration of bridges. This gives indirectly valuable information on the small branches that can be translated into the singular structure of the GF B(z). First, the relation that determines the branches of the characteristic curve can be put under the form z=

1 : P(u)

(39)

This shows that a branch can become in(nite only at z = 0; in fact, the corresponding solutions give rise precisely to the large branches v1 ; : : : ; vd . By general principles (the inverse of an analytic function at a point where the derivative is non-zero is analytic), relation (39) is invertible analytically in the neighbourhood of any point C such that P  (C) = 0. Accordingly, a singularity (in the sense of analytic functions) must occur at any value E such that P  (E) = 0. At u = @, with @ the structural constant, one has P  (@) = 0 by construction, while P  (@)¿0. Then, the local form of (39), reads 1 z = ? − P  (@)(u − @)2 + O((u − @)3 ); 2

? :=

1 : P(@)

This is readily inverted, yielding two local solutions P(@) 1 − z=? + · · · (z → ?− ): u(z) = @ ± 2  P (@)

(40)

(41)

In particular, the principal branch u1 (z) has a square root singularity; it takes as value the structural constant @ at the place ?=

1 P(@)

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√ and the − determination must be adopted in (41) since u1 (z) increases as z → ?− : P(@) u1 (z) = @ − 2  1 − z=? + · · · (z → ?− ): (42) P (@) Next, for z = 0, all singularities of the solutions of (39), since they correspond to (nite values of u, can only be (nite branch points E with a local expansion of the form a0 + b0 (z − E)1=r for some rami(cation index r¿1. (This is easily seen directly by a suitable generalization of (40) and (41) upon taking into account the (rst non-zero derivative of 1=P.) We can now confront the result of (42) with the the saddle-point estimation (36), remembering that one has by (9) B(z) = z

d log Y (z); dz

Y (z) := (u1 (z) · · · uc (z)):

First, Y (z) that is analytic near 0 must remain analytic throughout the disk |z|¡?, since otherwise B(z) would be singular for some value inside the disk and this would contradict the asymptotic growth (36) that is of type P(@)n for Bn . Next, Y (z) cannot have any (algebraic) singularity other than z = ? on the circle |z|= ?, since, by singularity analysis, 6 this would entail the presence of oscillating terms in the asymptotic expansion of Bn , again contradicting (36). Also, Y (z) can only have a branch point of rami(cation index r = 2 at z = ?, since otherwise some term of the form n−1+1=r would have been present in the expansion of Bn . Finally, the de;ated product Y1 (z) = u2 (z) · · · uc (z) must be analytic at ? since otherwise, being capable only of having a branch point with rami(cation index 2, one would reach a contradiction regarding the leading coeScient of Bn (as checked from comparing (36) against the consequences of (42) on coeScients). In other words, this sequence of indirect arguments shows the following: 7 The product of all the non-principal small branches Y1 (z) = u2 (z) · · · uc (z)

(43)

is analytic at all points of the closed disk |z|6?. It is now an easy matter to complete the estimate of the number of excursions by singularity analysis applied to (20) in Theorem 2. The unique dominant singularity of E(z) must be at z = ? where the local expansion (42) gives

(−1)c−1 P(@) E(z) ∼ E(?) − B0 1 − z=?; B0 = Y1 (?) 2  p−c ? P (@) 6

Singularity analysis [34,61] allows us to transfer a singular element of the form (1−z=&)F in the expansion of a function f(z) at a singularity & into a corresponding asymptotic element of the form &−n n−F−1 =G(−F) in the expansion of the coeScient [z n ]f(z) at in(nity. It is applicable unconditionally to algebraic functions. 7 An alternative argument based on the re(nement of domination relations evoked after the proof of Lemma 2 is possible; see Banderier’s thesis [3] for details.

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with Y1 given by (43). A full expansion of u1 (z) in powers of (1 − z=?)1=2 being available, and Y1 (z) being analytic on the whole of |z|6?, the proof of (37) is at last completed. Example 6. Asymptotics of tree codes. The case of walks with only one type of descending step equal to −1 corresponds to tree codes, as discussed in Example 3. In this very special case, there is only one small branch, and the GF of excursions is E(z) = u1 (z)=(p1− z). For aperiodic walks, result (37) of Theorem 3, or plainly estimate (41), gives us @:

P  (@) = 0;

1 1 √ En ∼ p−1 2 n3



P(@)3 P(@)n : P  (@)

(44)

In terms of trees, the principal branch u1 (z) is precisely the GF of trees corresponding to the degree set 1 + S with generating polynomial 0(u) := uP(u) and one has T (z) = p−1 zE(z) = u1 (z). Estimate (44) then coincides with the well-known asymptotic estimate of the number Tn of trees of size n, 0(@) − @0 (@) = 0;  n 0(@) 0(@) 1 Tn ∼ √ ; @ 2 n3 0 (@) @:

(45)

which was (rst discovered by Meir and Moon [55]. As soon as c¿1, there are several small branches, and, in this case, the algebraic constant Y1 (?) intervenes. Numerically, this constant can be determined easily as it only involves the product of the small solutions to the kernel equation taken at z = ?. Algebraically, since Y1 (?) is the product of c − 1 solutions to an algebraic equation of c+d degree c+d, it is an algebraic number of degree at most ( c−1 ) over Q(?) ≡ Q(@) that is computable by the techniques of Section 2.3 (upon changing c to c − 1 in the Platypus Algorithm). However, since @ is a double root of the kernel equation instantiated at z = ?, further simpli(cations accrue. This explains that constants involving radicals are often to be observed when analysing problems of relatively low “complexity”. The next example is typical of this state of aNairs. Example 7. Asymptotics of the {−2; −1; 0; 1; 2}-excursions. The walk introduced in Example 4 is symmetric, and like for any symmetric walk system, the structural constant is equal to 1 while the structural radius is the rational number, ? = 1=P(1) = 15 . The product of the non-principal small branches at ? reduces to u2 (?). This quantity is a priori one of the roots of an equation of degree 4 (Eq. (25) instantiated at z = ?), but since this equation has already @ = 1 as a double root, the equation satis(ed by

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u2 (?) is in fact of degree 2 (it is u2 + 3u + 1 = 0) so that 3 1√ u2 (?) = − + 5 2 2 and this quantity is precisely Y1 (?) of (38). Thus, we can conclude and get easily    √ 1 5 5n : 1+O En = (3 − 5) √ 4 n n3 The quality of the asymptotic approximation provided by the (rst term is 11% when n = 10 and 1.2% when n = 100, where the En are conveniently determined by (34). The estimate is also consistent with the nature of the singularity at ? = 15 of the diNerential equation (33). 3.2. Paths and meanders Now that the bulk of the work is done, asymptotic estimates of the basic counts of paths and meanders fall as a ripe fruit. The result for unconstrained paths is trivial, since the number of possibilities for size n is P(1)n , a fact consistent with the simple pole of W (z; 1) = (1 − zP(1))−1 . For meanders, three cases are to be distinguished depending upon the value of a quantity called the drift. Denition 5. Given a simple walk with characteristic polynomial P(u), the drift is de(ned by the quantity ; = P  (1): In the unweighted case, the drift is thus the sum of all the possible values of the jumps, which constitutes an indicator of the “tendency” for the walk to go up or down. In the probabilistic case (P(1) = 1), the drift represents exactly the expected movement in the y-direction of any single step. For a symmetric walk, the drift is ; = 0, while @ = 1. Theorem 4. Consider a simple aperiodic walk. The number of paths of length n, [z n ]W (z; 1), is P(1)n exactly. Set Y_ 1 (z) :=

c 

(1 − uj (z)):

j=2

The asymptotic number of meanders depends on the sign of the drift ; = P  (1) as follows: P(1)n  c1 c2 1+ ; = 0: Mn ∼ H0 √ + 2 + ··· n n n P(1) H0 := 2  Y_ 1 (?); ? = P(@)−1 = P(1)−1 ; P (1)

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; ¡ 0:

; ¿ 0:

  c− c− P(@)n 1 + 1 + 22 + · · · Mn ∼ H0− √ n n 2 n3 Y_ 1 (?) P(@)3 ; ? = P(@)−1 ; H0− := − 2  P (@) P(@) − P(1) P(@)n Mn ∼ H0+ P(1)n + H0− √ 2 n3

 1+

 c+ c1+ + 22 + · · · n n

H0+ := (1 − u1 (?1 ))Y_ 1 (?1 );

?1 := P(1)−1 :

The formulX have an intuitive meaning. In the case of a positive drift, a fraction close to H0+ of all the (unconstrained) walks is a meander, in accordance with the natural tendency of walks to go up. For negative drift, most paths tend to go down and the proportion of meanders is exponentially √ small, roughly like (P(@)=P(1)) n . For zero drift, the proportion becomes as large as 1= n, while the walks tend to oscillate not too far from the horizontal axis. Proof. The discussion is based on the formula of Corollary 1 rewritten as M (z) = F(z; 1) =

1 − u1 (z) _ Y 1 (z); 1 − zP(1)

Y_ 1 (z) :=

c 

(1 − uj (z)):

j=2

It suSces to examine the position of the zeros and the dominant singularity of the numerator in relation to 1=P(1) that is always a zero of the denominator. By proof, arguments similar to Lemma 2, the quantity Y_1 (z) being a symmetric function of small branches each of which is dominated by u1 , must remain analytic throughout |z|6?. In the case ; = 0, one has P  (1) = 0, @ = 1, and ? = 1=P(@) = 1=P(1). Thus, (1 − u1 ) contributes a term of the form (1 − z=?)1=2 at z = ? while the denominator (1 − zP(1)) √ has a simple zero there. Globally, the singularity of F(z; 1) is thus of type 1= , and the result follows. For a negative drift, meaning P  (1)¡0, one must have @¿1, since P  (u) increases from −∞ to +∞ when u ranges from 0+ to +∞. With ? = 1=P(@) (the structural radius) and ?1 := 1=P(1), one then has ?1 ¡?. In this case, the prefactor (1 − zP(1))−1 has a pole at ?1 ; this pole is however cancelled by a zero in the numerator induced by the numerator (1 − u1 (z)) (since u1 (?1 )= 1), so that ?1 is a removable singularity of F(z; 1). Consequently, the dominant singularity of F(z; 1) is at ?, where F(z; 1) is of the square-root type. For a positive drift, one must have @¡1, so that the prefactor induces a pole at ?1 := 1=P(1) before Y_1 or 1 − u1 become singular. The argument concludes by “subtracting singularities”, since the function, F(z; 1) −

Y_ 1 (?1 )(1 − u1 (?1 )) ; 1 − zP(1)

?1 :=

1 P(1)

now has a dominant singularity of the square-root type at ?.

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The earlier discussion about the algebraic character of asymptotic constants applies: quantities like Y_1 (?1 ) and Y_1 (?) can be determined by adapting the Platypus Algorithm of Section 2.3. Should the degrees of the algebraic numbers involved become fairly large, one can always resort to numerical analysis as the next example illustrates. Example 8. Lucky periods in die casting. In [63, p. 45], PWolya introduces the following problem: “En jetant 2n dAes aB la fois, on peut obtenir diCAerentes sommes de points de 2n aB 12n. Le cas le plus probable est celui de 7n points. DAesignons par An le nombre de combinaisons oBu se produit cet eA vAenement.” Imagine that at each of n rounds two dice are cast and the score of the round is the sum of the two dice’s values. PWolya thus considers the number of ways An (and probability An =36n ) of reaching the balanced score 7n at the end of a game of dice consisting of n rounds. PWolya proceeds by an integral representation (precisely of the type used in the proof of Theorem 1) from which he concludes that the GF A(z) has the character of an algebraic function, but does not make the calculation explicit. By centring around the mean score of a round, which equals 7, it is easily realized that the problem is equivalent to a walk whose characteristic polynomial is P(u) = u−5 (1 + u + u2 + u3 + u4 + u5 )2 : Let Bn be the number of bridges. (The quantity Bn is exactly PWolya’s An .) Here, c =−5, d = +5; also @ = 1 as the walk is symmetric, and ? = 1=36. The asymptotic number of bridges is simply Bn ∼ √

6 · 36n 22 · 3 · 5 · 7 n

;

which is nothing but an avatar of the local limit gaussian law. Consider next the modi(cation of PWolya’s problem where we ask for the number of “lucky” games, in the sense that at any time t the score is at least 7t. This is equivalent to (nding the number of meanders. Excursions surface if we further impose the (nal 1 score to be 7n exactly. We have @ = 1 and ? = 36 . One should then examine the kernel equation at z = ?, u5 −

1 5 u P(u) = 0 36

as this gives all the values of the small branches there. We (nd that there are 10 roots, amongst which @ = 1 is a double root. The eight other go by pairs of complex conjugates, with : E = −0:36381 + 0:22924i; : E = −1:96746 + 1:23976i;

: E = 0:06208 + 0:47622i; : E = 0:26919 + 2:06476i:

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Then, the quantity Y1 (?) is determined numerically as the product of the roots of : modulus less than @ = 1, namely, E E_ E E . We (nd Y1 (?) = 0:42648, so that the constant in the asymptotic formula for excursions can be determined to great accuracy: 36n En ∼ C · √ ; n3

: C = 0:35865 42111 34518 86172:

(46)

: _ − E )(1 − E ) to be Y_1 (?) = In the same vein, we determine Y_1 (?) = (1 − E)(1 − E)(1 2:11615, and C 1 n [z ]F(z; 1) ∼ √ ; n 36 n

: C  = 0:93071 59694 87799 20216

gives the probability of a lucky game (a meander). PWolya’s example is interesting structurally. For instance, the excursion constant C in (46) involves Y1 (?) that is a root of a self-reciprocal polynomial I(y) of degree 16 (found by Platypus Algorithm and factorization), itself equivalent to a resolvent of degree 8 that turns out to be irreducible, ˆ + y−1 ); I(y) = y8 I(y ˆ I(v) = v8 − 17v7 − 152v6 + 34v5 − 551v4 − 12053v3 + 8038v2 + 38692v + 12664; but algebra stops there. In contrast, analysis based on the decomposability devolving from the kernel method provides fully satisfactory numerical answers. 3.3. Periodicities The discussion above has been conducted under the assumption of aperiodicity. As we explain now, similar results hold for periodic walks provided suitable congruence conditions are imposed on the indices of coeScients of generating function. For reasons explained after De(nition 4, we freely assume the set of jumps to be at least reduced, as this implies no loss in generality. Take a set S corresponding to period p. We sketch the discussion in the case of excursions, with E(z) the corresponding GF. Then, E(z) is periodic with period ˆ p ) for some E(z) ˆ p, meaning that it is of the form E(z) = E(z that is analytic at 0. The foregoing discussion of small branches continues to apply as long as |z| stays inside the disk |z|¡?, and the local analysis (42) of u1 continues to hold as z → ?. However, it appears now that there are p conjugate dominant singularities at the points ?j := ?Jj ;

J = e2i

=p

:

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Indeed, E(z) satis(es E(z) = E(Jz), while Eq. (42) describes the behaviour of u1 (z) at ?j upon changing z into z=J j . Then, each of the p singular elements cumulate and contribute jointly to [z n ]E(z) provided n ≡ 0 mod p. One (nds in this way that P(@)n En ∼ pB0 √ ; 2 n3

n = p3;

3 ∈ Z¿0 ;

where B0 is (still) given by (38). The analysis easily adapts to the other types of paths considered, and is summarized by a simple rule: For a system of jumps of period p, the asymptotic form of the count of index n must be restricted to a suitable congruence class of n mod p in order for objects to exists; then the corresponding asymptotic formula is obtained from the estimate of the aperiodic case through multiplication by a factor of p. Example 9. Asymptotics of generalized Duchon’s clubs. We return to Example 5. The kernel equation is 1 − z(u−c + ud ) = 0, which gives the structural constant @=

 c 1=e d

;

e = c + d:

The period is equal to e. The number of excursions of length n is non-zero only if n ≡ 0 (mod e) and it satis(es (with r = ?e ) −3 −3=2 Ee3 ∼ Dc;d rc;d 3 ;

rc;d =

c c dd ee

for some computable constant Dc; d . This generalizes the estimate of Duchon [22] who determined D2; 3 by a particular grammar construction followed by a speci(c algebraic elimination.

4. Basic parameters and limit laws The singular structure of basic generating functions of paths, bridges, meanders, and excursions is well-established by Section 3. On the other hand, many parameters “decompose” combinatorially, so that their GFs are expressible in terms of the basic generating functions, or equivalently, they lie in Q(z; X ; u1 ; : : : ; uc ) for some set X of markers. In this paper, we only exhibit few sample cases of application of this methodology. As pointed by Philippe Robert (private communication), the whole combinatorial-analytic apparatus largely parallels what probabilists do by means of Wiener–Hopf decompositions (this is analogous to the separation between small and large branches) and Tauberian theorems (instead of singularity analysis that aNords greater asymptotic accuracy through complete asymptotic expansions).

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4.1. Arches and contacts De(ne an arch as an excursion of size ¿0 whose only contact with the horizontal axis is at its end points and let A be the set of arches. The set E of excursions satis(es the combinatorial equation E∼ = S{A}; where S denotes the combinatorial construction that freely forms sequences. By wellknown mechanisms this translates directly into the GF equation E(z) =

1 1 − A(z)

or equivalently

A(z) = 1 −

1 : E(z)

(47)

The singular form of A(z) then reads immediately: E(z) ∼ E(?) − B0

 1 − z=?;

implying A(z) ∼

1−

1 E(?)

 −

B0 1 − z=?: E(?)2

Thus, the number of arches An is asymptotically proportional to ?−n n−3=2 , hence also to the number of excursions En . De(ne a vertex of an excursion not equal to one of the end points to be a contact if its altitude is 0. Then, A(z)k+1 is the GF of excursions having k contacts. For any (xed k, the function Ak+1 has again a singularity of the square root type that is amenable to singularity analysis. An easy calculation then gives: Theorem 5. The probability that a random excursion of size n has k contacts is for any ;xed k of the form k    1 1 1 : (k + 1) 1 − +O E(?)2 E(?) n The number of contacts is thus asymptotically distributed like the sum of two independent geometric random variables with parameter 1 − E(?)−1 . In particular, An ∼

1 En : E(?)2

The constant E(?) is expressible in terms of the quantity Y1 (?) and is thus a close relative of '0 introduced in Theorem 3. On the relation between bridges and excursions. We brie;y discuss here a construction that relates excursions to arches. Consider a bridge and let m (with m60) be the minimal altitude of any vertex. Any non-empty bridge ' decomposes uniquely into a walk ’1 of size ¿1 from 0 to m that only reaches level m at its right end, followed by an excursion ” (this is the part where one wanders around but above level m), followed by a path ’2 of size ¿0 from m to 0 that only touches level m at its beginning. By rearrangement, one can write ' = ” · (’2 |’1 ), where the glueing of ’2 ’1 is an arch and

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71

the bar keeps track of where the splitting should occur. This construction is illustrated by the following diagram:

In other words, the set of non-empty bridges is combinatorially isomorphic to the product of the set of excursions by the set of arches with a split step that is distinguished. This construction is then nothing but the combinatorial re;ex of the identity excursions

 

split arches

      d B(z) − 1 = E(z) · z A(z) ; dz bridges

(48)

which, in view of (47) is equivalent to   1 E  (z) d 1− =z : B(z) − 1 = E(z) · z dz E(z) E(z) (Thus, combinatorics of arches gives back Corollary 2.) Such relations are ubiquitous in the theory of paths, the most famous ones being known by the names of Spitzer and Sparre Andersen: see Kittel’s appendix to [35] and Lothaire’s book [52, Section 5.3] for a summary. Raney’s classic [65] and Gessel’s papers [38,39] make use of similar ideas (inter alia, the “cycle lemma”) in combinatorial proofs of the Lagrange inversion formula. One of the many consequences of this orbit of ideas is for instance the possibility of analysing the number of times a bridge attains its minimum value by adapting the decomposition (48) and closely mimicking the proof of Theorem 5. Louchard’s analyses in [53] provide many striking illustrations of such an interplay between probabilistic and combinatorial properties. 4.2. Final altitude of a meander The ;nal altitude of a path is the abscissa of its end point. For unconstrained paths, the usual local and central limit theorems for discrete random variables apply [40, Chapter 9], so that the limit law, after normalization, is Gaussian, the underlying technology being plainly the saddle-point method. We consider now meanders. The

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random variable associated to (nite altitude when taken over the set of all meanders of length n is denoted by Xn , and it satis(es Pr(Xn = k) =

[z n uk ]F(z; u) : [z n ]F(z; 1)

We state: Theorem 6. The ;nal altitude of a random meander of size n admits a limit distribution, with the limit law being dictated by the value of the drift ;. (i) For a negative drift, ;¡0, the limit distribution is a discrete one characterized in terms of the large branches: lim Pr(Xn = k) = [uk ]$(u);

n→∞

where $(u) =

(1 − @)2  1 − v‘ (?) : (u − @)2 ‘¿2 u − v‘ (?)

(ii) In the case of zero drift, ; = 0, the normalized random variable P  (1) Xn √ ; #= P(1) # n 2

converges in law to a Rayleigh distribution de;ned by the density xe−x =2 :   2 Xn √ 6 x = 1 − e−x =2 : lim Pr n→∞ # n (iii) In the case of a positive drift, ;¿0, the standardized version of Xn ,    2  P  (1) P  (1) P  (1) Xn − Hn P (1) 2 √ ; H= ; P = + − P(1) P(1) P(1) P(1) P n converges in law to a Gaussian variable N(0; 1):    x 2 1 Xn − Hn √ 6x = √ e−y =2 dy: lim Pr n→∞ P n 2 −∞ In the case of a negative drift, the limiting distribution admits an explicit form
c‘ v‘ (?)−k [uk ]$(u) = @−k (c0 + c1 k) + ‘¿2

for a set of constants cj that can be made explicit by a partial fraction expansion of $(u). Proof. (i) For a negative drift, one directly shows that the probability generating function of Xn at u converges pointwise to a limit that precisely equals $(u), the convergence holding for u ∈ (0; 1). By the fundamental continuity theorem [27, p. 280]

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for probability generating functions (PGFs), this entails convergence in law of the corresponding discrete distributions. We now (x a value of u taken arbitrarily in (0; 1) and treated as a parameter. The PGF of Xn is [z n ]F(z; u) ; [z n ]F(z; 1) where F(z; u) is given by Theorem 2. In the case of a negative drift we know from the proof of Theorem 4 that @ = v1 (?) satis(es @¿1 while the radius of convergence of F(z; 1) coincides with the structural radius ?. Then, the quantity Y1 (z; u) =

d  ‘¿2

1 u − v‘ (z)

is analytic in the closed disk |z|6?: being a symmetric function of the non-principal large branches, it has no algebraic singularity there; given the already known domination relations between the large branches (Lemma 2), the denominators cannot vanish. It then suSces to analyse the factor containing the principal large branch v1 . This factor has a branch point at ?, where 1 P(@) 1 1 2  1 − z=? ∼ + 2 u − v1 (z) u − @ (u − @) P (@) as follows directly from (42) and the fact that v1 is conjugate to u1 at z = ?. Singularity analysis then gives instantly the fact that, for some non-zero constant C, [z n ]F(z; u) ∼ C?−n n−3=2 /(u);

where /(u) =

1 Y1 (?; u) (u − @)2

and the result follows after normalization by [z n ]F(z; 1). For the remaining two cases, it will prove convenient (rst to estimate the mean value (expectation E(·)) of Xn , E(Xn ) =

[z n ]Fu (z; 1) ; [z n ]F(z; 1)

(49)

where Fu indicates diNerentiation with respect to u. Logarithmic diNerentiation gives Fu (z; 1) = F(z; 1)

d
‘=1

1 ; 1 − v‘ (z)

(50)

from which one attains singularities easily. (ii) In the case of a zero drift, the value of the structural constant is @ = 1 and the radius of convergence of F(z;

1) is ? = 1=P(@) = 1=P(1). Then, the singularity at ? of Fu (z; 1) combines a factor 1= 1 − z=? that arises from F(z; 1) and another similar factor that arises from the term (1 − v1 (z))−1 . This singularity is thus, to (rst-order

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asymptotics, similar to a simple pole. A √ computation based again on (42) reveals that the mean value of Xn is of the order of n. Precisely, one (nds  n P  (1) ; #= : E(Xn ) ∼ # 2 P(1)

(Note that =2 is the mean of the standard Rayleigh distribution.) The formula of Corollary 3 then suggests that Fk (z) should behave very much like v1k , implying that the √ coeScients should resemble, up to scaling, the coeScients in the large power [z n ](1 − 1 − z)k . Such a situation is known to be conducive to Rayleigh laws: it is covered extensively in Drmota and Soria’s study [21] and revisited in the paper [5]; see also [20]. In particular, Theorem 1 of [21] gives us the convergence in distribution to the Rayleigh law, while a simple adaptation of the results of Appendix B in [5] provides corresponding density estimates (a “local” limit law). We omit the tedious but routine details. (iii) For a positive drift, probabilistic intuition indicates that there are relatively few chances for a walk to ever come under the negative axis, and when this happens, it only tends to do so early in the history of the walk. Consequently, the (nal altitude should be only marginally aNected by the meander conditioning. In this case, one has @¡1 and the radius of convergence of F(z; 1) is ?1 = 1=P(1) while the structural radius satis(es ?¿?1 . By de(nition, one has v1 (?1 ) = 1. Consequently, the function Fu (z; 1) in (50) admits a double pole at ?1 , with Fu (z; 1) ∼ F(z; 1)

1 v1 (?1 )(z

− ?1 )

:

so that (one has v1 (?1 ) = −(?21 P  (1))−1 ), E(Xn ) =

[z n ]Fu (z; 1) P  (1) =n + O(1): n [z ]F(z; 1) P(1)

In the probabilistic case, the coeScient of n in the estimate reduces to the drift, and this estimate does agree with the probabilistic argument sketched above. Similarly, the variance is found to satisfy    2  P  (1) P  (1) P (1) n + O(1): Var Xn = + − P(1) P(1) P(1) Finally, the Gaussian law is established from the power-sum form of Corollary 3 upon applying Cauchy’s coeScient formula. One has  dz 1 [z n ]Fk (z) = .1 (z)v1 (z)−k−1 n+1 + Rn;k : 2i z |z|=?1 The error term Rn; k that arises from all the non-principal branches is exponentially smaller than ?1−n because of the domination properties of 1=v1 (z) (see the proof of Lemma 2, once more). The main integral is then treated by the saddle-point method

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√ in the range considered, k = Hn + O( n) with H := P  (1)=P(1). The saddle-point of the integrand is at ?1 , very nearly. The Gaussian density then comes out from a standard saddle-point perturbation analysis. 5. Directed two-dimensional models The kernel method is generally well suited to problems where all the jumps are of the form (aj ; bj ) with aj ¿0. In this case, each choice of a step implies progression along the horizontal axis. One considers the trivariate GF
Fn;p;q z n xp yq ; F(z; x; y) := n;p;q

where Fn; p; q is the number of meander paths in Z¿0 × Z¿0 with size (number of steps) equal to n that connect the origin to the point of coordinates (p; q). The walk is thus directed in the sense of Section 1. As we now explain such enumeration problems, though formulated in two-dimensional space, are in fact fake one-dimensional problems amenable to the kernel method. In the directed case, the method of “adding a slice” encountered in Eqs. (14) and (16) gives rise to the fundamental equation F(z; x; y)(1 − zP(x; y)) = 1 − z{y¡0 }(P(x; y)F(z; x; y));

(51)

where the characteristic polynomial is now
aj bj x y ; P(x; y) := j

which is entire in x but of Laurent type with respect to y. The parameters of size (marked by z) and horizontal displacement (marked by x) are bound by linear inequalities, and one of them can be treated as the basic variable, the other as an auxiliary parameter or even the constant 1. Then, the adaptation of the kernel method consists in binding the Laurent variable, here y, to the basic variable chosen (x or z) by 1 − zP(x; y) = 0:

(52)

Newton’s polygon then shows that, for the bound equation, the number of “small” roots of the kernel equation coincides with the maximum negative vertical span, namely, c := | minj bj |, and this number is precisely the number of unknown functions on the right-hand side of (51). We let uj represent these small branches. The treatment of walks and bridges adapts easily from what has been done earlier. Regarding excursions and meanders, substitution of the uj then shows the following: The GF of excursions (de;ned by ;nal altitude 0) and the BGF of meanders (de;ned by ;nal altitude ¿0) depend rationally on the variables z; x and the set of small branches {uj } of the associated “kernel equation” (52). Example 10. Chess moves of Labelle and Yeh. In two papers [49,50], Labelle and Yeh develop an interesting set of decompositions for generalized knight moves on a

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chessboard. The standard version of the problem is: Consider the Z¿0 ×Z¿0 chessboard. How many sequences of Eastbound knight moves (S= {(1; 2); (1; −2); (2; 1); (2; −1)}) are there from (0; 0) to (n; 0)? By de(nition, the moves are not allowed to involve points with negative coordinates. As size is not needed, we take x as the independent variable and set z = 1. The kernel equation is then 1 − (xy2 + xy−2 + x2 y + x2 y−1 ) = 0; so that the characteristic curve is a quartic. The vertical symmetry of the moves implies that the kernel equation can be rewritten as a combination of two quadratic equations, 1 − x(W 2 + xW − 2) = 0;

W := y +

1 : y

There results that the four branches of the characteristic equation are given by

1 1 y± (W ) = (W ± W 2 − 4); W± (x) = (−x2 ± x4 + 8x2 + 4x): 2 2x It appears that the two small branches u1 ; u2 correspond to taking opposite signs in the determinations of y(W ) and W (x), and one (nds for the GF of excursions (i.e., paths terminating at altitude 0), in complete analogy to the simple walk, 1 1 E(x) = − (u1 (x)u2 (x)) = − y− (W+ (x)) · y+ (W− (x)) x x = 1 + x2 + 3x4 + 2x5 + 12x6 + 14x7 + 54x8 + 86x9 + · · · : This is the sequence (an ) of [49] and also EIS A005220. Decomposability renders especially easy the asymptotic analysis of the number of excursions and of corresponding parameters. More general knight moves can be treated similarly by the kernel method. In particular, the equation satis(ed by the excursion generating functions tends to be of a degree exponential in c; see [49,50]. Here, the kernel method yields a reduction to an equation of degree 2c, which even reduces to a resolvent of degree c when symmetry is taken into account via the W -parameterization. This illustrates a sharp contrast between the exponential blow-up in combinatorial complexity and the linear character of the analytic complexity. 6. Conclusion In this paper, we have aimed at illustrating the analytic tractability of many onedimensional path problems, a boon of the kernel method. The reduction in the asymptotic–analytic complexity of the problem is often spectacular, as exempli(ed by Duchon’s clubs or the Labelle–Yeh knight moves. Parameters that are easily readable on paths lead to generating functions whose singularities arise simply from the branches of a characteristic curve of low degree. The method applies to all one-dimensional problems as well as to two-dimensional problems provided they remain directed. For

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a thorough discussion of the algebraic power of the kernel method, we refer once more to the study by Bousquet-MWelou and PetkovYsek [13]. (The kernel technique is also reminiscent of Tutte’s quadratic method much of use in the enumerative theory of planar maps [42]; see Bousquet-MWelou’s paper [11] for a perspective.) The case of undirected two-dimensional problems, where one can go back and forth in all four cardinal directions, is appreciably harder. Even in the case of movement of amplitude 61, Fayolle et al. show [26] that stationary solutions involve elliptic functions and integrals. Some directed path problems in dimension higher than 2 can however still be successfully treated by speci(c combinatorial decompositions; see [12] for an example. A tribute to Maurice Nivat. As is apparent from the bibliography of this paper, many papers directly relevant to our study have been published in the journal Theoretical Computer Science along the years. We owe much for this to the Editor-in-Chief, Maurice Nivat. His openness of mind has been a constant help in the emergence and shaping up of sub-communities within theoretical computer science. Examples are the GASCOM (Generation of Random Combinatorial Objects) and AofA (Analysis of Algorithms) communities which have greatly bene(tted from special issues of TCS, this at the invariably encouraging initiative of Maurice. In view of this and of Maurice’s long-standing interest in similar discrete geometrical objects (see, e.g., [6,7,9,17]), we kindly dedicate this study to him. Acknowledgements This work was supported in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). The authors are grateful to Philippe Robert for many insightful discussions on probabilistic aspects of random walks as well as to Mireille Bousquet-MWelou and an anonymous referee for a careful scrutiny of the paper that greatly helped us improve our presentation. Thanks (nally to Christian Krattenthaler for valuable bibliographical remarks. References [1] S.-S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, 1990. [2] C. Banderier, Combinatoire analytique: application aux marches alWeatoires, D.E.A. memoir, UniversitWe Paris VII, July 1998. [3] C. Banderier, Combinatoire analytique des chemins et des cartes, Ph.D. thesis, UniversitWe Paris VI, June 2001. [4] C. Banderier, M. Bousquet-MWelou, A. Denise, P. Flajolet, D. Gardy, D. Gouyou-Beauchamps, Generating functions of generating trees, Tech. Report ALCOM FT-TR-01-17, Alcom-FT Project, February 2001, 26 pages, Accepted for publication in Discrete Mathematics. [5] C. Banderier, P. Flajolet, G. SchaeNer, M. Soria, Random maps, coalescing saddles, singularity analysis, and Airy phenomena, Random Structures & Algorithms. 19 (2001) 194–246. [6] E. Barcucci, S. Brunetti, A. Del Lungo, M. Nivat, Reconstruction of discrete sets from three or more X-rays, Discrete Math. 241 (2001) 65–78.

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Theoretical Computer Science 281 (2002) 81 – 97

www.elsevier.com/locate/tcs

Groups and tilings Dani#ele Beauquiera; ∗ , Jean-Claude Fournierb a Laboratory

of Algorithmics, Complexity and Logic, Department of Informatics, University Paris-12, 61 Av. du G$en. de Gaulle, 94010 Creteil, France b Equipe Combinatoire, CNRS Paris-6, Department of Informatics, University Paris-12, 61 Av. du G$en. de Gaulle, 94010 Cr$eteil, France

Abstract We prove that the word problem for the group of dominoes is equivalent to the existence of a directed tiling for the corresponding closed curve in the plane, which, in turn is equivalent to the fact that the curve is “balanced”. This last property beeing decidable, the word problem is also decidable. Moreover we prove that this result is transposable in any other regular grid c 2002 Elsevier Science (hexagonal or triangular) and we partially extend it to “exact” tiles.
B.V. All rights reserved.

1. Introduction Conway [2] introduced a technique using in7nite 7nitely presented groups which permits to answer the question whether a region of the plane is tilable by translation of given tiles. This idea was developed later in numerous papers [7,5,3]. Consider a group Gs (s for square) de7ned by two generators a; b and relations abba−1 b−1 b−1 = 1 and baab−1 a−1 a−1 = 1 which represent the contour words of vertical and horizontal dominoes, respectively. It is easy to prove that if a region R of the Z2 -grid is tilable by dominoes then the contour word of R belongs to the group Gs . In [2], it is proved that the converse is not necessarily true and the authors introduce the notion of signed tiling which turns out to be the appropriate geometrical interpretation of the fact that a region has a contour word equal to 1 in the group Gs . Actually, [2] proves that a region R of the plane has a contour word equal to 1 in the group Gs i= it admits a signed tiling.

∗ Corresponding author. E-mail addresses: [email protected] (D. Beauquier), [email protected] (J.-C. Fournier).

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 0 8 - 7

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O

Fig. 1. The chessboard like plane.

In this paper, we extend this result in several directions. Firstly we prove that this result holds also for general closed curves. The case of general closed curves was never considered before. Secondly, we extend the result to the di=erent regular lattices of the plane, triangular and hexagonal ones. More precisely we prove that for the three regular lattices, the three following properties are equivalent: (1) w = 1 in the considered associated group, (2) there exists a signed tiling with boundary w, (3) w de7nes a balanced closed curve. As a consequence of the last property we get that the word problem is decidable for each of the three considered groups. As far as we know, it is a new result for the hexagonal case. Thirdly we propose a generalization of the set of possible used tiles and get partial results in this case. Section 2 is devoted to the square lattice and Section 3 to the hexagonal one. Section 4 proposes a generalization and states a conjecture which provides an in7nite class of groups for which the word problem should be decidable.

2. The 2-squares group G s 2.1. Preliminaries Let a Z2 -grid which de7nes a tiling of the euclidian plane with unit squares called cells. The plane is colored like a chessboard (see Fig. 1). Consider the four letters alphabet A = {a; a ; b; b } encoding, respectively, a unit move on the right, on the left, upwards and downwards on the Z2 -grid. A vertex A ∈ Z2 and a word w on the alphabet A de7nes a curve on the Z2 -grid starting in A and following the moves given by the sequence of letters of w. The end of this curve is denoted by A + w. If O is the origin of the grid, the curve is denoted by Cw . If w is a word on this alphabet, and x is a letter, we denote by |w| the length of w and by |w|x the number of occurrences of x in w. The empty word is denoted 1. If w is a word, w˜ is de7ned inductively as follows: 1˜ = 1;

a˜ = a ;

b˜ = b ;


x = x˜w: if x is letter; w ˜

 a˜ = a; b˜ = b;

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83

b

a -1

a b

-1

An horizontal domino with (direct) contour word aaba-1a -1b -1

A vertical domino with (direct) contour word abba -1b -1b -1

Fig. 2. The two dominoes.

Two words w1 and w2 ar called conjugate if there exist words u and v such that w1 = uv and w2 = vu. The conjugation is an equivalence relation and the equivalence class of w will be denoted by [w]. A word w de7nes a closed curve i= |w|a = |w|a
and |w|b = |w|b
. A vertical domino has a direct contour word which is a conjugate of abba b b depending on the starting vertex and an horizontal domino has a direct contour word which is a conjugate of aaba a b . A directed domino is a domino with a 7xed choice of orientation of the contour (Fig. 2). Let us consider the 2-squares group Gs de7ned by two generators a and b and the two relations abba−1 b−1 b−1 = 1;

aaba−1 a−1 b−1 = 1

which correspond, respectively, to direct contour words of vertical and horizontal dominoes. There is a morphism from A∗ onto Gs which maps letters a; b, respectively, onto generators a; b, and a ; b respectively, onto the inverse of generators a; b. From now on, we identify a ; b , respectively, to their image in Gs , namely a−1 ; b−1 . For two words w1 and w2 we write w1 ≡ w2 i= 1 and w2 have the same image in Gs by this morphism. Let =df aba−1 b−1 . This word is a direct contour word of a cell. In a standard way, one associates to Gs a rewriting system, whose rules are the following ones: Rules of type (I) aa−1 → 1;

a−1 a → 1;

bb−1 → 1;

b−1 b → 1

and the reverse ones. Rules of type (II) w → w ∗

˜ if ww˜  ∈ [ ] ∪ [ ]:

We write w → w i= w is obtained from w using the rewriting rules of Gs , and by ∗  w i= w ≡ w . We are interested in the word problem for de7nition of Gs we have w →

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the group Gs , namely is the property w ≡ 1 decidable? In order to solve this problem we introduce some notions. 2.2. Balanced closed curves First of all, let us de7ne the index of a cell and the index of a color according to a closed curve C. Afterwards, “closed curve” means “bounded closed curve”. The index of a cell c according to a bounded closed curve C, denoted by IC (c), is the number of turns of C around cell c, taking into account the direct or undirect sense of the turns. This index can be computed as follows: consider a vertical line  going up from in7nity to the center of the cell and add the weights of all the directed unit segments of C crossed by , a segment directed to the right or upwards has weight 1, a segment directed to the left or downwards has weight −1. We can generalize this notion to a 7nite set of directed unit segments of the Z2 -grid instead of a closed curve. Let S be a set of directed unit segments of the Z2 -grid. The index of a cell c according to S, denoted by IS (c), is the number of segments of S crossed by a vertical line  going up from in7nity to the center of the cell, each segment being counted with its weight. For simplicity we will denote by Iw (c) the index ICw (c). The index of a color i ∈ {black; white} according to a closed curve C, denoted by IC (i) is the sum of the indices of the cells colored by i: IC (i) =

 color(c)=i

IC (c):

Notice that since the curve is bounded only a 7nite set of cells have a nonnull index, so the above sum is 7nite. A bounded closed curve C is balanced if IC (black) = IC (white): The notion of balanced closed curve is introduced in [2] in a slightly di=erent but equivalent way. 2.3. Directed tiling with boundary a closed curve C Let C be a closed curve. A directed tiling with boundary C is a set D of directed dominoes such that for every unit segment s of the Z2 -grid, the sum of weights of the directed segments of C with support s (with their multiplicity), denoted by WC (s), is equal to the sum of weights of the directed segments of D with support s, denoted by WD (s). An example is shown in Fig. 3. If w is a word, for simplicity we will write Ww (s) instead of WCw (s).

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3

85

4

2

5

1 Fig. 3. A closed curve C and a directed tiling with boundary C.

2.4. Relations between tilings, balance, and the word problem Theorem 1. Let w be a
A (i)u ≡ u A+u (i);

−1

A−1 (i)u ≡ u A+u (i):

(ii) If Cu is a closed curve,

A (i)u ≡ u A (i);

A−1 (i)u ≡ u A−1 (i):

(iii)

A (black) A (white) ≡ A (white) A (black) ≡ 1: Proof. (i) For symmetry reasons we have only to consider A = O and i = black. Thus A (i) = aba−1 b−1 . The proof is by induction on the length of u. For |u| = 0, the property holds. For |u| = 1, there are four cases to consider. • u=a We have to prove (aba−1 b−1 )a ≡ a(ba−1 b−1 a). Obvious. • u = a−1 We have to prove (aba−1 b−1 )a−1 ≡ a−1 (ba−1 b−1 a). Notice that for any letters x; y we have xyx−1 ≡ x−1 yx. Thus (aba−1 b−1 )a−1 = (aba−1 )b−1 a−1 ≡ (a−1 ba)b−1 a−1 = a−1 b(ab−1 a−1 ) ≡ a−1 b −1 −1 (a b a). QED

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• u=b We have to prove (aba−1 b−1 )b ≡ b(ba−1 b−1 a). (aba−1 b−1 )b ≡ aba−1 ≡ bb−1 aba−1 ≡ b(b−1 ab)a−1 ≡ b(bab−1 )a−1 . QED • u = b−1 We have to prove (aba−1 b−1 )b−1 ≡ b−1 (ba−1 b−1 a). (aba−1 b−1 )b−1 ≡ a(ba−1 b−1 )b−1 ≡ a(b−1 a−1 b)b−1 ≡ ab−1 a−1 ≡ b−1 b(a−1 b−1 a). QED For |u|¿1, let u = xv with x ∈ A. By induction hypothesis: u A (i) = xv A (i) ≡ x A+v (i) ≡ (A+v)+x (i)x = A+xv (i)x = A+u (i)x. QED The second equivalence of (i) is obtained straightforwardly from the 7rst one, computing inverses of each member. (ii) Clear from (i) since in the case when Cu is a closed curve, A (i) = A+u (i). (iii) C A (i) is a closed curve, so from (ii) we get:

A (black) A (white) ≡ A (white) A (black): Then A (black) A (white) has two possible values depending on the position of point A with respect to black cells. First case:

A (black) A (white) = (aba−1 b−1 )(ba−1 b−1 a) ≡ aba−1 a−1 b−1 a ≡ 1. Second case:

A (black) A (white) = (ba−1 b−1 a)(aba−1 b−1 ) = ba−1 b−1 a(aba−1 )b−1 ≡ ba−1 b−1 a(a−1 ba)b−1 ≡ 1. The second lemma expresses an invariant property of the rewriting operation Lemma 2. For every word w such that Cw is a closed curve, every rewriting w → w satis<es: (i) Cw
is a closed curve (ii) Cw
is balanced i= Cw is balanced. Proof. (i) is clear, since every rewriting rule keeps invariant the di=erences between the number of occurrences of a letter x and its inverse x. ˜ (ii) One has to consider every rule. We just give the proof for one of them, the other cases are similar. Consider the rule a → b−1 b−1 abb. Let w = w1 aw2 and w = w1 b−1 b−1 abbw2 . As shown in Fig. 4, only two cells have their index modi7ed in this transformation, namely the cells c1 and c2 . We have Iw
(c1 ) = Iw (c1 ) + 1 and

Iw
(c2 ) = Iw (c2 ) + 1:

But one of the cells is black and the other one is white, so Cw
is balanced i= Cw is balanced. The following lemma expresses the link between the number of white and black cells in a region and the possible reduction of its contour word.

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87

a c1

b

-1

c1

b

c2

b -1

c2

b

a Fig. 4. Modi7cation of indices.

T

O

Fig. 5. A spanning tree.

Lemma 3. If Cw is a direct (resp. undirect) simple closed curve whose interior contains p black cells more than white ones, then w ≡ p (resp. w ≡ −p ). In other words, if Cw is a simple closed curve, then w ≡ Iw (black)−Iw (white) . Proof. Recall that = aba−1 b−1 . Suppose 7rstly that Cw is direct and let R be the region inside Cw . Let D be a maximal set of disjoint dominoes inside Cw and let T be the domain covered by dominoes of D. If R = T , it means that there is an exact tiling of R, and it is well known [2,7] that in that case, w ≡ 1. Actually, in that case, Cw is balanced and p = 0. If T ⊂ R, using only the edges of dominoes of D one can draw a tree which joins the origin O and the noncovered cells of R (Fig. 5). If we “cut” the domain T along the edges of the tree we get a word w such that ww is a direct contour word of T starting in O (Fig. 6). Since T is tiled by dominoes, one has ww ≡ 1. Thus w ≡ w−1 : We are to prove w−1 ≡ p . In order to make the reduction easier, we replace w by w such that w ≡ w and  w contains the undirect contour of each cell of R not covered by D as it is shown in Fig. 7. Now we use Lemma 1 to shift all the cells of R not covered by D at the origin O, following the path drawn by w . If there are n + p black cells of R not covered by

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D. Beauquier, J.-C. Fournier / Theoretical Computer Science 281 (2002) 81 – 97

Fig. 6. A direct contour word of T .

Fig. 7. Modi7cation of w .

D and n white cells of R not covered by D, we get: w ≡ 0−(n+p) (black) 0−n (white)w where w is reducible to 1 using relations xx−1 ≡ 1 for x ∈ A. We obtain 7nally: w ≡ w−1 ≡ w−1 ≡ 0n+p (black) 0n (white) ≡ 0p (black) = p : Suppose now that Cw is an undirect simple closed curve. Then Cw˜ is a direct simple closed curve and thus: w˜ ≡ p ; So w ≡ −p . ∗

Proof of Theorem 1. (i) ⇒ (ii). Suppose w → 1. Each rewriting rule u → u of type (I) de7nes a directed domino whose directed contour word is uu−1 . Thus the reduction ∗ w → 1 de7nes a set of directed dominoes S corresponding to rules of type (I) used in this derivation. We claim that S is a directed tiling with boundary Cw . Let wn be the word obtained after using n rules, and Sn be the corresponding set of n directed dominoes obtained after n rules. Thus w0 = w and S0 = ∅. As a matter of fact, for each segment s the value of Wwn (s)+WSn (s) is independent of n. Clearly a rule of type (I) keeps Wwn (s) and WSn (s) unchanged. Concerning rules of type (II), take for example the rule bab−1 → b−1 ab (other cases are similar). There are six segments for which Wwn (s) and WSn (s) may change, namely the boundary segments of the domino de7ned by the rule. It is clear from Fig. 8 that for each of these six segments, the value of Wwn (s) + WSn (s) is unchanged.

D. Beauquier, J.-C. Fournier / Theoretical Computer Science 281 (2002) 81 – 97

89

a b -1

b

-1 b

b a

Fig. 8. Modi7cation of segment weights. ∗

Let N be the length of the derivation w → 1. The sum Ww0 (s) + WS0 (s) is thus equal to WwN (s) + WSN (s). But wN = 1, so WwN (s) = 0. Moreover SN = S. Therefore, Ww (s) = WS (s) for every s and the claim is proved. (ii) ⇒ (iii). Let D be a directed tiling with boundary Cw . We will prove that the closed curve Cw is balanced, i.e. Iw (black) = Iw (white). Since for every segment s we have WD (s) = Ww (s), we can write:   Iw (c) − Iw (c) color(c)=black

=

 d∈D



color(c)=white

Id (cd ) − Id (cd ) +

 c=cd ∧c=cd


 Id (c) ;

(1)

where d represents a domino of D; cd ; cd , respectively, its black and white cell, and Id (x) represents the index of cell x according to the set of directed segments of domino d. For each domino d we have Id (c) = 0 for c = cd ∧ c = cd . On the other hand, for each domino d; Id (cd ) = Id (cd ). Thus from (1) we get the result. (iii) ⇒ (iv). Firstly a notation: for a non empty word u, we denote by
q , and w ≡ u. Thus using Lemma 2, Iu (black) − Iu (white) = Iv (black) − Iv (white) = 0. On the other hand, Iu (black) − Iu (white) = q. So q = 0 and then w ≡ 1. 

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a b c

O

c a b

Fig. 9. The hexagonal grid.

T1

T2

B1

B2

B3

Fig. 10. Five tiles.

3. The 3-hexes group G h In this section, we are interested in tilings of the hexagonal grid in the plane. We prove that the results obtained for the square grid hold also for the hexagonal one. The plane is here colored with three colors black; white; gray (Fig. 9) and the used tiles are all the balanced 3-hexes [4]: 2 “triangles” and 3 “bars” (see Fig. 10). The corresponding group Gh is de7ned by three generators a; b; c and the following relations: a2 = b2 = c2 = 1; (ca)2 (bc)2 (ab)2 = 1 (T1 ); (ab)2 (ca)2 (bc)2 = 1 (T2 ); (ca)3 b(ca)3 b = 1 (B1 ); (ab)3 c(ab)3 c = 1 (B2 ); (bc)3 a(bc)3 a = 1 (B3 ): Actually this set of relations is not minimal, relations T2 ; B2 and B3 for example can be deduced from the other ones. Fig. 11 gives a geometric view of these dependencies (hexagons are represented as black points). The notions de7ned in Section 2 can be easily transposed. Since there are three colors, there are three indices of a closed curve, one for each color.

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91

B2

B2 B2 T1

B3

T1 T2

T1

B1 B1 from B 2 and T 1

T 2 from B 2 and T 1

B3 from B 2 and T 1

Fig. 11. Dependencies between tiles.

A closed curve is balanced if IC (black) = IC (white) = IC (gray). Here is the hexagonal version of Theorem 1: Theorem 2. Let w be a
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bac c b a b b ac b c b a

B2

T1 Fig. 12. Shift of a cell.

Proof of Lemma 3 in the hexagonal case. The hexagonal version of the Lemma is as follows: If Cw is a direct (resp. undirect) simple closed curve whose interior contains p black cells more than white ones, and q white cells more than gray ones then w ≡ $0p+q (black)$0q (white) (resp: w ≡ $0−q (white)$0−p−q (black)): The idea of the proof is the same: every cell is “shifted” to the origin and Lemma 1 is used. As a corollary of Theorem 2, we get: Proposition 1. One can decide whether a given word w is equal to 1 in the group G h . 4. Generalization: the group Gp There exist three regular polygons which tile the plane by translation: triangles, squares and hexagons. If we consider a triangular tiling as in Fig. 13 and the associated group Gt with generators a; b; c and relations ab = ba; bc = cb; ca = ac, the results of Theorems 1 holds too, and are easy to prove since Gt is an abelian group. At this point, one can conjecture that these three results for Gs ; G h ; Gt are particular cases of a more general one. Consider a regular tiling of the plane, for example the square one. A polyomino is a simply connected set of unit squares (two squares are connected if they share at least one vertex). Choose a polyomino tile p which tiles the plane by translation. Such a tile is called an exact tile. In [1], it is proved that the exact tile p admits a contour word of the form uvwu˜v˜w˜ (see Fig. 14). Consider a periodic tiling T of the plane with translated copies of p such that p is adjacent to 6 copies of itself (the adjacency may be reduced to a vertex). Such a tiling is called a 6-periodic p-tiling. Provide colors 1; 2; : : : ; n to the n squares of p and extend this coloration to the tiling T (Fig. 15). We have the following property:

D. Beauquier, J.-C. Fournier / Theoretical Computer Science 281 (2002) 81 – 97

93

Fig. 13. A triangular grid.

u ~ w

v

~ v w

~ u Fig. 14. An exact tile.

Lemma 4. Let p be an exact tile. Consider a plane colored by a 6-periodic p-tiling. Every polyomino q of the colored plane which contains exactly one square of each color is also an exact tile. Proof. Let W be a translation vector of T. Denote by qW the translated image of q in translation W . Then q and qW have no common unit square. Actually, suppose a square c belongs to q and qW . Then squares c and cW belong to qW , but these squares have the same color, impossible. Let q be a translated copy of q which intersects p. Consider the sets A = p ∩ q ; B = p\q and C = q \p. Thus p = A ∪ B and q = A ∪ C. By [1], we know that there exist two independent vectors U and V such that the plane is the disjoint union of the copies pkU +k
V = (A ∪ B)kU +k
V for k; k  ∈ Z. It is easy to  prove that the plane is also the disjoint union of the copies qkU +k
V = (A ∪ C)kU +k
V   for k; k ∈ Z. Thus q is also an exact tile. Let p be an exact tile with direct contour word uvwu˜v˜w; ˜ T be a 6-periodic p-tiling of the plane and let ( be the starting point of u on the boundary of p. Let c be a

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1

3

4

6

7

8

(a)

5

1 1 2

2

3

4

5

6

7

1

8

2

3

4

1

2

3

4

5

6

7

1

2

3

5

3

4

6

7

1

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4

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7

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2

3

4

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2

3

4

5

6

7

8

2

3

4

5

6

7

6

7

5 5

8

8

8

(b)

Fig. 15. (a) An exact tile p; (b) a 6-periodic p-tiling.

cv

Ω c

V

U

cu

Fig. 16. c; cU and cV .

square of p which contains the vertex (. There are two vectors U and V such that translated copies of c, respectively, cU and cV , are adjacent to p (see Fig. 16). Thus using Lemma 4, pU = p\{c} ∪ {cU } and pV = p\{c} ∪ {cV } are exact tiles. Denote by p the set of three tiles {p; pU ; pV } (Fig. 17). Consider the group Gp with generators a; b and relations q = 1 for the contour words q of the three tiles in p. The notions of directed tiling by p with boundary Cw , closed balanced curve Cw , are de7ned as before. It is easy to prove using the same technics as in theorem 1 that: Lemma 5. Let C be a closed curve with contour word w. (1) w = 1 in the group Gp implies there exists a directed tiling with boundary Cw ,

D. Beauquier, J.-C. Fournier / Theoretical Computer Science 281 (2002) 81 – 97

u

p:

v

~ w

95

w u~

~ v

pV :

pU :

Fig. 17. {p; pU ; pV }.

c+

+ cU Fig. 18. A one step move of a colored cell.

(2) if there exists a directed tiling with boundary Cw then Cw is a closed balanced curve. Finally we prove the following statement: Lemma 6. Consider a plane colored by a 6-periodic p-tiling. If a closed curve C is balanced, there exists a directed tiling by p with boundary C. Proof. Consider the square c used to de7ne p. Denote by c+ (resp. c− ) the square c oriented in a positive (resp. negative) way, and use the same notation for general tiles. + Starting with c+ , if we cover it with the oriented tile p− and cover p− with pU + as shown in Fig. 18 we get cU , in the sense that for each segment s in the plane: Wc+ ∪p− ∪pU+ (s) = WcU+ (s). The same holds for vector V . It means that iterating the process, one can “move” an oriented square from one place into any other place with the same color in the plane (with the same orientation). Now, consider the simpler case when C is a simply closed balanced curve with positive orientation. Let r be the number of squares with a given color inside C.

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With respect to the weight of segments, C can be looked as the union of its squares oriented positively. Thus each square inside C is “moved”, using tiles of p to the unique cell of the same color in p. Let P be the directed tiling composed with the directed tiles of p used for these moves. We complete P by adding r copies of p− over p. We have W (C ∪ P)(s) = 0 for every segment s. Thus C is the boundary of the directed tiling P− obtained from P by changing the orientation of tiles in P. If we consider now the general case for C, we decompose C into a union of loops by the following “loop algorithm” similar to the one used in Section 2, where u is a contour word of C: Begin ) := 1; * := u; D := ∅ While * = 1 Do ) := )
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97

References [1] D. Beauquier, M. Nivat, Tiling the plane with one tile, in: Proc. 6th Annu. Symp. on Computational Geometry, ACM Press, New York, 1990, pp. 128–138. [2] J.H. Conway, J.C. Lagarias, Tiling with polyominoes and combinatorial group theory, J. Combin. Theory 53 (1990) 183–208. [3] J.C. Fournier, Pavage des 7gures sans trous par des dominos: fondement graphique de l’algorithme de Thurston, parallSelisation, unicitSe, dSecomposition, Theoret. Comput. Sci. 159 (1–2) (1997) 105–128. [4] B. GrTunbaum, G.C. Shepard (Eds.), Tilings and Patterns, W.H. Freeman and Company, New York, 1987 [5] C. Kenyon, R Kenyon, Tiling a polygon with rectangles, in: 33rd Symp. on Foundations of Computer Science, IEEE Computer Society, Silver Spring, MD, 1992, pp. 610 – 619. [6] R.C. Lyndon, P.E. Schupp (Eds.), Combinatorial Group Theory, Inst. fTur Technische Informatik und Kommunikationsnetze, ETH ZTurich, 1977. [7] W.P. Thurston, Conway’s tiling group, Amer. Math. Monthly (1990) 757–773.

Theoretical Computer Science 281 (2002) 99 – 107

www.elsevier.com/locate/tcs

Coding rotations on intervals Jean Berstela , Laurent Vuillonb; ∗ a Institut

Gaspard Monge (IGM), Universite de Marne-la-Vallee, 5, boulevard Descartes, 77454 Marne-la-Vallee Cedex 2, France b Laboratoire d’informatique algorithmique: fondements et applications (LIAFA) and CNRS, Universite Denis-Diderot (Paris VII), 2, place Jussieu, 75251 Paris Cedex 05, France

Abstract We show that the coding of a rotation by  on m intervals can be recoded over m Sturmian c 2002 Elsevier Science B.V. All rights reserved. words of angle .


1. Introduction The coding of rotations is a tool for the construction of in,nite words over a ,nite alphabet. Consider a rotation R  , given by an angle , and de,ned for a point x by R  (x) = {x + }, where {y} denotes the fractional part of y. Consider next a partition of the unit circle in m + 1 half open intervals {B0 ; B1 ; : : : ; Bm }. For any starting point x with 06x¡1, one gets an in,nite word u by I (x)I (R  (x))I (R2 (x)) · · · I (Rn (x)) : : : ; where I (y)= i if y ∈Bi . In the special case where  is irrational and the partition is B0 = [0; [ and B1 = [; 1[, this construction produces exactly the well-known Sturmian words. These words appear in various domains as computer sciences [2], Physics, Mathematical optimization and play a crucial role in this article. It is remarkable that Sturmian words have a combinatorial characterization. Thus, they are exactly aperiodic words with (subword) complexity p(n)= n + 1 where the complexity function p : N → N counts the number of distinct factors of length n in the in,nite word u [2]. The same general construction allows also to compute Rote words with complexity p(n) = 2n by using an irrational rotation and the partition B0 =[0; 12 [ and B1 = [ 12 ; 1[ (see [7]). More generally, one can ∗

Corresponding author. E-mail address: [email protected] (L. Vuillon).

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 0 9 - 9

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J. Berstel, L. Vuillon / Theoretical Computer Science 281 (2002) 99 – 107

0 0 0

0 0

1 1

0 0

1 1 1

0 0 0

0 0 1 0 1 0

1 1 1 1

0 0 0 0

1

0

1 1 0

1 0 0

0 0 1

0 1 1 1

1 0 0 0

2 3

0 0 0

1 1 1 1

1

1 0 1

0 1 1

1

0 0 0 0

0 1 0 0

0 1 0

1 1

1 1 1

1 1 0 0

1 1 1

1 0 0 1

0 1 1 0 1 1 1 0

1 0 1 1

1 1 0 1

0 0 1 0

0 0 1 1

2

0 0 0 1 0 0 0 0

1 1 1 1

0 0 0 0

1 1 1 1

Fig. 1. Automata for m = 1; 2; 3.

obtain in,nite words with complexity p(n) = an + b; where a and b are real, by coding of rotation [1,3]. In addition, codings of rotation with an irrational value of  and the partition B0 = [0; [ and B1 =[; 1[ are intimately related to Sturmian words. Indeed, the ,rst sequence is the diBerence term by term of two Sturmian words [6]. Didier gives a characterization of the coding of rotation with a partition of m intervals of length greater than  by using Sturmian words and cellular automata [5]. Finally, Blanchard and Kurka study the complexity of formal languages that are generated by coding of rotation [4]. The goal of this article is to show that the coding of a rotation by  on m + 1 intervals can be recoded over m + 1 Sturmian words of angle  by the use of an automaton. Let the partition of [0; 1[ be given for j = 0; : : : ; m, by [j ; j+1 [, where 0 = 0¡1 ¡2 ¡ · · · ¡j ¡ · · · ¡m+1 = 1, then the kth Sturmian word is given by Jk (x)Jk (R (x))Jk (R2 (x)) · · · Jk (Rn (x)) · · · ; where Jk (y)=1 if y ∈Ik =[k ; k + [ and 0 otherwise (thus the m + 1 Sturmian words have the same language). The automaton for the recoding is a universal automaton (i.e. it is the same for all (m + 1) tuples of Sturmian words). It has m + 1 states. Its edges are (m + 1) tuples of 0 and 1. The edge labeled by the vector composed of the ith letters of each Sturmian word gives the ith letter of the coding of rotation (see Fig. 1).

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101

Fig. 2. Partition of the unit circle.

2. Examples Fig. 2 shows a partition of the unit circle into 4 intervals of the form [j ; j+1 [. It shows also the positions of i +  and the coding by 8 intervals associated with binary vectors called XK in Section 3. We can ,nd the coding of the interval [j ; j+1 [ by the automaton for m = 3 applied to the binary vector values. As an example, using the universal automaton for m = 3, the four following Sturmian words can be recoded on a word on a four letter alphabet: 10001001000100010 · · · 01000100100010001 · · · 00100100010001001 · · · 00010001001000100 · · · is recoded on the following word: 01230220123012302 · · · : 3. Notation We will consider subsets of [0; 1[ that we call intervals. Let x; y be in [0; 1[. Then we set  if x¡y;   {z | x 6 z ¡ y} if x = y; [x; y[= ∅   {z | x 6 z ¡ 1} ∪ {z | 0 6 z ¡ y} if x¿y: In particular, [x; y[ =[0; y[ ∪ [x; 1[ if x¿y. This is precisely the notion of an interval on the torus T =R=Z. Let ; 1 ; : : : ; m be numbers in the interval ]0; 1[, with 1 ¡ · · · ¡m . It will be convenient to set 0 = 0 and m+1 = 1. The m + 1 intervals Bk = [k ; k+1 [;

k = 0; : : : ; m

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are a partition of [0; 1[. We consider the rotation of angle  de,ned by R  (x) = x +  mod 1. De,ne intervals Ik by (all values are computed modulo 1) Ik = [k ; k + [;

k = 0; : : : ; m:

We will be specially interested in the nonempty intervals   XK = Ik ∩ Ik : k∈K

k ∈K =

Here, K is a subset of M = {0; : : : ; m}, and Ik = [0; 1[\Ik is the complement of Ik . Observe that, for any nonempty interval I = [x; y[, one gets IH = [y; x[. 4. Circular order We want to compute intersections of intervals. Although the geometric approach is easy to understand, it is error prone because points are usually not in general position. Therefore, we consider a more combinatoric approach. Given numbers x1 ; : : : ; xn ∈[0; 1[, the sequence (x1 ; : : : ; xn ) is circularly ordered, or c-ordered for short, if there exists an integer h with 16h6n such that 0 6 xh 6 xh+1 6 · · · 6 xn 6 x1 6 · · · 6 xh−1 ¡ 1:

(1)

If (1) holds, then either x1 = · · · =xn , or the integer h is unique. Also, if (x1 ; : : : ; xn ) is c-ordered, then clearly (x2 ; : : : ; xn ; x1 ) is c-ordered. Any subsequence of a c-ordered sequence is c-ordered. Observe also that if (x1 ; : : : ; xn ) is c-ordered and x1 ¡xn then x1 6 · · · 6xn . Indeed, if (1) holds for h = 1, then xn 6x1 . Two rules are useful. Lemma 4.1. (i) Translation Rule If (x1 ; : : : ; xn ) is c-ordered and yi ≡ xi + mod 1, then (y1 ; : : : ; yn ) is c-ordered. (ii) Insertion Rule If (x1 ; : : : ; xn ) and (y1 ; : : : ; ym ) are c-ordered, if furthermore y1 = ym and xi =y1 ; xi+1 =ym , then (x1 ; : : : ; xi ; y2 ; : : : ; ym−1 ; xi+1 ; : : : ; xn ) is c-ordered. Proof. (i) We may assume 06x1 6 · · · 6xn ¡1. The real numbers xi +  satisfy x1 + 6 · · · 6xn + ¡1 + x1 + . If xn + ¡1, then yi = xi +  and (y1 ; : : : ; yn ) is c-ordered. Otherwise, let h be the smallest integer such that xh + ¿1. Then x1 +  6 · · · 6 xh−1 +  ¡ 1 6 xh +  6 · · · 6 xn + : If h = 1, one gets 1¡x1 + 6 · · · 6xn + ¡2 and clearly (y1 ; : : : ; yn ) is c-ordered. If h¿1, then xn +  − 1¡x1 +  implies yh 6 · · · 6 yn ¡y1 6 · · · 6 yh−1 : (ii) There are two cases. If xi = max{x1 ; : : : ; xn }, then xi+1 6 · · · 6xn 6x1 6 · · · 6xi . From xi =y1 ; xi+1 =ym , it follows that ym ¡y1 . Let h = 1 be the integer such that 06yh 6 · · · 6ym ¡y1 6 · · · 6yh−1 . Then 0 6 yh 6 · · · 6 ym = xi+1 6 · · · 6 xn 6 x1 6 · · · 6 xi = y1 6 · · · 6 yh−1 :

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103

If xi ¡ max{x1 ; : : : ; xn }, then xi =y1 ¡ym = xi+1 and consequently xi = y1 6y2 6 · · · 6ym = xi+1 . We observe that the insertion rule does not hold if y1 = ym . Consider the two c-ordered sequences (x; x; y) and (x; y; x), where 0¡x¡y¡1. Inserting the second into the ,rst give the sequence (x; y; x; y) which is not c-ordered. We prove another useful formula. Lemma 4.2. Let ¡ 12 . If (x; y; x + ) is c-ordered, then (x; y; x + ; y + ) is c-ordered. Proof. The condition ¡ 12 implies that (x; x +; x +2) is c-ordered. By the translation rule, we get that (x + ; y + ; x + 2) is c-ordered. The insertion rule shows that (x; x + ; y + ; x + 2) is c-ordered and, again by the insertion rule, one gets that (x; y; x + ; y + ) is c-ordered. 5. Intersection Circular order is useful in considering intersections of intervals. Let I = [x; y[ be a nonempty interval. Then x ∈[x; y[ iB (x; x ; y) is ordered. Let I = [x; y[ and I  = [x ; y [ be nonempty intervals. Then x ∈I iB (x; x ; y) is c-ordered. Since I ∩ I  = ∅ iB x ∈I or x ∈I  , the intervals I and I  are disjoint iB (x; y; x ) and (x ; y ; x) are c-ordered. Consequently, we have shown Lemma 5.1. Let I =[x; y[ and I  =[x ; y [ be nonempty intervals. Then I ∩ I  = ∅ if and only if (x; y; x ; y ) is c-ordered. The length s of an interval I =[x; y[ is the number s = y − x if x6y, and is s = 1 − (x − y) if y¡x. In both cases, y ≡ x + s mod 1 so that, knowing the length, we may write I = [x; x + s[. Lemma 5.2. Let I =[x; y[ and I  =[x ; y [ be intervals of the same length 0¡¡ 12 . If I and I  intersect, then (x; x ; y; y ) or (x ; x; y ; y) is c-ordered. In the ;rst case, I ∩ I  = [x ; y[, in the second case, I ∩ I  = [x; y [. Observe that if the length of I and I  is greater than 1=2, then the intersection needs not to be an interval. Proof. The discussion before Lemma 5.1 shows that I and I  intersect if and only if (x; x ; y) or (x ; x; y ) are c-ordered. From Lemma 4.2, it follows that (x; x ; y; y ) or (x ; x; y ; y) is c-ordered. Moreover, y = x and x = y since otherwise (x; y; x ; y ) is c-ordered and the intervals are disjoint by Lemma 4.2. If x =x (or equivalently if y =y ), then I = I  . Thus, we may assume that the numbers x; y; x ; y are distinct. Assume the ,rst ordering holds. The formula for the intersection is straightforward if 06x¡x ¡y¡y ¡1. If 06x ¡y¡y ¡x¡1, then I = [0; y[ ∪ [x; 1[ and I ∩ I  = [x ; y[. The two other cases are proved in the same way.

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The previous lemma will be applied to the intervals Ik = [k ; k + [. They all have same length . We write the conclusion for further reference. Lemma 5.3. Let ¡ 12 . Let Ik =[k ; k + [ and I‘ = [‘ ; ‘ + [ be two intervals. If Ik and I‘ intersect then (k ; ‘ ; k + ; ‘ + ) or (‘ ; k ; ‘ + ; k + ) is c-ordered. Moreover, Ik ∩ I‘ =[‘ ; k + [ in the ;rst case, and Ik ∩ I‘ = [k ; ‘ + [ in the second case. The following observation is the basic step for analyzing the coding induced by a rotation. Recall that for K ⊂{0; : : : ; m},   Ik ∩ Ik : XK = k∈K

k ∈K =

We assume from now on that ¡ 12 . Proposition 5.4. Assume XK = ∅ for some K ⊂ {0; : : : ; m} and assume (i1 ; i2 ; i3 ; i4 ) is a c-ordered sequence. If i1 ; i3 ∈K, then i2 ∈K or i4 ∈K. = Since XK = ∅, the interval Proof. Arguing by contradiction, suppose that i2 ; i4 ∈K. Ii1 ∩ Ii3 is not empty, therefore by Lemma 5.3 (i1 ; i3 ; i1 + ; i3 + ) or (i3 ; i1 ; i3 + ; i1 + ) is c-ordered (or the sequence obtained by exchanging i1 and i3 ). Consider the ,rst case, the second is the same by exchanging i2 and i4 . Since (i1 ; i2 ; i3 ) is c-ordered, the translation rule shows that (i1 + ; i2 + ; i3 + ) is c-ordered which gives, applying twice the insertion rule, that (i1 ; i2 ; i3 ; i1 + ; i2 + ; i3 + ) is c-ordered. From this, we get that (i1 ; i2 ; i1 +; i2 +) is c-ordered. From Lemma 5.1, we know that Ii1 ∩ Ii3 =[i3 ; i1 + [, and this is then disjoint from Ii2 = [i2 + ; i2 [. Proposition 5.5. If XK is not empty, then there exist integers k; ‘ with 06k¡‘6m such that {K; M \K} ={{k; : : : ; ‘ − 1}; {‘; : : : ; m; 0; : : : ; k − 1}}. Proof. This is a direct consequence of the preceding discussion. It follows that there are only (m + 1)(m + 2) intervals XK to be considered. In fact, consider the numbers 0; 1 ; : : : ; m ; 1 and ; 1 + ; : : : ; m + . They partition [0; 1[ into exactly 2m + 2 intervals. Each of these intervals is contained in one and only one of the XK (but X∅ may be scattered over several of the small intervals). This means that, among the (m + 1)(m + 2) possible intervals XK , there are only 2m + 2 that are used in a particular setting of the values of ; 1 ; : : : ; m . Theorem 5.6. Assume K = ∅; M , and XK = ∅. Then XK = [‘−1 ; k + [ ∩ [k−1 + ; ‘ [ where k and ‘ are de;ned in Proposition 5:5. If K = {k} is a singleton, then the formula still holds with ‘ − 1 = k. Proof. Supposethat K ={k; : : : ; ‘ − 1} with k¡‘.  The other case is symmetric. We ,rst prove that n∈K In =[‘−1 ; k + [: Set YK = n∈K In .

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Since XK = ∅, the interval Ik ∩ I‘−1 is not empty. By Lemma 4.3 there are two cases: either (k ; ‘−1 ; k +; ‘−1 +) is c-ordered, or (‘−1 ; k ; ‘−1 +; k +) is c-ordered. We show that this second case cannot happen. Indeed in this case, YK ⊂ Ik ∩ I‘−1 = [k ; ‘−1 + [: Moreover for each n ∈M \K, the sequence (‘−1 ; n ; k ) is c-ordered. By translation and insertion, the sequence (‘−1 ; n ; k ; ‘−1 +; n +; k +) is c-ordered. This shows that In ⊃ Ik ∩ I‘−1 , and consequently Ik ∩ I‘−1 ∩ In = ∅ for each n in M \K, contradicting the assumption that XK = ∅: Thus, (k ; ‘−1 ; k + ; ‘−1 + ) is c-ordered. This implies that Ik ∩ I‘−1 = [‘−1 ; k +[: If i ∈K then (k ; i ; ‘−1 ) is c-ordered. By translation, (k +; i +; ‘−1 +) is c-ordered. By insertion of (k ; i ; ‘−1 ) into (k ; ‘−1 ; k + ; ‘−1 + ) one gets (k ; i ; ‘−1 ; k + ; ‘−1 + ) is c-ordered. Again by insertion of (k + ; i + ; ‘−1 + ), the sequence (k ; i ; ‘−1 ; k + ; i + ; ‘−1 +) is c-ordered. Thus YK = [‘−1 ; k + [: The second part of the proof deals with n∈M \K In : In this intersection the index n runs through the set {0; : : : ; k − 1; ‘; : : : ; m}. The set M \K is partitioned into three possibly empty subsets as follows: n∈N iB In ⊃ YK ; n ∈P iB In ∩ YK = [‘−1 ; n [ and ,nally n ∈Q iB In ∩ YK =[n + ; k + [: Of course,    (In ∩ YK ) ∩ (In ∩ YK ) ∩ (In ∩ YK ): XK = n∈N

n∈P

n∈Q

If one of the  sets N; P; Q is empty it does  not contributeto the intersection. Clearly, (I ∩Y )= Y . Next ;  [. If P is not K n∈N n n∈P (In ∩YK ) = n∈P [ ‘−1 n  K empty then ‘ is in P and n∈P (In ∩ YK ) = [‘−1 ; ‘ [. Finally, n∈Q (In ∩ YK ) = n∈Q  [n +; k +[. If Q is not empty then k−1 is in Q and n∈Q (In ∩ YK ) = [k−1 +; k +[.  To ,nish the proof, we just have to verify that in each case, XK = n∈N (In ∩ YK ) ∩   (In ∩ YK ) is equal to [‘−1 ; k + [ ∩ [k−1 + ; ‘ [. n∈P (In ∩ YK ) ∩ n∈Q If P = ∅ then n∈P (In ∩ YK )=[‘−1 ; ‘ [ and the sequence (‘−1 ; ‘ ; k + ) is c-ordered (case  P1 ). If P =∅ then the sequence (‘−1 ; k + ; ‘ ) is c-ordered (case P0 ). If Q = ∅ then n∈Q (In ∩ YK )=[k−1 +; k +[ and the sequence (‘−1 ; k−1 +; k +) is c-ordered (case Q1 ). If Q =∅ then the sequence (‘−1 ; k + ; k−1 + ) is c-ordered (case Q0 ). Case (P1 Q1 ): If P and Q are nonempty then XK = [‘−1 ; k +[ ∩ [‘−1 ; ‘ [ ∩ [k−1 + ; k + [. As the sequences (‘−1 ; ‘ ; k + ) and (‘−1 ; k−1 + ; k + ) are c-ordered, by the insertion rule either the sequence (‘−1 ; ‘ ; k−1 + ; k + ) or (‘−1 ; k−1 + ; ‘ ; k + ) is c-ordered. The ,rst case is impossible because XK is not empty. The second case implies that XK =[‘−1 ; k + [ ∩ [k−1 + ; ‘ [. Case (P0 Q1 ): If P = ∅ and Q = ∅, then XK = [‘−1 ; k + [ ∩ [k−1 + ; k + [ and the sequences (‘−1 ; k + ; ‘ ), (‘−1 ; k−1 + ; k + ) are c-ordered. By insertion the sequence (‘−1 ; k−1 +; k +; ‘ ) is c-ordered. Thus, XK = [‘−1 ; k +[ ∩ [k−1 +; ‘ [. Case (P1 Q0 ): is symmetric to case (P0 Q1 ). Case (P0 Q0 ): If P = ∅ and Q = ∅, then XK = [‘−1 ; k + [ ∩ [k−1 + ; ‘ [ and the sequences (k + ; k−1 + ; ‘−1 ), (k−1 + ; ‘ ; ‘−1 ) are c-ordered. By insertion rule either the sequence (k +; k−1 +; ‘ ; ‘−1 ), or (k +; k−1 +; ‘ ; ‘−1 ) is c-ordered. The ,rst case is impossible because XK is non empty. The second case implies that XK = [‘−1 ; k + [ ∩ [k−1 + ; ‘ [.

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Remark. As an additional property, the preceding proof shows that XK is an interval and the interior of XK does not contain any i or {i + }:

6. Main result Proposition 6.1. If x ∈Bi and x +  ∈Bj ∩ XK , then j ≡ i + |K| mod m + 1. Proof. If K = ∅ or K = M then XK =[‘−1 ; k + [ ∩ [k−1 + ; ‘ [: As x ∈[i ; i+1 [; by translation rule we have y =x +  ∈[i + ; i+1 + [. Furthermore, y ∈[j ; j+1 [: By the preceding remark and by identi,cation, the only possibility is j = ‘ − 1 and i = k − 1: It follows that j =i + |K| mod m + 1: If K =∅ then XK = n∈M In : By hypothesis we have y =x+ ∈Bj : But y ∈XK implies that y ∈[ = j ; j + [= Ij . If |Ij |¿|Bj | then Bj ∩ XK should be empty in contradiction with the hypothesis. Thus |Ij |¡|Bj |. The interval Bj is equal to Ij ∪ [j + ; j+1 [: That is x +  ∈[j + ; j+1 [ and x +  ∈[i + ; i+1+ [: By identi,cation, we have i =j: If K = M then XK = n∈M In : As x ∈In implies x + ∈I = n , by contraposition x + ∈XK implies x ∈I = n for all n. Thus (x; n ; x+) is c-ordered for all n ∈M . As x ∈Bi and x is not in Ii the sequence (i ; i + ; x; i+1 ) is c-ordered. Thus (x; i+1 ; : : : ; i+m−1 ; i ; x + ) is c-ordered. Consequently, XK is equal to [i ; i+1 + [: As x +  ∈[j ; j+1 [ by identi,cation we ,nd j =i. From this proposition, we get the following automaton A (Fig. 1 gives the automata for m= 1; 2; 3). Its set of states is the set M in bijection with the intervals Bk . The alphabet is the set of subsets of M corresponding to the nonempty intervals XK . As already mentioned, there are (m+1)(m+2) of them. The transitions or edges are given by the proposition: (i; K; j) is a transition if j ≡ i + |K| mod m + 1. Observe that the automaton is deterministic. Also, it is universal in the following sense: for a particular setting of ; 1 ; : : : ; m , if the i and the j +  are two by two distinct, there are only 2m+2 of the edges that are used. Indeed they are exactly 2m+2 intervals in the partition and between i and i+1 the coding is uniquely determined for all i. Moreover, the automaton has the noteworthy property that each label provided it is neither (0; : : : ; 0) nor (1; : : : ; 1) appears at only one edge. Consequently, both the starting state and the arrival state of such an edge is determined by is label. We denote by A(v0 ; : : : ; vm ) the arrival state of the edge with label (v0 ; : : : ; vm ). We now explain the result in terms of in,nite words. Recall that Bi = [i ; i+1 [ and Ii = [i ; i + [. Let u be the in,nite word de,ned by I (x)I (R (x))I (R2 (x)) · · · I (Rn (x)) · · · ;

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where I (y)= i if y ∈Bi . Let sk be the kth Sturmian word de,ned by Jk (x)Jk (R (x))Jk (R2 (x)) · · · Jk (Rn (x)) · · · ; where Jk (y)=1 if y ∈Ik =[k ; k + [ and 0 otherwise. Theorem 6.2. The in;nite word u and the m + 1 Sturmian words si are related by A(s0 (n); : : : ; sm (n)) = u(n) for all n¿n0 , where n0 is the smallest integer such that {s0 (n0 ); : : : ; sm (n0 )} contains both 0 and 1. For n¡n0 the value u(n) is the starting state of the edge labeled {s0 (n0 ); : : : ; sm (n0 )}. Remark that the result is written for  irrational in order to manipulate Sturmian words. The construction of the automaton is independent of the rationality of  (this case leads to in,nite periodic words). References [1] P. Alessandri, V. BerthKe, Three distance theorems and combinatorics on words, Enseig. Math. 44 (1998) 103–132. [2] J. Berstel, P. SKeeK bold, Sturmian words, in: M. Lothaire (Ed.), Algebraic Combinatorics on Words, Chap. 2, Cambridge University Press, Cambridge, to appear. [3] V. BerthKe, Etude arithmKetique et dynamique de suites algorithmiques, Habilitation aM diriger des reserches, UniversitKe d’Aix-Marseille II, 1999. [4] F. Blanchard, P. Kurka, Language complexity of rotations and Sturmian sequences, Theoret. Comput. Sci. 209 (1998) 179–193. [5] G. Didier, Combinatoire des codages de rotations, Acta Arith. 85 (1998) 157–177. [6] P. Hubert, PropriKetKes combinatoires des suites dKe,nes par billard dans les triangles pavants, Theoret. Comput. Sci. 164 (1996) 165–183. [7] G. Rote, Sequences with subword complexity 2n, J. Number Theory 46 (1994) 196–213.

Theoretical Computer Science 281 (2002) 109 – 130

www.elsevier.com/locate/tcs

Noninterference for concurrent programs and thread systems  G%erard Boudol ∗ , Ilaria Castellani INRIA, Sophia-Antipolis, 2004 Route des Lucioles, BP 93, 06902 Sophia-Antipolis, Cedex, France

Abstract We propose a type system to ensure the property of noninterference in a system of concurrent programs, described in a standard imperative language enriched with parallelism. Our proposal is in the line of some recent work by Irvine, Volpano and Smith. Our type system seems more natural and less restrictive than that originally presented by these authors for the concurrent case. Moreover, we show how to extend the language in order to formalise scheduling policies for systems of sequential threads. The type system is extended to the new constructs, and we show c 2002 Elsevier that noninterference still holds, while remaining in a nonprobabilistic setting.
Science B.V. All rights reserved.

1. Introduction The aim of this paper is to study the property of secure information 3ow, and more speci4cally of noninterference (a notion 4rst introduced by Goguen and Meseguer in [6]) in the setting of concurrency. Here noninterference means that variables of a given security level do not interfere with those of lower or incomparable levels. More precisely, the values of variables at some level are not dependent on the values of variables of higher or incomparable level. Noninterference is meant to model the absence of information 3ow from any security level
0 to another one
1 , except when this is explicitly allowed, that is
0 6
1 . Such information 3ow is considered insecure, as it amounts to the disclosure of secret information into the public domain. In this introduction, and in the examples given in the paper, the security levels will simply be high (H ), or secret, and low (L), or public. However, all results will be given  Research partially funded by the EU Working Group CONFER II and by the french RNRT Project MARVEL. ∗ Corresponding author. E-mail address: [email protected] (G. Boudol).

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 0 - 5

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 : if PIN = 0 then t := tt else t := tt  : while t
= tt do nil ; r := 0 ; t := tt  : while t
= tt do nil ; r := 1 ; t := tt PIN; t ; t : boolean variables of type H r : boolean variable of type L Fig. 1. Information 3ow through control 3ow.

for an arbitrary lattice of security levels. For some concrete examples using a more sophisticated lattice of security levels, see for instance [3]. Our starting point is the paper [21] by Volpano et al. and the subsequent paper [18] by Smith and Volpano, where noninterference is enforced by means of a simple type system in an imperative language with security levels. The language considered in [21] is purely sequential, and is extended in [18] with asynchronous parallelism (interleaving). In such a language, insecure 3ow can be explicit, when assigning the value of a high variable to a low variable, or implicit, when testing the value of a high variable and then assigning to a low variable a value depending on the result of the test, for instance. In the approach of [21,18], explicit 3ow is prevented by requiring that the level of the assigned variable be at least as high as that of the source variable, while implicit 3ow is prevented by asking that the level of the commands in the branches of a conditional (the level of a command being that of its lowest assigned variables) be at least as high as that of the tested variable. Implicit 3ow can also arise in while-loops, and is prevented by a similar condition on the type of the body of the loop. In fact, because of while-loops, the de4nition of noninterference is more precise than what is stated above: it says that no change in the values of low-level variables should be observed as a consequence of a change in high-level variables, provided that the program terminates successfully. Using subscripts to explicitly indicate the security level of a variable, let us consider the following program, that terminates if x H
= 0 and loops forever (doing nothing) otherwise: while xH = 0 do nil; yL := 1

(1)

Should this program be accepted, that is, should it be typable? According to the above de4nition of noninterference the answer is “yes”, since whenever the program terminates it produces the same value yL = 1 for its low-level variable. Indeed, this program is typable in the original system of Volpano et al. [21], since the loop is typable and the sequential composition of typable programs is always typable. However, accepting such a program leads to problems when parallelism is introduced in the language. These problems can be concisely described as “disguising information 3ow as control 3ow”. Let us illustrate the problem by means of an example, which is a simpli4ed version of the PIN example given by Smith and Volpano in [18]. In this example, given in Fig. 1, three threads ,  and  are run (asynchronously) in parallel. There are four variables, a high-level variable PIN tested by thread , two high-level

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111

variables t and t serving as “triggers” for threads  and , and a low-level variable r written by  and . As can be easily seen, with initial values t = t = 3 the eKect of this program is to copy the value of the secret variable PIN into the public variable r. The illicit information 3ow from PIN to r is implemented through the control 4ow from  to  or . However, if we assume that a system of concurrent threads is typable provided each component is typable, as would be natural, this particular system is to be accepted. To circumvent this problem, Smith and Volpano propose in [18] to forbid the use of high-level variables in guards of while-loops, and more speci4cally 1 to accept only while-loops of low level. While ruling out the program in (1), and also the threads  and  of the PIN example, this solution seems a bit drastic. It excludes inoKensive programs such as while x H = 0 do nil. We shall propose here a diKerent solution to the problem raised by while-loops in the presence of parallelism, which allows this program to be typed, while ruling out the programs of example (1) and Fig. 1. Our solution is based on the observation that a program such as while xH = 0 do nil should indeed be considered with some care in a concurrent setting, but only as a “guard”, that is, as regards what may follow it. In the context of concurrent threads, if the control comes back to this while loop, this may be with a value for x H diKerent from 0, contrarily to what happens in a sequential setting. In other words, this program may observe the behaviour of other, concurrent components, in the course of their execution, and in3uence accordingly the behaviour of the thread in which it participates. Our aim is then to ensure a stronger form of noninterference, where the course of values —not just the 4nal value—of a low-level variable does not depend upon the value of high variables. Typically, program (1) is no longer interference-free in this stronger sense. In order to reject it, we introduce a re4nement of the notion of type, and a new type system, where the level of loop guards—the expressions tested by while loops— is taken into account in sequential composition. The basic idea is that assigning to a low variable must not depend upon consulting the value of a high variable. Then also conditional guards—the expressions tested by conditional statements—will be taken into account (we will see an example in the next section). We will also examine the situation where a scheduling policy is in force in a thread system of sequential programs: we will introduce a couple of new programming primitives to describe formally such a situation, and show how to extend the type system to this new setting, where new interference phenomena arise. We show that our type system still ensures the noninterference property in this case, while imposing a less severe restriction than that pre4gured in [18]. The rest of the paper is organised as follows. In Section 2 we introduce the language, its operational semantics and its type system. Section 3 presents the properties of typed programs, including subject reduction and noninterference. Finally, in Section 4 we consider the extended language with scheduling policies for sequential threads. 1

Assuming, in the case of an arbitrary lattice of security levels, that there is a lowest level ⊥.

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2. The language and type system The language we consider is essentially that of [18] (where e stands for a boolean or arithmetic expression, whose syntax we do not detail here). We use the following two-level syntax, where U; V denote sequential programs, while P; Q denote general (concurrent) programs: U; V : : : ::= nil | x := e | U ; V | if e then U else V | while e do U P; Q : : : ::= U | U ; P | if e then P else Q | (P  Q) Note that on the left of a sequential composition, we must have a sequential program. Thus programs of the form (P  Q); R are not allowed. With this restriction, our language is still more general than that of [18], which describes concurrent systems as collections of threads, thus allowing only top-level parallelism, while we allow the dynamic spawning of new threads (although in a limited way). The operational semantics of the language is given in terms of transitions between con4gurations (P; ) → (P
; 
) where P; P
are programs and ; 
stand for memories, that is mappings from variables to values. These mappings are extended in the obvious way to expressions, whose evaluation is assumed to be atomic as in [18]. We use the notation [v=x] for memory update. The rules specifying the operational semantics of programs are presented in Fig. 2. The semantics used here is a small step semantics, as opposed to the big step semantics of [21]. 2 The rules are fairly standard, and we shall not comment on them. In the introduction we argued that, in a concurrent setting, program (1) should be treated as another case of implicit information 3ow. Intuitively, when exiting a loop one gets some information about its guard; it seems then appropriate to require that what follows the loop–its “continuation”–have level at least as high as that of the loop guard. This will be the basic idea of our new type system, which is closely inspired by that given by Volpano et al. [21]–however as suggested by the above Example (1) it will be more restrictive than that of [21] on the sequential sublanguage, because of our more detailed observation of programs, where an attacker is allowed to read the low part of the memory at any time. The types of data and expressions are security levels, that is elements of a lattice (T;6). We denote the operations of meet and join, respectively, by  and  . These types are ranged over by ;
: : : : In the examples, the lattice of security levels will simply be {L; H }, with L¡H . The types of variables (when used in the left-hand side of an assignment) are of the form  var. Our 4rst point of departure from [21] concerns the types for programs. Type judgements in [21] are of the form  P :  cmd, where  is a mapping from variables to types of variables, i.e. elements of { var |  ∈ T}. The meaning of  P :  cmd is that in the type environment , the type  is a lower 2

In [18], the semantics is a mixture of small and big step semantics: transitions are given between con4gurations but there are two kinds of con4gurations, intermediate and 4nal ones, suggesting that termination should be observed.

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(ASSIGN-OP) (SEQ-OP1) (SEQ-OP2) (COND-OP1) (COND-OP2) (WHILE-OP1) (WHILE-OP2) (PAR-OP1) (PAR-OP2)

113

(x := e; ) → (nil; [(e)=x]) (U; ) → (U
; 
) (U ; P; ) → (U
; P; 
) (P; ) → (P
; 
) (nil; P; ) → (P
; 
) (e) = tt (if e then P else Q; ) → (P; ) (e)
= tt (if e then P else Q; ) → (Q; ) (e) = tt (while e do U; ) → (U ; while e do U; ) (e)
= tt (while e do U; ) → (nil; ) (P; ) → (P
; 
) (P  Q; ) → (P
 Q; 
) (Q; ) → (Q
; 
) (P  Q; ) → (P  Q
; 
)

Fig. 2. Operational semantics for parallel programs.

bound for the level of the assigned variables of P. In line with this intuition, subtyping for programs is contravariant, that is  cmd6 
cmd if 
6. Thus for instance any program of type H cmd can be downgraded to type L cmd. A program of type H cmd is guaranteed not to contain any assignment to a low variable. As we said previously, we have to take into account the level of loop guards in typing. To deal with possible divergence (and, as we shall see, with the scheduling of threads) we also have to record the level of the tested expression in conditional branching, which we regard as a guard as well. Thus we shall use here more re4ned types (;
) cmd, where the 4rst component  plays the same role as in the type  cmd, while the second component
is the guard type, an upper bound on the level of the loop and conditional guards occurring in a program. Accordingly, the subtyping for programs is contravariant in its 4rst component and covariant in the second: (;
)cmd 6 (
;

) cmd if 
6  and
6

: The guard type will be set up by while-loops and conditional branchings, and looked up by sequential composition. The complete type system for programs is shown in

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(NIL) (ASSIGN) (SEQ) (COND) (WHILE) (PAR) (SUBTYPING)

 nil : (;
) cmd  e : ;

(x) =  var

 x := e : (;
) cmd  U : (;
) cmd;

 P : (
;

) cmd;


6


 U ; P : (  
;


) cmd  e : ;

 Pi : (;
) cmd;

6

 if e then P0 else P1 : (;  
) cmd  e : ;

 U : (;
) cmd;

 
6

 while e do U : (;  
) cmd  Pi : (;
) cmd  P0  P1 : (;
) cmd  P : (;
) cmd;


6;


6



 P : (
;

) cmd Fig. 3. Typing rules for concurrent programs.

Fig. 3, where we assume given a system allowing us to infer judgements  e :  about the security level of an expression. These are used in the (ASSIGN), (COND) and (WHILE) rules. Notice that the guard type plays no particular rˆole in rules (NIL) and (ASSIGN), which are plain adaptations of the ones in [21]. As in that paper, the side condition 6 in the rules (COND) and (WHILE) is meant to avoid implicit insecure 3ow. Let us comment a little on the rules for conditional branching, while-loops and sequential composition, which are the main novelty w.r.t. [21,18]. As explained, the guard type is at least  for a conditional branching or a while-loop testing an expression of level , and from then onwards it should remain at least  to prevent concatenation with programs of a lower or incomparable level. Rule (SEQ) is precisely designed to avoid sequencing “low” assignments after a program with “high” guards. This rules out the kind of implicit 3ow exhibited by program (1). It also rules out the (semantically equivalent, in a sequential setting) program if xH = 0 then while tt do nil else nil ; yL := 1

(2)

which is not interference-free in our sense. Indeed, this example shows why the level of conditional guards should be recorded as a guard type. The side condition
6 in the rule (WHILE) is needed for the preservation of types during execution, as one can

G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

115

see from the unfolding of while e do P into if e then P; while e do P else nil. Notice that if P0 and P1 are both typable in the typing context , say with respective types (0 ;
0 ) cmd and (1 ;
1 ) cmd, then their parallel composition (P0  P1 ) is typable, since both P0 and P1 have type (0  1 ;
0 
1 ) cmd, thanks to the (SUBTYPING) rule. 3. Properties of typed programs In this section we prove some desired properties of our type system. The 4rst property, subject reduction, states that types are preserved along execution. Theorem 3.1 (Subject reduction). If  P : (;
) cmd and (P; ) → (P
; 
), then  P
: (;
) cmd. Proof. By induction on the inference of  P : (;
) cmd, and then case analysis on the last rule used in this inference. If this rule is the subtyping rule, we simply use the induction hypothesis. We examine the main cases: (ASSIGN). Here we have P = x := e with  e :  and (x) =  var. Moreover, it must be that P
= nil and 
= [(e)=x]. The theorem is trivial in this case, since  nil : (;
) cmd. (SEQ). Here P = U ; Q with  U : (1 ;
1 ) cmd,  Q : (2 ;
2 ) cmd,  = 1  2 and
=
1 
2 with
1 62 . There are two possibilities for a transition (P; ) → (P
; 
). If this results from (U; ) → (U
; 
), by means of (SEQ-OP1), we have P
= U
; Q, where  U
: (1 ;
1 ) cmd by induction, and we conclude using the rule (SEQ). If (SEQOP2) is used, we have U = nil with (Q; ) → (P
; 
). By induction  P
: (2 ;
2 )cmd, and we use the subtyping rule to conclude. (WHILE). Here P = while e do U and  U : (;

) cmd, with
=  

and
6. If (e) = tt we have P
= U ; P, and since

6 we may use the rule (SEQ) to conclude  P
: (;
) cmd. Otherwise P
= nil, and this case is trivial. All the other cases are similar (in the case of (COND) we use subtyping). We shall use the following assumptions about expressions: Assumption 3.2 (Termination of expression evaluation). For any memory  and expression e, the value (e) is de7ned. Notice that this implies for instance that for any sequential program U
= nil ; · · · ; nil and for any  there exist U
and 
such that (U; ) → (U
; 
). Assumption 3.3 (Simple security). If  e : , then every variable occurring in e has type 
var in , with 
6. We introduce now a notion of equality on memories, depending on a given type environment  and a given set L ⊆T of security levels, which are to be understood as “low” security levels. In the following, L will always denote a downward-closed

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G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

set of security levels, that is a set satisfying  ∈ L &
6  ⇒
∈ L: Two memories are (; L)-equal if they coincide on variables to which  assigns a level in L. Intuitively, such memories are indistinguishable for an observer which has access only to information of level in L. Denition 3.4 ((; L)-Equality of memories).  =L   ⇔def ∀x: (x) =  var & ∈L ⇒ (x) = (x). {L}

In the examples we implicitly used the particular case = , which is denoted ∼ in [18]. As it has become standard, we use the notion of bisimulation [11,13] to (state and) establish the security properties of concurrent programs. We shall actually use several kinds of bisimulations. To de4ne these notions, it is convenient to introduce some notations. First, we denote by  the re3exive closure of → , that is, (P; )  (P
; 
) ⇔def P
= P & 
= 

or

(P; ) → (P
; 
):

∗

As usual, → is the re3exive and transitive closure of → . We say that P
is a derivative ∗ of P, denoted P P
, if for some  and 
we have (P; ) → (P
; 
). Then, given a typing context  and a set L of (low) security levels, we denote by P ; L the set of “(semantically) high” programs, that is programs that never modify the low part of the memory: P ∈ P;L ⇔def ∀P
; ; P

; 
: P

∗

P
& (P
; ) →(P

; 
) ⇒ 
=L  :

Clearly, if P ∈ P ; L and P
is a derivative of P then P
∈ P ; L . Denition 3.5 ((; L)-Bisimulation). A relation R on con4gurations is a (; L)bisimulation if R is symmetric and (P; ) R (Q; ) implies (i)  =L   (ii) (P; ) → (P
; 
) ⇒ ∃Q
; 
: (Q; )  (Q
; 
) & (P
; 
) R (Q
; 
). The relation R is a strong (; L)-bisimulation if it satis4es the conditions above, where  is replaced by → . The (; L)-bisimulation equivalence on con4gurations, L noted ≈L  (resp. the strong (; L)-bisimulation equivalence, noted ∼ ) is the largest (; L)-bisimulation (resp. the largest strong (; L)-bisimulation). L Clearly, we have ∼L  ⊆ ≈ , and the inclusion is strict, since for instance, for any memories  and  such that  =L  , we have

(nil; ) ≈L  (while tt do nil; ): This is a consequence of the following: Lemma 3.6 (Bisimilarity of high programs). The relation R0;L = {((P; ); (Q; )) | P ∈ P ; L ; Q ∈ P ; L &  =L  } is a (; L)-bisimulation.

G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

117

Proof. This is immediate, because if (P; ) → (P
; 
) then P
∈ P ; L and 
=L   =L  . For technical reasons we introduce one last notion of bisimulation: Denition 3.7 (Quasi-strong (; L)-bisimulation). A relation R on con4gurations is a quasi-strong (; L)-bisimulation if R is symmetric and (P; ) R (Q; ) implies (i)  =L  : (ii) P ∈ P ; L and Q ∈ P ; L , or (P; ) → (P
; 
) ⇒ ∃Q
; 
: (Q; ) → (Q
; 
) & (P
; 
) R (Q
; 
): The quasi-strong (; L)-bisimulation equivalence on con4gurations, noted L  , is the largest quasi-strong (; L)-bisimulation. It should be clear that a quasi-strong (; L)-bisimulation is also a bisimulation, and that a strong (; L)-bisimulation is also a quasi-strong L L bisimulation, and therefore we have ∼L  ⊆  ⊆ ≈ . As in [16], we shall abuse of language, that a relation S on programs is a (strong, quasi-strong) bisimulation if the relation

(; L)(; L)say, by (; L)-

{((P; ); (Q; )) | PSQ &  =L  } is a (strong, quasi-strong) (; L)-bisimulation. For instance, Lemma 3.6 shows that the relation S0; L given by P S0;L Q ⇔def P ∈ P;L & Q ∈ P;L is a (; L)-bisimulation. Observe also that quasi-strong bisimulation is not preserved by parallel composition, as shown by (nil  yL := 0)
L  (while tt do nil  yL := 0): One should note that although there exists a largest (; L)-bisimulation on programs (since the union of a family of such relations is a (; L)-bisimulation), that we still denote ≈L  , this is not a re3exive relation, since for instance the program yL := x H is not bisimilar to itself. However, the relation ≈L  on programs is a nonempty (as shown by Lemma 3.6) partial equivalence relation. The domain of ≈L  characterizes the secure programs: Denition 3.8 (Secure programs). A program P is secure in the typing context  if and only if P ≈L  P for any downward-closed set L of security levels. We let the reader check that, for instance, the program of Example (1) is not secure in this sense. To establish the noninterference result, stating that a typable program is secure, we need some preliminary lemmas. The following two lemmas con4rm the intuition,

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G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

discussed earlier, behind the type judgements  P : (;
) cmd. Both results are easily proved by induction on the inference of type judgements. Lemma 3.9 (Con4nement). If  P : (;
) cmd then every variable assigned to in P has type  var in , with 6. Lemma 3.10 (Guard safety). If  P : (;
) cmd then every loop or conditional guard in P has type  in , with 6
. Denition 3.11 (L-Boundedness). A program P is L-bounded in  if  P : (;
) cmd implies  ∈ L. We shall often use the complementary notion of non L-boundedness: P is not Lbounded in  if there exist ;
, with  ∈= L, such that  P : (;
) cmd. Denition 3.12 (L-Guardedness). A program P is L-guarded in  if there exist
∈ L and  such that  P : (;
) cmd. In the following, we shall most often use the terminology “(non)L-bounded” and “L-guarded” without any explicit reference to the current typing context . As a consequence of subject reduction, both non L-boundedness and L-guardedness are preserved by execution. Note that for any downward-closed L, by the Con4nement Lemma a program which is not L-bounded cannot write on variables of low level, that is in L, and therefore such a program is “high”, that is in P ; L . Note however that, in general, a high program is not necessarily non L-bounded. For instance, assume that the lattice of security levels is {L; A; B; H } with L = A  B and H = A  B. Then if L = {L}, the program xA := 0; yB := 0 is high and L-bounded in this case. Similarly, by the Guard Safety Lemma, a program which is L-guarded does not contain loop guards of high level, that is not in L. Lemma 3.13 (Behaviour of L-guarded programs). If P is L-guarded in  and 




L
=L  , then (P; ) → (P ;  ) implies (P; ) → (P ;  ), with  =  . Proof. By induction on the inference of  P : (;
) cmd, where
∈ L, and by case analysis on the last rule used in this inference. We examine the main cases. (ASSIGN). Here P = x := e with  e : , (x) =  var, P
= nil and 
= [(e)=x]. By
Assumption 3.2, (e) is de4ned. Let 
= [(e)=x]. It is easy to see that 
=L   ,
L
since if  ∈= L then  =  by the De4nition 3.4, while if  ∈ L then (e) = (e) by Assumption 3.3. (SEQ). Here P = U ; Q,  U : (1 ;
1 ) cmd and  Q : (2 ;
2 ) cmd with
=
1 
2 and
1 62 . If the transition (P; ) → (P
; 
) is proved by means of (SEQ-OP1), that is P
= U
; Q with (U; ) → (U
; 
), then we use the induction hypothesis, since
1 

2 ∈ L ⇒
1 ∈ L. This will give us (U; ) → (U
; 
), with 
=L   , and by (SEQ-OP1)

we conclude that (P; ) → (U ; Q;  ), as required. In the case where (SEQ-OP2) is used, with U = nil and (Q; ) → (P
; 
), we simply use the induction hypothesis.

G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

119

(COND). Here P = if e then P1 else P2 with  e : ,  Pi : (;

) cmd, with 6 and
=  

. Since
∈ L, also  ∈ L, and thus (e) = (e) by Assumption 3.3. Hence a transition (P; ) → (Pi ; ) is matched by (P; ) → (Pi ; ). (WHILE). Here P = while e do U and  e : ,  U : (;

) cmd, with
6 and
=  

. Clearly
∈ L implies  ∈ L, therefore (e) = (e) by Assumption 3.3, and thus a transition of the loop, which is necessarily of the form (P; ) → (P
; ), will be matched by (P; ) → (P
; ). The case of (PAR) is easy, since the components have the same type as the whole program, and the case of (SUBTYPING) is immediate by induction. This lemma is crucial for the proof of our main results, namely that typable programs (and, in the next section, scheduled thread systems) are secure. Obviously, it could not be stated in the original system of Volpano et al. [21], where the level of guards is not recorded in the type. Now, given  and L, let us de4ne inductively the relation S1; L on sequential programs as follows: U S1; L V if and only if U and V are typable in , and one of the following holds: 1. U S0; L V , that is U ∈ P ; L and V ∈ P ; L , or 2. V = U and U is L-bounded in , or 3. U = U0 ; W and V = V0 ; W , where U0 S1; L V0 and W is not L-bounded in . Lemma 3.14. (i) The relation S1; L is symmetric. (ii) If U is typable in  then U S1; L U . (iii) If nil S1; L V then V ∈ P ; L . Proof. The 4rst point is easy to check, by induction on the de4nition of S1; L . For the second point, if U is typable in  then either U is L-bounded in , in which case U S1; L U by clause 2, or not, in which case U ∈ P ; L , as we have seen, hence U S1; L U by clause 1. Finally, observe that if nil S1; L V , this must be by clause 1 or 2, and in both cases this implies V ∈ P ; L . Theorem 3.15 (Sequential noninterference). The relation S1; L is a quasi-strong (; L)-bisimulation, and therefore if U is typable in  then U is secure in . Proof. Let R1;L = {((U; ); (V; )) | U S1;L V &  =L  }: By construction, (U; ) R1; L (V; ) implies  =L  . We show, by induction on the de4nition of S1; L , that either U and V are both in P ; L , or if (U; ) → (U
; 
) and (U; ) R1; L (V; ) then there exist V
and 
such that (V; ) → (V
; 
), with (U
; 
) R1; L ; (V
; 
). The case where U and V are both in P ; L , that is the case of clause 1 is trivial, and therefore for the rest of the proof we assume that U and V are not both in P ; L . In this proof we implicitly use the Subject Reduction Theorem, which is needed to check that we are dealing with typable terms.

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G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

Clause 2. We proceed by induction on the structure of U . Assignment: Here U = x := e, U
= nil and 
= [(e)=x]. Then the matching move is
L
(U; ) → (nil; 
), where 
= [(e)=x]. Indeed from  =L   we deduce  =  since U is L-bounded, and thus by Assumption 3.3 (e) = (e). Sequence: Here U = U0 ; W . We distinguish two cases, according to which of the rule (SEQ-OP1) or (SEQ-OP2) is used to deduce the transition (U; ) → (U
; 
). 1. If (U; ) → (U
; 
) is proved by means of (SEQ-OP1), that is U
= U0
; W with (U0 ; ) → (U0
; 
), then there are two possibilities: 1.1. W is L-bounded in : then  W : (2 ;
2 ) cmd implies 2 ∈ L, therefore since U is typable there exist 1 ;
1 such that  U0 : (1 ;
1 ) cmd, with
1 ∈ L (this is where we use the side condition of the rule (SEQ)). Thus U0 is
L-guarded. Then by Lemma 3.13 we have (U0 ; ) → (U0
; 
), with 
=L   .

Therefore (U; ) → (U0 ; W;  ) by (SEQ-OP1), which is the required matching move by clause 2 again since U0
; W is L-bounded (because 2 ∈ L implies 1  2 ∈ L). 1.2. If W is not L-bounded, then W ∈ P ; L . Since we assumed U ∈= P ; L , U0 must be L-bounded, and by induction on the structure of the programs, there
exist U0

and 
such that (U0 ; ) → (U0

; 
) with U0
S1; L U0

and 
=L   .


Then (U; ) → (U0 ; W;  ) by (SEQ-OP1), which is the required matching move by clause 3. 2. Let (U; ) →(U
; 
) be proved by means of (SEQ-OP2), with U0 = nil and (W; ) → (U
; 
). Since we assumed U ∈= P ; L , W must be L-bounded, and by induction on the structure of the programs, there exist U

and 
such that (W; ) → (U

; 
)



with U
S1; L U

and 
=L   . Then (U; ) → (U ;  ) by (SEQ-OP2), which is the required matching move. Conditional: Here U = if e then U1 else U2 with  e : ,  Ui : (;

) cmd, with 6 and
=  

. Then  ∈ L implies  ∈ L, and by Assumption 3.3 we have (e) = (e). Thus the move (U; ) → (Ui ; ) is matched by (U; ) → (Ui ; ), since S1; L is re3exive. The case of a while-loop U = while e do W is similar. Clause 3. Here U =U0 ; W and V =V0 ; W , where U0 S1; L V0 and W is not L-bounded in . We assumed that U and V are not both in P ; L . Then U0
= nil, since otherwise we would have V0 ∈ P ; L by Lemma 3.14(iii), hence U ∈ P ; L , and V ∈ P ; L , for W ∈ P ; L . Therefore the transition (U; ) → (U
; 
) must be proved by means of (SEQOP1), that is U
= U0
; W with (U0 ; ) → (U0
; 
), and we use the induction hypothesis related to U0 S1; L V0 (notice that, due to our assumption about U and V , U0 and V0 cannot be both in P ; L ). To extend this result to concurrent programs, we de4ne inductively the relation S2; L as follows: P S2; L Q if and only if P and Q are typable in , and one of the following holds: 1. P S0; L Q, that is P ∈ P ; L and Q ∈ P ; L , or 2. Q = P and P is L-bounded in , or

G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

121

3. P = U ; R and Q = V ; R, where U S2; L V and R is not L-bounded in , or 4. P = P1  P2 and Q = Q1  Q2 with Pi S2; L Qi . It is easy to see that Lemma 3.14 holds, replacing sequential programs by concurrent ones, for this relation too. Theorem 3.16 (Concurrent noninterference). The relation S2; L is bisimulation, and therefore if P is typable in  then P is secure in .

a

(; L)-

Proof. For the clauses 1, 2 and 3, the proof is mostly the same as in the previous theorem (using the proof of Lemma 3.6). The remaining cases only concern parallel composition (in clauses 2 and 4), and are trivial, using the induction hypothesis. Observe that if 1 and 2 are incomparable levels in (T;6), they may be separated by a downward-closed set L, namely L = { | 61 }, and therefore a typable program cannot yield any 3ow from level 2 to 1 (and conversely). In other words, information 3ow arising from typable programs is indeed of the kind allowed by the security lattice, that is from  to
where 6
. As we have seen, (P  Q) is typable as soon as both P and Q are. Therefore, the previous result means that there can be no information leakage from one secure program to another, concurrent one. Then one may say that our result shows that typed programs are “internally secure”, that is they are guaranteed against attacks coming from other, concurrent programs. Notice that, assuming that there is a lowest security level ⊥, a program that has direct access only to the lowest information is always typable, with type (⊥; ⊥) cmd. Such a program may be regarded as a potential attacker. As is well-known, bisimulation equivalence is a strong relation, and therefore our noninterference result is quite sharp. In particular, bisimulation equivalence is usually stronger than trace semantics, and therefore we can draw some consequences of our result regarding the trace behaviour of programs. To this end, let us introduce a nota


L tion: we write (P; ) L  (P ;  ) if and only if 
=  and there exist P0 ;  0 ; : : : ; Pn ;  n L (possibly with n = 0) such that  i =  for all i and (P; ) = (P0 ; 0 ) → (P1 ; 1 ) → · · · (Pn ; n ) → (P
; 
): Given an initial memory  0 , let us de4ne the set, denoted TraceL  (P;  0 ), of (L; )traces of the program P as the set of 4nite or in4nite sequences  0 ; 1 ; : : : ;  n : : : such that there exist P1 ; : : : ; Pn : : : with L L (P; 0 ) L  (P1 ; 1 )  · · · (Pn ; n )  · · ·

Note that, by de4nition,  i+1
=L   i for all i. The (; L)-equality of memories is easily extended into a (; L)-equality of traces of the same length, as follows: L 0 ; 1 ; : : : ; n : : : =L  0 ; 1 ; : : : ; n : : : ⇔def ∀i: i = i :

Then we can show that a typable program has the same traces (up to =L  ) when execution starts from (; L)-equal memories:

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G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

Corollary 3.17 (Trace equivalence). Let L be a downward-closed set of security levL els. If P is typable in  and  =L   then for any trace t ∈ Trace (P; ) there exists L

L t ∈ Trace (P; ) such that t = t. Proof. By the Theorem 3.16, we have (P; ) ≈L  (P; ), and it is easy to see that

L (P ;  ) and (P; ) ≈ (Q; ) implies that there exist Q
and 
such that (P; ) L   L

(Q; )  (Q ;  ). 4. Scheduling sequential programs As pointed out by Smith and Volpano in [18], their noninterference result for concurrent threads relies on the hypothesis of a purely nondeterministic execution. This result breaks down if particular scheduling policies are enforced. We recall the example given in [18]. Assume a round robin time slicing scheduler, with a time slice of t steps, t¿2, and consider the composition P = U1  U2 of the following two threads: U1 = if xH = 0 then V else nil ; yL := 0 U2 = yL := 1

(3)

Then, supposing that V is a convergent program that takes at least t − 1 steps to execute, and that the scheduler gives precedence to U1 , the value of yL will depend on that of x H . The solution proposed in [18] to preserve noninterference in the presence of an arbitrary scheduler consists in forbidding conditionals with high guards, 3 that is, again assuming that there is a lowest security level, to accept only conditional branching on low-level expressions. This condition, combined with the exclusion of loops with high guards, required for multi-threading, resulted in [18] in a very severe limitation: the impossibility for any program to test a variable, except at the lowest level. Let us immediately point out that, with the typing we propose in this paper, we cannot accept any scheduling policy. For instance, if the scheduling of two concurrent threads consists in performing the 4rst, up to completion, and then the second, we may reproduce the insecure 3ow exhibited by program (1), simply by executing (while x H = 0 do nil  yL := 1) under this scheduling. Therefore, we have to reject some scheduling policies, and the way we do that is by encoding these policies into programs, and reject the non typables ones. One may observe that the program U1 of Example (3) is actually ruled out by our typing rule for conditional branching, which ensures that low assignments cannot be performed after a high test (cf. Example 2). To show that our typing is indeed adequate for dealing with scheduling, we 4rst formalise what it means for a system of concurrent programs to be controlled by a scheduler. Essentially, this means running the system 3

It is also suggested there that a better approach to scheduling would be probabilistic. Indeed, a whole line of research on probabilistic noninterference has been developed, but this will not be our concern here, where we stick to a possibilistic setting.

G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

(WHEN-OP1) (WHEN-OP2) (CONTROL-OP1) (CONTROL-OP2)

123

(e) = tt; (U; ) → (U
; 
) (when e do U; ) → (when e do U
; 
) (e) = tt; (U; ) 9 (when e do U; ) → (when e do U; ) (P; ) → (P
; 
); (T; ) → (T
; 

) (P[T ]; ) → (P
[T
]; 
  

) (P; ) → (P
; 
); (T; ) 9 (P[T ]; ) → (P
[T ]; 
)

Fig. 4. Additional operational rules for systems.

in lockstep with a program that implements the scheduling policy. To describe controlled execution, we use a construction P[Q], which makes P and Q move hand in hand, but allows the controller, P, to proceed by itself whenever Q is unable to move. Then a system consisting of n sequential programs Ui controlled by a scheduler Sched will be described as Sched[T1  · · · Tn ]; where the Ti are adaptations of the Ui , so that the threads can be triggered and suspended by the scheduler. To this end we introduce a new construct when e do U , whose semantics is that U is allowed to proceed, for one step, when the condition e holds. It is technically convenient to introduce two more levels in the syntax: besides the sequential and concurrent programs U and P, written according to the grammar given in Section 2, there is a set of “thread systems” T , and a set of “controlled thread systems” S built as follows: T ::= when e do U | (T T ); S ::= P[T ]: Letting w(P) denote the set of variables written (assigned to) by P, and similarly for w(T ), the construct P[T ] is only legal under the condition w(P) ∩ w(T ) = ∅. Notation 4.1. We use (T; ) → to mean ∃T
; 
such that (T; ) → (T
; 
), and (T; ) 9 for the negation of (T; ) → (and similarly for the other syntactic categories). The semantics of the new constructs is given in Fig. 4, where 
  

represents the memory  with the conjunction of the updates operated by P and by T , that is 
\ ∪ 

\ ∪ (
∩ 

). As we said, the scheduled sequential programs are written Ti = when ei do Ui where the expression ei is the “proceed” signal for program Ui ,

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G. Boudol, I. Castellani / Theoretical Computer Science 281 (2002) 109 – 130

(WHEN) (CONTROL)

 e : ;

 U : (;
) cmd;

6

 when e do U : (; ) cmd  P : (;
) cmd;

 T : (;
) cmd;

¿


 P[T ] : (;
) cmd Fig. 5. Additional typing rules for systems.

set up by the scheduler. For instance, assuming that each ei is a variable si , initially false, the following program Rtn = i := 0 ; while tt do k := 0; (while k ¡ t do si+1 := tt; si+1 := 3 ; k := k + 1); i := [i + 1]mod n describes a scheduler for a system of n threads, implementing round robin with time slice t. It is easy to imagine how to program other scheduling policies in a similar style. For instance, assuming that there is a random function in the language, such that the evaluation of random(n) consists in randomly choosing an integer i such that 16i6n, then while tt do (i := random(n); si := tt; si := 3 ) describes a uniform probabilistic scheduler. Notice also that, if the controlled threads have the form when s ∧ si do Ui , the scheduling enforced by s1 := tt; · · · ; sn := tt; s := tt; while tt do nil is—assuming that s is initially false—simply the nondeterministic interleaving of the Ui ’s. We see from these examples that the scheduler usually needs some time to determine the index of the next thread to execute, and this is why we need the rule (CONTROL-OP2). This rule also allows one to ignore a terminated thread, and therefore the rule (WHEN-OP2), which is technically convenient, is harmless. The typing rules for the new operators are given in Fig. 5. The side-conditions in rules (WHEN) and (CONTROL) need some comments. First, note that a when statement can induce an implicit 3ow, just like the conditional and while statements, as for instance in the system: when xH = 0 do yL := yL + 1: This explains the requirement 6 in rule (WHEN). Now let us discuss the side condition in the rule (CONTROL). In a controlled thread system P[T ] where T = (when e1 do U1  · · · when en do Un );

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125

the conditions ei are normally managed by the controller process P. Then if the Ui ’s have types (i ;
i ) cmd, to type P[T ] the conditions ei must be chosen so that they have types i satisfying  = 1  · · ·  n 61  · · ·  n = : If that is the case, P[T ] may be typed with (; ) cmd, regardless of the
i ’s, thanks to the side condition in rule (WHEN), and to the fact that we only record the level of the condition as a guard in this rule. Therefore, the side condition ¿
in the rule (CONTROL) is more a constraint for the scheduler P than for the thread system T . Typically, programs of (L; ) cmd type can only be controlled by schedulers of type (L; L) cmd. One can see for instance that the round robin scheduler Rnt has type (L; L) cmd in a context where the variables i, k and si have type L var. Therefore, for any typable Ui ’s (in this context 4), the system Rtn [when s1 do U1  · · · when sn do Un ] is typable. The side condition ¿
in the rule (CONTROL) precludes, for instance, schedulers whose unique type is (L; H ) cmd. Indeed, it would be unsafe to accept programs of this kind as schedulers. For instance, if P = s := tt ; while xH = 0 do nil; U = yL := 0 ; yL := yL + 1; then the system P[when s do U ] is not interference-free, and would be typable without the side condition in the rule (CONTROL), with (s) = L var. Let us see another, similar example. Suppose for instance that we wish to design a scheduler which is able to detect the termination of a thread Ui , and then to skip it from execution. Then we may transform Ui into when ei do (Ui ; ti := tt) where the variable ti is a termination signal, initially false. Now if Ui has type ( ; H ) cmd, then ti must be high. Certainly, the scheduler will have to test the value of this variable, and therefore it must have type ( ; H ) cmd, but then, due to the condition ¿
in the (CONTROL) rule, it must have type (H; H ) cmd, and it can only control programs of type (H; ) cmd. Indeed, checking the termination of a program with high guards may be dangerous, as shown by the system—with a mistaken, still typable scheduler: P[when s1 do (U ; t1 := tt)  when s2 do (yL := 0 ; yL := yL + 1)]; where P = s1 := tt ; s2 := tt ; while not t1 do nil: It is easy to check that the Subject Reduction Theorem and the Con4nement Lemma, and therefore also Lemma 3.6, extend to the new language. Similarly, the de4nitions of (strong, quasi-strong) (; L)-bisimulation and L-boundedness remain formally the 4 Note that the controlled programs U usually do not contain the control variables manipulated by the i scheduler. Indeed, i and k should be local to Rnt —for a treatment of local variables, see [21].

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same as those for the base language (modulo the replacement of programs by systems), as well as the de4nition of secure systems. Given a typing context  and a downwardclosed set L of security levels, let us de4ne the relation S3; L on controlled thread systems as follows: S 0 S3; L S1 if and only if S 0 and S1 are both typable in , and S0 = P[when e1 do U1  · · ·  when en do Un ]; S1 = P[when e1 do V1  · · ·  when en do Vn ] with 1. P is L-guarded in , and 2. for all i there exists i ∈ L such that  ei : i , and 3. Ui L  Vi for all i. Lemma 4.2. The relation S3; L is a strong (; L)-bisimulation.

Proof. Let  =L   and (S 0 ; ) → (S 0 ;  ). There are two cases: 1. If this transition is proved by means of (CONTROL-OP1) then

S0
= P
[· · ·  when ei do Ui
 · · ·]; where (P; ) → (P
;  0 ), (when ei do Ui ; ) → (when ei do Ui
; 1 ) and 
=  0   1 . Since P is L-guarded in , by Lemma 3.13, there exists 0 such that (P; ) → (P
; 0 ) with 0 =L   0 . Regarding the transition of the thread system, there are again two cases: 1.1. The transition is proved by means of (WHEN-OP1), that is (e) = tt and (Ui ; ) →
(Ui
; 1 ). Since Vi L  Ui , either there is a transition (Vi ; ) → (Vi ; 1 ) such that
L
L ; L Vi  Ui and 1 = 1 , or both Ui and Vi are in P . By the Assumption 3.3 we have (ei ) = (ei ) = tt (since ei is low in ), and therefore in the 4rst case (when ei do Vi ; ) → (when ei do Vi
; 1 ) by (WHEN-OP1), hence (S1 ; ) → (P
[· · ·  when ei do Vi
 · · ·]; 
);
where 
= 0   1 =L   , and this is the required matching move. Otherwise, ; L ; L Ui ∈ P and Vi ∈ P , and we distinguish again two cases: either there exist Vi
L L and 1 such that (Vi ; ) → (Vi
; 1 ), and then 1 =L   =  = 1 , and we conclude as in the previous case, or (Vi ; ) 9 . Then (when ei do Vi ; ) → (when ei do Vi ; ) L by (WHEN-OP2), with  =L   = 1 , and

(S1 ; ) → (P
[· · · when ei do Vi  · · ·]; 0 ) is the required matching move in this case. 1.2. The transition is proved by means of (WHEN-OP2), that is (e) = tt, (Ui ; ) 9 and ; L , and we distinguish two Ui
= Ui and 1 = . Since Vi L  Ui , we have Vi ∈ P cases, depending on whether (Vi ; ) may perform a transition or not. We conclude as in the previous case. 2. If this transition is proved by means of (CONTROL-OP2) then S 0
= P
[T0 ] with (P; ) → (P
; 
) and (T0 ; ) 9 , where T0 = (when e1 do U1  · · · when en do Un ):

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It is easy to see that (T0 ; ) 9 implies (ei ) = 3 for all i, and therefore also (ei ) = 3 by Assumption 3.3 since the ei ’s are low in . Then (T1 ; ) 9 where T1 = (when e1 do V1  · · ·  when en do Vn ): Since P is L-guarded in , by Lemma 3.13, there exists 
such that (P; ) → (P
; 
)


with 
=L   , and therefore (S1 ; ) → (P [T1 ];  ), and this is the required matching move. We are now ready to prove our main result: Theorem 4.3 (Noninterference for controlled thread systems). If S is a controlled thread system typable in  then S is secure in . Proof. Given a downward-closed set L of security levels, let us still denote by P ; L the set of high programs and systems (of any kind), and by S0; L the cartesian product P ; L × P ; L . It is easy to see (as in Lemma 3.6) that this is a (; L)-bisimulation, and therefore the relation S4;L = S0;L ∪ S3;L is a (quasi-strong) (; L)-bisimulation, by the previous lemma. We show that if S is typed in  then S S4; L S. This is obvious if S is not L-bounded in , by the (extended) Con4nement Lemma, since S is in P ; L in this case. If S is L-bounded in , and S = P[T ] where T = (when e1 do U1  · · · when en do Un ); then a typing  S : (;
) cmd may be inferred either using the (SUBTYPING) or the (CONTROL) rule. More precisely, there exist 
and

such that  S : (
;

) cmd by the (CONTROL) rule, that is  P : (
;

) cmd and  T : (
;

) cmd with 6
¿

and

6
. Since S is L-bounded in , we have

∈ L, and therefore P is L-guarded in . Now  T : (
;

) cmd may be proved either using the (SUBTYPING) or the (PAR) rule—or the (WHEN) rule if n = 1. Then it is not diUcult to see that there exist i , i and
i for each i such that  ei : i and  Ui : (i ;
i ) cmd with 
61  · · ·  n and 1  · · ·  n 6

and therefore i ∈ L for all i. Finally, we have Ui L  Ui for all i by Theorem 3.15, and this concludes the proof. 5. Conclusion and related work We have addressed the question of secure information 3ow in systems of concurrent programs. This covers one of the security problems that can arise, for instance, when a mobile program visits diKerent sites, namely that of preserving the con7dentiality of the visited sites’ private data. In fact, in [5], it is shown how a form of noninterference called nondeducibility on composition may be used to model also other security properties like authenticity, nonrepudiation and fairness. Noninterference thus appears as a

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rather interesting notion to study when security is concerned. On the other hand, it may be argued [10] that covert channels, that is implicit information 3ows, are unavoidable in practice, as they can arise also at the hardware level. Thus the aim of statically ensuring the absence of covert channels might be a hard one to realise. We certainly do not claim here to cover the whole range of possible attacks from a hostile party. The issue of noninterference has been largely studied in the literature, using diKerent models, and it is not our intention to review the various approaches. We focused on the approach of Volpano et al., as it gives an elegant treatment for a fairly standard language, which can be assumed to be the kernel of more sophisticated practical languages. The question of secure information 3ow and noninterference has also been investigated in the setting of process calculi [5,15]. More recently, there have been studies on noninterference for functional languages [14] and mobile process calculi [9,8,7]. The latter papers are closer to our work, as they use a type system to enforce noninterference. The treatment in [9,8], however, seems overly restrictive: it amounts (at least in the core calculus) to forbid all control 3ow from actions on high channels to actions on low channels. In [9], the core calculus is extended with more sophisticated constructs; in the extended calculus some actions may be classi4ed as “innocuous”, and the restriction on control 3ow may be relaxed when these actions are involved. The papers [14,7] are less restrictive and closer in spirit to our approach, as they try to distinguish the dangerous control 3ow (implementing information 3ow) from the harmless control 3ow which should not be restricted. Another related paper is [1], which studies secrecy properties in security protocols expressed in the spi-calculus. There again, a type system is used to ensure the security properties. As concerns noninterference in the presence of scheduling policies, the most popular approach has been so far the probabilistic one, taken for instance in [20,16]. Our intention here was to handle scheduling entirely within a possibilistic setting. It should be noted, also, that in [20,16] scheduling is introduced at the semantic level (adding probabilities to the transitions), while we express scheduling policies—which, after all, are programs—at the syntactic level. This has the advantage of allowing us to type the schedulers themselves, and express through typing conditions the restrictions one needs to impose on any reasonable scheduling policy for multilevel systems. This is in line with the view expressed by Sabelfeld and Sands [16], where a scheduler is described as a “mechanism for selecting threads which itself satis4es some noninterference property”. Moreover, [16] also identi4es a few characteristics which are deemed necessary for schedulers to be well-behaved in a multilevel security setting. Essentially, schedulers should be allowed to have access to the low part of the memory (to allow some interaction with the controlled threads), and be able to use the history of thread creation=generation along the computation, which may be of high level. In our setting we are able to express more precise conditions, for instance that schedulers allowed to test the termination of high threads should themselves be high-level programs. However, it would be interesting to compare more precisely our syntactic approach with the probabilistic approach of [20,16]. As we have seen, conditional branchings combined with scheduling can introduce new interferences due to the unbalanced duration of the branches. Some proposed

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solutions to this problem rely on a controlled execution time for conditional branchings with high guards. For instance, in [20], a “protect” construct is used to encapsulate these conditional branchings as atomic operations. Another solution was presented by Agat [2], and proved correct in [16], where the branches of high-guarded conditionals are padded so that they always have the same duration. It should be noted that both these approaches allow loops of low-level only. Here we have adopted a diKerent solution to this speci4c problem, which allows us to deal uniformly with various notions of guards. For instance, with our notion of type it is very easy to deal with the standard wait=signal primitives used for synchronisation and cooperative scheduling. By the time we started revising this paper, and thanks also to one of the anonymous referees, we became aware of an independent paper by Smith [17] which was about to appear (indeed, within a similar time frame as our extended abstract [4]). As it turns out, Smith’s paper proposes a type system which is identical to ours, with the additional possibility of recording the running time of programs when this is known statically, thus incorporating the above-mentioned timing techniques for conditional branchings in the type system. Indeed, Smith’s concern is in controlling time leaks, of which scheduling leaks are a particular case. The absence of time leaks is formalised as a property of probabilistic noninterference. Since our type system is slightly more restrictive than Smith’s one (in the sense that the class of typable programs is smaller), it also implies a form of probabilistic noninterference. Finally, let us mention that the type system presented here is diKerent from the one of [4], where the level of conditional guards does not need to be taken into account as long as scheduling is not considered. Instead, the typing rule for loops is more restricted: loops are only allowed to have uniform types of the form (; ) cmd. This alternative type system also guarantees noninterference for concurrent programs (without scheduling), and we conjecture that it allows a larger class of programs to be typed. On the other hand, the solution for scheduling we propose here is more generous than that of [4], as long as we are only interested in scheduling sequential threads. An issue which has not been addressed here, but is planned for future work, is the feasibility of checking noninterference using a type inference algorithm, in the line of [19]. Current work is also oriented towards the treatment of more realistic languages, as advocated for instance in [12], including exceptions and higher-order constructs. Acknowledgements We would like to thank GeoKrey Smith for his interest in, and comments on the submitted version of this paper, and the anonymous referees, whose questions and suggestions led us to largely revise and improve the last section of the paper. References [1] M. Abadi, Secrecy by typing in security protocols, Journal of the ACM 46 (5) (1999) 749–786. [2] J. Agat, Transforming out timing leaks, in: 27th ACM Symposium on Principles of Programming Languages (POPL), 2000, pp. 40 –53.

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[3] P. Bieber, J. Cazin, V. Wiels, G. Zanon, P. Girard, J.-L. Lanet, Electronic purse applet certi4cation (extended abstract), in: Workshop on Secure Architectures and Information Flow, ENTCS, Vol. 32, 2000. [4] G. Boudol, I. Castellani, Noninterference for concurrent programs, in: ICALP’01, Lecture Notes in Computer Science, Vol. 2076, 2001, pp. 382–395. [5] R. Focardi, R. Gorrieri, F. Martinelli, Non interference for the analysis of cryptographic protocols, in: Proceedings ICALP’00, Lecture Notes in Computer Science, Vol. 1853, 2000. [6] J.A. Goguen, J. Meseguer, Security policies and security models, in: Proceedings 1982 IEEE Symposium on Security and Privacy, 1982, pp. 11–20. [7] M. Hennessy, The security )-calculus and noninterference, Computer Science Technical Report 2000:05, University of Sussex, 2000. [8] M. Hennessy, J. Riely, Information 3ow vs. resource access in the asynchronous pi-calculus (extended abstract), in: Proceedings ICALP’00, Lecture Notes in Computer Science, Vol. 1853, 2000. [9] K. Honda, V. Vasconcelos, N. Yoshida, Secure information 3ow as typed process behaviour, in: Proceedings ESOP’00, Lecture Notes in Computer Science, Vol. 1782, 2000, pp. 180 –199. [10] J. Millen, 20 years of covert channel modeling and analysis, in: IEEE Symposium on Security and Privacy, 1999. [11] R. Milner, A Calculus of Communicating Systems, Lecture Notes in Computer Science, Vol. 92, Springer, Berlin, 1980. [12] A. Myers, J3ow: practical mostly-static information 3ow control, in: 26th ACM Symposium on Principles of Programming Languages (POPL), 1999. [13] D. Park, Concurrency and automata on in4nite sequences, in: Proceedings 5th GI-Conf. on Theoretical Computer Science, Lecture Notes in Computer Science, Vol. 104, 1981. [14] F. Pottier, S. Conchon, Information 3ow inference for free, in: Proceedings ICFP’00, 2000. [15] P.Y.A. Ryan, S.A. Schneider, Process algebra and non-interference, Journal of Computer Security 9 (1=2) (2001). [16] A. Sabelfeld, D. Sands, Probabilistic noninterference for multi-threaded programs, in: IEEE, 13th Computer Security Foundations Workshop, 2000. [17] G. Smith, A new type system for secure information 3ow, in: 14th IEEE Computer Security Foundations Workshop, 2001. [18] G. Smith, D. Volpano, Secure information 3ow in a multi-threaded imperative language, in: ACM, Proceedings POPL ’98, ACM Press, 1998, pp. 355 –364. [19] D. Volpano, G. Smith, A type-based approach to program security, in: TAPSOFT’97, Lecture Notes in Computer Science, Vol. 1214, 1997, pp. 607– 621. [20] D. Volpano, G. Smith, Probabilistic noninterference in a concurrent language, Journal of Computer Security 7 (2–3) (1999). [21] D. Volpano, G. Smith, C. Irvine, A sound type system for secure 3ow analysis, Journal of Computer Security 4 (3) (1996) 167–187.

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www.elsevier.com/locate/tcs

Dynamic connectors for concurrency
Roberto Bruni ∗ , Ugo Montanari Dipartimento di Informatica, Universita di Pisa, Corso Italia 40, I-56125 Pisa, Italy

Abstract We propose a metatheoretic approach to the de,nition of operational and abstract concurrent semantics for systems where name handling is a key issue. The main ingredients of our framework are connectors and tile logic. The former allow for the modeling of various kinds of graph-like and term-like structures, which can be useful in representing system distribution and causal histories. The latter is used to combine the use of connectors in space and in time (i.e., according to con,gurations and computations). As a detailed case study, we show how to de,ne a concurrent operational semantics and causal weak bisimilarity for a 0avor of basic parallel c 2002 Elsevier Science B.V. All rights reserved. processes.
Keywords: Distributed systems; Concurrent and causal semantics; Tile logic; Algebras of connectors; Wirings; Process rewrite systems; Basic parallel processes

0. Introduction New technologies and the di6usion of wide area networks have had a great impact on the role of nowadays concept of computation, where issues like distribution, interaction, reactivity and coordination have become central and call for adequate formal models. The growth of many successful proposals for new calculi, each with its own distinguishing features, is a faithful witness of this situation:
-calculus [69] for name mobility, HO
[77] for process mobility, spi-calculus [1] with encryption primitives for the analysis of secure protocols, join calculus [42] for guaranteeing the locality principle in distributed implementations, action calculi [68] for separating name handling from controls and handling encapsulation of agents, ambients [25] for studying the distribution of processes, mobile and dynamic nets [3,22] for network recon,gurability, just to


Research supported by the EC TMR Network GETGRATS and by the Italian MURST Project TOSCA. Corresponding author. Tel.: +39-050-2212-768; fax: +39-050-2212-726. E-mail addresses: [email protected] (R. Bruni), [email protected] (U. Montanari). URLs: http://www.di.unipi.it/∼bruni, http://www.di.unipi.it/∼ugo

∗

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 1 - 7

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R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

cite a few well-known examples. However, the operational semantics of many of the aforementioned calculi require ad hoc techniques for dealing with interesting issues like, distribution, name handling, causal histories, which can hardly be automatically handled via standard speci,cation formats. This is also the case of located CCS [9], whose abstract semantics requires a specialized bisimulation, and of calculi equipped with history preserving bisimilarities. In this paper, we focus on a mathematical setting that exploits analogies in the representation of names, locations and causal links and that thus can serve as a general purpose metamodel to be instantiated to several cases of interest. We then show in detail an application to the simple case study of basic parallel processes with weak synchronization, de,ning a concurrent operational semantics and a causal abstract semantics, which is shown to coincide with bisimilarity via causal trees [33]. Finally, we informally discuss the treatment of other features like mobility and restriction on the basis of [40,70]. To characterize the kernel of properties, which are common to the features we want to model, and for which a uniform mathematical treatment is feasible, we rely on our joint work with Gadducci [14,15], where a small set of connectors is proposed, which can be used for modeling, e.g., communication infrastructures, spatial distribution of system components, partial orders of actions. In particular, these connectors can interact according to a small number of algebraic laws, which precisely characterize di6erent kinds of mathematical entities (all closed under sequential composition), like relations, partial orders, and partitions. Given this correspondence, when modeling a system, we can choose which subset of connectors to employ and ,x a few axioms for ,nitely generating the mathematical setting of connections we are interested in. It is worth remarking that all the structures considered in [15] admit normal form representations. Moreover, the various classes for connectors can be easily represented and manipulated in logical languages like Maude [28] (of the OBJ family [49]). In fact, Maude (as well as other languages supporting algebraic speci,cations and rewriting mechanisms) allows for eJcient rewriting over a mixing of associativity, commutativity, identity, idempotency and -conversion, which are the common axioms involved in the algebras of connectors. The process of speci,cation and prototyping can then be accelerated and conveniently tested. As semantic framework, we ,nd it convenient to take tile logic (TL) [45]. In fact, TL is to some extent parametric with respect to the algebras of states and observable actions, and it is adequate for expressing (and exploiting) the analogies between these two algebras. Like process calculi (PC), which usually consist of a term algebra for processes (possibly up to some axiomatic structural congruence), together with a labeled transition system (LTS) or a reduction system (in the style of Berry and Boudol’s chemical abstract machine [6]) inductively de,ned on the state structure, TL reconciles the algebraic structure of states with the coalgebraic view of computations. Moreover, analogously to the theory of SOS speci,cation formats [35,7,50], several tile speci6cation formats [45,12] have been developed for de,ning compositional abstract semantics. As argued in [12], TL is reminiscent of context systems [6] and hence designed to handle both closed terms and open terms in a uniform way (also analogously to Rensink’s conditional transition systems [75]).

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133

Our aim is to combine connectors and TL to de,ne a general speci,cation paradigm based on algebraic representations of term-like and graph-like structures, which can o6er a valid semantic framework for concurrency in distributed, reactive and open systems. We end up with a taxonomy of categories of connectors, which can be extended to the two-dimensional tile paradigm in a standard way, allowing us to model di6erent features of distributed settings in a uniform way. Moreover, we can apply known results and proof mechanisms from tile theory for studying abstract properties such as observational congruences and adequacy of the concurrent semantics with respect to interleaving. 0.1. An algebra of connectors The key notion we need is that of link to names. In fact, names are to some extent multi-purpose tags that can be attached to e.g. resources, locations, objects, functions, channels, messages, memory cells and even to relevant events in the past history of the computation. The information about sharing is often more important than names themselves (think e.g. of bound names that can be freely -converted, and of canonical naming as in the De Brujin notation). Having multiple links to the same name means that the name is shared, while having at disposal no link to a name means that the name is private and cannot be accessed from the outside. Many mathematical structures have been recently proposed for expressing graphlike representations of name-based systems. They range from the :ownomial calculus of SMtefNanescu [26,78], to the bicategories of processes of Walters [56,57], to the interaction combinators of Lafont [60], to the pre-monoidal categories of Power and Robinson [73,55], to the action structures of Milner [68], to the interaction categories of Abramsky [2], to the sharing graphs of Hasegawa [51] and to the gs-monoidal categories of Corradini and Gadducci [29,30], just to mention a few (other references are [31,44,47,53,76]). Similarly to [29,30], we ,x a ,rst-order term-like notation which is reminiscent of Lawvere’s theories [62], but requires a di6erent axiomatization of the auxiliary operators involved in the construction. It is noteworthy that most of the above structures can be seen as enrichments over symmetric monoidal categories, which form the basis for the description of distributed environments in terms of the so-called “wire and box” diagrams. The informal wire and box notation gives an intuitive, visual understanding of the name sharing mechanism: Wires represent links from and to names, while boxes, labeled with operators in a signature , represent the modules interconnected in the system. Connection points between wires and boxes are marked by the sort of the corresponding name (i.e., the type of the wire). When composing systems, it is essential to know which names are private and which ones are public, therefore public names form an input and an output interface, represented by untied left-end or right-end of wires. Moreover, the basic operations of sequential and (disjoint) parallel composition for building larger diagrams arise naturally (see Fig. 1). Fig. 2 shows three examples of wire and box diagrams: on the left, a component C has two identical subcomponents p; in the center, D has two connections to the same subcomponent q. Both systems have empty input interfaces and unary output interfaces.

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input

.. .

.. .

output

.. .

. ..

.. .

. ..

.. .

. ..

. ..

Fig. 1. A generic wire and box diagram and two ways of composing diagrams.

p

.

.

. C

p

.

.

.

q

x

D

.

.

.

p

.

q

.

.

y

r

.

.

z

x

Fig. 2. Examples: three wire and box diagrams.

For denoting these situations as terms, we can write C(p; p) in the ,rst case, but we must write e.g. let x = q in D(x; x), instead of D(q; q), to faithfully describe the sharing of q (the same resource q, denoted by a local name x, is referenced twice in the context D). This distinction plays a signi,cant role when dealing with distributed systems, as distinct instances of the same resource can evolve independently (possibly nondeterministically), while the evolution of a shared component must be coordinated with all the contexts it is linked to. The rightmost diagram in Fig. 2 represents a situation which we shall exploit further in our case study: a system is composed by three components (i.e., p, q and r) that are attached via several connectors to the output interface. We can think of the interface
x; y; z as a list of places, encoding some information about the past history of the system (e.g., which process spawned the components). It is evident that p is linked to x and y, while q is connected to x, y and z, and r is linked to z. But what about p and z? If links were undirected, then there is a path connecting p to z, but if instead links were directed (e.g., from left to right), then p cannot see z. The answer lies in the algebraic laws of connectors: we shall see that both situations are possible and that the di6erence is given by just one axiom. A unambiguous notation for denoting the third diagram can be p[x; y] | q[x; y; z] | r[z] (for directed links), which is di6erent from p[x; y; z] | q[x; y; z] | r[x; y; z] (associated with undirected links). The wire and box notation clearly separates the signature (i.e., the basic modules) from the auxiliary structure which can be used for coordination (i.e., wires or connectors). In particular, in [15] it is shown that by imposing suitable wiring formats (e.g., that at most one wire can exit from a name) it is possible to characterize very di6erent frameworks which can encode, e.g., Petri net processes, contextual net processes, relations, partitions, partial orders, and multi-sets of traces. At the algebraic level, they correspond to di6erent ways of combining four auxiliary operators: duplicator ∇ (for

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135

initial configuration initial input interface trigger final input interface

initial output interface

s a

α

b

t

effect final output interface

final configuration Fig. 3. A generic tile.

sharing names), discharger ! (for hiding names), co-duplicator T (for matching names), and co-discharger ¡ (for creating names). 0.2. A two-dimensional logic Connectors can be employed both in “space” (to describe the topology of the system, i.e., its structure or syntax) and in “time” (to express behavioral observation, i.e., for abstract semantics). Nevertheless, the common underlying algebraic representation of connectors suggests that some interplay might naturally occur between their structural and semantic usages, which is one of the main issues in moving from rewriting logic to tile logic. Rewriting logic (RL) [65] gives the basis for reduction semantics, lifting the structure of states to the level of (concurrent) computations. It o6ers both a semantic and logical framework, where many di6erent languages and systems can be straightforwardly encoded and many other logics can be naturally represented. Notably, several programming languages based on rewriting logic have been developed (e.g., Maude [28], ELAN [74] and CafeObj [37]). However, as noticed in [46], RL is not designed around interaction and coordination issues, which must be recovered via suitable encodings. Tile logic extends rewriting logic (in the unconditional case) by taking into account state changes with side e6ects and rewriting synchronization. Basically, a set of rules describes the behavior of open con6gurations, which may interact through their interfaces. The name “tile” is due to the graphical representation of such rules, which have a the form shown in Fig. 3, also written as  : s → t, stating that the initial con6guration b

s of the system evolves to the 6nal con6guration t via the tile , producing the e;ect b. However, such a step is allowed only if the subcomponents of s (i.e., the arguments of the context s) evolve to the subcomponents of t, producing the e6ect a, which acts as the trigger for the execution of . Triggers and e6ects are called observations and tile vertices are called interfaces. The arrows s, a, b and t give the border of . The interfaces shared between con,gurations and observations represent the way in which the two dimensions are connected. In fact, con,gurations and observations are just arrows in two distinct monoidal categories having the same set of objects. In particular, the categories we consider are those freely generated starting from suitable signatures and enriched with the auxiliary connectors of interest. Tiles can be composed horizontally, in parallel, and vertically to generate larger steps. Horizontal composition yields rewriting synchronization: the e6ect of the ,rst tile acts

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β α

α

β α

β

Fig. 4. Horizontal, parallel and vertical tile compositions.

as trigger for the second tile, and the resulting tile expresses the synchronized behavior of both. The operation of parallel composition corresponds to building concurrent steps, where two (or more) disjoint con,gurations can concurrently evolve. Of course, the border of a concurrent step is the parallel composition of the borders of each component of the step. Vertical composition models the execution of a sequence of steps starting from an initial con,guration. It corresponds to sequential composition of computations. The three compositions are illustrated in Fig. 4. Given a set of basic tiles, the associated tile logic is obtained by adding some canonical “auxiliary” tiles and then closing by (the three kinds of) composition both auxiliary and basic tiles. Analogously to rewrite rules in rewriting logic, tiles can be seen as sequents in tile logic, with tile compositions yielding the inference mechanism for deriving new sequents. Thus, tile logic o6ers a two-dimensional Curry–Howard analogy, where borders are types, i.e. formulas, and tiles are terms, i.e. proofs: Type inhabitance means ,nding a tile with a certain border (and of course, it may happen that di6erent tiles have the same border). Auxiliary tiles are the two-dimensional counterparts of (one-dimensional) connectors. Therefore, when the same connectors are used on the horizontal and vertical dimensions, auxiliary tiles model the “sliding” of connections from one dimension to the other through tile vertices. As an example, auxiliary tiles may be introduced that accommodate consistent sharing of names in the two dimensions. In this paper, we shall give just an informal view of the categorical models of tile logic (expressed by suitable monoidal double categories [45,18,19]), and of the equational algebra of proof terms decorating tile sequents. We shall instead pursue the view of tile logic as a concurrent operational machinery, where trigger=e6ect pairs are labels that provide a natural way of expressing abstract semantics of open systems (e.g., via traces, tile bisimulation and coalgebraic techniques [32]). By analogy with the theory of SOS formats, several speci,cation formats have been de,ned for tiles, e.g.: (a) the monoidal tile format [67] and the process tile format [16], which have monoidal structures of both con,gurations and observations (i.e., linear terms); (b) the algebraic tile format [45], which has cartesian con,gurations and monoidal observations; and (c) the term tile format [16], which has cartesian con,gurations and observations. Moreover, the so-called basic source syntactic constraint guarantees that tile bisimilarity is a congruence for (a) and (c), while for the algebraic format the congruence is guaranteed for closed terms only [12]. Intuitively, the fact that the basic source works ,ne for the monoidal and the cartesian tile format but not for the algebraic tile format suggests that having analogous horizontal and vertical structures allows for proving stronger results. Therefore, the

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idea is to encourage the use of the same algebra of horizontal and vertical connectors, so that a two-dimensional connector (also dynamic connector) is just any tile that expresses the consistency between its corresponding horizontal and vertical connections. In other words, the categorical perspective and the graphical representation of tiles both suggest that two-dimensional connectors must be commuting squares in the category of connectors. Informally, two-dimensional connectors have the main task of propagating e6ects through horizontal connectors, preserving the correct correspondences. Driven a by the results in [16,18,19], we take as connectors all tiles  : s → t, where all s, a, b b

and t are obtained as compositions of one-dimensional basic connectors and such that s; b
= a
; t, for a
and b
the horizontal counterparts of the vertical connectors a and b (uniquely determined by the analogy between horizontal and vertical connectors). As a ,rst result, we show that two-dimensional connectors can be ,nitely generated and axiomatized according to a very simple scheme whenever one-dimensional connectors can. Since this is the case of all the structures considered in [15], then we are con,dent that the resulting metamodel (which is parametric with respect to horizontal and vertical signatures, the algebra of connectors, and the operational moves de,ned by basic tiles) is general enough to cover several cases of interest. 0.3. An example from basic process algebras In the literature, tiles have been already applied to many formalisms, e.g., CCSlike calculi, located CCS,
-calculus and actors. Nevertheless, no uniform approach to provide such calculi with a causal and concurrent operational semantics has been de,ned. In this paper, we focus on such an approach for systems whose abstract view of states is similar to the one illustrated in the rightmost diagram of Fig. 2: There are several components (which are to some extent similar to the tokens of a Petri net) and an output interface of the system (essentially a list of names). Between them, suitable connectors establish the internal interaction between components and the external interaction between components and any environment attached to the interface. An input interface can also be present, whose names can be of course connected with components and names in the output interface. Operationally, each component must be able to express all of its possible behaviors, also those that require some coordination with other components. In doing this, suitable e6ects can be produced that must be propagated through horizontal connectors and eventually coordinated; a task which is assigned to two-dimensional connectors, as we already noticed. The simplest example that ,ts our aim is based on a description of in,nite state concurrent systems in the style of PA-processes [5,63] and process rewrite systems [64], where a smaller set of basic operators (namely parallel and sequential composition) is considered with respect to CCS-like calculi, while action pre,x, (guarded) choice and recursion are directly encoded in the operational semantics, in the form of basic labeled rewrite rules. In particular, we rely on the subclass of basic parallel processes (BPP), whose processes t are just the empty process , constants in a countably in,nite set Const = {X; Y; : : :}, or the (associative, commutative and with unit ) parallel composition t1  t2 of two other processes t1 and t2 . The operational semantics

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is obtained by closing a set  of basic rules (each having the form X → t for a an action taken in a countably in,nite set Act = {a; b; : : :}) with respect to free concurrent application in any parallel composition, i.e., according to the classi,cation of [64], we rely on BPP(1,P) processes (the symbols in parentheses denote the forms of sources and targets of basic transitions, hence 1 means that the sources of basic transitions are single constants, while P means that any ,nite parallel composition of constants can be taken as target). Note that since  is ,nite, only a ,nite number of actions and constants must be considered, although Act and Const can be countably in,nite. Pre,xes are encoded in transition labels, guarded choice is given by inserting multiple a b basic transitions having the same constant as source (e.g. X → Z  W and X → Y  Y ), and (guarded) recursion is obtained by allowing the re-use in the transition target of a the same symbol in the source, as in X → Y  X  Y . The idea is to combine the use of connectors and tiles to equip BPP(1,P) with: (1) synchronization between otherwise communication-free processes; (2) causality information based on abstract locations; (3) weak observational semantics, giving a satisfactory account of causality, in the style of Darondeau and Degano’s causal trees [33,34]; and (4) a concurrent operational semantics, i.e., a concurrent implementation (as opposed to concurrent abstract semantics). We call WS-BPP(1,P) the resulting process paradigm (the pre,x “ws” stands for weakly synchronized). We shall state a formal correspondence between ordinary labeled transitions of WS-BPP(1,P) processes and tiles and prove that tile bisimilarity coincide with the causal weak bisimilarity of Darondeau and Degano [33]. 0.4. Structure of the paper Section 1 introduces the basic notation and concepts about our algebraic view of connectors (Section 1.1) and tile logic (Section 1.2). Section 2 presents the original results about the characterization of dynamic connectors which will constitute the basis of our metamodel. Section 3 focuses on the application of our paradigm to the study of causal and concurrent semantics for WS-BPP(1,P) processes. Section 4 discusses further applications of our metamodel, sketching a comparison with related work in the literature.

1. Background 1.1. Normal forms for algebras of connectors We recall here a categorical view of connectors, along the lines traced in [15]. Essentially, connectors represent suitable enrichments of monoidal categories and, as it is argued by various authors (cf. surveys in [15,30]), these structures allow to recast the usual notion of term over a signature in a more general setting. Moreover, the progressive enrichment of a basic theory with additional connectors generates several di6erent models, where the notions of relation, partial order, partition, and many others

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139

can be represented and compared. In particular, a key role is played by the way in which the various connectors can interact. A category C can be seen as a graph with nodes in O and arrows in A, together with a total map id : O → A that associates an arrow id x : x → x with each object x ∈ O, and a partial, associative operation for sequential composition of arrows ; : A × A → A that is de,ned if and only if the target of its ,rst argument matches the source of its second argument: whenever f : x → y and g : y → z are arrows of C then f; g : x → z is an arrow, and for all f : x → y we have id x ; f = f = f; idy . As usual, we will often denote the arrow id x by the object name x itself. The obvious notion of morphism between categories (mapping arrows into arrows and objects into objects) must preserve the categorical structure (source, target, identities and sequential composition) and is called functor. In what follows, we will sometimes refer to the opposite category C op of a category C as the category having the same objects as C, but where the direction of the arrows is reversed (e.g., if f : x → y and g : y → z are arrows of C, then f op : y → x, g op : z → y and g op ; f op = (f; g)op ). When no confusion can arise, we shall omit the superscript (·)op from arrow symbols. For simplicity, we restrict to categories strictly preserving the additional structure; and, in particular, to monoidal structures where the tensor product is strictly associative and with unit, instead of being so only up to a canonical isomorphism. This allows us to focus on the overall presentation of the monoidal structure itself as a suitable data type, at the same time easing the notation. Denition 1. A (strict) monoidal category (MC) is a triple
C; ⊗; e, where C is the underlying category, the tensor product ⊗ : C × C → C is a functor satisfying the associative law (t1 ⊗ t2 ) ⊗ t3 = t1 ⊗ (t2 ⊗ t3 ), and the unit e is an object of C satisfying the identity law t1 ⊗ e = t1 = e ⊗ t1 , for all arrows t1 , t2 and t3 . As a matter of terminology, given two functors F; G : C → D, we use the term transformation (from F to G) to denote a family $ of arrows in D indexed by the objects of C, such that $x : F(x) → G(x) for all objects x in C. It is a natural transformation if F(f); $y = $x ; G(f) for any arrow f : x → y in C. Denition 2. A symmetric monoidal category (SMC) is a 4-tuple
C; ⊗; e; %, where
C; ⊗; e is a MC, and % :−1 ⊗−2 ⇒−2 ⊗−1 is a natural transformation satisfying the coherence axioms %x⊗y; z = (x ⊗ %y; z ); (%x; z ⊗ y) and %x; y ; %y; x = x ⊗ y, for all x; y; z. Note that, the naturality of symmetries in a SMC
C; ⊗; e; % amounts to say that for all arrows f : x → y and g : z → w we have (f ⊗ g); %y;w = %x; z ; (g ⊗ f). We shall consider connectors which can all be expressed as suitable categorical transformations. Symmetries and connectors can also be nicely illustrated via the wire and box notation. For example, the symmetry %x; y can be seen as the crossing of wires in Fig. 5 (left), and consequently the coherence axioms can be drawn as in Fig. 5 (center and right), while the naturality of % is represented in Fig. 6. (The labels x, y and z denote typing information associated with wires.)

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R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

x y

x

x

z

x

z

y

z

x = y

x

z

y

y

y

y

y

x y

x

z

x y

x

=

x

x

y

y

Fig. 5. Symmetries: the wire and box notation.

x

f

y

w

x

z

g

w

z

x

f

y

= z

g

w

y

Fig. 6. Wire and box representation of the naturality axiom for symmetries.

The constructive de,nition of algebraic theories [62] as symmetric monoidal categories, enriched with two natural transformations (duplicator and discharger), dates back to the mid-seventies [52,72], even if it has received a new stream of attention in these days. With respect to the usual presentation via universal properties (or equivalently, the ordinary description of terms by means of the metaoperation of substitution), it emphasizes and separates very nicely the “link structure” from the term signature . The naturality axioms for these transformations express a controlled form of data sharing and garbaging. This intuition is further con,rmed by the results in [30], where it is shown that, if these axioms are missing (i.e., if duplicators and dischargers are just transformations), then the corresponding gs-monoidal theory is the natural framework for the representation of term graphs [4] (the pre,x “gs” comes indeed from graph substitution). In this section we propose a taxonomy for these enriched structures, which are essentially built on duplicators, dischargers, and their “opposite” versions. A duplicator ∇ is a transformation from the identity functor on C to the functor D; ⊗ =−1 ⊗−1 obtained by pre-composing the monoidal tensor product by the diagonal functor D : C → C × C (with D(f) = (f; f) for all f ∈ C). A discharger ! is a transformation between the identity on C and the constant functor that sends everything into the unit e. Moreover, the following coherence axioms must be satis,ed: ∇x⊗y = (∇x ⊗ ∇y ); (x ⊗ %x;y ⊗ y) !x⊗y =!x ⊗!y ∇x ; (∇x ⊗ x) = ∇x ; (x ⊗ ∇x )

∇e =!e = id e

∇x ; %x;x = ∇x ∇x ; (!x ⊗ x) = x

This situation is sometimes referred to in the literature by saying that each object is equipped with a co-monoid structure [43], while, after [30], we denote such a structured category as gs-monoidal. If ∇ and ! are natural (i.e., if they satisfy f; ∇y = ∇x ; (f ⊗ f) and f; !y = !x for all f : x → y), then the category is cartesian. We shall omit the subscripts from connector components (e.g., by writing ∇ instead of ∇x ) when the object name is obvious from the context. Similarly, co-duplicators Tx : x ⊗ x → x and co-dischargers ¡x : e → x must satisfy dual coherence axioms, and we denote the resulting category as cogs-monoidal; when nat-

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x x

∆

x

!

∆x =

x

!

!x =

x = x

x =

!

x

141

x

x

Fig. 7. Four connectors in the wire and box notation: ∇x , !x , Tx and ¡x .

urality is also satis,ed, it yields co-cartesianity. The four connectors are illustrated in Fig. 7. Wire and box representations for their coherence axioms can be found in [15]. There are various ways for the gs-monoidal and the cogs-monoidal structures to interact. In particular, the following laws have been pointed out in the literature (see, e.g., [24,27,15]): ∇x ; Tx = x;

(1)

Tx ; ∇x = (∇x ⊗ ∇x ); (x ⊗ %x;x ⊗ x); (Tx ⊗ Tx );

(2)

Tx ; ∇x = (x ⊗ ∇x ); (Tx ⊗ x);

(3)

Tx ; ∇x = (∇x ⊗ x); (x ⊗ Tx );

(4)

¡x ; !x = e;

(5)

¡x ; ∇x = ¡x ⊗ ¡x ;

(6)

Tx ; !x =!x ⊗!x :

(7)

As a set, these laws are of course redundant. For example, laws (3) and (4) are equivalent, and they subsume law (2). At the same time, they are linked with naturality: law (2) is e.g. satis,ed when the monoidal product is both cartesian and co-cartesian, i.e., if the underlying category has biproducts. Several subsets of the eight laws listed above have been studied in the literature, usually as a model for various algebras of systems. Case studies range from the causal streams of Ferrari and Mowbray (in joint work with the second author) [41] and the :ownomials calculus proposed by CNazNanescu and SMtefNanescu [27,78], to the bicategories of relations introduced by Carboni and Walters [23,24], and to the (co)span categories considered in [13] by Gadducci (in joint work with the ,rst author). The relevant axioms are graphically illustrated in Fig. 8, where axiom (4) is omitted because equivalent to law (3). Table 1 summarizes the taxonomy of categories obtained via several mixing of constructors. In particular, the ,rst four rows show whether (co)duplicators and (co)dischargers are required, while the remaining six rows illustrate the laws that regulate the interplay between duplicators, dischargers and their opposites. The columns are devoted to each one of the categories presented. Entries are marked with the symbol + if the objects of the category in the corresponding columns (which are categories themselves) must satisfy the structural requirement associated with that row. The symbol - is used otherwise. Note that, for the sake of readability, we presented a graphical view of axioms (1) – (3) and (5) – (7) in the ,rst column, which is reminiscent of the corresponding

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R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

x (1) x = x

x

x

x

x

x

x

!

x

x

!

(5) =e

x x

x

x

x x

!

(2) =

x x

x

x

x

x

x

x

x

!

x

x x

x x

(7) x = x

x

x

(3) =

x

(6) =

x

x

x

x

!

x

!

x

x

x

x

!

!

x

Fig. 8. Axioms (1) – (3) and (5) – (7) as wires and boxes.

Table 1 A taxonomy of connectors

ShCat GSCat coGSCat RMCat MShCat NBCat PMCat DGSCat TRMCat ∇

T ! ¡

+

+

−

+

−

− − −

−

− − X=Z − − |−| = e |−≺ = |= − −| = =| −

− − − − − −

=− X=,

−

+ + − − − − − −

+ + + + + +

−

+ + +

+ +

− −

+ + +

− − −

− −

+ +

− − −

+

− −

+ + + +

+ + + +

+ + + +

+ + +

− −

− − −

+ +

−

+

+ +

− −

+

−

wire and box diagrams in Fig. 8. For example, the category ShCat has share categories as objects and share functors between them as arrows. The corresponding column tells us that a share category is a SMC with duplicators. Share functors are just SMC functors strictly preserving duplicators. A more interesting example is given by RMCat, whose objects are r-monoidal categories and whose arrows are R-monoidal functors. r-Monoidal categories have a very rich structure, which includes all connectors, satisfying laws (1) – (2) and (5) – (7). The name r-monoidal comes from the word relation, because Rel the category of sets (as objects) and relations (as arrows) has that structure.

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Leaving aside gs-monoidal categories (i.e., the objects of GSCat), cogs-monoidal categories (in coGSCat = GSCatop ), match-share categories (in MShCat) and the new-bang categories (in NBCat), let us mention also the p-monoidal categories of PMCat, which are very similar to r-monoidal categories, except for the fact that the stronger axiom (3) holds rather than just law (2), but that cannot be compared with r-monoidal categories because laws (6) –(7) are not required. The name p-monoidal comes from the word partition: in fact, the valid laws state that it is not important the way in which two names are connected, but just the fact that they are connected, i.e., the paths formed by connectors de,ne a partitioning of input and output names. To some extent, axiom (3) imposes a sort of transitive (and re0exive) closure on paths, which corresponds to the di6erence between a relation and the partition it generates. Finally, p-monoidal categories are a particular instance of dgs-monoidal categories (see row DGSCat). Though any of the above structures has its own merit and can be exploited in the two-dimensional tile setting, we shall focus on TRMCat which suits the case study presented in the Introduction. The category TRMCat di6ers from RMCat in that dischargers are not required, hence the name TRMCat, which stands for total relation. Moreover, since ! is not considered, then axioms (5) and (7) must be relaxed. We conclude this overview with an example, taken from [15], which illustrates how to model (,nite) labeled partial orders by means of connectors. This representation is in fact used for the observations of tiles in Section 3. Example 3. Let L be a set of labels and L = {l : 1 → 1 | l ∈ L} be a special signature of unary operators associated with L. For any l ∈ L, let us denote by el the arrow ∇1 ; (id1 ⊗ l); T1 taken in the free r-monoidal category RM(L ) generated by L . Given the symmetric monoidal functor F : RM(L ) → RM(L ) de,ned by F(l) = el and behaving as the identity on objects and connectors ∇, T, ! and ¡, we have that the set F(RM(L )) [ 0; 0] = F(RM(L ) [ 0; 0]) (i.e., the set of arrows from 0 to 0 in the subcategory F(RM(L )) is isomorphic to the set of ,nite partial orders (E; ) labeled over L. The functor F has the task of selecting only arrows built using el as generators, instead of l. In fact, by joining the source and target of l with those of an identity in parallel with l via ∇1 and T1 , the arrow el provides a direct connection from the source of l to its target, and this short-circuit is transitively closed through sequential composition thanks to axiom (2) of r-monoidal connectors. Thus, given a generic arrow f ∈ F(RM(L )) [ 0; 0], the generators l appearing in f become the events of the partial order (labeled by l), and two di6erent events e1 , e2 are ordered as e1  e2 if there is a direct connection between the target of e1 and the source of e2 . We just take arrows from 0 to 0, because otherwise source and target ranks should also be taken into account. 1.2. Tile logic The tile model has been introduced in [45] as an extension of rewriting logic along the intuition of context systems. While ordinary (unconditional) rewrite rules do not rely on the interaction with the environment (they can be freely instantiated with any

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term and also contextualized inside any context), the goal of tiles is a reactive view of the system, so that components can be designed in isolation and then composed via their behavior. In fact, triggers and e6ects allow for modeling interaction where also the interface of the component can be modi,ed, and this is possible only if also the environment can accomplish this change, or vice versa, the component might be required to o6er di6erent services on demand, which may involve some rearrangement of its local network topology. It is straightforward that rewrite rules are just a special case of tiles, whose triggers and e6ects are identities. Moreover, being designed for application to concurrency, tiles require a monoidal operation on states and observations, for parallel composition of components. Varying the algebraic structure of con,gurations and observations, tiles have been used to model many di6erent aspects of dynamic and reactive systems, ranging from synchronization of net transitions [20], to causal dependencies for located calculi and ,nitely branching approaches for asynchronous
-calculus [40], to name handling in synchronous
-calculus [19] (where a higher-order version of tiles is used), to actor systems [70]. Though at present no automated veri,cation tool is available which is based directly on TL, some prototyping has been made possible for suitable TL speci,cation classes, via a conservative encoding in RL [16,17]. Denition 4. A tile system is a tuple R = (H; V; N; R) where H, V are monoidal categories with the same set of objects OH = OV , N is the set of rule names and R : N → H × V × V × H is a function such that for all  ∈ N , if R() =
s; a; b; t, then s : x → y, a : x → z, b : y → w, and t : z → w for suitable objects x, y, z and w. We a shall write such rule either as  : s → t, or as the tile b

The category H is called horizontal and its arrows con6gurations. The category V is called vertical and its arrows observations. The objects of H and V are called interfaces. Starting from the basic tiles R() of the system, more complex tiles can be constructed via horizontal, vertical and parallel compositions. Moreover, the horizontal and vertical identities are always added to the system and composed with the basic tiles. All this is illustrated in Fig. 9. Depending on the chosen tile format, H and V must satisfy certain constraints and suitable auxiliary tiles are added and composed with basic tiles and identities in all the possible ways. The set of resulting tiles (called :at a sequents) de,ne the :at tile logic associated with R. We say that s → t is entailed by a

a

b

b

b

the logic, written R f s → t, if the sequent s → t can be expressed as the composition of basic and auxiliary tiles.

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145

R() =
s; a; b; t

t:x→y∈H

a:x→z∈V

a

x

x −→ z

s −→ t a

y

b

s −→ t b

a

s −→ t

h −→ f a

a

t −→ t

b

c

b

s; h −→ t; f

a

c

a

s −→ t

t : −→ h a;c

d

b

s −→ h

c

c

h −→ f a⊗c

d

s ⊗ h −→ t ⊗ f

b;d

b⊗d

Fig. 9. Inference rules for 0at tile logic: (from left to right, top to down) basic sequents; vertical and horizontal identities; horizontal, vertical and parallel compositions.

The usual abstract equivalences of LTS can then be straightforwardly rede,ned by considering the pairs (trigger, e6ect) as the labels of the transition system. Denition 5. Let R = (H; V; N; R) be a tile system. A symmetric relation ∼t on a con,gurations is called tile bisimulation if whenever s ∼t t and R f s → s
, then t
b a exists such that R f t → t
, and s
∼t t
. b

The maximal tile bisimulation is denoted by t , and two con,gurations s and t are tile bisimilar if s t t. Tile bisimilarity is a congruence if whenever s t t and s
t t
, then s ⊗ s
t t ⊗ t
and s; s
t t; t
(if composable). A semantics being a congruence is of course fundamental for having a compositional framework, since it guarantees that any component of the system can be safely replaced by any semantically equivalent component. Note that the congruence property must be veri,ed against the two basic operations of tile con,gurations: sequential and parallel compositions (ordinary congruence with respect to operators in the con,guration signature follows as an easy implication). The monoidal structure of V makes tile bisimilarity suitable for expressing true-concurrent semantics in the style, e.g., of [48]. We are particularly interested in considering tile systems where the monoidal categories of con,gurations and observations are freely generated from suitable horizontal and vertical signatures, respectively, e.g., they are initial models in the categories discussed in the previous section. The tile format originally proposed in [45] is the so-called algebraic tile format that recollected the perspective of ordinary transition system speci,cations: con,gurations are terms over a certain signature, and observations are the arrows of the monoidal category freely generated by certain labels (regarded as unary operators). Auxiliary tiles lift the horizontal cartesian structure to the horizontal composition of tiles. In the a ⊗···⊗a algebraic tile format, basic tiles have the form s −1−−−−→n t, where ai ’s and a can be a

either labels or identities and s; t ∈ T (X ). A typical auxiliary tile for the algebraic a format is ∇ −→ ∇, which duplicates the observation a (trigger of the tile). We refer a⊗a

to [45] for more details. The algebraic tile format is not uniform in the two dimensions, since H is cartesian, whereas V is only monoidal (nonsymmetric). The results in [12] suggest that a more

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Basic proof sequents: Generators and Identities: a  : s −→ t ∈ R(N ) a : x −→ z ∈ S(V ) b

s : x −→ y ∈ S(H )

 : s −→ t ∈ Pp (R)

1s : s −→ s ∈ Pp (R)

a

a

idy

1a : id x −→ id z ∈ Pp (R) a

b

idx

x; y ∈ S ∗

Auxiliary proof sequents: Symmetries:

%x; y

2x; y : %x; y −→ idy⊗x ∈ Pp (R) idy ⊗ x

Composition rules: Parallel and Sequential compositions: a

 : s −→ t; b




⊗ : a

 : s −→ t; b

b

a⊗a s ⊗ s
−→ t b⊗b

3 : s
−→ t
∈ Pp (R)  a

b

 ∗ 3 : s; s
−→ t; t
∈ Pp (R)  b

a


: s
−→ t
∈ Pp (R)  b

⊗ t
∈ Pp (R) a

 : s −→ t; b

a

3 : t −→ t
∈ Pp (R)  a;a

b

 · 3 : s −→ t
∈ Pp (R) b;b

Fig. 10. Inference rules for process tile logic.

convenient framework can be obtained if we renounce to the cartesian structure altogether, resorting to the simpler monoidal tile format [67] where only linear contexts are allowed (and observed), or if we consider the more general term tile format [16], where also V is cartesian. Since here we are not interested in naturality axioms for ∇ and !, the term tile format is not adequate; on the other hand, the monoidal tile format is not suJcient for our purposes (all categories of connectors have at least the symmetries). We thus consider process tile logic [11,16]. Denition 6. A process tile system (PTS) R is a tile system, whose categories of con,gurations and observations are the free SMC’s S(H ) and S(V ) generated by two signatures H and V (over the same set of sorts S). Without loss of generality, we represent the objects of S(H ) and S(V ) as ,nite strings of sorts, i.e. as elements of S ∗ , which is indeed the free monoid on S. In process tile logic, the consistent permutation of wires is introduced for free. Starting from a PTS, we derive all the tiles obtained by applications of some deduction rules, de,ning the free compositions of the basic and auxiliary tiles in R according to parallel, horizontal, and vertical compositions. We decorate the sequents with proof terms, that can be axiomatized in order to capture equivalent proofs. The resulting equivalence classes make fewer identi,cations than those induced by the undecorated 0at version. Denition 7. Let R be a PTS. We say that R entails the class Pp (R) of process sequents, obtained by a ,nite number of applications of the inference rules in Fig. 10. For any  ∈ Pp (R) we write R p  (while f is used for 0at sequents).

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

147

Fig. 11. A composition of two-dimensional symmetries.

Entailment in the logic can be expressed using the algebraic structure on proof sequents. Due to space limitations, we refer the interested reader to [11] for the detailed description of the axiomatization of proof sequents. We say that R entails the class Ap (R) of abstract process sequents, whose elements are equivalence classes of Pp (R) modulo such axiomatization. This makes Ap (R) a symmetric monoidal double category [18], de,ning the initial model of PTL. 1.2.1. A digression on symmetric monoidal double categories and PMEqtl This informal digression introduces concepts to be used in conjunction with Propositions 16 and 17. If skipped, this material does not otherwise compromise the readability of the rest of the paper. Monoidal double categories come equipped with the operations of sequential, vertical and parallel compositions, thus providing a convenient class of models for tile logic. Essentially, the axioms de,ne the associativity (and the obvious unit elements) for all compositions, plus the functoriality of each composition with respect to the other two. The axiomatization of two-dimensional symmetries is more complex and not discussed here. However, the idea is that any two tiles with the same border that are obtained by the composition of auxiliary tiles and horizontal and vertical identities are equated. Thus we have e.g., (see Fig. 11): (2x;y ∗ 1%y;x ) · (1%y;x ∗ 2y;x ) = 1id x ⊗id y = 1id x ⊗id y : Formally, a double category is an internal category (i.e., a cat-object) in Cat, the category of small categories (as objects) and functors (as arrows). Given two double categories D and E, a double functor F : D → E is a 4-tuple of functions mapping objects to objects, horizontal and vertical arrows to horizontal and vertical arrows, and cells to cells, preserving identities and compositions of all kinds. Thus, monoidal double categories are double categories together with a double functorial tensor product ⊗ and a unit object e. An alternative characterization of monoidal double categories can be given in terms of product theories in partial membership equational logic (PMEqtl) [66], as proposed in [67]. The features of PMEqtl (partiality, poset of sorts, membership assertions) offer a natural framework for the speci,cation of categorical structures, ,rstly because arrow composition is a partial operation, and secondly because membership predicates over a poset of sorts allow, e.g., the objects to be modeled as a subset of the arrows,

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and the arrows as a subset of the cells. Moreover, the tensor product construction of theories can be conveniently formulated in PMEqtl, and yields a concise de,nition of monoidal double categories. In fact, the tensor product construction works at the syntax level of the theories, combining the conditional sentences (equations and membership assertions) according to a simple scheme. Essentially, if we let CAT and MON be, respectively, the theories of categories and of monoids, then the theory MONCAT of monoidal categories is just the tensor product CAT ⊗ MON  MON ⊗ CAT, the theory DCAT of double categories is the tensor product CAT ⊗ CAT and the theory of monoidal double categories is given by any of the tensor products MONCAT ⊗ CAT  MON ⊗ DCAT  MON ⊗ CAT ⊗ CAT. We refer to [67,16] for more information on the tensor product construction. For dealing with both horizontal and vertical symmetries and their consistent interaction on the two dimensions, the two-dimensional analogous of natural transformations has been presented in [16,18], by introducing the notion of generalized natural transformations between four double functors, in the style of Ehresmann work on hypertransformations [38,39].

2. Auxiliary connectors as tiles From the mathematical viewpoint, the two-dimensional extension of symmetries is completely satisfactory. Similarly, the extension of one-dimensional cartesianity has been discussed extensively in [16,18,11]. But for the connectors we are interested in, the naturality axioms are not required. Therefore, we must exploit some other interpretation of two-dimensional connectors. Informally, we would like: • to ,x suitable categories of observations and con,gurations having not only the same objects, but also the same auxiliary structure, i.e., the same connectors according to the taxonomy in Section 1.1; • to have at disposal all the tiles whose border contains only connectors and no primitive module and that would commute if horizontal and vertical connectors were arrows in the same category; such tiles are called auxiliary tile connectors, dynamic connectors or also two-dimensional connectors; • from the point of view of sequent decorations, we also would like to identify all the auxiliary tile connectors having the same border. Denition 8. A theory of connectors is a pair T = (C; L), where C ⊆ {∇; T; !; ¡} and L ⊆ {1; 2; 3; 5; 6; 7} (any connector used in the laws of L must be present in C). Denition 9. Given a theory of connectors T = (C; L), a T-category of connectors is a symmetric monoidal category with all transformations in C which must respect the coherence axioms plus the axioms in L. We call T-Cat the category whose objects are T-categories and whose arrows are symmetric functors that strictly preserves the additional T-structure. For example, GSCat is ({∇; !}; ∅)-Cat and RMCat can be equivalently de,ned as ({∇; !; T; ¡}; {1; 2; 5; 6; 7})-Cat.

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

x

s

y

t

idy

1t

idy

y

t

1b

b

β

α

a y b z

idz

z

idz

149

z idz z idz z

Fig. 12. The diagonal composition  / 3.

Denition 10. Let T = (C; L) be a theory of connectors. A T-tile system is a tile system R whose categories of con,gurations and observations are the free T-categories generated by two signatures H and V (over the same set of sorts S). According to the above de,nition, a process tile system is a (∅; ∅)-tile system. Denition 11. Let T = (C; L) be a theory of connectors and R be a T-tile system. We say that R entails the class PT (R) of T-tile sequents, obtained by a ,nite number of applications of the inference rules for process tile logic, plus any of the following two rules for any connector 5 ∈ C: -auxiliary proof sequents:

x ∈ S∗

x ∈ S∗

5x : 5x → id ∈ PT (R)

5x⊥ : id → 5x ∈ PT (R)

5x id

id

5x

a

For any sequent  ∈ PT (R) we write R T . We also write R f s → t to say that the b

a

0at sequent s → t is entailed by R. b

From the “0at” viewpoint, the 5-auxiliary proof sequents are enough for generating all the tiles we are interested in. In fact we have the following lemmata: Lemma 12. Let T = (C; L) be a theory of connectors and let s : x → y be any parallel and sequential compositions of identities, symmetries and connectors in C. Then, s R f s −→ idy for any T-tile system R. idy

Proof. The proof is analogous to the one used in [21] for dynamic closure of arbitrary contexts. The idea is to exploit the diagonal composition / which is de,ned for any tiles  and 3 such that (1) their horizontal and vertical targets are identities, and (2) the ,nal output interface of  is the initial input interface of 3. The composition  / 3 is obtained by completing the tiling with two identities, as illustrated in Fig. 12. In fact, we can proceed inductively on the structure of the arrow s. The base cases are identities, symmetries and constructors, for which the corresponding tiles trivially s1 ⊗s2 exist. If s = s1 ⊗ s2 , then the tile s1 ⊗ s2 −→ id is the parallel composition of the tiles id

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R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

x idx x

idy

s

y

1s

idy

α

s

y

b

1s;b

idx x a z

b w idw

a

z

t

β

idz

1t

z

idz

y

w idw w

t

Fig. 13. Graphical proof of Lemma 14. s ;s

1 2 inductively associated with s1 and s2 . Finally, if s = s1 ; s2 then the tile s1 ; s2 −→ id is

the diagonal composition of the tiles inductively associated with s1 and s2 .

id

Lemma 13. Let T = (C; L) be a theory of connectors and let s : x → y be any parallel and sequential compositions of identities, symmetries and connectors in C. Then, idx s for any T-tile system R. R f id x −→ s

Proof. The proof exploits a second diagonal composition ., which is de,ned between any two tiles  and 3 such that (1) their horizontal and vertical sources are identities, and (2) the ,nal output interface of  is the initial input interface of 3. For the rest, the proof is analogous to that of Lemma 12 and thus omitted. a

Lemma 14. Let T = (C; L) be a theory of connectors and let s → t be any sequent, b

where s, t, a, and b can be expressed as the parallel and sequential compositions of identities, symmetries and connectors in C, such that s; b = a; t holds in the free a T-category over the empty signature. Then, R f s → t for any T-tile system R. b

Proof. By construction s; b can be seen as a horizontal arrow, and as such, it generates id

x s; b. Since s; b = a; t in the initial algebra, it follows that the an identity tile s; b −→

idw

id

x a; t. Then we can exploit Lemmata 12 and above tile can also be written as s; b −→

idw

idy

a

b

id z

13 to prove the existence of two tiles  : idy −→ b and 3 : a −→ id z , which can be composed as in Fig. 13, proving the thesis.

For example, if we select the theory of connectors ({∇; T}; {1; 2; 3}), i.e., the theory id ⊗∇

x of match-share categories, then R f Tx −− −→x Tx ⊗ id x , because Tx ; ∇x = (id x ⊗ ∇x );

∇x

∇

x (Tx ⊗ id x ) by axiom (3). Likewise, R f id x −→ ∇x , because ∇x ; Tx = id x ; id x = id x by

idx

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

151

axiom (1). Note that the e6ect of the ,rst tile is equal to the trigger of the second tile. id ⊗∇

x −→x (Tx ⊗ id x ); Tx , for which it is in fact By horizontal composition, we get R f Tx −−

idx

easy to verify that Tx = (id x ⊗ ∇x ); (Tx ⊗ id x ); Tx (by coherence of T, we know that (Tx ⊗ id x ); Tx = (id x ⊗ Tx ); Tx , but then (id x ⊗ ∇x ); (id x ⊗ Tx ) = id x ⊗ (∇x ; Tx ) = id x ⊗ id x ). Regarding the decorated sequents, we should ,nd an axiomatization of the basic auxiliary sequents which is able to identify exactly all dynamic connectors with identical border. We call this property “coherence of dynamic connectors”. Denition 15. Let T = (C; L) be a theory of connectors and R be a T-tile system. Given an equational theory E on the class PT (R) of T-tile sequents entailed by R, we say that the dynamic connectors of R are coherent under E if E   = 3 for any two tiles ; 3 ∈ PT (R) with the same border and built without using the basic tiles of R. Thanks to the diagonal compositions illustrated in the proofs of Lemmata 12 and 13, and to the results in [16,18,11], such an axiomatization can be de,ned by stating: • the axioms of process tile logic (including the axiomatization of horizontal, vertical and parallel compositions and the laws for coherence of symmetries); • the coherence axioms with respect to the two diagonal compositions (stating the coherence of auxiliary transformations); • the axioms in L with respect to the two diagonal compositions (which, together with coherence axioms, state the uniqueness and appropriateness of all combinations of dynamic connectors); • the following two-dimensional coherence axioms for any connector 5 ∈ C: 5x⊥ · 5x = 15x ;

(8)

5x⊥ ∗ 5x = 15x :

(9)

For example, suppose the match-share theory ({∇; T}; {1; 2; 3}) is selected. Then, the coherence axiom ∇x ; %x; x = ∇x generates the axioms:   ∇ x / %x;x = ∇x ; ⊥ ⊥ ∇⊥ x . %x;x = ∇x ; id

⊥ %x; y where % ∗ (2y; x · 1%y; x ) : id −→ %x; y . Note that the laws x; y = 2x; y and %x; y = 1 %x; y

1∇x ; 1%x;x = 1∇x ; 1∇x ; 1%x;x = 1∇x hold because of the functoriality of horizontal and vertical identities.

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R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

Similarly, axiom (1) for connectors yields the axioms  id x = 1id x ; ∇ x / Tx = 1 ⊥ id x ∇⊥ = 1id x : x . Tx = 1

Finally, the two-dimensional coherence laws for the selected connectors are:  ∇⊥ x · ∇x = 1∇x ;  ∇x ∇⊥ x ∗ ∇x = 1 ;  T⊥ x · Tx = 1Tx ;  Tx T⊥ x ∗ Tx = 1 :

Any T-tile logic extends process tile logic with additional connectors (and their coherence axioms). However, any such extension is conservative, in the sense that: (1) any valid theorem of process tile logic is provable in (any) T-tile logic without using additional axioms; and (2) the additional axioms of (any) T-tile logic cannot prove more theorems than process tile logic with respect to the formulas of process tile logic (i.e., two tiles carrying di6erent proofs in process tile logic cannot be identi,ed by T-tile logic). Proposition 16. T-tile logic is a conservative extension of process tile logic for any theory of connectors T. The proof requires the application of simple PMEqtl techniques for specifying the theories of T-tiles and process tiles, then showing that there is an obvious signature morphism from process tiles to T-tiles which induces a forgetful functor U from the category of T-tile models to the category of process tile models. The functor U has a left adjoint, and the unit of the adjunction is conservative, i.e., injective. Proposition 17. Let T be a theory of connectors. With the axiomatization sketched above, dynamic connectors are coherent for any T-tile system R. These results allow us to lift the taxonomy of connectors in Table 1 to the level of dynamic connectors, i.e., to the two-dimensional case. 3. Connectors for concurrency It is nowadays commonly accepted that concurrent processes should be distinguished from nondeterministic sequential processes, especially when dealing with distributed implementations. The most common example is given by the CCS processes P = a:nil | b:nil and Q = a:b:nil + b:a:nil: process P models a concurrent system with two independent sequential components, while Q is a sequential system with a centralized choice point.

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176 a

t1 −→ t1
∈ T

a

t1  t2 −→ t1
 t2 ∈ T

153

a

X −→ t ∈ 

a

X −→ t ∈ T Fig. 14. Inference rules for T .

Of course, P and Q are bisimilar, and therefore equivalent for what concerns ordinary observations, but they are di6erent when viewed as concurrent systems. Thus, a more precise description of their behaviors should state this di6erence in a formal way. Moreover, according to [58,71], we can distinguish between causality-based (see e.g. [8,33]) and locality-based approaches [9,10,36]: In a causality-based semantics, taking inspiration from Petri nets, the interest concerns the gradual development of each concurrent subcomputation (e.g., which events cause certain events), which is relevant in 0ow analysis; in a locality based approach, the 0avor is more on the (logic) spatial distribution of the system, which is a relevant issue in the implementation. In our metamodel, like in [58], causal and spatial information are dealt with uniformly. Thus, rather than saying either that “two actions happen at the same location” or that “the action a is caused by action b”. We can express that “actions a and c happen at the location l, which has been spawned by the execution of b”. Moreover, we take into account the fact that synchronized communication is represented as an internal action, not observable from the outside environment, and therefore that spawned events or locations must refer to their closer observable antecedents. As announced in the Introduction, we put our metamodel to the test on a simple extension of BPP(1,P) processes, which equips them with (weakly observational) synchronization, and focuses on causality and locality aspects. Denition 18. Let Act = {a; b; : : :} and Const = {X; Y; : : :} be countably in,nite sets of actions and process constants, respectively, where Act is structured with CCS-like V with (V·) an injective function s.t. AV = co-actions, i.e., Act can be partitioned as A ∪ A, V {aV | a ∈ A} and aV = a. Then BPP(1,P) processes are generated by the grammar: t::= |X | t  t where  is a distinguished (inactive) constant, X ∈ Const, and the operator  is the associative and commutative parallel compositions of processes, with unit . Note that associativity, commutativity and unit axioms of parallel composition introduce a structural equivalence on processes. Denition 19. The dynamics of BPP(1,P) systems consists of a ,nite set  of rules a in the form X → t for X ∈ Const, a ∈ Act and t any process. All process constants for which no operational rule can be found in  are considered equal to the empty process . The set  induces an LTS T , which is the least one generated by the inference rules in Fig. 14.

154

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176 a

X −→ t ∈  a

X −→ t ∈ Tws

a

aV

t1 −→ t1
∈ Tws ; t2 −→ t2
∈ Tws 9

t1  t2 −→ t1
 t2
∈ Tws

:

t1 −→ t1
∈ Tws :

t1  t2 −→ t1
 t2 ∈ Tws

Fig. 15. Inference rules for Tws .

Note that parallel composition is commutative and thus the symmetric inference rule for parallel composition can be safely omitted. Denition 20. The language WS-BPP(1,P) consists of BPP(1,P) processes with a dynamics , but whose LTS Tws is generated by the rules in Fig. 15, with 9 a distinguished silent action and : ranging over Act ∪ {9}. a

9

a

9

9

Denition 21. We write t ⇒ t
if t →∗ t1 → t1
→∗ t
for some t1 and t1
, where →∗ denotes any (possibly empty) sequence of 9 moves. Given w = a1 a2 : : : an ∈ Act ∗ we an
a2 a1 w t1 ⇒ t2 · · · tn−1 ⇒ t for suitable t1 ; t2 ; : : : ; tn−1 . write t ⇒ t
if t ⇒ Then, ordinary observational equivalences can be straightforwardly de,ned for WSprocesses (e.g., trace equivalence or strong and weak bisimilarities). To handle causal and spatial information, we follow a tile-based approach, which gives emphasis to the analogy between the underlying algebraic usage of event names and locations. The building blocks of our approach are the following: (i) Each constant corresponds to a distinct basic con,guration <X = with empty input interface and unary output interface (when no confusion can arise, we shall use the symbol X to denote the con,guration <X =). (ii) Parallel composition is modeled via connectors that go from the involved constants into the same abstract location (viewed as a name). (iii) Each action a corresponds to a distinct basic observation
BPP(1,P)

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

155

. .

. X

. a Syncha

a

[[X → t ]]

a

]]

t ]]

.

]]

ε]]

a-

. .

.

.

. .

. .

Fig. 16. Wire and box tiles for BPP.

place-holder for the otherwise pending reference to the location where X lived. For example, after the application of the rule to <X  Y =, the component Y will be attached to the location associated with <=, which is logically di6erent from that of
R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

.

Y

.

.

.

.

.

.

.

Y

.

. . .

[[

Y

.

[[ε

Z

.

Z

a

.

.

a

.

.

.

Y

X

.

X

.

156

b

.

.

[[ε

.

.

[[

a

a

.

.

.

W

.

Z

.

b

Fig. 17. <X Y −→ Z Y = on the left and <X Y −→ Z Y −→ Z W = on the right. a

b

with rules X → Z and Y → W . Then, X  Y is represented by

If X moves, then the e6ect a can be propagated via the connector exiting from X (which is matched with Y ), while Y stays idle. The step is shown in Fig. 17 (left), a where the place-holder <= produced by <X → Z = can be safely omitted because Y is attached to the same location. Then, if Y moves, the action b happens at a di6erent location than that of Z, see Fig. 17 (right). The e6ect of the two consecutive moves, after some graphical rearrangement of connectors, is in Fig. 18 (left), which is clearly di6erent from the situation where the action b is causally dependent on a (Fig. 18, right). The latter causal dependency can arise e.g. in the system with basic steps a b b a X
→ Z
, Z
→ , X
→ Y
, and Y
→ , starting from the con,guration X
. (Note that,
under the rules above, X is to some extent the BPP representation of a:b:nil + b:a:nil). Hence, <X  Y = cannot be bisimilar to <X
=. Moreover, in <X  Y = the order in which the two moves are performed is not important, thus a commuting diamond is obtained (e.g., X  Y and X
are not even trace equivalent in our setting). It is worth noting that, by the axioms of total relations, a further dependency arise in the sequential execution of a and b, which is drawn as a dotted line in Fig. 18 (it is implicit, in the sense that

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

.

.

.

.

.

a

b

a

.

.

.

.

157

.

b

.

.

.

.

Fig. 18. Parallel and sequential execution of actions a and b. s

n −→ m a↑ ↑b k −→ m t

Fig. 19. A tile with “reversed” observations.

the two observations, with and without the additional wire, are identi,ed by the laws of connectors). Formally, the ,rst step in our approach consists of ,xing the categories of con,gurations and observations. This amounts to de,ning: (1) the set of sorts; (2) the horizontal signature; (3) the vertical signature; (4) the horizontal and vertical theories of connectors. Denition 22. There is only one sort (for processes). Therefore, operator arities are just natural numbers, and the objects of the categories of con,gurations and observations will be denoted by underlined natural numbers (with n ⊗ m = n + m). The signature for WS-BPP(1,P) con,gurations contains a constant <X = : 0 → 1 for each X ∈ Const with X = . The signature for WS-BPP(1,P) observations contains a unary operator
of s; b = a; t (see Fig. 19). This is necessary for maintaining the correct interpretation of dynamic connectors. At the semantic level, this corresponds to state that con,guration connectors can be observed but not arbitrarily added. We hope the reader will not be confused by this assumption, which does not create any particular problem, and in fact, categorically, it just corresponds to standard duality consideration, taking the opposite

158

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

vertical category. (A less intuitive view is to consider the category of observations as horizontal and the category of con,gurations as the vertical one, e.g., we should rotate clockwise by 90◦ the tile in Fig. 19.) Denition 23 (From processes to con6gurations). The mapping of constants to con,gurations can be extended to processes as below: • <= = ¡1 •
(by de,nition)

= (
(by coherence of T)

= %0;0 ; (
(by naturality of %)

= (
(by coherence of %)

=
(by de,nition):

The proofs for associativity and unit are analogous. Note that any t is mapped to an arrow from 0 to 1. The converse also holds. Proposition 25. Given a con6guration f : 0 → 1, a unique such that
WS-BPP(1,P)

process t exists

Proof. Thanks to the results in [15], it is easy to see that a generic con,guration has the form (id n ⊗ M ); r, where M is the monoidal composition X1 ⊗ · · · ⊗ Xm for suitable not necessarily distinct constants (assuming an arbitrary total order on Const such that Xi 6Xi+1 for i ∈ [1; n−1]), and r : n + m → k is a total relation r ⊆ [1; n+m] × [1; k] such that if Xi = Xj then r(i) 4 r( j), where r(x) = {y | (x; y) ∈ r} and 4 is an arbitrary total order on 2[1; k] . Thus, when n = 0 and k = 1, then r = T1m , where, as a matter of notation, we let Txi+1 = (Tix ⊗ id x ); Tx for i¿0, with T1x = id x and ∇xi+1 = ∇x ; (∇xi ⊗ id x ) for i¿0, with ∇x1 = id x be convenient abbreviations for expressing consecutive matchings and duplications of the object x. By taking t = X1  · · ·  Xm the hypothesis trivially follows. Uniqueness follows by the fact that each process t is uniquely determined by the multi-set of constants it contains. To ,nd a term-like notation for con,gurations which is closer to processes, we can exploit the above-mentioned normal form representation.

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

Denition 26. Let N be a countably in,nite set of location names. A located state is a term generated by the grammar

159

WS-BPP(1,P)

t::= | X [@] | t  t where @ ⊆ N is a nonempty, ,nite set of names and the parallel composition operator is AC1. We denote by n(t) the set of names used in t. Given a ,nite list A of distinct names, a located state t is acceptable in A, written A  t, if n(t) ⊆ |A|, where |A| is the set of names in A. If t is acceptable in A and the cardinality of |A| is n, we say that t is n-acceptable. Acceptable located states are taken up to renaming of names in the environment A, letting B(A)  B(t) = A  t for any injective name substitution B : |A| → N. We shall abbreviate the notation X [{x1 ; : : : ; xn }] by writing X [x1 ; : : : ; xn ]. Proposition 27. The set of n-acceptable the con6gurations from 0 to n.

WS-BPP(1,P)

located states is isomorphic to

Proof. Informally, the environment A of an acceptable located state A  t represents the interface of the con,guration (remind that names are not important). Then, the name set @ associated with each constant X expresses the wiring between the location of <X = and the names in the interface. The precise correspondence can be formally stated as
mfollows. Given an arbitrary con,guration from 0 to n, we consider its normal form i=1 Xi ; r, with r : m → n representing a total relation r ⊆ [1; m] ×
[1; n]. We ,x the environment A = [x1 ; : : : ; xn ] and m let the located state associated with i=1 Xi ; r be the acceptable term A  X1 [@1 ]  · · ·  Xm [@m ], where @i = {j | (i; j) ∈ r} for any i ∈ [1; m]. For the converse correspondence, given a generic acceptable located state A  X1 [@1 ]  · · ·  Xm [@m ], let n be the cardinality of |A|, and let ki ¿0 be the cardinality of @i for i ∈ [1; m]. We denote by Aj the name in the jth position of A. As a matter of notation, let
i; j be equal to id1 when Aj ∈ @i , and equal to ¡1 otherwise. Then the con
m ,guration with A  X1 [@1 ]  · · ·  Xm [@m
] is f = i=1 (Xi ; ∇1ki ); C; Tm n , where
m associated
n m C = i=1 j=1
i; j . In fact, f can be written as i=1 Xi ; r, where r ⊆ [1; m] × [1; n] is the total relation r = {(i; j) | Aj ∈ @i }. It is easy to verify that the two mappings de,ne a bijective correspondence. We let < = denote the component of the above bijection that maps acceptable located states to con,gurations, and let < =−1 denote its inverse. Denition 28. The underlying process of a located state X1 [@1 ]  · · ·  Xm [@m ] is X1  · · ·  Xm . For an acceptable located state A  t, we denote its underlying process by u(t). Proposition 29. Let A  t be an acceptable located state; then
160

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

Proof. The proof is a straightforward application of the axioms in the theory of connectors under consideration. As a matter of notation, for A  t acceptable and @ ⊆ A, we de,ne A  t[@] by def

def

[@] = 

(t[@1 ])[@] = t[@1 ∪ @]

def

(t1  t2 )[@] = t1 [@]  t2 [@]:

By choosing the theory Tws we also ,xed one of the parameters of the metamodel proposed in Section 2. The ,nal step of the speci,cation is de,ning the basic tiles of the Tws -tile system for WS-BPP(1,P) processes. To shorten the notation, we let Ea denote the observation ∇1 ; (
a

<X → t= : X

→0

(Ea ⊗id 1 );T1

WS-BPP(1,P)

is denoted by R ws and contains


is considered (remind that (Ea ⊗ id1 ); T1 goes from the ,nal output interface to the initial output interface, because we are considering an opposite vertical category); • for any a ∈ Const, the below synchronization tile Syncha is introduced: (id 1 ⊗Ea ⊗EaV⊗id 1 );(T1 ⊗T1 )

Syncha : id 1 ⊗ id 1 −−−−−−−−−−−−−−→(id 1 ⊗ (T1 ; ∇1 ) ⊗ id 1 ); (T1 ⊗ T1 ) id 1 ⊗id 1

We let {[a]} denote the observation (Ea ⊗ id1 ); T1 associated with a. We have already shown some examples of tile computations. Here we illustrate how the mechanism for e6ect propagation works. In the following examples, we shall a aV b consider the process X  Y , with rules X → Z, Y → Z, and Z → X  Y . a

Example 31. Suppose we want to apply the tile <X → Z = to the X component in the initial con,guration (X ⊗ Y ); T1 = X ; (id1 ⊗ Y ); T1 . Thus, we must synchronize the a e6ect {[a]} of <X → Z = with the con,guration (id1 ⊗ Y ); T1 . This can be done by exploiting the dynamic connector T ⊗id

1 Prop : T1 −−1−−→ id 1 ⊗ T1

T1

in Fig. 20 (left). This connector exists by Lemma 14, because (T1 ⊗ id1 ); T1 = (id1 ⊗ T1 ); T1 is a (one-dimensional) coherence axiom for matching (we remind that since vertical arrows are directed from bottom to top, the two-dimensional coherence for dynamic connectors requires that the trigger composed with the initial con,guration be equal to the ,nal con,guration followed by the e6ect). Tile Prop can be sequentially pre-composed with 1T1 ⊗ 1Y , and the result can be vertically composed with 1Ea ⊗ 1(id 1 ⊗Y );T1 . The ,nal result  = ((1T1 ⊗ 1Y ) ∗ Prop)·

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

.

.

1

Prop .

2

. .

2 .

id1⊗ id1 ⊗Y

3

∆1 Prop id1⊗ ∆1

1λa ⊗1(id1 ⊗Y); λ1

λa ⊗ id1 .

2

1∆1 ⊗1Y ∆1 ⊗id1

∆1

.

id1 ⊗ Y

(id1⊗id1 ⊗Y); ∆1

161

1 ∆1 2 λa⊗ id1 2

Fig. 20. Dynamic connectors for e6ect propagation.

(1Ea ⊗ 1(id 1 ⊗Y );T1 ) is illustrated in Fig. 20 (right). Then, the cell  can be used to a coordinate the activity of X , because the e6ect of <X → Z = matches the trigger of , and we have a

<X → Z= ∗  : (X ⊗ Y ); T1

id

→0

(Ea ⊗id 1 );T1

Z ⊗ Y;

because we have (Z ⊗ <= ⊗ Y ); (id1 ⊗ T1 ) = Z ⊗ Y by !1 = <= and coherence of T. Thus, in the ,nal con,guration Z and Y are associated with two distinct locations: the one of Z has been spawned by the action a, while the one of Y is the original one, where also X was allocated. It follows that any subsequent action performed by Z will be causally dependent on action a, because it will be performed in the location spawned by a. The acceptable located state representing Z ⊗ Y is [z; y]  Z[z]  Y [y]. Example 32. More interestingly, in the con,guration <X  Y =, the two subprocesses a aV can concurrently evolve. In fact, the tile <X → Z = ⊗
1 id 1 ⊗ T1 ⊗ id 1 ; Comb: T1 −−1−−→

T31

which is illustrated in Fig. 21 (left). In fact, Comb can be vertically composed with 1Ea ⊗ 1T1 ⊗ 1EaV , yielding the coordinator we were looking for, which is illustrated in Fig. 21 (right). Note that a two-dimensional symmetry %1;1

Sym : id 2 −→%1;1 id 2

is needed for matching the observation {[a]} ⊗ {[a]}, V (because {[a]} V = (EaV ⊗ id1 ); T1 = (EaV ⊗ id1 ); %1;1 ; T1 = %1;1 ; (id1 ⊗ EaV ); T1 by coherence of T and % and by naturality of %). As a ,nal result, we get a tile 3 with e6ect (Ea ⊗ id1 ⊗ EaV ); T13 and ,nal con,guration (Z ⊗ ¡1 ⊗ Z ⊗ ¡1 ); (id1 ⊗ %1;1 ⊗ id1 ); (id1 ⊗ T1 ⊗ id1 ) = (Z ⊗ ¡1 ⊗ ¡1 ⊗ Z); (id1 ⊗ T1 ⊗ id1 ) = Z ⊗ ¡1 ⊗ Z, where the two components Z are attached to the locations spawned by a

162

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

Fig. 21. Dynamic connectors for concurrent propagation of e6ects.

and a, V while ¡1 is a place-holder for the original location, where X and Y were allocated (which is now empty). The located state associated with the ,nal con,guration is [z1 ; x; z2 ]  Z[z1 ]  Z[z2 ]. It is worth remarking that the useless location x can be cut from the prosecution of the computation via the dynamic connector id

1 id ; Collect : ¡1 −→ ¡1 1 vertically composing 3 with the tile 1Z ⊗ Collect ⊗ 1Z . From the viewpoint of located states, this corresponds to move from [z1 ; x; z2 ]  Z[z1 ]  Z[z2 ] to [z1 ; z2 ]  Z[z1 ]  Z[z2 ] (with e6ect id1 ⊗ ¡1 ⊗ id1 ).

As a ,nal example, we show how weak causality is handled by the rule for synchronization. The idea is that 9 actions should not be observed, but that, when synchronizing two actions which happens at two locations ‘1 and ‘2 spawning fresh locations ‘1
and ‘2
, then each process attached to ‘1
or ‘2
should also keep causal links to both ‘1 and ‘2 . This mechanism extends the spatial information from a single location to a set of locations, which are inherited from the (unobservable) synchronized actions. Example 33. The concurrent actions a and a, V executed, respectively, by X and Y , can synchronize via the tile Syncha . To see this, we can vertically compose Syncha with (Sym ⊗ 1id 2 ) ∗ 1r , where r = (id1 ⊗ (T1 ; ∇1 ) ⊗ id1 ); (T1 ⊗ T1 ) is the ,nal con,guration of Syncha , and then pre-compose the resulting tile with the concurrent step a

aV

id

<X → Z = ⊗
con,guration has been simpli,ed by applying twice the coherence axiom (id1 ⊗ ¡1 ); T1 = id1 for eliminating the two place-holders produced by the basic rules. Note that the relation T1 ; ∇1 in the ,nal con,guration expresses the fact that each component Z is attached to both locations in the ,nal output interface, i.e., the ,nal con,guration is the located state [z1 ; z2 ]  Z[z1 ; z2 ]  Z[z1 ; z2 ].

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

163

Since the global e6ect is id 2 , then no observation must be propagated through the T1 of the initial state, i.e., we can just horizontally compose G with 1T1 , obtaining id

0 (Z ⊗ Z); T1 ; ∇1 ; T1 . But ∇1 ; T1 = id1 by axiom (1) in the the tile H : (X ⊗ Y ); T1 −→

id1

theory of connectors; hence the ,nal con,guration is (Z ⊗ Z); T1 =
Theorem 34. If t ⇒ t
, then a con6guration f : 0 → 2 exists such that
0 f. R ws f
{[a]}

The proof easily follows from the following lemmata: a

Lemma 35. If t → t
, then a con6guration f : 0 → 2 exists, such that
0 f. R ws f
{[a]} a

Proof. If t → t
, then, by a simple inspection of the inference rules for T ws , it must a be the case that t = X  s for some constant X and process s such that X → s
∈  and t
= s
 s (let s =  if t = X ). Then, by a construction analogous to that employed in Example 31, where this time s plays the role of Y , we can prove that id

0 <s
= ⊗ <s= and conclude by taking f = <s
= ⊗ <s=, as
{[a]}


esis, and <s  s= = (<s
= ⊗ <s=); T1 by de,nition. a

Lemma 36. If u(A  t) → t
, then two natural numbers m and n, two total relations id

0 −−−− −−→ f and r and r
and a con6guration f : 0 → m exist, with R ws f
r ;(idn ⊗{[a]});r


The proof is analogous to that of Lemma 35, except that we must ,rst exploit the normal form representation of
id

0 Lemma 37. If t → t
, then R ws f
id1

9

Proof. If t → t
, then, by a simple inspection of the inference rules for T ws , it must be the case that t = X  Y  s for suitable constants X; Y and process s such that a aV X → s1 ∈ , Y → s2 ∈  and t
= s1  s2  s (let s =  if t = X  Y ). Then, by a construction analogous to that employed in Example 33, where this time there is an adid

0 <s1 = ⊗ <s2 = ditional s in parallel with X  Y , we can ,rst prove that R ws f X ⊗ Y −→

id 2

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R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

and then conclude by composing with the vertical identity of (T1 ⊗ <s=); T1 , since
Lemma 38. If u(A  t) → t
, then a con6guration f : 0 → m exists, with m the cardiid

0 f and
id m

The proof is analogous to that of Lemma 37, except that we must propagate the e6ect id 2 of the synchronization through a generic relation, but this can be done exploiting dynamic connectors. The result then follows by Proposition 29. Proof of Theorem 34 (sketched). By straightforward induction on n, we prove that Lemmata 37 and 38 can be extended to any ,nite sequence of 9 moves. We arbitrarily a ,x a ,nite sequence of transitions which corresponds to t ⇒ t
(there can be more than one), and then we apply: (1) the above result on simulating any sequence of 9 moves (on the 9 transitions pre,xing the execution of a); (2) Lemma 35 to the transition with label a; and (3) again the above result on simulating any sequences of 9 moves (on the 9 transitions post,xing the execution of a). Similarly, we can prove the following result: a

Theorem 39. If u(A  t) ⇒ t
, then two relations r and r
, two natural numbers n id

0 −−−− −−→ f and
r ;(idn ⊗{[a]});r

f; T1m .

For the converse correspondence, note that the concurrent moves of tiles cannot be simulated in one step by the interleaving transition system Tws . Therefore, tile bisimilarity on R ws is ,ner than ordinary (interleaving) bisimilarity on Tws . We ,rst state the complete concurrency of our tile system, and then show that any linearization of a concurrent tile-computation can be simulated by Tws . id

0 Proposition 40 (Complete concurrency). Let R ws f f −→ g with : = :
; (id n ⊗ {[a]});

:

r for some relation r, natural number n and generic observation :
. Then, a con6gid

id

(idn ⊗{[a]});r

:

0 0 −→ g
and R ws f g
−→ g. uration g
exists such that R ws f f −−−−− 

The functoriality of tensor product allows for selecting any (top-level) action {[a]} out of a generic concurrent execution : (concurrent actions are pushed in :
). The id

0 g, allowing to cut the partial process can then be iterated to the tile R ws f g
−→ 

:

order of observations in the e6ect at any consistent point, hence the name complete concurrency for the above proposition. id

Theorem 41. If R ws f
and con6guration g : 0 → m, then u(t) ⇒ t with
R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

.

(β ,{1})

. .

(β,∅) (α,{1})

. .

(α ,∅) (β,∅)

.

. .

(α, ∅)

. .

165

(β,∅) (α,∅)

Fig. 22. Causal trees for :3:nil + 3::nil (left) and :nil | 3:nil (right).

3.1. Tile bisimilarity and causal weak bisimilarity Next we want to prove an interesting correspondence result with (the analogous for processes of) the causal weak bisimilarity of [33], which can be summarized by saying that:

WS-BPP(1,P)

The tile bisimilarity on R ws coincide with the causal weak bisimilarity of Darondeau and Degano de,ned via causal trees [33]. We remind that causal trees are a special kind of synchronization trees, whose labels are pairs
action, set of positive numbers such that the second element is a set of relative backward pointers to past events (each positive number in the set counts the number of arcs that separate the ancestor event from the one under consideration). Thus the CCS terms :3:nil + 3::nil and :nil | 3:nil yield the trees in Fig. 22. To de,ne the causal tree semantics of WS-BPP(1,P), we ,rst need to extend the syntax of the calculus for pre,xing processes by the sets of their causes, obtaining the socalled causal processes. Beside parallel composition and inactive process  (on which this time we relax AC1 axioms), we introduce a new operator ⇒ to this aim. Hence, a causal process of the form K ⇒ X tells that the actions of X have the set of activating causes K. We let ⇒ distribute over all the operators, and let K1 ⇒ K2 ⇒ t = K1 ∪ K2 ⇒ t. This way, causal processes can always be represented in a form where only constants are pre,xed by causes. As a special case an ordinary process t can be viewed as the causal process ∅ ⇒ t. We denote by K(t) the set of all causes appearing in t. The transition rules for the upgraded calculus must also dynamically update the causal information contained in the states, and it is convenient to introduce two special operations for this task: G(K) increases by one every element of a nonempty set K; while $(H; K) joins the sets H and K only if 1 ∈ K and otherwise returns K. These def operations are trivially extended to causal processes by letting G(K ⇒ X ) = G(K) ⇒ X def and $(H; K ⇒ X ) = $(H; K) ⇒ X and assuming distributivity over parallel composition. The strong causal operational semantics Tsc can then be de,ned as in Fig. 23, by adapting the rules for CCS. We recall that  is the set of basic moves for constants. When a generic constant X with activating causes K makes a move a, then the set of causes is observed in the transition label and the causes of the spawned process t are given by {1} (because it depends on the last action executed, i.e., a) and G(K) (because the old relative pointers must now be increased by one to skip the last executed action). When a process t1 performs an asynchronous move, then the causes of any idle process

166

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176 :; K

a

t1 −→ t1
∈ Tsc

X −→ t ∈  a; K

:; K

K ⇒ X −→ {1} ∪ G(K) ⇒ t ∈ Tsc a; K

t1  t2 −→ t1
 G(t2 ) ∈ Tsc :; K

a; VK

t1 −→1 t1
∈ Tsc ; t2 −→2 t2
∈ Tsc

t2 −→ t2
∈ Tsc

9; K ∪K

:; K

1 2 t1  t2 −→ $(G(K2 ); t1
)  $(G(K1 ); t2
) ∈ Tsc

t1  t2 −→ G(t1 )  t2
∈ Tsc

Fig. 23. Inference rules for Tsc .

a; K

t −→ t
∈ Tsc

9; K

a; K

t −→ † (t
) ∈ Twc

t −→ t
∈ Tsc

9

t −→ t
∈ Twc Fig. 24. Inference rules for Twc .

t2 in parallel with t1 must be increased by one to skip the action executed by t1 . In the case of synchronization, the causes observed with the invisible action are the union of the causes of the two actions involved in the communication. Moreover, we must add the causes G(K2 ) to the constants in t1
spawned by the move
a; K1 , which are precisely characterized by having 1 as cause. Similarly, G(K1 ) must be added to the processes spawned by
a; V K2 . The two updates are delegated to the operation $. Note that no inference rule is necessary for generic K ⇒ t with t not a constant, because causes can always be distributed to constants. The weak causal operational semantics Twc is de,ned on the basis of Tsc by the obvious rules in Fig. 24, where † (read :exa) is a special operator distributing over pardef allel composition, such that †(K ⇒ X ) = †(K) ⇒ X , with †(K) returning ∅ if K ⊆ {1}, and {k − 1 | k ∈ K; k¿1} otherwise. Flexa essentially removes references to 9 events from all sets of activating causes. Note that the operational semantics de,ned via causal trees is interleaving: we cannot change the order in which two transitions are executed without changing the set of activating causes in the reached states. 9 We write t ⇒ t
if there exists a sequence of n¿0 9-transitions in Twc , leading from a; K

9

a; K

9

t to t
. We write t =⇒ t
if t1 and t2 exist with t ⇒ t1 −→ t2 ⇒ t
.

Denition 42. A symmetric relation ∼w on causal processes is a causal weak bisimulation if for all t1 ; t2 such that t1 ∼w t2 , then whenever & ∈ {9} ∪ {
a; K | a ∈ Act; K ⊆ N} & & and t1
exists with t1 → t1
then t2
exists such that t2 ⇒ t2
and t1
∼w t2
. We denote by w the maximal causal weak bisimulation and call it causal weak bisimilarity. We show that tile bisimilarity and causal weak bisimilarity coincide on WS-BPP(1,P) processes.

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

167

To this aim we ,rst consider a correspondence between con,gurations and causal processes and then extend the correspondence to operational steps of the two approaches. Proposition 43. The structural congruence on causal processes given by associativity, commutativity and unit of parallel composition is a causal weak bisimulation. Proof. It is immediate to show that the AC1 axioms de,ne a strong bisimulation with respect to Tsc and therefore, also a causal weak bisimulation (because Twc is de,ned on top of Tsc ). We show the proof for the commutativity axioms, the other proofs :; K

:; K

are analogous. We need to prove that if t1  t2 −→ t1
 t2
, then t2  t1 −→ t2
 t1
. If the :; K

transition t1  t2 −→ t1
 t2
has been obtained by a left-asynchronous move of t1 , then :; K

t1 −→ t1
and t2
= G(t2 ), but then we can apply the rule for right-asynchronous moves :; K

to t2  t1 to obtain t2  t1 −→ G(t2 )  t1
. Likewise, if the transition has been obtained by a right-asynchronous move of t2 . Instead, if the transition is the result of a synchroa; K a; VK nization between t1 −→1 t1

and t2 −→2 t2

, then : = 9, K = K1 ∪ K2 , t1
= $(G(K2 ); t1

) and a; VK

t2
= $(G(K1 ); t2

), thus we can apply the synchronization rule to the premises t2 −→2 t2

VV K a;

9; K

and t1 −→1 t1

to conclude that t2  t1 −→ t2
 t1
. We say that a causal process t is closed if K(t) is a possibly empty interval of natural numbers [1; n] for some n¿0. Proposition 43 guarantees that we can take causal processes up to AC1 axioms without colliding against causal weak bisimilarity. This allows us to fully exploit the evident syntactic analogy between located states and closed causal processes (taken up to AC1 axioms) de,ned via the bijection: def

(K1 ⇒ X1  · · ·  Kn ⇒ Xn ) = A  X1 [@1 ]  · · ·  Xn [@n ]; where |A| = {xj | j ∈ K} for K = {1} ∪ G(K1 ∪ · · · ∪ Kn ) and xi precedes xj in the list A whenever i¡j, @i = {xmax(A) } ∪ {xj | j ∈ Ki ). Since the Ki in Ki ⇒ Xi can also be empty, while @i in Xi [@i ] must be nonempty, we are forced to create a ,ctitious initial event xmax(A) , which acts e.g. as the location of all constants with empty set of activating causes. The closure property of the causal process serves for uniquely determining the environment, i.e., to give a bound to the number of names. We are only interested in closed causal process because these are the only one reachable from ordinary processes, as the following results demonstrate: a; K

Lemma 44. If t −→ t
∈ Twc , then K(t
) = {1} ∪ G(K(t)). Proof. Obvious rule induction on the inference rules for Twc . 9

Lemma 45. If t → t
∈ Twc , then K(t
) = K(t). Proof. Obvious rule induction on the inference rules for Twc .

168

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176 &

Proposition 46. If t → t
∈ Twc and t is closed, then also t
is closed. Proof. Easy consequence of Lemmata 44 and 45. Lemmata 44 and 45 are also useful in understanding the way in which events correspond to names. In fact, if we regard a cause i as a location xi , then when a new event a is performed, all the existing locations are renamed (i.e. their references are increased by one) and a fresh location is generated which becomes x1 . On the other hand, if a 9 event occurs, then locations are left unchanged, but the constants spawned by the synchronization have now the same set of causes, which is given by the union of the causes of the synchronized processes. Another consequence is that for reasoning on ordinary processes (i.e., causal processes t with K(t) = ∅) it makes no sense to discuss the equivalence between causal processes with di6erent sets of activating causes, as they will never be compared in the bisimulation game, i.e., we can consider a version of causal weak bisimilarity ranked by the set of activating causes: We write t ≈w t
for t w t
and K(t) = K(t
). Denition 47. For t a closed causal process, we let
id

Proposition 48. If t −→ t
∈ Twc , then R ws f
r : n + 1 → n is the (total) relation {(1; k) | k ∈ K ∪ {n}} ∪ {(i + 1; i) | 16i6n}. Proof. Without loss of generality, we can assume that (1) t = K ⇒ X  t1 ; a (2) X → t2 ∈ ;
(3) t = {1} ∪ G(K) ⇒ t2  G(t1 ). By (1) we have
id

0 X ⊗ g. r −→ id n+1 , i.e., r can be observed. Thus R ws f 1X ⊗g ∗ r ⊥ :
a

id

0 <X −→ t2 = : X −−−−−−−− −−−−→
(Ea ⊗id 1 );T1

exists, which can be vertically composed with the tile 1
id

1 where ¡⊥ −−− →id 1 1 : ¡1 − ¡1

id

0 is the basic dynamic connector 1 associated with ¡1 . Thus, R ws f 3 = X −→
Ea

1 We recall that we are using a dual version of tiles, where the direction of vertical arrows is reversed, hence the rotation of the border with respect to the notation in De,nition 11.

R. Bruni, U. Montanari / Theoretical Computer Science 281 (2002) 131 – 176

169

The tile 1X ⊗g ∗ r ⊥ can then be vertically composed with 3 ⊗ 1g , yielding id

0 −−−−→
(Ea ⊗id n );r

By (3) it is easy to check that
9

0
idn

By repeated application of Proposition 49 and by the properties of identities, we trivially obtain the following corollary. id

9

0
idn

By Proposition 48 and Corollary 50, we also get Corollary 51. a; K

id

Corollary 51. If t =⇒ t
∈ Twc , then R ws f
r = {(1; k) | k ∈ K ∪ {n}} ∪ {(i + 1; i) | 16i6n}.

Note that the tile observation associated with the move is determined by the rank K(t), by the executed action a and by the set of causes K, which are the criteria on which ≈w is based. We are now ready to prove the correspondence theorem. Theorem 52. Given two ordinary only if t ≈w t
.

WS-BPP(1,P)

processes t and t
, then
Proof (sketched). Thanks to Propositions 48 and 49, together with Theorem 41, we know that t distinguishes as much as ≈w . It remains to prove that dynamic connectors do not introduce any step that allows one to distinguish otherwise equivalent processes. In fact, dynamic connectors allow to observing part of the state, an operation that cannot be directly performed in Twc . We have three cases to analyze: • The ,rst case concerns the co-discharger id

0 ¡⊥ 1 : ¡1 → id 1 : ¡1

It allows for observing inactive processes, but we have the main assumption that all constants have at least an admissible move, i.e., that  is the only inactive processes and cannot be equivalent to any other process unless they are structurally equivalent. Since <= = ¡1 , the tile ¡⊥ 1 creates no problem. id

2 id 2 . It allows for observing that two processes • The second case concerns T1⊥ : T1 −→

T1

have a common cause, but since inactive processes can always be spawned by

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structural equivalence, the information is not important until some action is executed by the two processes. But when an action is executed we are indeed interested in observing the causes. Hence, also T1⊥ creates no problem. id

1 id1 . It allows for observing that a certain process • The third case concerns ∇1⊥ : ∇1 −→

∇1

has two causes, but since new causes are introduced only by the execution of certain events, this information could have been as well retrieved when observing the last action which spawned the process.   The tiles ¡ 1 , T1 and ∇1 allow only for moving to con,gurations some information which has already been observed, and therefore cannot a6ect tile bisimilarity.

We emphasize that one of the advantages with respect to [33] is that our approach for weak bisimilarity not only de,nes a concurrent (and causal) abstract semantics, but, thanks to Proposition 40, it also yields a truly concurrent operational machinery, which is analogous to that of Petri nets.

4. Related work Summarizing the results, the combined use of connectors and tile logic allows us: (1) to de,ne a causal and concurrent semantics for WS-BPP(1,P) processes; (2) to express the semantics via two straightforward operational rules, with causal dependencies automatically handled by the theory of (dynamic) connectors; (3) to have at disposal a concurrent operational machine instead of just a denotational model; (4) to recast our case study inside a more general concurrent framework, which is parametric with respect to the theory of connectors, and thus suitable for embedding and comparing di6erent systems, models and logics. Our metamodel, and its instantiation to the case study of WS-BPP(1,P) processes, gives evidence of the analogy between names and events, and provides a uniform framework for both. Moreover, di6erent kind of names can be distinguished, if necessary: it suJces to have many di6erent sorts, one for each name kind. We think that the case study in Section 3 further motivates our approach, illustrating the novelties with respect to other previous results in the literature. In fact, it allows for having, for the ,rst time to our knowledge, at the same time: (a) states with structural axioms aW la CHAM for parallel composition; (b) a concurrent operational semantics. (c) partial orders of observed events (in the style of causal trees); (d) a weak abstract semantics with automatic propagation of causal dependencies due to invisible internal actions; It is also worth noting that these features arise naturally from the choice of dynamic connectors. A comparison with other well-known LTS-based and reduction-based approaches can be sketched as follows: (i) CHAM is apt for structural axioms like associativity, commutativity and identity of parallel composition, but it is less suited for abstract observational semantics;

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(ii) Causal trees and located calculi [33,9] o6er a truly concurrent denotational semantics, but not a concurrent machine (the underlying LTS relies on actions interleaving); (iii) In comparison with other approaches focusing on time–space issues of concurrency in process algebras (e.g., [8] that reconciles three noninterleaving semantics for CCS, respectively, based on permutation equivalence, 0ow event structures and a special kind of Petri nets called 0ow nets), here we de,ne a concurrent operational model, where guarded recursion is dealt with a compact state representation and where 9 actions are invisible; (iv) the extended labels of the LTS for taking care of locations and events are automatically generated by the tile framework, in a systematic way, which is especially convenient for de,ning the weak semantics (where 9 actions disappears, but not their in0uence on the history and distribution of the system). The last point is particularly relevant, because location names are not important (only the links to them matter) and fresh locations can be generated concurrently. As a consequence, we neither need to consider side-conditions and renaming of idle components in the inference rules of the LTS (e.g., for avoiding name clashes), nor to de,ne specialized notions of bisimulation (e.g., where certain moves are not to be simulated because fresh location names are generated that already appear in the other process). Let us now discuss analogies and di6erences with other applications of tiles in the literature. In [40], ad hoc tile systems have been de,ned for located CCS [71] and asynchronous
-calculus [54], whose con,gurations and observations are based on gs-monoidal structures. According to the de,nitions given in this paper (cf. Section 2), the tile systems in [40] are just T-tile system, for T the theory of (co)gs-monoidal connectors (i.e., T and ¡), and they can be straightforwardly recast in our metamodel. The application to
calculus shows the use of two di6erent kinds of names, one for channel names, and one for the events (analogous to the one used in our case study). Sharing and name passing are then handled by many coordination rules (e.g., twelve ad hoc rules are needed for action propagation), which extensively use connectors for modeling such features. Dynamic connectors allow for a simpler and systematic de,nition of coordination rules. Moreover, the gs-monoidal structure is not suJcient to give a satisfactory account of weak causal semantics, because many-to-many dependencies cannot be expressed (∇ is missing). Our case study encompasses the above limitation by introducing a richer relationallike structure (and a standardization of coordination rules). We emphasize that the tile system in [40] is an instance of our framework, and thus the techniques for name mobility presented there can be easily transferred to and re-used in other models. For example, though we cannot illustrate the issue here in full detail for space limitations, name restriction can be modeled by introducing a suitable constructor M in the signature together with operational rules analogous to those in [40]. Finally, let us mention the work in [19], where a higher-order 0avor of tiles (whose models are cartesian closed double categories) is used to model the synchronous
calculus. In that case, name creation and name passing are conveniently handled by the

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higher-order features of tiles, but the cartesian structure imposes naturality axioms on connectors. An interesting issue would be to reconcile the theory of dynamic connectors presented here with monoidal-closedness. 5. Conclusion We have proposed a metamodel for concurrent semantics, where causal and spatial aspects can be suitably analyzed. Our approach can be summarized by the formula connectors + tiles = concurrent causal semantics: The theory of connectors makes the framework parametric with respect to the way in which the components of a distributed system can interact and communicate, i.e., with respect to the network realization. The use of tile logic allows us to extend structural connectors to the semantics in a uniform way. From this originates the terminology dynamic connectors. Connectors and tiles together, allow for “,ne tuning” our metamodel around di6erent issues of interest. The case study of weakly synchronized and distributed BPP(1,P) processes illustrates the main steps of the methodology we are proposing: (1) Fix the signature of con,gurations and the connectors needed for the particular framework we model; (2) Fix the signature of observation, but re-use the same connectors introduced for con,gurations, or at most a closed subset of them; (steps (1) and (2) can be iterated possibly many times for ,nding a common structure of connectors); (3) De,ne the basic tiles of the system but, if necessary, go back to step (1) to change the horizontal and vertical structures to suit the operational rules; (4) Dynamic (two-dimensional) connectors are automatically added depending on the choices at steps (1) and (2). As a result, the system comes automatically equipped with a concurrent abstract semantics, namely tile bisimilarity. Congruence proofs for such equivalence can be either guaranteed by suitable formats for basic tiles, or obtained by slightly re,ning the proof schemes in [45,12,59] (if necessary, we can always resort to the dynamic tile bisimilarity of [21], characterizing the coarsest congruence which is also a tile bisimulation). In particular, the tile decomposition property of [45] implies that tile bisimilarity is a congruence but is often easier to prove, it is format-independent and has been used with success in many cases. Analogous to the auxiliary tiles of [21], dynamic connectors can be conveniently used in graphical proofs (tile pastings) of tile decomposition. Acknowledgements We thank the anonymous referees for the very detailed comments and suggestions, which have helped us in preparing the ,nal version.

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Theoretical Computer Science 281 (2002) 177 – 206

www.elsevier.com/locate/tcs

The evaluation of rst-order substitution is monadic second-order compatible Bruno Courcellea; ∗ , Teodor Knapikb a LaBRI

(CNRS laboratory 5800), Universite Bordeaux 1, 351 Cours de la Liberation, 33405 Talence Cedex, France b ERMIT, Universit e de la Reunion, BP 7151, 97715 Saint Denis Messageries, Cedex 9, La Reunion, France

Abstract We denote rst-order substitutions of nite and in nite terms by function symbols indexed by the sequences of rst-order variables to which substitutions are made. We consider the evaluation mapping from in nite terms to in nite terms that evaluates these substitution operations. This mapping may perform in nitely many nested substitutions, so that a term which has the structure of an in nite string can be transformed into one isomorphic to an in nite binary tree. We prove that this mapping is monadic second-order compatible which means that a monadic second-order formula expressing a property of the output term produced by the evaluation mapping can be translated into a monadic second-order formula expressing this property over the input term. This implies that, deciding the monadic second-order theory of the output term reduces to deciding that of the input term. As an application, we obtain another proof that the monadic second-order properties of the algebraic trees, which represent the behaviours of recursive applicative program schemes, are decidable. This proof extends to hyperalgebraic trees. These in nite trees correspond to certain recursive program schemes with functional parameters of arbitrary high c 2002 Elsevier Science B.V. All rights reserved. type.
Keywords: Monadic second-order logic; First-order substitution; Algebraic tree; Hyperalgebraic tree; Decidability of logical theories

∗

Corresponding author. E-mail addresses: [email protected] (B. Courcelle), [email protected] (T. Knapik). URLs: http://www.labri.fr/∼courcell, http://www.univ-reunion.fr/∼knapik

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 2 - 9

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1. Introduction For investigating the recursive applicative program schemes considered in the 1970s by Nivat and others [11,12,17,18] (see [5] for a survey), it is useful to associate with every such a scheme an in nite term that represents its computations in all semantical domains. Useful informations about a scheme can be obtained from the corresponding tree and used, for instance, for rewriting the scheme in a simpler or in a standard way [17]. It is thus useful to know which properties of these terms (called algebraic trees in the above quoted works) are decidable. A fundamental theorem by Rabin [20] (see [21]) states that the properties expressible in monadic second-order logic (MS logic in short) are decidable over regular trees, a proper subclass of that of algebraic trees. Monadic second-order logic is quite powerful (it subsumes temporal logics) but regular trees are insuCcient to represent computations of recursive applicative program schemes. However, the monadic second-order properties of algebraic trees are decidable [8]. We give a new proof of this result which, furthermore, extends to the hyperalgebraic trees that correspond to certain higher-level program schemes, with parameters of function and functional types, already considered by Damm [15]. A rst result about the decidability of the MS theory of hyperalgebraic trees is given in [19] where, so called safe applicative schemes of level 2 are considered. In these schemes procedure names carry functional parameters in addition to individual ones but, according to the safety restriction, a parameter of the basic type should not occur within a functional argument of a procedure call. In an essential way, the proof of [19] uses paths in -graphs. The latter are borrowed form the geometry of interactions [1] and the theory of optimal reductions [2]. The present paper is independent of these -calculus related theories. We develop several tools for the MS logic of higher-level trees. Our rst tool is a signature of graph operations studied by Courcelle, Olariu, Engelfriet and others [10,16,6,9,13]. We use operations which allow to add edges to a vertex labelled graph, from every vertex labelled by, say, a to every vertex labelled by, say, b. Other operations rename labels and a binary operation builds the disjoint union of two labelled graphs. Using these operations, graphs are generated from constants, each of which represents a single labelled vertex. In such a way, a graph may be represented by a nite or in nite term over the corresponding signature of graph operations. The set of such terms is organized into a continuous algebra and the unique continuous morphism from this algebra into the algebra of graphs is called the value morphism. The graph represented by the term is the image of the term under the value morphism. Our rst result shows that this value morphism is a monadic second-order transduction (an MS transduction in short). MS transductions are transformations of logical structures speci ed in MS logic; see [7,9]. This implies that the value morphism preserves the decidability of the MS theory of the structures to which it is applied. Hence the MS theory of a graph de ned by an in nite regular term is decidable. Our second main tool is an explicit operation of rst-order substitution of terms. This operation speci es the variables to which substitutions are made. The arguments are thus: the term on which the substitution acts and the terms which are substituted. We show that the evaluation of substitutions is an MS-compatible operation.

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A transformation of structures f is MS-compatible if for every MS formula ’, one can eGectively construct an MS formula such that for every structure S; S satises iG f(S) satis es ’. Every MS-transduction is MS compatible, but some MScompatible mappings, like the unfolding of a graph into a tree (by a result of [14], see Theorem 2.11), are not MS transductions. Our proof uses in a crucial way the MS-compatibility of unfolding. We also consider second-order substitutions in the context of recursive applicative program schemes. By reducing second-order substitutions to rst-order ones, we reduce the functionality of the types of the variables in recursive de nitions. In such a way, we establish that every algebraic tree may be obtained from a regular tree with rst-order substitution operators by evaluating these substitution operators (a result known from [15]). This gives another proof of the decidability of the MS theory of every algebraic tree. Since the evaluation of rst-order substitutions is MS-compatible, by evaluating them on algebraic trees, we get more complex trees with decidable MS theories. This process may be iterated. We obtain that each tree in the strict hierarchy de ned by Damm in [15] has a decidable MS theory. It is not clear yet whether these trees represent all lambda-schemes, i.e., all systems of typed mutually recursive de nitions using lambdaabstraction in the right-hand sides of equations.

2. Graphs, terms, monadic second-order logic: basic denitions In Sections 2.1–2.5, we review from previous articles de nitions and notation concerning relational structures and MS logic, nite and in nite terms, graphs and graph operations. In Section 2.5 we prove that the value mapping associating a graph to an innite term is an MS-transduction, hence is MS-compatible. In Section 2.6 we introduce notation for new graph operations derived from those of Section 2.5. In Section 2.7 we review graph unfolding. 2.1. Structures and monadic second-order logic We let R be a nite set of relation symbols, each of them, say r, being given with an arity (r) in N+ . We denote by S(R) the set of nite or countable R-structures, i.e., of tuples of the form S =
DS ; (rS )r∈R  where rS ⊆DS(r) for r ∈R. For two structures S and S  in S(R), we write S ⊆S  (read S is included in S  ) if DS ⊆DS
, and rS ⊆rS
for each r in R. We recall that monadic second-order logic (MS logic for short) is rst-order logic augmented with (uppercase) variables denoting subsets of the domain of the considered structure, and new atomic formulas of the form x ∈X expressing the membership of x in a set X . We denote by MS(R; X) the set of MS formulas over R with free variables in X (the set of all individual and set variables.) A property of structures (or of elements and=or of sets of elements of a structure) is MS-de:nable if it can be expressed by an MS formula.

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We denote by S ≡S  the existence of an isomorphism between two structures S and S . If L; L ⊆S(R), we write L≡L if every structure of L is isomorphic to a structure in L and vice-versa. 

2.1.1. MS-transductions A transduction of structures is a multivalued mapping: S(R) → S(R ), formally handled as a mapping f : S(R) → ˝(S(R )) such that S ≡S  implies f(S) ≡f(S  ), where ˝(−) denotes a powerset operation. We say that f as above is MS-compatible [14] if there exists a total recursive mapping f # : MS(R ; ∅) → MS(R; ∅) such that S |= f # (’) iG S  |= ’ for some S  ∈f(S). We call f # (’) the backwards translation of ’ relative to f. Let L ⊆S(R). We say that a transduction f : S(R) →S(R ) is deterministic on L if for every S ∈L; f(S) is not empty and its elements are pairwise isomorphic. We now consider transductions de ned by MS formulas. A parameterless MSde:nable transduction f : S(R) →S(R ) is de ned as follows, from k ∈N+ and MS formulas  in MS(R; ∅); 1 ; : : : ; k in MS(R; {x}); r; i1 ; :::; in in MS(R; {x1 ; : : : ; xn }), for r ∈R ; n = (r); 16i1 ; : : : ; in 6k: (1) f(S) is non-empty iG S |=. (2) Assuming S |=, then f(S)={T } where T is constructed as follows: • DT =D1 × {1}∪ · · · ∪Dk × {k} ⊆DS × {1; : : : ; k}; • each Di is {x ∈DS | S |=i (x)}; • each relation rS , for r ∈R is de ned on T as the union of the sets of tuples of the form {((x1 ; i1 ); : : : ; (xn ; in )) | S |= r; i1 ;:::; in (x1 ; : : : ; xn )}, for all i1 ; : : : ; in ∈ {1; : : : ; k}. The tuple
; 1 ; : : : ; k ; (r; i1 ; :::; i(r) )r∈R; 16i1 ; :::; i(r) 6k  is called a de:nition scheme. An MS-de nable transduction is k-copying, if k¿1 (and noncopying if k = 1). In all cases, f is deterministic: L → S(R ) where L = {S∈S(R) | S |= } and its domain is of course MS-de nable. A parameterless MS-de nable transduction f is MS-compatible [7,9]. One can even de ne a backwards translation f # (’) for MS formulas ’ over R with free variables. If ’ has p free variables, then f # (’) has kp free variables. We refer the reader to [7,9] for details. 2.1.2. MS-transductions with parameters We now extend the previous de nitions in order to de ne (by MS formulas) transductions: S(R)→S(R ) that are not deterministic. Let p1 ; : : : ; pn be n unary relation symbols, that we will call parameters (p1 ; : : : ; pn ∈R = ∪R ). We let R : S(R ∪ {p1 ; : : : ; pn }) →S(R) be the mapping that “forgets” the relations p1 ; : : : ; pn . (It is actually a noncopying MS-de nable transduction.) A transduction f : S(R)→ S(R ) is MS-de:nable with parameters (we also say that it is an MS-transduction) if there exists an MS-de nable subset L of S(R ∪ {p1 ; : : : ; pn }) and a parameterless MS-de nable transduction g : L → S(R ) such that: f(S) = {g(S  ) | S  ∈ L; R (S  ) = S}:

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It is k-copying or noncopying if g is so. The set {S∈S(R) | f(S) = ∅} is MS-de nable: it is de ned by the formula ∃X1 ; : : : ; X n : [X1 =p1 ; : : : ; X n =pn ] where  ∈MS(R ∪ {p1 ; : : : ; pn }; ∅) de nes L and Xi =pi denotes the substitution of Xi for pi in . (We replace pi (x) by x ∈Xi for every i and x.). An MS-transduction f as above is MS-compatible: for every ’ ∈MS(R ; ∅) one takes f # (’) equal to ∃X1 ; : : : ; Xn :( ∧ g# (’))[X1 =p1 ; : : : ; Xn =pn ]: If f : S(R)→ S(R ) and g :S(R )→ S(R ) are two MS-transductions then the transduction (g ◦ f): S(R)→S(R ) de ned by
(g ◦ f)(S) = {g(S  ) | S  ∈ f(S); S ∈ S(R)} is an MS-transduction. It is noncopying if f and g are so. See [7,9]. We now consider as an example and for later use the mapping S → S=≈ which associates with a structure S its quotient by an MS-de nable equivalence relation ≈. We will see that this mapping is a deterministic noncopying MS-transduction, and its de nition uses one parameter. Let S =
DS ; (rS )r∈R  be a structure in S(R), and ≈ be an equivalence relation on DS . We denote by [d] the equivalence class of d ∈DS . We let S  = S=≈ be the structure in S(R) de ned as follows: DS
= DS = ≈ rS
([d1 ]; : : : ; [dn ]) holds iG rS (d1 ; : : : ; dn ) holds for some d1 ≈ d1 ; : : : ; dn ≈ dn : We call it the quotient structure of S by ≈. Lemma 2.1. Let be a formula in MS(R; {x; y}) such that, for every S∈S(R) the relation {(x; y)∈DS2 | S |= (x; y)} is an equivalence relation ≈. The mapping S → S=≈ is a deterministic noncopying MS-transduction. Proof. We let p be a parameter. We let  ∈MS(R ∪ {p}) be such that S |= iG every equivalence class of ≈ contains one and only one element of pS . The set pS can be taken as domain for the structure S=≈. Hence there exists a parameterless noncopying MS-transduction g : L →S(R) such that L={S  ∈S(R ∪ {p}) | S  |= }, and g(S  )= R (S  )=≈ for S  ∈L, where ≈ is the equivalence relation on R (S  ) de ned by . It follows that the mapping S → S=≈ is an MS-transduction. For every S there are several possible sets pS that satisfy , hence several structures S  such that R (S  )= S. But the structures g(S  ) are all isomorphic. Although the transduction S → S=≈ is deterministic, we need a parameter for de ning it as an MStransduction. 2.2. Finite and in:nite terms We review de nitions and notation from [3,11,12]. We let F be a nite set of function symbols, each of them, f, given with an arity (f) ∈N. We let X be a nite

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set of variables. We denote by T (F; X ) (resp. T ∞ (F; X )) the set of nite (resp. nite or in nite) well-formed terms written over F ∪X . We let kF = Max{(f) | f ∈F} and Fk = {f ∈F | (f)= k}. The occurrences in t ∈T ∞ (F; X ) of the symbols of F ∪X can be formalized as Dewey words, i.e. as elements of {1; : : : ; kF }∗ . A tree-domain is a subset L of {1; : : : ; kF }∗ such that: (i) L is pre x-closed, (ii) if ui ∈L with u ∈ {1; : : : ; kF }∗ and j ∈ {1; : : : ; kF }; 16j¡i then uj ∈L. A term t ∈T ∞ (F; X ) can be formalized by a mapping Symt : Dom(t) → F ∪X where Dom(t)⊆{1; : : : ; kF }∗ is a tree-domain and for each u ∈Dom(t); Symt (u) is the symbol occurring at u in t. This mapping is subject to the following condition: for all u ∈Dom(t) and i ∈ {1; : : : ; kF }, we have ui ∈Dom(t) iG i6(Symt (u)). (For a variable x ∈X , we let (x) = 0.) Finite terms will be, as usual, denoted by nite words. For an example the term t = f(x; f(x; g(a))) has the corresponding mapping Symt that associates f with ” and 2, x with 1 and 21, g with 22 and a with 221. Two terms t and t  are equal iG Dom(t) = Dom(t  ) and Symt = Symt
. A term t is nite iG Dom(t) is nite. If t ∈T ∞ (F; X ) and u ∈Dom(t), we denote by t=u the subterm of t issued from u. It is de ned by: Dom(t=u) = {v ∈ {1; : : : ; kF }∗ |uv ∈ Dom(t)}; Symt=u (v) = Symt (uv)

for v ∈ Dom(t=u):

We now recall that T ∞ (F; X ) is a complete partial order. We assume that F contains a special nullary symbol denoted by +. We de ne a partial order on T ∞ (F; X ) by letting: t ≺ t  iG Dom(t) ⊆ Dom(t  ) and for every u ∈ Dom(t); either Symt (u) = + or Symt (u) = Symt
(u): Every increasing sequence of terms t0 ≺t1 ≺t2 ≺ · · · ≺tn ≺ · · · has
a least upper bound in T ∞ (F; X ) denoted by t = supn¿0 (tn ). (We have Dom(t) = n¿0 Dom(tn ).) Finally, every in nite term t ∈T ∞ (F; X ) is the least upper bound of an increasing sequence of nite terms (t (n) )n¿0 de ned as follows: Dom(t (n) ) = {u ∈ Dom(t) | |u| 6 n};  Symt (u) if |u| ¡ n; Symt (n) (u) = + if |u| = n: (Hence t (0) =+.) We will use this fact in order to extend to in nite terms a k-ary mapping, say m: T (F; X ) k → D where D is a complete partial order and m is monotone, by letting: m(t ˆ 1 ; : : : ; tk ) = sup (m(t1(n) ; : : : ; tk(n) )): n¿0

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In this way we de ne an !-continuous mapping mˆ : T ∞ (F; X ) k → D that extends m (“!-continuous” means “monotone and continuous over in nite increasing sequences”, sometimes also called “!-chains”; see [3,4,5,11]). 2.3. Graphs All graphs will be directed and at most countable. An edge e of a graph G has a source srcG (e), a target tgtG (e) and a label in a nite set of edge labels, usually denoted by A. All graphs will be simple which means that no two edges have same source, same target and same label. We denote by VG the set of vertices of a graph G and by EG its set of edges. (An element of EG is a triple in VG ×A ×VG ). Furthermore, vertices may have labels from a nite set, say P. A vertex may have several labels or none. Hence, a graph G can be identi ed with the relational structure:
VG ; (edgaG )a∈A ; (labpG )p∈P ; where labpG is the set of vertices labelled by p; edgaG = {(srcG (e); tgtG (e)) | e is an edge labelled by a}.We denote by G(A; P) the class of graphs with A and P as respective sets of edge and vertex labels. A walk in G from x to y is a sequence of edges (e1 ; : : : ; en ) such that tgtG (ei ) = srcG (ei+1 ) for every i =1; : : : ; n − 1; x = srcG (e1 ) and y =tgtG (en ). A walk as above is a path if tgtG (ei )= srcG (ej+1 ) for 16i¡j6n − 1. A circuit is a path from x to x for some x. If B ⊆A, we say that an edge is a B-edge if its label is in B. A B-walk, a B-path, a B-circuit is a walk, a path or a circuit, all edges of which are B-edges. If G ∈G(A; P), and u ∈VG , we let Acc(G; u) be the set of vertices of G that are the end of a walk beginning at u. We let G Acc (u) denote the subgraph of G induced by the set of vertices Acc(G; u): Let G; H ∈G(A; P). We say that G is a subgraph of H , if G ⊆H where G and H are considered as structures (see Section 2.1). We say that G is an induced subgraph of H , written G ⊆i H , if G ⊆H and labpG = labpH ∩VG and edgaG = edgaH ∩(VG ×VG ) for all p and a. A homomorphism h :G → H is a mapping VG → VH such that: (i) h(labpG )⊆labpH for each p ∈P; (ii) ((h(x); h(y))∈edgaH if (x; y)∈edgaG ; a ∈A. It is an isomorphism iG it is bijective, we have = instead of ⊆ in condition (i) and we have “iG ” instead of “if ” in condition (ii). If G ∈G(A; P) and ∼ is an equivalence relation on VG , then the quotient graph G=∼ is de ned as a quotient structure (see Section 2.1). In particular we have a canonical surjective homomorphism: G →G=∼. For B ⊆A we denote by ContrB the mapping from G(A; P) to G(A − B; P) that contracts all B-edges and removes the resulting loops. Formally, ContrB (G) = H where H is the graph G=∼ with all its B-edges deleted, and ∼ is the equivalence relation on VG generated by the union of the relations edgbG for b ∈B.

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2.4. Terms and rooted graphs A rooted graph is a graph G (directed as are all our graphs) given with a distinguished vertex called the root from which every vertex is accessible by a directed path. We denote by rootG the root of G. Since graphs may have circuits, several vertices may be chosen as the root. We will use a vertex label rt to distinguish the root (labrtG = {rootG }). We denote by D(A; P) the set of rooted graphs with set of edge labels A and set of vertex labels P. We denote by D (A; P) the set of those such that labrtG is singleton, but do not satisfy necessarily the accessibility condition. Hence D(A; P) ⊆D (A; P) ⊆G(A; P ∪ {rt}). A graph G ∈D(A; P) is a tree iG it has no circuit and has at most one path between any two vertices. We will handle terms in T ∞ (F; X ) as trees as well as mappings on tree-domains. Let t belong to T ∞ (F; X ) with Dom(t), and Symt as in Section 2.2. We let G(t) be the tree in D({1; : : : ; kF }; F ∪X ) de ned by G =
Dom(t); (edgiG )16i6kF ; (labpG )p∈P ; where P =(F|{+})∪X ∪ {rt}, and: edgiG = {(u; ui) | u; ui ∈ Dom(t)}; 1 6 i 6 kF ; labpG = {u | Symt (u) = p};

if p ∈ F ∪ X;

labrtG = {”}: Note that G(+) is a single vertex labelled just by rt. In the graph G(f(+; +)) the vertices corresponding to the two occurrences of + have no label. From a structure G isomorphic to G(t) for some t ∈T ∞ (F; X ), one can determine t in a unique way, as one can check easily. Hence, in order to prove that t1 ; t2 ∈T ∞ (F; X ) are equal, it is enough to prove that the structures G(t1 ) and G(t2 ) are isomorphic. Let t ∈T ∞ (F; X ) and u ∈Dom(t); u = ”. Let G(t) Acc (u) be the subgraph of G(t) induced by Acc(G(t); u) according to the de nition given in Section 2.3. Let H =G(t)rtAcc (u) be this graph augmented with a “root label” i.e., we let labrtH = {u}. Then it is not hard to see that H is isomorphic to G(t=u). The isomorphism h : G(t=u) → H is given by h(w) = uw (we recall that the sets of vertices of H and G(t=u) are subsets of {1; : : : ; kF }∗ ). Remark 2.2. If t1 ≺t2 then G(t1 )⊆G(t2 ). For expressing properties of terms t in MS logic, we will use the corresponding structures G(t). In particular, the relation w is an ancestor of u which is exactly w6u, (w is a pre x of u) is expressible in MS logic in G(t) because the transitive closure of an MS de nable relation is MS-de nable (see [9, Lemma 1.7]).

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2.5. Operations on graphs We will use operations on graphs which are very close to those used in [13] for multilabelled graphs (see pp. 96 – 97 of that article). We x A and P as in Section 2.3. The rst operation is disjoint union which is binary. For G; H ∈G(A; P) we denote by G ⊕H their disjoint union. If necessary, we replace H by a copy disjoint with G and one de nes K = G ⊕H as follows: VK = VG ∪ VH

(with VH ∩ VG = ∅);

edgaK = edgaG ∪ edgaH

for a∈A;

labpK = labpG ∪ labpH

for p ∈P:

The next operation is unary. It adds edges as follows. For p; q ∈P; a ∈A; G ∈G(A; P) we let H = addp; q; a (G) be such that VH = VG ; labrH = labrG

for all r ∈ P;

edgaH = edgaG ∪ (labpG × labqG ); edgbH = edgbG

if b ∈ A; b = a:

The third operation, also unary, modi es vertex labels as follows. For p ∈P; Q ⊆P; G ∈G(A; P), we let H =renp→Q (G) be de ned by VH = VG ; edgaH = edgaG

for all a ∈ A;

labrH = labrG ∪ labpG labrH = labrG labpH = ∅

if r = p; r ∈ Q;

if r = p; r ∈= Q;

if p ∈= Q;

labpH = labpG

if p ∈ Q:

In words, this means that each label p of a vertex of G is replaced by the set of labels Q. This set Q may be empty; it may contain p. For each p ∈P we let p be a nullary symbol denoting the graph with one vertex labelled by p and no edge. Finally, we also introduce a nullary symbol + denoting the empty graph ∅. (It will be useful for dealing with in nite terms denoting countable graphs.) We let FA; P denote this set of operations and constant symbols. Every term t in T (FA; P ) de nes a nite graph in G(A; P) that we will denote by val(t). Because of the need to take disjoint copies in the evaluation of G ⊕H , the graph val(t) is only de ned up to isomorphism. This makes complicated to designate precise vertices of val(t). In

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order to overcome this diCculty we present an alternative, more precise construction of val(t), where the set Vval(t) is xed in a unique way. Construction 2.3. We let t ∈T (FA; P ) and we construct from it a graph G as follows. The graph G is intended to be isomorphic to val(t). We let VG be the set of “leaves” of t labelled by an element of P (as opposed to by +). Hence VG = {u | u ∈ Dom(t); Symt (u) ∈ P}:

(2.1)

Our next task is to determine the set LABt (u) = {q ∈P | q labels u in val(t)} for u ∈VG . Let R be the set of operations of the form renp→Q for p ∈P; Q ⊆P. We :rst consider the special case of t ∈T (R∪P), i.e. of t ∈R∗ P (it is convenient to identify r1 (r2 (· · · rn (p) · · ·)) ∈T (R ∪P) with the word r1 r2 : : : rn p). The graph val(t) has then a unique vertex say s. (Because of the renaming operations ri , we need not have p in LABt (s).) The sets of labels of the unique vertex of val(p); val(rn p); : : : ; val(r1 r2 : : : rn p) can be computed by a :nite automaton reading r1 r2 : : : rn p from right to left. The set of the states of such automaton is the powerset ˝(P); ∅ is the initial state, Q is the renp
→Q


p


:nal state and the transitions are of the form P  −→ (P  \{p }) ∪Q or ∅ −→{p }. Hence for every Q ⊆P the set of words r1 : : : rn p ∈R∗ P such that LABr1 ::: rn p (s) = Q is a regular language. It is written L Q . We now go back to the general case where t ∈T (FA; P ) and u ∈Dom(t). We let u =u1 u2 : : : un (with ui ∈ {1; : : : ; kF }) and SYMt (u) be the word in (FA; P )∗ de:ned as Symt (”)Symt (u1 )Symt (u1 u2 ) : : : Symt (u): (Hence SYMt (u) is the sequence of operations seen on the path from the root to u in the tree representing t:) Then, if u is a leaf of t having a label in P; we have: LABt (u) = Q ⊆ P

iG y ∈ LQ ;

(2.2)

where y is the word in R∗ P obtained from SYMt (u) by removing all symbols not in R. Removing from SYMt (u) the symbols not in R corresponds to the fact that only the operations from R modify the labelling of vertices. We now determine the existence of edges between two vertices of the graph G (intended to be val(t)); given as elements of Dom(t). We extend as follows the function SYMt de:ned above. If u ∈VG and w ∈Dom(t); w6u (i.e., w is “above u in t”), we let u = wv1 v2 : : : vn (with v1 ; : : : ; vn ∈ {1; : : : ; kF }; n61), and SYMt (w; u) be the word: Symt (w)Symt (wv1 )Symt (wv1 v2 ) : : : Symt (wv1 : : : vn ) ∈ (FA;P )∗ : Hence SYMt (u)= SYMt (”; u). Let us consider u; u ∈VG . We put in G an edge from u to u labelled by a iB there is in t an occurrence w of the operation addp; q; a and p ∈LABt=w1 (v); q ∈LABt=w1 (v ) where u =w1v; u =w1v . Intuitively, these conditions mean that in the subgraph of G de:ned by t=w1, the vertex u has label p, the vertex u has label q, so that addp; q; a introduces an edge from u to u labelled by a; this edge remains in G. (Since addp; q; a is unary,

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1 follows w in both u and u :) We denote by C(t; w; u; u ; p; q; a) this condition. Hence, in G we have, if u; u ∈VG ; a∈A: (u; u )∈edgeaG iB: there exist w ∈ Dom(t) with w 6 u; w 6 u ; Symt (w) = addp;q;a for some p; q ∈ P; such that C(t; w; u; u ; p; q; a) holds:

(2.3)

Condition C(t; w; u; u ; p; q; a) is written as follows: There exist Q; Q ⊆ P such that p ∈ Q; q ∈ Q ; y ∈ LQ and y ∈ LQ
; where y; y are the words in R∗ P obtained from SYMt (w; u) and SYMt (w; u ) by removing the symbols not in R: This ends Construction 2.3. We claim that the graph G de ned in this way from t is isomorphic to val(t). This follows actually from the observations we made along with the de nition. From now on and unless speci ed otherwise val(t) will denote the graph de ned from t as explained above. We have in particular: Lemma 2.4. If t; t  ∈T (FA; P ) and t ≺t  then val(t) is an induced subgraph of val(t  ). Proof. Every occurrence in t of a symbol p ∈P is also one in t  . Hence Vval(t) ⊆Vval(t
) . The labelling of u in Vval(t) and the edges from u to u for u; u ∈Vval(t) depend only on the operation symbols in SYMt (w) for the elements w of Dom(t) such that w6u or w6u . And SYMt
(w)= SYMt (w) for these w. Hence val(t) ⊆i val(t  ). If t ∈T ∞ (FA; P ) and t = supn¿0 (tn ) where tn is an increasing sequence of nite terms, then we have an increasing sequence of induced subgraphs: val(t0 ) ⊆i val(t1 ) ⊆i · · · ⊆i val(tn ) ⊆i · · · and we de ne val(t) as its union. Note that for each n ∈N; Vval(tn ) ⊆Dom(t) ⊆ {1; : : : ; kF }∗ . Hence we take the countable union of an increasing sequence of graphs with sets of vertices all included in Dom(t). It is not hard to verify that if t = supn¿0 (tn ) where tn is another increasing sequence of nite terms, then the union of the graphs val(tn ) is the same. Hence val(t) is well-de ned. Furthermore, Construction 2.3 works for t in nite as well as nite. We have de ned val as a mapping from T ∞ (FA; P ) to G(A; P). Since a term in T ∞ (FA; P ) is de ned in a unique way from the rooted graph G(t), we can extend the notation by letting val(G(t)) =val(t). Proposition 2.5. The mapping that associates val(t) with G(t) for t ∈T ∞ (FA; P ) is an MS-transduction. Proof. It is straightforward to translate Construction 2.3 into an MS-transduction. By (2.1), Vval(G(t)) is the set of vertices u of G(t) such that labpG(t) (u) holds for some p ∈P.

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An MS-formula Q (u) can be constructed that translates condition (2.2) of Construction 2.3. This is possible by the following two observations: rst one can de ne in G(t) the sequence of ancestors of a given u (simply by using the relations edgiG(t) ) so that we can de ne the word SYMt (u); the second observation is that nite automata on words can be translated into  MS formulas [21]. Having thus the formulas (u) for Q ⊆P, we let  (u) de ned as Q p p∈Q⊆P Q (u). It expresses that u has label p in val(t). For the same reasons, condition (2.3) of Construction 2.3 can be translated into an MS formula a (u; u ) expressing the existence of an a-edge in val(t) from u to u . Hence we obtain a noncopying parameterless de nition scheme:
; ’(u); (a (u; u ))a∈A ; (p (u))p∈P ;  where ’(u) is p∈P labp (u) and  is an MS formula (easy to construct) expressing that a given structure is of the form G(t) for some t ∈T ∞ (FA; P ). This concludes the proof. 2.6. Interpretations We have de ned on G(A; P) a structure of FA; P -algebra. If F is a set of function symbols and for each f ∈F of arity k we de ne a term I(f) in T (FA; P
; {x1 ; : : : ; xk }) (where P  is a nite superset of P) intended to de ne a k-ary total function on G(A; P), then we make G(A; P) into an F-algebra GI (A; P). Each operation of it is called a derived operation [3,4] of the FA; P -algebra G(A; P). In order to simplify the notation, we will not distinguish between the term I(f) in T (FA; P ; {x1 ; : : : ; xk }) and the total k-ary function on G(A; P) it de nes, namely the function which maps each tuple (G1 ; : : : ; Gk ) of G(A; P) k to the graph val(I(f))(G1 ; : : : ; Gk ). We will furthermore use I to de ne for every t ∈T ∞ (F) its interpretation Int(t)∈G(A; P). We let Int(t): T (F)→ G(A; P) be de ned in a natural way by Int(f) = I(f) if f ∈ F0 ; I(+) = ∅

(the empty graph);

Int(f(t1 ; : : : ; tk )) = I(f)(Int(t1 ); : : : ; Int(tk )); for f ∈ Fk ; k ¿ 1 and t1 ; : : : ; tk ∈ T (F): Note that Int(t)= val(t  ) where t  is obtained from t by substituting I(f) for each f ∈F. This substitution, called second-order substitution, will be recalled from [3] in Section 4.2. Since the operations of FA; P are monotone for graph inclusion we have Lemma 2.6. If t; t  ∈T (F) and t ≺t  then Int(t) ⊆i Int(t  ). Hence we can de ne Int(t) for t in nite in T ∞ (F) as the union of the graphs Int(tn ) which form an increasing sequence for inclusion, where (tn )n¿0 is an increasing sequence of nite terms such that t = supn (tn ).

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Lemma 2.7. The mapping t → Int(t) is an MS-transduction from T ∞ (F) into G(A; P). Proof. This mapping is the composition of two MS-transductions. The rst one transforms t into a term over FA; P , and the second one is the mapping val of Proposition 2.5. 2.7. Coverings and unfoldings We recall the unfolding operation considered in [14] as well as the notion of covering. From now on, A; P are xed nite sets of edge and vertex labels. If G ∈G(A; P); x ∈VG ; we denote by SucaG (x) the set {y ∈VG | (x; y) ∈edgaG }. Since graphs are simple, this set is in bijection with the set of a-edges with source x. Denition 2.8. Let G; H ∈D(A; P). A homomorphism h : G → H is a covering, if (i) it is surjective, (ii) h(labpG )= labpH for all p ∈P, (iii) h(rootG )= rootH , (iv) for every a ∈A and x ∈VG ; h is a bijection of SucaG (x) onto SucaH (h(x)). We also say that G is a covering of H . If h :G →H is a covering and ? is a walk in H from x to y and h(x ) = x; then there exists y ∈VG and a walk ? in G from x to y the image of which by h is ?. The proof is easy by induction on the length of ?. Proposition 2.9. Every graph H ∈D(A; P) has a covering G that is a tree. Proof. We let VG be the set of nite walks in H the origin of which is rootH . We put in VG the empty walk ” from the root to itself; it has no edge. We let rootG =” and edgaG ={(?; ? )∈VG2 | ? is ? extended by one {a}-edge}. We let h : G → H map ? ∈VG to x; where ? goes from rootH to x: Hence h(”) = rootH and labpG = h−1 (labpH ) for each p: It is straightforward to verify that G is a tree and that h : G → H is a covering. This tree is denoted by Un(H ), and, as in [14], we call it the unfolding of H (Fig. 1). Proposition 2.10. Let T; G; H ∈D(A; P) such that T is a tree, t :T →H and g : G → H are coverings. There exists a unique homomorphism k : T →G such that g ◦ k = t. This homomorphism is a covering. If G is a tree, it is an isomorphism. Proof. For every n we let Tn be the subtree of T induced by the set of vertices at distance at most n from the root. We prove by induction on n that for every n there is a unique homomorphism kn : Tn → G such that g(kn (x)) = t(x) for all x ∈VTn : The case of n = 0 is clear since T0 reduces to rootT : Let us assume that kn is known, and that y is in Tn+1 but not in Tn : We have an edge x → y in T , labelled by, say, a∈A where x ∈Tn . Since t is a bijection of SucaT (x)

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Fig. 1. A graph and its unfolding

onto SucaH (t(x)) and g is one of SucaG (kn (x)) onto SucaH (t(x)) we can de ne kn+1 (y) as g−1 (t(y)) for y ∈SucaT (x). Of course we take kn+1 (z) = kn (z) for every z ∈VTn . Hence we have the desired homomorphism kn+1 : Tn+1 → G. For k; we take the common extension of all the homomorphisms kn ; n¿0. At each level n, we have no choice for de ning kn since we want g ◦ kn to coincide with t on Tn . Hence k is the unique homomorphism T → G such that g ◦ k =t. Moreover, it is clear from the construction that it is a covering. If G; G  ∈D(A; P), and m :G  → G is a covering where G is a tree, then G  is a tree (otherwise G would have circuits or distinct paths between a same pair of vertices) and m is a bijection (for the same reason) and, moreover, an isomorphism. By applying this remark to T , we get that k is an isomorphism: T → G. It follows that, up to isomorphism, Un(H ) is the only covering of H that is a tree. We extend the mapping Un to graphs in D (A; P) as follows. We let Un(H ) = Un(HrtAcc (rootH ); i.e., we apply Un to the subgraph of H induced by the vertices accessible from the root. Theorem 22 of [14] can be reformulated as follows with the notation of the present paper: Theorem 2.11. For all :nite sets A and P, the mapping Un from D (A; P) to D(A; P) is MS-compatible. 3. The evaluation of rst-order substitutions In this section, we establish our main theorem, saying that the evaluation of order substitutions is an MS-compatible mapping.

rst-

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3.1. First-order substitutions As in Section 2.2 we let F and X be nite sets of function symbols and of variables, respectively. For each n-tuple xO =(x1 ; : : : ; xn ) of pairwise distinct variables in X , we introduce an (n + 1)-ary operation subxO such that subxO (t0 ; t1 ; : : : ; tn ) is the result of the simultaneous substitution of ti for xi in t0 for all i = 1; : : : ; n. A common notation for substitution used in [3] is t0 [t1 =x1 ; : : : ; tn =xn ]. This operation is the textual substitution for nite terms t0 ; t1 ; : : : ; tn (represented by words, say in Polish pre x notation) and is extended by continuity to in nite terms (because it is monotone in all its arguments [3, Proposition 3.3.3]). Other characterizations are given in [3]. If t ∈T ∞ (F; X ), we let Var(t)= {x ∈X | Symt (u) = x for some u ∈Dom(t)}. We have: Var(subxO(t0 ; t1 ; : : : ; tn )) = (Var(t0 ) − {x1 ; : : : ; xn })
∪ {Var(ti ) | xi ∈ Var(t0 ); 1 6 i 6 n}: We let Sub(X ) be the set of operation symbols subxO where xO is a nonempty sequence of pairwise distinct variables in X and (subxO ) = |x| O + 1. Every term t ∈T (F ∪Sub(X ); X ) can be evaluated into a term Eval(t)∈T (F; X ) just by performing the substitutions as prescribed by the operations subxO . For t =f(t1 ; : : : ; tk ); f ∈F; (f) = k; ti ∈T (F ∪Sub(X ); X ) we have of course: Eval(f(t1 ; : : : ; tk )) = f(Eval(t1 ); : : : ; Eval(tk )) (hence functions in F are not evaluated). O Since subxO : T (F; X )|x|+1 → T (F; X ) is monotone, the mapping Eval : T (F ∪Sub(X ); X ) →T (F; X ) is monotone and extends into an !-continuous mapping T ∞ (F ∪Sub(X ); X )→ T ∞ (F; X ) by Eval(t) = supn (Eval(tn )) where (tn )n¿0 is an increasing sequence of nite terms with least upper bound t. (We recall that “!-continuous” means “monotone and continuous over in nite increasing sequences”.) Example 3.1. Let t be the in nite term over symbols f (binary), x and subx de ned by t = subx (f(x; x); subx (f(x; x); : : : :) : : :) i.e., the in nite term such that: t = subx (f(x; x); t): Then T = Eval(t) satis es T = f(T; T ) hence is isomorphic to the complete in nite binary tree. Remark 3.2. We have subxO (+; t1 ; : : : ; tn ) = +. It follows that Eval(t)= + if t = subxO (subyO (subzO (· · ·); · · ·) where t, considered as an in nite tree, has a left-most branch with only symbols in Sub(X ). This corresponds to

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the fact that a recursive de nition like w = w[f(x)=x] where w ∈T ∞ (F; X ) de nes the “bottom” element, here + in the case of terms. See also the last example in Section 4.2. Our next objective is to prove that the mapping Eval : T ∞ (F ∪Sub(X ); X ) → ∞ T (F; X ) is MS-compatible, where a term t is represented by the tree G(t) as explained in Section 2.4. 3.2. Evaluating substitutions We denote by D1 (A; B; P) the set of graphs in D(A; P) such that every vertex has at most one outgoing B-edge. We denote by D1 (A; B; P), the corresponding subsets of D (A; P) for which we waive the accessibility condition (see Section 2.4). The main theorem of this section is the following: Theorem 3.3. Let F; X be :nite. (1) One can de:ne two :nite sets A; P where ” ∈A, and an interpretation Int : T ∞ (F ∪Sub(X ); X )→ D1 (A; {”}; P) such that for every t ∈T ∞ (F ∪Sub(X ); X ) we have: G(Eval(t)) = Contr” (Un(Int(t))): (2) The mapping Eval: T ∞ (F ∪Sub(X ); X ) → T ∞ (F; X ) is MS-compatible. Proof. We rst prove (2) assuming (1). The mappings Int and Contr” are MS-transductions (by Lemmas 2.7 and 2.1) hence are MS-compatible. The mapping Un is MS-compatible according to Theorem 2.11. Hence Eval is MS-compatible. We now start the proof of (1). We rst de ne A and P as follows: A = {1; : : : ; kF } ∪ {”}; P = F ∪ X ∪ Paux ∪ {rt}: where Paux is the set {auxi | 1 6 i 6 Card(X )} ∪ {rti | 0 6 i 6 Max{kF ; Card(X )}} of auxiliary “new” labels. In order to make easily understandable our construction, we show immediately an example. Example 3.4. If t = subx; y; z (t0 ; t1 ; t2 ; t3 ) where t0 = f(x; g(x; h(z; u))); t1 = f(x; h(y; u)); t2 = f(a; a); t3 = x;

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then Int(t) is the following graph:

By Un the subgraph G(t1 ) is duplicated (because x has two occurrences in t0 ), and G(t2 ) disappears (because it is not linked to G(t), since y has no occurrence in t0 ). It is clear that Contr” (Un(G)) is the graph G(Eval(t)) and Eval(t) = f(f(x; h(y; u)); g(f(x; h(y; u)); h(x; u))): In order to shorten the de nition of Int we introduce an auxiliary operation built from the basic ones. If q1 ; : : : ; qm ∈P, we let, for any graph G, remq1 ;:::;qm (G) = renq1 →∅ (renq2 →∅ (: : : (renqm →∅ (G)) · · ·)): This operation removes all labels qi . We are now ready to specify Int : T (F ∪Sub(X ); X ) → G(A; P). We will see later that the object graphs are actually in D1 (A; {”}; P). Denition 3.5. We de ne I: (i) I(+)= rt (we do not set I(+)= ∅ as in Section 2.6) (ii) I(w)= renrt→{w; rt} (rt) if w ∈X ∪F0 , (iii) For f ∈Fk we let I(f)(G1 ; : : : ; Gk ) = remrt1 ;:::;rtk (addrt;rt1 ;1 (· · · (addrt;rtk ;k [renrt→rt;f (rt) ⊕ renrt→rt1 (G1 ) ⊕ · · · · · · ⊕ renrt→rtk (Gk )]) · · ·)) Intuitively, the graph denoted by I(f)(G1 ; : : : ; Gk ) is obtained as follows: one takes the union of disjoint copies of G1 ; : : : ; Gk , and a new vertex say s that will be the root; one links by an i-edge s to the root of Gi , for each i = 1; : : : ; k. The labels rt1 ; : : : ; rtk

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Fig. 2.

are “temporary”. They are used for insuring correct connection of s with the roots of G1 ; : : : ; Gk by 1-; : : : ; k-edges. Before going on we can make an observation: Claim 3.6. For t ∈T ∞ (F − {+}; X ) we have Int(t) = G(t). It remains to de ne I(subxO ). We illustrate rst the mapping I(subx; y ). With rooted graphs G0 ; G1 ; G2 ; it associates the graph shown in Fig. 2. The ”-edges towards G1 originate from the vertices of G0 labelled by x (the rst variable of x), O and those to G2 from the vertices of G0 labelled by y. The labels x; y in G0 are removed. However G0 ; G1 ; G2 may contain vertex labels in X − {x; y} and G1 ; G2 may contain vertices labelled by x or y. We now give the general de nition of I(subxO ) where xO = (x1 ; : : : ; xn ). We use auxiliary labels rt0 ; : : : ; rtn and aux1 ; : : : ; auxn . We de ne I(subxO)(G0 ; G1 ; : : : ; Gn ) = remaux1 ; : : : ; auxn ; : : : ; rt0 ; : : : ; rtn (addrt;rt0 ;” (addaux1 ;rt1 ;” (· · · (addauxn ;rtn ;” [ rt ⊕ renrt→rt0 (renx1 →aux1 (· · · renxn →auxn (G0 )) · · ·)) ⊕ renrt→rt1 (G1 ) ⊕ · · · ⊕ renrt→rtn (Gn )]))) · · ·))): We recall that we apply the operator Un to directed graphs having a unique vertex labelled by rt, (still called the root) but such that not necessarily all vertices are reachable from the root by a directed path. Lemma 3.7. For every t ∈T ∞ (F ∪Sub(X ); X ) we have G(Eval(t)) = Contr” (Un(Int(t))): Before proving this lemma, we show that Int(t) is well-de ned (which does not follow from Section 2.6) because we let I(+) be rt and not ∅.

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Fact 3.8. For every t ∈T (F ∪Sub(X ); X ) we have Int(t) ∈D1 (A; {”}; P). Proof. By induction on the structure of t. This is true for t =+ since we have I(+) = rt. The other cases follow from the de nition of I(w) for the other symbols in F ∪ Sub(X )∪X . Fact 3.9. If t; t  ∈T (F ∪Sub(X ); X ) and t ≺t  then Int(t) ⊆Int(t  ). Proof. By induction on the structure of t using the fact that rt = I(+) ⊆G for every G ∈D1 (A ∪ {”}; P). Hence we can de ne Int(t) for t ∈T ∞ (F ∪Sub(X ); X ) as the union of the increasing sequences of graphs Int(tn ) where t0 ≺t1 ≺; : : : ; ≺ tn ≺ : : : ; each tn is nite and t = supn (tn ). Proof of Lemma 3.7. First part: By induction on t for t :nite. The cases where t ∈ {+} ∪X ∪F0 are clear. Two cases remain. Case 1: t =f(t1 ; : : : ; tk ). We have Eval(t)= f(Eval(t1 ); : : : ; Eval(tk )) hence, by the de nition of I(f) we have G(Eval(t)) = I(f)(G(Eval(t1 )); : : : ; G(Eval(tk ))):

(3.1)

On the other hand: Un(Int(t)) = Un(I(f)(Int(t1 ); : : : ; Int(tk ))) = I(f)(Int(t1 ); : : : ; Int(tk )) by the way I(f) is de ned (intuitively, it is a tree-construction operation, hence is “invariant by unfolding”); similarly, since I(f) introduces no ”-edge, it is invariant by ”-edge contractions, hence: Contr” (Un(Int(t))) = Contr” (I(f)(Un(Int(t1 )); : : : ; Un(Int(tk )))) = I(f)(Contr” (Un(Int(t1 ))); : : : ; Contr” (Un(Int(tk )))): Hence Contr” (Un(Int(t))) = I(f)(G(Eval(t1 )); : : : ; G(Eval(tk ))) by using induction, hence is equal to G(Eval(t)) by (3.1). Case 2: t = subxO (t0 ; t1 ; : : : ; tn ). From the de nition of I(subxO ) it is clear that G(Eval(t)) = Contr” (Un(I(subxO)(G(Eval(t0 )); : : : ; G(Eval(tk ))))):

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Hence, using induction, we need only prove that for H0 ; : : : ; Hk ∈D1 (A; {”}; P) we have Contr” (Un(I(subxO)(H0 ; H1 ; : : : ; Hk ))) = Contr” (Un(I(subxO)(Contr” (Un(H0 )); : : : ; Contr” (Un(Hk ))))):

(3.2)

We have actually: Un(I(subxO)(H0 ; : : : ; Hk )) = Un(I(subxO)(Un(H0 ); : : : ; Un(Hk ))):

(3.3)

(clear from the de nition of I(subxO ) and the construction of Un used in Proposition 2.9). No more diCcult is to prove that Contr” (Un(I(subxO)(Un(H0 ); : : : ; Un(Hk )))) = Contr” (Un(I(subxO)(Contr” (Un(H0 )); : : : ; Contr” (Un(Hk ))))):

(3.4)

Hence (3.3) and (3.4) yield (3.2), which completes the proof of the rst part. Second part: Extension to t in:nite. We let t = supn (tn ) where tn is an increasing sequence of nite terms, tn ∈T (F ∪Sub(X ); X ). By the de nition of Eval we have Eval(t) = supn (Eval(tn )). The graphs G(Eval(tn )) form an increasing sequence for inclusion and G(Eval(t)) is the union of these graphs. On the other side, we have to check that Contr” (Un(Int(t))) is the least upper bound of the graphs Contr” (Un(Int(tn ))) which also form an increasing sequence since they are equal respectively to G(Eval(tn )) for each n. Since for G ∈D1 (A; P); Un(G) has for vertices the nite paths in G with origin rootG , it is clear
that if G ⊆G
 , then Un(G)⊆Un(G  ). Furthermore, if G0 ⊆G1 ⊆ · · · ⊆Gn ⊆ · · · then Un( n¿0 Gn )= n¿0 Un(Gn ). Hence Un(Int(t)) is the union of the increasing sequence of graphs Un(Int(tn )); n¿0. It is not true in general that Contr” (G) ⊆Contr” (G  ) if G ⊆G  (because if G  is G augmented with more ”-edges, Contr” (G  ) may have less vertices than Contr” (G)). However this is true for G; G  if they are trees, as one checks easily. Furthermore, Contr” is also !-continuous on trees. Hence the least upper bound (for inclusion) of the increasing sequence Contr” (Un(Int(tn ))) is Contr” (Un(Int(t))) as was to be proved.

4. Regular, algebraic and hyperalgebraic trees We review the de nitions of regular, algebraic and hyperalgebraic trees, and we apply the main result of Section 3 to these trees and to the recursive de nitions they represent.

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4.1. Regular trees We review results from [3,4]. A regular system of equations over T ∞ (F; X ) is an n-tuple of the form:
ti = subxO(si ; t1 ; : : : ; tn ); 1 6 i 6 n;

(4.1)

where xO =(x1 ; : : : ; xn ); si ∈T (F; X ) and each ti is an unknown. It has a least solution in T ∞ (F; X ). (We assume that + ∈F so that T ∞ (F; X ) is partially ordered by ≺ and !-complete.) Note that here subxO is evaluated and handled as a mapping on T ∞ (F; X ). A regular system of the form (4.1) above has also a least (and actually unique) solution in T ∞ (F ∪Sub(X ); X ) where the operations subxO are unevaluated and treated as those in F. We let (t1 ; : : : ; tn ) be this solution. By a fundamental result of Mezei and Wright [4, Lemma 5.3], least solutions of regular systems are preserved under !-continuous mappings. Applying this to Eval we obtain that Eval(ti ) = ti for each i = 1; : : : ; n. A term t ∈T ∞ (F; X ) is regular iG it is a component of the solution of such a system. We denote by REG(F; X ) the set of these terms. We will call them regular trees in order to keep the well-known terminology. Other characterizations of regular trees are given in [3]. By routine transformations of regular systems, one can de ne regular trees in REG(F; X ) by regular systems in normal form i.e., of the form
ti = ei ; 16i6n, where for each i, • either ei ∈X ∪F0 • or ei = f(ti1 ; : : : ; tik ) for some k, some f ∈Fk , some i1 ; : : : ; ik ∈ {1; : : : ; n}. Example 4.1. The system t1 = subu;v (f(x; f(u; v)); t1 ; t2 ); t2 = subu;v (g(z; f(u; v)); t1 ; t2 ); can be written more concretely as t1 = f(x; f(t1 ; t2 )); t2 = g(z; f(t1 ; t2 )) and, can be replaced by the following system in normal form: t1 = f(t3 ; t4 ); t2 = g(t5 ; t4 ); t3 = x; t4 = f(t1 ; t2 ); t5 = z: The monadic second-order theory of a regular tree t is decidable (by Rabin’s theorem, see [21]). This means that one can decide whether a given MS-formula ’ holds in t.

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Since we represent a term t by the structure G(t), we write G(t) |=’ although G(t) is not the standard structure for representing t. However, G(t) and the classical structure with kF successors are de nable in each other by MS-formulas. Hence, our de nition of “t has a decidable MS theory” is equivalent to the classical one. Some nonregular in nite trees also have decidable MS theories: see the introduction for references. Our objective is precisely to de ne such trees. 4.2. Second-order substitution We recall from [3] the notion of second-order substitution. We let B be a set of function variables, each of them given with a xed arity ((’) ∈N for ’ ∈B). Our intention is to de ne the result of the substitution in t ∈T ∞ (F ∪B; X ) of s for ’ where the variables x1 ; : : : ; xm of s correspond to the 1st; : : : ; mth argument of ’. In order to specify this sequence we will write that we substitute x1 ; : : : ; xm ·s for ’ or that we substitute s for
’; x1 ; : : : ; xm . We let C be a sequence

’1 ; w1 ; : : : ;
’n ; wn  where ’1 ; : : : ; ’n ∈B and are pairwise distinct and for each i; wi is a sequence of pairwise distinct variables in X of length (’i ). From C we de ne a second-order substitution operation: SUBC = T ∞ (F ∪ B; X )n+1 → T ∞ (F ∪ B; X ) as follows. To simplify the notation we assume that B ={’1 ; : : : ; ’n }. We rst de ne SUBC (t; t1 ; : : : ; tn ) for t ∈T (F ∪B; X ); ti ∈T ∞ (F ∪B; X ) by induction on the structure of t: SUBC (a; t1 ; : : : ; tn ) = a

if a ∈ {+} ∪ X ∪ F0 ;

SUBC (f(s1 ; : : : ; sk ); t1 ; : : : ; tn ) = f(SUBC (s1 ; t1 ; : : : ; tn ); : : : ; SUBC (sk ; t1 ; : : : ; tn )) if f ∈ Fk ; k ¿ 1; SUBC (’i (s1 ; : : : ; sm ); t1 ; : : : ; tn ) = subwi (ti ; SUBC (s1 ; t1 ; : : : ; tn ); : : : ; SUBC (sm ; t1 ; : : : ; tn )); where
’i ; wi  is the ith pair in C for 16i6n. Example 4.2. We let C =

’; x; z; u;
; y; z; u; t = f(’(a; (a; x; g(x; z)); b)); t1 = h(x; z); and t2 = k(m(x; y); p(z; u)):

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199

Then SUBC (t; t1 ; t2 ) is the term t  = f(subx;z;u (t1 ; SUBC (a; t1 ; t2 ); SUBC ((a; x; g(x; z)); t1 ; t2 ); SUBC (b; t1 ; t2 ))) = f(h(SUBC (a; t1 ; t2 ); SUBC ((a; x; g(x; z)); t1 ; t2 ))) = f(h(a; suby;z;u (t2 ; SUBC (a; t1 ; t2 ); SUBC (x; t1 ; t2 ); SUBC (g(x; z); t1 ; t2 ))) = f(h(a; suby;z;u (t2 ; a; x; g(x; z)))) = f(h(a; k(m(x; a); p(x; g(x; z)))): We make a few observations. The term t1 that we substitute for ’ has no occurrence of u. Hence the third argument of ’ in t, namely b disappears. The variable x does not appear in the argument list of  (in C). Hence it is treated in t2 as a constant, and we nd it as rst argument of m, as it is in t2 : The other occurrences of x, as arguments of p and of g “come from” the second and the third argument of  in t. Lemma 4.3. For each C the mapping SUBC is monotone in its :rst argument, and !-continuous in the other ones. Proof. Standard proof by induction on the structure of the rst argument. It follows that SUBC extends into a continuous mapping: T ∞ (F ∪B; X ) n+1 → T (F ∪B; X ) in a standard way. ∞

Remark 4.4. Let ’ be unary, let ’! be the term ’(’(’(· · ·))) ∈T ∞ ({’}), let C =

’; x. Then SUBC (’! ; x)= +. It is easy to check that SUBC (’n (+); x) = + for every n whence the observation. Hence, even if + does not appear in any argument of SUBC , it may appear in the result. 4.3. Algebraic trees For each m¿1, we denote by Xm the “standard” set of variables Xm = {x1 ; : : : ; xm }. An algebraic system of equations on T ∞ (F; Xm ) is a system, written with a set B ={’1 ; : : : ; ’n } of function symbols with (’i )6m, of the form: S =
’i (x1 ; : : : ; x(’i ) ) = si ; 1 6 i 6 n; where for each i; si ∈T (F ∪B; X(’i ) ). A solution of S is an n-tuple of terms (t1 ; : : : ; tn ), where ti ∈T ∞ (F; X(’i ) ) such that for each i: ti = SUBC (si ; t1 ; : : : ; tn ); where C =

’1 ; w1 ; : : : ;
’n ; wn  and for each j = 1; : : : ; n we have wj = (x1 ; : : : ; x(’j ) ). Such a system has a least solution in T ∞ (F; X n ) n . An algebraic tree is a component of the least solution of an algebraic system. We denote the corresponding set by ALG(F; Xm ); (it is a subset of T ∞ (F; Xm )).

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Algebraic systems have been introduced as syntactic base of recursive applicative program schemes (see [5,11]). Proposition 4.5. Every algebraic tree t ∈ALG(F; X ) is Eval(t  ) for some regular tree t  ∈REG(F ∪Sub(X  ); X  ), with X  :nite X  ⊇X . A regular system de:ning t  can be constructed from an algebraic system de:ning t. Proof. Let t ∈ALG(F; X ) be given as the rst component of the least solution of an algebraic system. It is also the rst component of the least solution of an algebraic system of the special “normal” form (that one can construct from the given one) S =
’i (x1 ; : : : ; x(’i ) ) = si ; 1 6 i 6 n such that each si is: (i) either xj (16j6(’i )), (ii) or f(’i1 (x1 ; : : : ; x(’i1 ) ); : : : ; ’ik (x1 ; : : : ; x(’ik ) )), (iii) or ’j (’i1 (x1 ; : : : ; x(’i1 ) ); : : : ; ’ik (x1 ; : : : ; x(’ik ) )) with 16j; i1 ; : : : ; ik 6n; k = (’j ). A solution of S is thus an n-tuple (t1 ; : : : ; tn ) of terms in T ∞ (F; Xm ) (with Var(ti ) ⊆ X(’i ) ) which satis es the equations:  (if we are in case (i)); ti = xj    (if we are in case (ii)); S  = ti = f(ti1 ; : : : ; tik )    ti = subx1 ;:::;xk (tj ; ti1 ; : : : ; tik ) (if we are in case (iii)): Note that the terms t1 ; : : : ; tn de ned by S  belong to T ∞ (F; Xm ), not to T ∞ (F ∪ Sub(Xm ); Xm ), hence that the operations subx1 ; :::; xk are evaluated. Let us denote by (t1 ; : : : ; tn ) the unique solution of S  in T ∞ (F ∪Sub(Xm ); Xm ) (without “evaluating” subx1 ; :::; xk , see Section 4.1). Hence each ti belongs to REG(F ∪Sub(Xm ); Xm ) and its image in T ∞ (F; Xm ) under Eval is ti according to the Mezei–Wright Theorem. As a corollary we obtain another proof of Theorem 5.2 of [8]. Theorem 4.6. The monadic second-order theory of an algebraic tree is decidable. Proof. Let t1 ∈ALG(F; X ) be de ned by an algebraic system. We have t1 = Eval(t1 ) where t1 ∈REG(F ∪Sub(X ); X ) is de ned by a regular system that one can construct from the one de ning t1 . The MS theory of t1 is thus decidable. Since Eval is MScompatible by Theorem 3.3, the MS theory of t1 reduces to that of t1 hence is decidable. Example 4.7. Consider the equation ’1 (x1 ; x2 ) = f(x1 ; ’1 (g(x2 ); ’1 (x1 ; x2 ))):

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201

We rst transform it into a system of the appropriate “normal form”: ’1 (x1 ; x2 ) = f(’2 (x1 ); ’3 (x1 ; x2 )); ’2 (x1 ) = x1 ; ’3 (x1 ; x2 ) = ’1 (’4 (x1 ; x2 ); ’1 (x1 ; x2 )); ’4 (x1 ; x2 ) = g(’5 (x1 ; x2 )); ’5 (x1 ; x2 ) = x2 : The corresponding regular system over T ∞ (F ∪Sub(X2 ); X2 ) is thus: t1 = f(t2 ; t3 ); t2 = x1 ; t3 = subx1 ;x2 (t1 ; t4 ; t1 ); t4 = g(t5 ); t5 = x2 : Of course the regular tree t1 ∈REG(F ∪Sub(X2 ); X2 ) could be de ned directly by: t1 = f(x1 ; subx1 ;x2 (t1 ; g(x2 ); t1 )): The notion of a system in normal form has been used to simplify the formal construction. Here is another example. Example 4.8. Consider the following equation: ’1 (x1 ) = ’1 (f(x1 )): Any term t ∈T (F; X ) without an occurrence of x1 is a solution. The least solution is +. The above equation can be translated into the regular equation: t1 = subx1 (t1 ; f(x1 )): The regular tree t1 corresponding to its unique solution in T ∞ (F ∪ {subx1 }; X ) has a left-most in nite branch with only the symbol subx1 . Hence t1 evaluates by Eval into +. We now state the following proposition which is the converse of Proposition 4.5. Proposition 4.9. For every :nite set X of variables, we have Eval(REG(F ∪ Sub(X ); X )) ⊆ ALG(F; X ): Proof. Let r ∈REG(F ∪Sub(X ); X ), where X = {x1 ; : : : ; xn }. As a regular tree, r is a component of the unique solution in T ∞ (F ∪Sub(X ); X ) of a nite system of equations
ti = ei ; 1 6 i 6 m;

(4.2)

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where for each i, • either ei ∈X; • or ei ∈F0 , • or ei = f(ti1 ; : : : ; tik ) for some k¿1, some f ∈Fk , some i1 ; : : : ; ik ∈ {1; : : : ; n}, • or ei = subxO (tj ; ti1 ; : : : ; tik ) for some sequence xO of pairwise distinct elements of X , some j; i1 ; : : : ; ik ∈ {1; : : : ; n}. Without loss of generality we can assume that xO is an increasing subsequence of (x1 ; : : : ; xn ), because if necessary, we can permute the sequence (i1 ; : : : ; ik ) in order to replace xO by an increasing sequence. Moreover, we can also assume that xO = (x1 ; : : : ; xn ). Indeed, if a variable xj is missing, we add it and we let xj to be the term that should be substituted to it. We illustrate this construction on the following example. Let n = 5. An equation of the form t =subx5 ; x1 ; x3 (t0 ; t1 ; t2 ; t3 ) is replaced by the equation t =subx1 ; x2 ; x3 ; x4 ; x5 (t0 ; t2 ; x2 ; t3 ; x4 ; t1 ). Once we agreed on this transformation, the last case of the de nition a system of equations is ei =subxO (tj ; ti1 ; : : : ; tin ) and xO = (x1 ; : : : ; xn ). For each i =1; : : : ; m, we introduce a function variable ’i of arity n and we de ne the algebraic system
’i (x1 ; : : : ; xn ) = si ; 1 6 i 6 m;

(4.3)

where in the above four cases we have now, respectively, • si = ei if ei ∈X; • si = ei if ei ∈F0 ; • si = f(’i1 (x); O : : : ; ’ik (x)) O if ei =f(ti1 ; : : : ; tik ); • si = ’j (’i1 (x); O : : : ; ’ik (x)) O if ei = subxO (tj ; ti1 ; : : : ; tik ); where xO is xed as (x1 ; : : : ; xn ) by the above agreement. Let (r1 ; : : : ; rm ) (resp. (r1 ; : : : ; rm )) be the unique solution of (4.2) (resp. (4.3)) in T ∞ (F ∪Sub(X ); X ) (resp. T ∞ (F; X )). It is clear that Eval(ri ) = ri . It follows in particular that t is an algebraic tree. 4.4. Damm’s hierarchy We have seen that Eval(REG)= ALG (leaving functions and variables unspeci ed). Provided suitable alphabets are used, we can de ne the family of level-n hyperalgebraic (called n-rational in [15]) trees roughly as ALG n = Evaln (REG): More precisely, we have already, for a set X of variables: 

nite set F of function symbols and a 


 ∞ Eval(REG(F ∪ Sub(Y ); Y ))  ALG(F; X ) =    ∩ T (F; X ): Y ⊇X Y nite

nite

B. Courcelle, T. Knapik / Theoretical Computer Science 281 (2002) 177 – 206

203

(In function de nitions we must allow for sets Y of “parameters” that are larger than X but still nite.) We de ne ALG 0 (F; X ) = REG(F; X ); 



 
Eval(ALG n−1 (F ∪ Sub(Y ); Y ))  ALG n (F; X ) = T ∞ (F; X ) ∩  ;  Y ⊇X nite

so that ALG 1 (F; X )= ALG(F; X ). Theorem 4.10. Every tree in ALG n has a decidable MS theory. Proof. This is straightforward since Eval is MS-compatible. One uses an induction on n. These trees have been considered by Damm in [15] where it is established that ALG n forms a strict hierarchy. The next question is to understand which types of recursive de nitions these trees represent. To this end, we consider explicit composition of functions of arbitrary arity. This is an extension of the usual composition of unary functions “◦”. For each n¿1 we denote by compn the overloaded functional operator such that h = compn (f; g1 ; : : : ; gn ) is well-de ned iG f : Bn → A

and

gi : C → B

for each i = 1; : : : ; n

are total functions and A; B; C are sets. The term compn (f; g1 ; : : : ; gn ) denotes the function x·f(g1 (x); : : : ; gn (x)). It should be clear that if t ∈T ∞ (F; {x1 ; : : : ; xn }); si ∈T ∞ (F; {x1 ; : : : ; xk }) and A; B; D are appropriate domains such that val(t) : Bn → A and val(si ) : D k → B, then val(sub(x1 ;:::;xn ) (t; s1 ; : : : ; sn )) = compn (val(t); val(s1 ); : : : ; val(sn )): Hence compn is the semantical meaning of the substitution operation subxO if xO has length n. Having this in mind, we can consider recursive de nitions as follows. We let b denote a “base” type (typically integers). We consider symbols of the following types: c: b

v; u; x; y : b

f: b×b → b

’; : b → b

g; h : b → b

H : (b → b) × (b → b) × b × b → b

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We let H be de ned by the following recursive de nition H (’; ; x; y) = f(x; ’(H (u:’( (u)); v:f(’(v); (h(v))); g(y); ’(c)))):

(4.4)

It involves parameters of two types: b and b → b. Hence it is not an algebraic system. Let us introduce a new symbol H  : (b → b) ×(b → b) → (b ×b → b) de ning a functional such that H  (’; ) = comp2 (f; ?1 ; comp1 (’; comp2 (H  (comp1 (’; ); comp2 (f; ’; comp1 (’; h))); comp1 (g; ?2 ); comp1 (’; c2 )))) where ?1 : b ×b →b ?2 : b ×b →b c2 : b ×b → b The de nition of

(4.5)

is the rst projection, is the second projection, is the constant function xy·c. H  involves the following types:

b = b → b b = b × b → b so that H  : b × b → b

f : b

comp1 : b × b → b

g; h : b

comp2 : b × b × b → b

?1 ; ?2 ; c2 : b

and clearly H  is de ned by an algebraic equation over basic functions f; g; h; comp1 ; comp2 ; ?1 ; ?2 ; c2 . The tree T (H )∈T ∞ ({f; g; h; ’; }; {x; y; c}) is thus Eval(T (H  )) where in addition • Eval(?1 )= x and Eval(?2 )= y, • Eval treats comp2 as subxO with xO =(x; y) and comp1 similarly with xO = (x). 1 Hence T (H )∈ALG 2 ({f; g; h; ’; }; {x; y; c}). It is not yet completely clear if any recursive de nition like (4.4) can be translated into an algebraic equation like (4.5). The problem comes with -schemes. Here is an example from [19] (called “unsafe” there). H (’; x) = f(x; ’(H (u · f(’(u); x); x))):

(4.6)

The underlined occurrence of x is free in the -subterm and this raises a problem with the translation. In the present example one can observe that (4.6) is equivalent to H (’; x) = H1 (uv · ’(u); x); H1 ( ; x) = f(x; (H1 (uv · f( (u; v); v); x); x)); 1

We write the de nition with comp in order to stress semantics.

(4.7)

B. Courcelle, T. Knapik / Theoretical Computer Science 281 (2002) 177 – 206

because H1 (with

205

: b ×b → b) is such that

H1 ( ; x) = H (u · (u; x); x): (This latter fact can be proved from (4.6) and (4.7) by the standard “Scott’s induction method”.) It follows that the tree associated with H de ned by (4.6) is in ALG:1 However, we leave as an open question whether a similar transformation can be applied for all “-schemes” with free variables in -terms. This question concerns level-2 schemes. What about higher levels? 5. Monadic second-order logic and program schemes Our initial motivation was to decide properties of program schemes expressible in MS logic. Consider a program scheme the semantics of which is represented by an in nite term T in T ∞ (F; X ). It may be the case that a variable x does not occur in T . This corresponds to the fact that the function de ned by the scheme does not actually depend on the argument corresponding to x. A simpli cation of the writing of the scheme may be obtained from this observation. Such transformations are considered in [17]. Whether or not x occurs in T is not necessarily easy to decide if the program scheme consists of several mutually recursive de nitions using parameters of function or functional type. Hence, even if the existence of an occurrence of x in T is a rst-order property over the relational structure representing T , its expression in terms of the syntax of the scheme is not necessarily easy. However, our result states that it is decidable for the schemes corresponding to level-n hyperalgebraic trees. Here is an example of a related monadic second-order but non rst-order property. We consider the niteness of the number of occurrences in T of a variable x (or of a function symbol). That this property is monadic second-order can be seen as follows. Let us say that a set of nodes W in an in nite tree is a cut if no two nodes of W are on a same branch and every maximal branch contains one (and thus only one) node of W . Since the trees corresponding to terms have nodes of nite (even bounded) degree, every cut in such a tree is nite (by Koenig’s Lemma). Hence a set of nodes U is nite iG there is a cut W such that every node of U is between the root and an element of W . This latter condition is expressible in MS logic. Hence, as an application of our result that is relevant to the static analysis of recursive de nitions, we get that, for every level-n hyperalgebraic tree, the niteness of the number of occurrences of a given symbol is decidable. References [1] A. Asperti, V. Danos, C. Laneve, L. Regnier, Paths in the lambda-calculus, in: Proc. 9th IEEE Symp. on Logic in Comput. Sci., 1994, pp. 426 – 436. [2] A. Asperti, S. Guerrini, The optimal implementation of functional programming languages, in: Cambridge Tracts in Theoretical Computer Science, Vol. 45, Cambridge University Press, Cambridge, 1998.

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[3] B. Courcelle, Fundamental properties of in nite trees, Theoretical Comput. Sci. 25 (1983) 95–169. [4] B. Courcelle, Equivalence and transformations of regular systems—applications to recursive program schemes and grammars, Theoretical Comput. Sci. 42 (1) (1986) 1–122. [5] B. Courcelle, Recursive applicative program schemes, in: J. van Leeuwen (Ed.), Formal Models and Semantics, Handbook of Theoretical Computer Science, Vol. B, Elsevier, 1990, pp. 459–492. [6] B. Courcelle, The monadic second-order logic of graphs, VII: Graphs as relational structures Theoretical Comput. Sci. 101 (1992) 3–33. [7] B. Courcelle, Monadic second-order de nable graph transductions: a survey. Theoretical Comput. Sci. 126 (1994) 53–75. [8] B. Courcelle, The monadic second-order theory of graphs IX: Machines and their behaviours. Theoretical Comput. Sci. 151 (1995) 125–162. [9] B. Courcelle, The expression of graph properties and graph transformations in monadic second-order logic, in: G. Rozenberg (Ed.), Handbook of Graph Grammars and Computing by Graph Transformation, vol. 1, World Scienti c, 1997, pp. 313–400. [10] B. Courcelle, J. Engelfriet, G. Rozenberg, Handle-rewriting hypergraph grammars, J. Comput. Syst. Sci. 46 (2) (1993) 218–270. [11] B. Courcelle, M. Nivat, Algebraic families of interpretations, in: IEEE Annual Symposium on Foundations of Computer Science, Houston, 1976, pp. 137–146. [12] B. Courcelle, M. Nivat, The algebraic semantics of recursive program schemes, in: J. Winkowski (Ed.), Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 64, Zakopane, 1978, pp. 16–30. [13] B. Courcelle, S. Olariu, Upper bounds to clique width of graphs, Discrete Applied Mathematics 101 (1999) 77–114. [14] B. Courcelle, I. Walukiewicz, Monadic second-order logic, graph coverings and unfoldings of transition systems, Annals of Pure and Applied Logic 92 (1998) 35–62. [15] W. Damm, The IO- and OI-hierarchies, Theoretical Comput. Sci. 20 (2) (1982) 95–208. [16] J. Engelfriet, Graph grammars and tree transducers, in: S. Tison (Ed), Trees in Algebra and Programming—CAAP’94, Edinburgh, Apr. 1994, Vol. 787, pp. 15 –36. [17] I. Guessarian, Program transformations and algebraic semantics, Theoretical Comput. Sci. 9 (1979) 39–65. [18] I. Guessarian, Algebraic Semantics, Lecture Notes in Computer Science, Vol. 99, Springer, Berlin, 1991. [19] T. Knapik, D. NiwiWnski, P. Urzyczyn, Deciding monadic theories of hyperalgebraic trees, in: S. Abramsky (Ed.), 5th Internat. Conf. on Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, vol. 2044, KrakWow, May 2001, pp. 253–267. [20] M.O. Rabin, Decidability of second-order theories and automata on in nite trees, Transactions of the American Mathematical Society 141 (1969) 1–35. [21] W. Thomas, Languages, automata and logic, in: G. Rozenberg, A. Salomaa (Eds.), Beyond Words, Handbook of Formal Languages, Vol. 3, Springer, Berlin, 1997, pp. 389–455.

Theoretical Computer Science 281 (2002) 207 – 217

www.elsevier.com/locate/tcs

Tilings as a programming exercise Guy Cousineau PPS Laboratory, Universite Denis Diderot, 175 rue du Chevalret, 75013 Paris, France

Abstract We investigate the problem of producing symmetric tilings by programs in a uniform way. By this, we mean that the construction of a tiling should be parameterized by the geometric model in which the tiling is de)ned (Euclidean, Hyperbolic, etc.), its formal symmetry group, the interpretation of this symmetry group in the geometric model and a speci)c tile that )ts this interpretation.This parameterization can be obtained at a basic level using higher order functions as described in Cousineau and Mauny (The Functional Approach to Programming, Cambridge University Press, Cambridge, 1997) but this fails to re2ect in the program the mathematical structure of tilings in an adequate way. We therefore explore the use of more abstract constructs such as modules and show that they allow to structure the program in a satisfying way. The Objective Caml programming language (Objective Caml, O’Reilly, 2000) allows us to place this c 2002 Elsevier Science B.V. All rights reserved. study in a convenient setting.
Keywords: Functional Programming; Modularity; Parametricity; Tilings

1. Introduction The use of computers in Mathematics is now widespread both in research and in education. Numerous applications such as Mathematica [9], or Maple [7] are very popular. Many more specialized applications such as GAP [5] are developed for research purposes and gather important communities of developers and users. These systems include a programming language in which each user can develop speci)c programs. This programming language is supposed to give access to mathematical theories in a way that is convenient from a programming point of view and this involves questions of abstraction, modularity, parametricity that are central also in the design of programming languages. Therefore, the design of such systems involves problems that are clearly relevant to Computer Science. When this aspect is ignored or underestimated, this can restrain the use of computer mathematics. This problem is indeed addressed by some E-mail address: [email protected] (G. Cousineau). c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter
PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 3 - 0

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G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

research groups in Computer Science (e.g. project FOC at University of Paris VI) but this has not led yet to widespread systems since the problems involved are indeed deep and diEcult. In this paper, our ambition is much more limited. We only intend to show how an application involving rather simple concepts from geometry and group theory can be programmed with a satisfying generality using a modern programming language with a well-designed module facility. We hope to convince the reader, that such a facility is indeed necessary to re2ect in programs the right level of abstraction for this kind of application. We address the problem of producing tilings [6] with the help of a computer. The tilings we consider are those with a non-trivial computable symmetry group acting transitively on the tiles. This includes periodic tilings of the Euclidean plane but also other kinds of tilings, for instance hyperbolic ones such as the Escherlike tilings in Fig. 4. The mathematical concepts involved in the design of such tilings are • Symmetry groups and their formal presentations • Geometric transformations that can interpret such groups in a given geometry • Geometric objects that can be submitted to such transformations such as points and lines or curves. Our general approach is that we produce a tiling by computing a )nite part of its symmetry group, interpret the obtained elements of this symmetry group as transformations in a given geometry and )nally apply these transformations to a basic tile or pattern to produce the tiling. 2. Design problems The main problem we have to face is parametricity. The objects we deal with are highly structured and refer to mathematical concepts that enable us to describe a tiling in an adequate way. Therefore, an implementation should re2ect this structure and our program should be parameterized by the involved mathematical objects such as a symmetry group or a geometry. Moreover, a tiling has many concrete aspects that are important for its )nal aspect such as colors, decorations, contour lines but that should not interfere with the overall generating process. This causes some diEculties. For instance, for a tiling to be colored in an adequate way, the transformations that are used to produce the diHerent copies of a tile should be paired with permutations that act on the colors. There are also variations that can be useful occasionally. For instance, it might be useful for pedagogical purposes to mark tiles by labels that indicate by which transformation a given tile has been produced. In that case, transformations should be paired with labels describing them as in Fig. 1. This sort of variation should be easy to make and should not imply modi)cations of the generating process. We shall )rst examine what this implies for the representation of tiles. To draw tiles, we must )rst assume that we can use a graphical library. We shall call pictures the graphical objects produced by that library. However, it would not be adequate to represent directly tiles by pictures because it is very unlikely that

G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

Ab

cb

209

Cb AB

BC

b B

bC

Ca

C a

ba Ba CC

c

Ac

A

Bc

CA

ac

Fig. 1. Escher’s Circle Limit III with labels.

the graphical library provides all the transformations that we might have to apply to tiles. For instance, when tiles are tagged and a tile is rotated, its tag should not be rotated. So a tile should have more structure than a picture. On the other hand, a tile should be something that can produce a picture when extra information is passed to it. A plausible solution is to consider that a tile is a function which takes that extra information and produces a picture. We shall call transformation such extra information which concerns position and shape (geometrical transformation), color (color permutation) or any other characteristics. We can summarize this point of view by a type statement tile: transformation -> picture However, as we already mentioned, transformation cannot be a )xed type because various kinds of transformations can be used for various tilings. So the type of tiles should depend on a speci)c transformation type which is part of the tile speci)cation. This is possible only if our programming languages allows us to specify such dependencies. We can also notice that a tile is in general a rather complex geometrical object. So, if we de)ne it as a function, this function will typically make use of some data structure related to it. It might be tempting to consider tiles as objects in the sense of object oriented languages of which the above function would be a method. But then

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the aforementioned type dependency should be expressed by a class dependency, a notion that does not exist in object-oriented languages. Now, let us take a look at tilings. To produce a tiling, we need both a tile (in the above sense) and a process that can generate transformations to be passed to the tile. We shall call generator such a process. Here again, the type of a generator should depend on the type of transformations it generates and moreover, when a generator is used together with a tile to produce a tiling, it is mandatory that both depend on the same transformation type. This kind of constraints should also be expressible in the programming language. Furthermore, generators will have to be constructed in a uniform way from a group presentation and an interpretation of that group in a given geometry and the transformation type it will use comes from the geometry. So we need to be able to de)ne some functions that take objects with rather complex type dependancies and produce objects with similar characteristics. We shall solve these problems by using the module system of the OCAML language which comes originally from the module system of the SML language with some additions. In this system, modules are complex entities which incorporate types and values. One can also de)ne functors which build new modules from arguments that are also modules and express constraints between arguments of functors.

3. Basic modules and functors for tilings Here is a module type de)nition for tiles: module type TILE = sig type transformation val visualize : transformation -> picture end This de)nition states that a tile (a module of type TILE also called a module with signature TILE) includes a type transformation on which the function visualize depends. A tile will contain other )elds but these two will be the only visible ones from the signature TILE. A generator will have the following type: module type GENERATOR = sig type element val generate : int -> element list end A generator includes a type element and a function generate that produces a list of elements given an integer. This integer will typically allow to specify how much of a symmetry group one wants to generate.

G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

211

A tiling will have the following type: module type TILING = sig val draw : int -> picture end Now, the following functor will produce a tiling from a generator and a tile. It includes a constraint saying that the elements generated by the generator should be transformations accepted by the tile. module Make_tiling (Gen: GENERATOR) (Tile: TILE with type transformation = Gen.element) = struct let draw n = group_pictures (map Tile.visualize (Gen.generate n)) end

4. Group representation One of the problems we have to face when generating symmetry groups of tilings is to do it in a non-redundant way. Fortunately, common symmetry groups have presentations with decidable word problems [4] and moreover, they have presentations with canonical rewriting systems as shown by Lechenadec in [8]. Therefore, the generation of normal forms are quite straightforward: the presentation of the group can be given as a function that takes a generator and a list of generators and adds the new generator to the list provided that this does not introduce a redex for the associated rewriting system. So, the module type for canonical groups is: module type CANONICAL_GROUP = sig type element val generators : element list val compose : element -> element list -> element list end For instance, the canonical presentation for Escher’s Circle Limit symmetry group is an hyperbolic group with 3 rotations of order 3, 3 and 4. If we denote A; B; C the 3 rotations and a; b; c their inverses, the corresponding rewriting system is the following (omitting trivial rules such as Aa → Id). AA → a CB → a aa → A bb → B ca → B

AC → b CCC → c ab → C bc → A cc → CC

BA → c aB → Cb bA → Bc cA → Ba

BB → b aC → Ab bCC → Ac cB → CCa

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G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

The corresponding module is de)ned below with the rotations written TA, TB and TC and their inverses Ta, Tb and Tc. module Group_hyp_3_3_4 = struct type element = TA | TB | TC | Ta | Tb | Tc let generators = [TA;TB;TC;Ta;Tb;Tc] let compose = function TA -> (function []->[TA] | ((TB::_)|(Tb::_)|(Tc::_) as tl’)->TA::tl’) | TB -> (function []->[TB] | ((TC::_)|(Ta::_)|(Tc::_) as tl’)->TB::tl’) | TC -> (function []->[TC] | ((TA::_)|(Ta::_)|(Tb::_)|((TC::(TA|TB|Ta|Tb|Tc)::_)) as tl’) -> TC::tl’) | Ta -> (function []->[Ta] | ((Tc::_) as tl’)->Ta::tl’) | Tb -> (function []->[Tb] | ((Ta::_)|((TC::(TA|TB|Ta|Tb|Tc):: _)) as tl’)->Tb::tl’) | Tc -> (function []->[Tc] | ((Tb::_) as tl’)->Tc::tl’) end

To each canonical group, we associate a generator using the following functor in which we do not detail the function power which produces an exponential from a product and its neutral element. module Make_canonical_generator (G:CANONICAL_GROUP) = struct type element = G.element list;; let power f x l = ... let generate n = power G.compose [] G.generators n end We also need a representation for concrete groups which are de)ned by module type GROUP = sig type element val compose : element -> element -> element val identity : element end In fact, we only use the monoidal structure of groups, so the inverse function is not included. 5. Mappings Next, we have to interpret elements of formal symmetry groups as geometric transformations using a morphism from formal groups to concrete transformation groups. A morphism is de)ned by giving a mapping from the generators of the formal group to geometric transformations and this mapping can then be automatically extended to a morphism. Here again, modules and functors are a great help in implementing these notions at a suitably abstract level. A mapping is just a function from a source to a destination, depending on the source and destination type.

G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

213

module type MAPPING = sig type source type dest val map : source -> dest end A useful construct is the pairing of mappings which is implemented by the functor: module MakePair (Map1: MAPPING) (Map2: MAPPING with type source = Map1.source) = struct type source = Map1.source type dest = Map1.dest * Map2.dest let map s = Map1.map s , Map2.map s end A morphism from canonical groups to concrete groups is of its generators by the following functor: module Make_morphism (G1:CANONICAL_GROUP) (T:MAPPING with type source (G2:GROUP with type element = struct type source = G1.element list;; type dest = G2.element;; let rec map = function [] -> G2.identity | a::l -> G2.compose (T.map end

computed from a mapping = G1.element) = T.dest)

a) (map l);;

What we obtain is a new mapping that involves lists of generators instead of simple generators. Note how the constraints on the arguments of the functors allow to specify in a simple way the type dependencies. It is now rather easy to take into account other aspects of a tiling. For instance, we can map the elements of a canonical group presentation to permutations using this functor: module Make_color_morphism (G1:CANONICAL_GROUP) (C:MAPPING with type source = G1.element and type dest = Permutation.permutation) = struct type source = G1.element list;; type dest = Permutation.permutation;; let rec map = function [] -> Permutation.identity | a::l -> Permutation.compose (C.map a) (map l);; end and pair the two obtained mappings using the functor MakePair.

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CANONICAL_GROUP

GROUP

Make_Morphism

Make_Canonical_Generator

MAPPING

Make_Generator_From_Canonical

TILE

GENERATOR

Make_Tiling

TILING

Fig. 2. Modules and functors.

The next step is to use such a morphism to transform a canonical generator into a generator for a concrete group. We just need a functor that takes a generator and a mapping and produces a new generator: module Make_generator_from_canonical (Gen: GENERATOR) (M: MAPPING with type source = Gen.element) = struct type element = M.dest;; let generate n = List.map M.map (Gen.generate n);; end All the ingredients to produce tilings are now de)ned and are shown in Fig. 2.

G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

215

6. Geometries To structure a bit more the construction of tilings, we introduce a last module type which corresponds to geometries. A geometry depends on a type point and a type transformation depending on point. It essentially introduces a transformation group which is a submodule and a function apply than applies a transformation to a point. This function is used by tiles that depend on that geometry to implement their function visualize. module type GEOMETRY = sig type point type transformation module Tgroup : GROUP with type element = transformation val apply : transformation -> point -> point end For instance, to de)ne hyperbolic geometries [3] (represented by its PoincarJe model in Euclidean geometry), the transformations are complex homographies of the form ha; = 

z+a 1 + az M

which we simply represent as a record with components  and a. module Hyperbolic_geometry = struct type point = Mlgraph.point type transformation = {iso_m:complex; iso_a:complex} let identity = {iso_m=cx_1;iso_a=cx_0} let compose {iso_m=mu1; iso_a=a1} {iso_m=mu2; iso_a=a2}= {iso_m= div_cx (mult_cx mu1 (add_cx mu2 (mult_cx a1 (conjugate a2)))) (add_cx cx_1 (mult_cx mu2 (mult_cx (conjugate a1) a2))); iso_a= div_cx (add_cx a1 (mult_cx mu2 a2)) (add_cx mu2 (mult_cx a1 (conjugate a2)))} module Tgroup = struct type element = transformation let compose = compose let identity = identity end let apply_cx {iso_m=mu; iso_a=a} z= ... let apply iso pt = point_of_cx (apply_cx iso (cx_of_point pt)) ...... end

The module contains other elements and in particular, de)nitions for speci)c transformations such as inversions and hyperbolic rotations, which are used to interpret the generators of formal hyperbolic groups.

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GEOMETRY GROUP

CANONICAL_GROUP

MAPPING

Construct_Tiling

TILE

TILING

Fig. 3. Modules and functors.

Now we can summarize the construction of tilings in one functor that takes all the necessary ingredients as parameters: module Construct_tiling (Group: CANONICAL_GROUP) (Geom: GEOMETRY) (GenMap: MAPPING with type source = Group.element and type dest = Geom.transformation) (GenColorMap: MAPPING with type source = Group.element and type dest = Permutation.permutation) (Tile: TILE with type transformation = Geom.transformation * Permutation.permutation) = Make_tiling (Make_generator_from_canonical (Make_canonical_generator(Group)) (MakePair (Make_morphism (Group) (GenMap) (Geom.Tgroup)) (Make_color_morphism (Group) (GenColormap))) (Tile) This leads to a simpli)ed graphical representation which appears in Fig. 3. 7. Conclusion What we have obtained is a very generic program that can produce any tiling with a computable symmetry group that operate transitively on the tiles. For instance, the two

G. Cousineau / Theoretical Computer Science 281 (2002) 207 – 217

217

Fig. 4. Escher’s Circle Limit III and a variant.

tilings in Fig. 4 are obtained by just switching the )sh Tile module to a kangaroo Tile module (the symmetry groups are the same though the topological aspects are quite diHerent). We can also obtain Euclidean tilings by changing the symmetry group into a Euclidean group, the geometry into Euclidean geometry and also changing the tile and the mappings. This 2exibility relies on the powerful programming constructs that have been used. Of course, this is only one speci)c example and we cannot infer from it how, more generally, mathematical structures should be implemented on computers. In particular, we had no need here for inheritance which would have been necessary if we had to deal with hierarchies of structures such as monoids, groups, rings, )elds etc. But nevertheless, we think it shows at least that advanced programming constructs can help a lot in designing and implementing mathematical applications. References [1] E. Chailloux, P. Manoury, B. Pagano, Objective Caml, O’Reilly, 2000. [2] G. Cousineau, M. Mauny, The Functional Approach to Programming, Cambridge University Press, Cambridge, 1997. [3] H.S.M. Coxeter, Introduction to Geometry, Wiley, New York, 1980. [4] H.S.M. Coxeter, W.O.J. Mauser, Generators and Relations for Discrete Groups, Springer, Berlin, 1972. [5] The GAP Group, GAP: Groups, Algorithms and Programming, Version 4.2, 2000. [6] B. GrQunbaum, G.C. Shephard, Tilings and Patterns, Wiley, New York, 1987. [7] M. Ko2er, Maple: an Introduction and Reference, Addison-Wesley, Reading, MA, 1997. [8] Ph. Lechenadec, Canonical Forms in Finitely Presented Algebras, Pitman, London, 1986. [9] S. Wolfram, The Mathematica Book, 4th Edition, Cambridge University Press, Cambridge, 1999.

Theoretical Computer Science 281 (2002) 219 – 233

www.elsevier.com/locate/tcs

Reduction de la non-linearite des morphismes d’arbres Recognizable tree-languages and non-linear morphisms Max Dauchet ∗ , Sophie Tison, Marc Tommasi LIFL, UPRESA 8022 CNRS UA 369, Bˆatiment M3, Universite de Lille I, 59655 Villeneuve d’Ascq Cedex, France

Abstract We try to understand certain complex phenomena which appear in the study of tree morphisms in the non-linear case. We prove that every transformation performed by a quasi-alphabetic homorphism that maps a recognizable tree language to another one can be done “without using non-linearity”. As a corollary, we get an extension of the cross-section theorem. R esum e Le but de ce papier est d’etudier certaines proprietes des morphismes d’arbres et d’eclairer certains phenom3enes complexes propres aux arbres. En particulier, on montre que si l’image par un morphisme quasi-alphabetique d’une forˆet reconnaissable est reconnaissable, on peut
reduire la non linearite du morphisme (en un sens que l’on precisera). Ce resultat nous permet de c 2002 Published by Elsevier Science B.V. generaliser le theor3eme de cross-section. 

1. Introduction Dans ce papier nous etudions certaines proprietes propres aux morphismes d’arbres. En e9et, si les principaux concepts—telle la reconnaissabilite par automate ;ni— de;nis sur les forˆets sont des generalisations naturelles du cas des langages de mots, de nombreuses proprietes ne se transposent pas: en particulier, un phenom3ene nouveau et important est celui de la non-linearite. Un terme t est dit non lineaire si il contient plusieurs occurrences d’une mˆeme variable. La manipulation de termes non lineaires est courante en programmation logique, fonctionnelle ou equationnelle mais ∗

Corresponding author. E-mail addresses: max.dauchet@li?.fr (M. Dauchet), tison@li?.fr (S. Tison), tommasi@li?.fr (M. Tommasi). c 2002 Published by Elsevier Science B.V. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 4 - 2

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M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

les
outils a3 notre disposition sont peu nombreux. Par exemple, l’ensemble des instances closes d’un terme non-lineaire n’est pas reconnaissable et donc l’ensemble des formes reductibles d’un syst3eme de reecriture n’est pas necessairement reconnaissable si le syst3eme n’est pas lineaire. Ou encore, pour citer un autre exemple qui marque la di9erence entre le cas lineaire et le cas general dans le domaine de la reecriture, un syst3eme de reecriture sans paire critique est con?uent dans le cas lineaire gauche, non con?uent dans le cas general. Face a3 cette situation, on peut envisager plusieurs approches; une premi3ere approche consiste a3 de;nir de nouveaux outils qui permettent de manipuler des
egalites, voire des di9erences; plusieurs formalismes ont ete proposes, comme les automates a3 tests [7,4,6,5] ou les grammaires synchronisees [14]. Une demarche complementaire consiste a3 reduire la non-linearite quand c’est possible. Le principe general de cette demarche peut s’enoncer grossi3erement par:
quand la reconnaissabilite est conservee, la non-linearite peut eˆ tre evitee (dans un sens a3 preciser). Par exemple, Kucherov et Tajine, Hofbauer et Huber, Gilleron et V3agvOolgyi etudient les syst3emes de reecriture dont l’ensemble des formes reductibles est reconnaissable et les transforment en syst3emes lineaires [11,12,13,17]. Dans cet article, nous appliquons cette demarche au cas des morphismes. En e9et, l’image homomorphe d’une forˆet reconnaissable est reconnaissable si le morphisme est lineaire, non-necessairement reconnaissable sinon—et donc la famille des langages reconnaissables d’arbres n’est pas close par morphisme, contrairement au cas des mots. Cependant, il existe des images par morphisme non-lineaire de forˆets reconnaissables— mˆeme in;nies—qui sont reconnaissables. C’est surtout cette situation que nous allons examiner ici. L’idee intuitive est la suivante: si une forˆet reconnaissable contient une in;nite d’arbres de la forme t(vi ; vi ), elle contient aussi des arbres de la forme t(vi ; vj ), avec vi di9erent de vj ; en exploitant ceci, on peut montrer le resultat de base suivant: Th eoreme. Soit  un morphisme quasi-alphabetique et K une forˆet reconnaissable; si (K) est reconnaissable, il existe une forˆet reconnaissable R et un morphisme lineaire, quasi-alphabetique, complet et strict tel que  ◦ soit lineaire, quasialphabetique, complet et strict, (R) ⊂K et  ◦ (R) = (K). (Les de8nitions des di9erentes classes suivent l’introduction.) Ce resultat technique permet d’etendre le theor3eme de transversale rationnelle (ou
cross-section): soit R une forˆet reconnaissable,  un morphisme, le theor3eme etablit l’existence d’une forˆet reconnaissable R
incluse dans R telle que  soit une bijection entre R
et (R). Le resultat a ete etendu des mots aux arbres quand  est lineaire [1]; il est en general faux pour un morphisme d’arbres quelconque. Nous conjecturons que l’hypoth3ese
 lineaire peut eˆ tre a9aiblie en
(R) reconnaissable; nous prouvons ici cette conjecture dans le cas o3u  est quasi-alphabetique. On peut appliquer aussi ce resultat aux bimorphismes et en particulier, aux bijections rationnelles [15]; on peut ainsi enoncer: Th eoreme. Toute bijection entre forˆets reconnaissables realisee par un bimorphisme quasi-alphabetique peut eˆ tre realisee par un bimorphisme lineaire. Nous conjecturons que l’hypoth3ese
quasi-alphabetique est super?ue.

M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

221

2. Notations et d enitions Soit un alphabet gradue et X = {x1 ; : : : ; xi ; : : :} un ensemble denombrable de variables. Nous notons X n l’ensemble ;ni des n variables {x1 ; : : : ; x n }. L’ensemble des termes sur indexes par X (resp. par X n ) est note T ( ; X ) (resp. T ( ; X n )) et T ( ) est l’ensemble des termes clos sur , i.e. sans variable. Un p-uple de termes est note ˜v = (v1 ; : : : ; vp ) et l’ensemble des p-uples de termes de T ( ; X n ) est T ( ; X n )p . L’ensemble des termes dont les occurrences de variables lues de gauche a3 droite sont exactement x1 ; : : : ; x n est designe par CTn ( ). Les termes de CTn ( ) sont dits d’arite n. Nous designons par n l’ensemble des lettres de d’arite n. Tout a de n s’identi;e alors a3 a(x1 ; : : : ; x n ) et x1 ; : : : ; x n sont appelees variables de a. L’ensemble des termes
n ( ). contenant exactement une occurrence de chaque variable dans X n est note CT Soient et  deux alphabets gradues. Un morphisme  de T ( ; X ) dans T (; X ) est de;ni par une application de dans T (; X ). Le morphisme  est lineaire (resp. complet) 1 si pour tout a de , chaque variable de a apparaˆRt au plus une fois dans (a) (resp. si chaque variable de a apparaˆRt au moins une fois dans (a)).  est strict si l’image d’une lettre contient au moins une lettre. Une quasi-lettre est un terme de la forme a(v1 ; : : : ; vn ) o3u a est une lettre et chaque vi est soit un terme clos, soit une variable. Un morphisme est alphabetique (resp. quasi-alphabetique ) si l’image d’une lettre est une lettre (resp. une quasi-lettre) ou une variable. Soit t un arbre de T ( ; X ); on note |t| sa hauteur, i.e. le nombre d’occurrences de lettres sur un plus long chemin. pour un morphisme , ||= max ∈ {|()|}. L’ensemble des variables qui apparaissent dans t est Var(t). En particulier, si  est un morphisme et a une lettre, Var((a)) sera confondu avec l’ensemble des indices correspondant aux variables apparaissant dans (a). Pour tout t de T ( ; Xp ) et tout p-uple ˜t
=(t1 ; : : : ; tp ), t:˜t
designe l’arbre obtenu en substituant dans t, pour tout i6p, ti a3 chaque occurrence de xi . Nous disons que v est facteur de t si t se decompose en u: v: w ˜ . Dans la suite, la notation u: v: w ˜ qui met en evidence un facteur v dans un arbre, suppose implicitement que u ∈ T˜ ( ; X1 ). L’ensemble F(t) designe tous les facteurs de t et si K est une forˆet, F(K) designe l’union des F(t) pour t dans K. Nous supposons le lecteur familiarise avec les automates d’arbres [10,8]. Un automate d’arbres est represente par ( ; Q; F; T ) avec F l’ensemble des etats de sortie, T l’ensemble des transitions. 3. Fid elit e et morphismes Dans cet article nous considerons des morphismes quasi-alphabetiques. Si  est un tel morphisme de T ( ; X ) dans T (; X ), chaque image de lettre f(x1 ; : : : ; x n ) de par  est de la forme xi ou encore u(x i1 ; : : : ; x ik ) o3u u ∈CTk () et chaque variable apparaˆRt a3 profondeur 1 dans u. En d’autres termes,  peut e9acer f et le
remplacer par un de ses ;ls, ou
transformer f en inserant des sous-termes clos, en supprimant, copiant ou echangeant ses ;ls directs. Pour etudier les transformations par , nous 1

Complet est aussi appele nondeleting par d’autres auteurs [9].

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M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

utiliserons des correspondances entre les facteurs d’un terme, et les facteurs de son image. Nous les enonTcons dans la remarque suivante. Remarque 1. Soit un v terme de T ( ) et soit p1 une position dans t = (v) et t1 le facteur de t a3 cette position; alors si t1 est de taille superieure ou egale a3 ||, on peut trouver au moins une correspondance de p1 dans v, i.e. il existe v0 , v1 , t 0 tels que v0 :v1 =v, t 0 : t1 =(v)= t et (v0 )= t 0 , (v1 ) = t1 , avec p1 correspondant a3 une occurrence de variable dans t 0 . Fid elit e. Rappelons qu’une relation  de U × V est dite 8d<ele si pour tout t de V , −1 (t) est ;ni. Nous allons montrer que pour tout morphisme  et toute forˆet reconnaissable R, on peut trouver R
reconnaissable inclus dans R tel que (R) = (R
) et (; R
) soit ;d3ele (on dit que (; R
) est ;d3ele si la relation {(t; (t)) | t ∈R
} est ;d3ele). D enition I.  est dit p-dele sur K si pour tout facteur u de K, |u| est inferieure a3 p × (1 + |(u)|). Il se peut que (; K) soit ;d3ele sans eˆ tre p-;d3ele pour aucun p (considerer K = {b(a i d; a j d) | i6j} et (b(x; y)) =y; (a(x)) = a(x); (d) = d). Par contre, si K est reconnaissable, alors les deux notions coORncident. Nous nous limitons ici a3 une simple proposition qui lie reconnaissabilite et p-;delite. Proposition 1. Soit ;  deux alphabets gradues,  un morphisme de T ( ; X ) dans T (; X ) et K une forˆet reconnaissable sur l’alphabet . On peut construire une forˆet reconnaissable K
incluse dans K tel que (K) = (K
) et telle que (; K
) soit p-8d<ele. D emonstration. Nous allons construire une forˆet K
a3 partir de K en oˆ tant les termes de K suVsamment grands, dont l’image peut eˆ tre obtenue par application de  a3 des termes plus petits. On distingue deux ensembles qui vont traiter les cas des r3egles e9aTcantes de la forme (a)= xi , et ceux des r3egles non compl3etes de la forme (f) = u(xi1 ; : : : ; xin ) pour f d’arite plus grande que n. Soit Ei = {t | ∀u ∈ F(t) |u| ¿ i ⇒ |(u)| ¿ 0} Ci = {t | ∀u ∈ F(t); ∀f ∈ ; [t = u:f(u1 ; : : : ; un ) ∧ xj ∈ Var((f))] ⇒ |uj | 6 i} Ei est l’ensemble des termes qui ne contiennent pas de facteurs e9aTcables par  de taille superieure a3 i. L’ensemble Ci designe les termes pour lesquels seuls des soustermes de hauteur inferieure a3 i peuvent eˆ tre elagues par des r3egles non-compl3etes. Soit A= ( ; Q; F; T ) un automate deterministe ascendant qui reconnaˆRt K. Soit p = 2 × |Q| (|Q| est le cardinal de Q). Soit K
= K ∩ Ep ∩ Cp .

M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

223

Nous montrons maintenant que (K
) =(K). Puisque K
⊆K, il suVt de montrer (K
) ⊇(K). Soit t dans (K), soit v le plus petit terme (en nombre de noeuds) de K tel que (v)= t. Nous montrons que v appartient a3 la fois a3 Ep et Cp et donc a3 K
. • Si v n’est pas dans Cp alors v = u: f(u1 ; : : : ; un ) et un sous-terme ui de hauteur superieure a3 p =2 × |Q| est e9ace 2 par . Soit q l’etat associe par A a3 ui . Sur une plus longue branche de ui , on trouve necessairement deux occurrences d’un mˆeme etat de A. On peut alors appliquer le lemme de l’etoile dans les arbres pour construire un terme ui
plus petit que ui et dont l’etat associe par A est q. On obtient alors une contradiction puisque u:f(u1 ; : : : ; ui
; : : : ; un ) est dans K, plus petit que v et de mˆeme image t par . • Si v n’est pas dans Ep , c’est qu’il existe un facteur u de v, de taille superieure a3 p et e9ace par . Alors v peut s’ecrire sous la forme v = v
: u: v

tel que t =(v) = (v
): (v

). Considerons un chemin & dans u de longueur maximale et la position ' de u a3 laquelle v

apparaˆRt dans v. Alors & = &1 : &2 o3u &1 est l’ensemble des positions inferieures a3 la position '. Puisque la longueur de & est superieure a3 2 × |Q|, soit &1 soit &2 est de longueur superieure a3 |Q|. Comme dans le premier cas, nous pouvons trouver en appliquant le lemme de l’etoile dans les arbres un terme ui
plus petit que ui pour lequel A associe le mˆeme etat qu’3a ui et tel que v
: u
: v

soit d’image t par . D’o3u la contradiction. En;n, nous montrons par recurrence sur la taille de tout facteur u de K
que  est bien p-;d3ele sur K
, donc que |u|6p(1 + |(u)|). • Soit u ∈K
, |u|¡p. Il est facile de voir que mˆeme si |(u)|= 0, alors |u|6p(1 + |(u)|). • Soit u ∈K
, |u|62 × p. Si |u|¿p alors |(u)|¿1 puisque u est dans Ep . Donc, |u|6p(1 + |(u)|). • Supposons que  est bien p-;d3ele pour l’ensemble des facteurs de K
de taille inferieure a3 k et soit un facteur u de K
de taille k + 1. Remarquons que u peut s’ecrire w:w
avec |u|= p + |w
|. Puisque t ∈Ep , nous avons |(w)| ¿ 1:

(1)

De plus, puisque |w
|¿p et que w
est un facteur de t dans Cp , alors w
n’est pas elague par . On obtient donc p(1 + |(u)|) ¿ p(1 + 1 + |(w
)|)

(d
apr es 3 (1))

¿ p + p(1 + |(w
)|) ¿ p + |w
|

(par hypothese 3 d
induction

¿ |u|:

2

Pour la condition d’appartenance a3 Cp , on pourrait se limiter a3 p = |Q|.

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M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

4. Non-lin earit e et reconnaissabilit e Soit  un morphisme et K un reconnaissable. Si  est lineaire, (K) est reconnaissable; la reciproque est bien sˆur fausse, comme l’illustre l’exemple suivant. Soit le morphisme non lineaire :

a(x) → c(x; x) b(x; y) → b(x; y) d→ d

et K l’ensemble de tous les arbres clos construits a3 partir de b, d, et a(d)). Alors (K) est l’ensemble (reconnaissable) de tous les termes construits a3 partir de b, d, et c(d; d). Cependant la proposition suivante correspond a3 une sorte de reciproque faible. Certes, il n’existe pas d’ensemble K
reconnaissable et inclus dans K tel que (K) = (K
) et tel que  soit lineaire sur K
. Neanmoins, nous pouvons dire informellement que presque tout peut eˆ tre obtenu de faTcon lineaire. Ainsi, dans l’exemple ci-dessus, le seul sous-arbre qui ne peut eˆ tre obtenu de faTcon lineaire est c(d; d) et si nous de;nissons R = T{a
; b; d} et & : b(x; y) → b(x; y) d→d a
→ c(d; d) alors  ◦ & est lineaire, &(R) ⊂ K et  ◦ &(R) = (K). La proposition suivante traduit cette idee. Elle suppose le morphisme  quasi-alphabetique mais nous conjecturons qu’elle reste vraie dans le cas non-alphabetique. Proposition 2. Soit  un morphisme quasi-alphabetique de T ( ; X ) dans T (; X ) et K une forˆet reconnaissable sur . Si (K) est reconnaissable alors il existe une forˆet reconnaissable R et un morphisme & lineaire, quasi-alphabetique complet et strict tel que  ◦ & soit lineaire, &(R) ⊂K et  ◦ &(R) = (K). La preuve de cette proposition se decompose ainsi : dans un premier temps nous exprimons que tout terme dans (K), sauf un nombre ;ni, peut eˆ tre obtenu sans pour autant que la non-linearite du morphisme soit necessaire. Pour cela, nous de;nissons la condition Cn (t 0 ; ˜t ) qui exprime que le terme t 0 :˜t de (K) peut eˆ tre obtenu en construisant chaque ti de ˜t independamment (voir Fig. 1). L’ensemble ♥ des termes dans ˜t (pour tout t 0 ) veri;ant la condition Cn (t 0 ; ˜t ) est ensuite montre co;ni (Lemme 3). Pour prouver ceci, nous construisons cet ensemble en utilisant le fait que K est reconnaissable. La preuve repose sur une induction sur le nombre n de variables dans le contexte t 0 . Dans un deuxi3eme temps, nous utilisons la propriete de p-;delite du morphisme  et la co;nitude de ♥ pour de;nir une extension (;nie) de l’alphabet qui encode en une simple quasi-lettre, tout terme de hauteur trop petite pour ne pas avoir l’assurance que son image est dans ♥. Cette transformation legitime l’apparition du morphisme &.

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225

Fig. 1. La condition Cn (t 0 ; t). Construire d’abord les termes de t de faTcon independante.

Fig. 2. Le decoupage selon dec de a(b(a(a(b(c); c); a(c; c))); c)). Les elements dans Ter sont a(c; c) et b(c).

La phase ;nale de la preuve de;nit un decoupage en couches de tout terme image dans (K) de la racine vers les feuilles (Fig. 2). La presence de termes de ♥ dans la couche la plus basse (vers les feuilles) de ce decoupage, conjuguee avec la condition Cn (:; :) va permettre d’exhiber une construction
lineaire de cette image. 5. Preuve de la Proposition 2 Montrons d’abord que nous pouvons supposer que (K) = T (). En e9et, supposons que la proposition soit vraie avec cette restriction. Maintenant, soit (; K) tel que (K) soit reconnaissable. Soit W une copie de  et i : W →  l’injection naturelle. On peut

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W K) W par: de;nir (; W • K = K ∪ i−1 (c (K)) o3u c A designe le complementaire de A dans T () • W : W = =; W=W =i; KW est reconnaissable; W conserve les proprietes (ˆetre ;d3ele et quasi-alphabetique) de W K)= W  et ( T (). Donc, d’apr3es notre hypoth3ese, la proposition s’applique a3 W et KW et nous pouvons construire & et RW qui satisfont les conditions de la proposition. Considerons maintenant R = RW ∩ &−1 (K) ; le lecteur peut alors veri;e que la proposition est vraie aussi pour (; &; R). Nous supposons donc, a3 partir de maintenant, que  est quasi-alphabetique, (; K) ;d3ele et (K)= T (). La condition Cn (t 0 ; ˜t ) est essentielle dans notre construction. Dans le Lemme 3 dit du coeur, notre raisonnement par induction repose sur la possibilite d’
etendre cette condition de n a3 n + 1. L’expression de la condition CF(t 0 ; t) et ses proprietes (Lemme 1) nous permet de le faire. Condition CF(t 0 ; t). Soient t 0 ∈CT1 () et t ∈T (). La condition CF(t 0 ; t) est de;nie par:    (v0 ) = t0 ∃v0 ∈ CT1 ( )   (CF(t0 ; t)) ∀v ∈ K; (v) = t 0 :t ⇒ : (v1 ) = t ∃v1 ∈ T ( ) v = v0 :v1 Intuitivement, CF(t 0 ; t) signi;e que mˆeme si t 0 : t peut eˆ tre obtenu d’une faTcon non lineaire, t est toujours obtenu independamment de tout sous-terme de t 0 . Condition Cn (t 0 ; ˜t ). Soit n un entier et t 0 ∈CTn (). Soit ˜t un n-uple de termes dans T ()n . La condition Cn (t 0 ; ˜t ) est de;nie par:

v v ∈ K 0 :˜ n n ( ); ∃˜v ∈ T ( ) (v0 ) = t0 Cn (t0 ; t) ∃v0 ∈ CT (Cn (t0 ; t)) (˜v) = ˜t: La condition Cn (t 0 ; ˜t ) est moins restrictive. Intuitivement, cela signi;e qu’il y a au moins une faTcon d’obtenir ˜t de faTcon lineaire dans t 0 :˜t, i.e. telle que chaque composante de ˜t soit obtenue independamment d’une part des autres composantes, d’autre part des sous-termes de t 0 . Comme (K)= T (), CF(t 0 ; t) implique de faTcon evidente Cn (t 0 ; t) pour tout couple (t 0 ; t) de CT1 () × T (). Le lemme suivant donne une condition suVsante pour CF(t 0 ; t): Lemme 1. Si |t|¿ max(||; |t 0 |), CF(t 0 ; t) est veri8ee. D emonstration. Si on pense a3 la signi;cation de CF(t 0 ; t), le lemme est evident : si la profondeur de t est superieure a3 ||, t ne peut-ˆetre un sous-terme de l’image d’une lettre ; si de plus |t| est superieure a3 |t 0 |, t ne peut eˆ tre egal a3 un sous-terme de t 0 , et donc est necessairement obtenu independamment de tout sous-terme de t 0 .

M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

227

Formellement, soit v tel que (v)=t 0 : t. Par la remarque 1, comme  est quasialphabetique et que la profondeur de t est superieure a3 ||, on peut trouver (v0 ; v1 ) dans CT1 ()1 × T () tel que: • v0 :v1 ∈K; • (v0 :v1 ) =t 0 : t; • (v0 ) = u: (x; : : : ; x); u ∈CTp (); p¿1; • (v1 ) = t. Alors t 0 : t =u(t; : : : ; t) pour u ∈CTp (); p¿1. Si p¿1 alors t est un sous-terme de t 0 . Or, |t|¿|t 0 | donc p = 1. La condition CF(t 0 ; t) est donc veri;ee. Nous utilisons cette propriete pour prouver le lemme suivant: Lemme 2. Soit un n-uple de termes ˜t =(t1 ; : : : ; tn ):  Cn (t0 :(x1 ; : : : ; x n ; tn+1 ); ˜t) ⇒ Cn+1 (t0 ; (˜t; tn+1 )): CF(t0 :(˜t; xn+1 ); tn+1 ) D emonstration. Remarquons d’abord que t 0 : (˜t; tn+1 ) = t 0 : (x1 ; : : : ; x n ; t n+1 ):˜t. Donc, si la condition Cn (t 0 : (x1 ; : : : ; x n ; tn+1 ); ˜t ) est veri;ee, il existe au moins une faTcon d’obtenir t 0 :(˜t; tn+1 ) par  telle que chaque composante de ˜t est obtenue independamment des autres et de chaque sous-terme de t 0 , i.e.:  (v0 ) = t0 :(x1 ; : : : ; x n ; tn+1 );    (˜v) = ˜t; ∃v0 ; ∃˜v tels que    v0 :˜v ∈ K: Mais on a aussi t 0 (˜t; tn+1 )= t 0 :(˜t; xn+1 ): tn+1 et donc par CF(t 0 :(˜t; xn+1 ); tn+1 ), tn+1 est obtenu independamment de chaque sous-terme de t 0 :(˜t; xn+1 ), i.e.:  (w0 ) = t0 :˜t;    ∃w0 ; w1 tels que (w1 ) = tn+1 ;    w0 :w1 = v0 :˜v: Mais alors, (w0 ∧ v0 )= t 0 (avec w0 ∧ v0 = inf {w0 ; v0 } pour l’ordre u¡v si v = u:w) et Cn+1 (t 0 ; (˜t; tn+1 )) est veri;ee. Nous enonTcons maintenant le lemme central (le lemme du coeur) qui exprime que les branches qui ne peuvent pas eˆ tre obtenues de faTcon lineaire appartiennent a3 une forˆet ;nie. L’idee de la preuve est la suivante: d’apr3es ce qui prec3ede, pour tout arbre t de CT1 (), CF(t; v) est veri;ee pour presque tout v. En utilisant le fait que (K) = T (), nous montrons qu’il existe une forˆet co;nie, i.e. dont le complementaire dans T () est ;ni, telle que pour tout arbre t de CT1 (), pour tout v de cette forˆet, CF(t; v) est veri;ee. Par recurrence, nous prouvons ensuite:

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Lemme 3 (du coeur). On peut construire une forˆet co8nie sur  qu’on notera ♥, telle que pour tout entier n, pour tout arbre t 0 de CTn (), et pour tout vecteur ˜t de ♥ n , Cn (t 0 ; ˜t ) est veri8ee. D emonstration. Soit A =( ; Q; F; T ) un automate ascendant qui reconnaˆRt K. Soit SQ
la forˆet reconnue par l’automate ascendant ( ; Q; Q
; T ). Notre forˆet ♥ sera l’intersection de toutes les forˆets (SR ) pour toute partie R de Q telle que (SR ) soit co;nie. Comme Q est ;ni et que chaque forˆet (SR ) est co;nie, ♥ est bien co;nie. • n = 1. Soit t 0 de CT1 (). Soit Q
l’ensemble des etats q tels qu’il existe un arbre v0 ∗ de CT1 ( ) avec v0 (q) → f, avec f ∈F et (v0 (x)) = t 0 . Soit la forˆet Ct 0 de;nie par Ct 0 =def {t ∈T () | CF(t 0 ; t)}. D’apr3es le Lemme 1, Ct 0 contient {t ∈T () | |t|¿ max(|t 0 |; |()|)} et est donc co;nie. Par construction, (SQ
) = Ct 0 et donc est co;nie. Par consequent, pour tout t 0 , Ct 0 contient ♥. Donc, pour tout t 0 de CT1 (), chaque arbre de ♥ appartient a3 Ct 0 , i.e. pour chaque arbre t 0 de CT1 (), pour chaque t de ♥, CF(t 0 ; t) est veri;ee, donc C1 (t 0 ; t) l’est aussi. • pas de recurrence: supposons la propriete vraie pour n. Soit t 0 dans CTn+1 (), ˜t dans ♥ n . Par hypoth3ese de recurrence, pour chaque arbre tn+1 de T (), Cn (t 0 :(x1 ; : : : ; x n ; tn+1 ); ˜t ) est veri;ee. Donc, par les Lemmes 1 et 2, si la profondeur de tn+1 est superieure a3 celle de t 0 :(˜t; xn+1 ) et a3 la profondeur maximale de l’image d’une lettre, Cn+1 (t 0 ; (˜t; tn+1 )) est assuree. Soit maintenant Q

la partie de Q de;nie par:    ∃i;              

Q = q ∈ Q               

 ∃v0 ∈ CTn+1 ( ); ∃(q1 ; : : : ; qn ) ∈ Qn tels que:     ∗   v0 (q1 ; : : : ; qi ; q; qi+1 ; : : : ; qn ) → F    (v0 (x1 ; : : : ; xi ; x; xi+1 ; :; x n )) = t0 :(xi1 ; : : : ; xin ; x) :     ∀j = k; ij = ik      ∀j; tj ∈ (Sqij )

Alors, tn+1 appartient a3 (SQ

) si et seulement si Cn+1 (t 0 ; (˜t; tn+1 )) est veri;ee; donc (SQ

) est co;nie et contient ♥ par construction; d’o3u Cn+1 (t 0 ; (˜t; tn+1 )) est veri;ee pour tout tn+1 de ♥. Par ce lemme on est maintenant en mesure de construire & et R satisfaisant aux conditions de la Proposition 2. Soit k = maxt ∈♥ = |t| + 1 (Comme ♥ est co;ni, k est bien de;ni). Comme (; K) est ;d3ele, nous savons d’apr3es la premi3ere partie qu’il existe un entier p tel que (; K) soit p-;d3ele. Soit la forˆet P de;nie par:     |s| 6 p × (1 + k)          P =def s ∈ CT1 ( )  (s) est une quasi-lettre :         (s) est lineaire

M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

229

Soit maintenant un nouvel alphabet P en bijection avec P : P =def { s | s ∈P},  s ayant l’arite de s. Alors & designe le morphisme de;ni par &( s ) = s, pour tout  s dans P et on de;nit R par R=&−1 (K). Alors: a. Par construction, & est lineaire, complet, quasi-alphabetique, strict et &(R) est inclus dans K. De plus,  ◦ & est lineaire par de;nition mˆeme de P. b. II faut montrer que  ◦ &(R)= T ( ); ici reside la diVculte de la preuve. Intuitivement il s’agit de montrer que les sous-arbres qu’on ne peut pas obtenir par  de faTcon lineaire et qui sont donc, d’apr3es le Lemme 3, le complement de ♥, sont images d’arbres de P. Nous allons pour cela de;nir pour chaque arbre t de T () un decoupage note dec(t) tel que chaque branche
terminale de t soit de profondeur k et donc appartienne a3 ♥. Nous de;nissons dec par: dec(a:(t1 ; : : : ; tn )) = (; dec(u1 ); : : : ; dec(un )) avec : a:(t1 ; : : : tn ) = :(u1 ; : : : un )  = a:(v1 ; : : : vn ) avec vi = ti si |ti | ¡ k; vi = ti sinon: Par exemple pour k = 2 avec 0 ={c}; 1 = {b}; 2 = {a}, on obtient: dec(a(b(a(a(b(c); c); a(c; c))); c)) = (a(x1 ; c); (b(x1 ); (a(x1 ; x2 ); (a(x1 ; c); b(c)); a(c; c)))): On peut remarquer que, pour tout terme t, chaque element de dec(t) est soit un terme clos de hauteur k, soit une quasi-lettre de hauteur au plus k. Maintenant, soit t un terme de T (). On va montrer qu’il existe un terme v de R tel que  ◦ &(v)= t. On distinguera deux cas: • |t|6k. Soit alors u dans K tel que (u) = t; comme  est p-;d3ele, u appartient a3 P et donc  u appartient a3 R avec  ◦ &( u ) = t. • |t|¿k. Alors chaque element de dec(t) est de profondeur inferieure ou egale a3 k. De plus, on peut decomposer t en t = to:˜t avec soit ˜t correspondant aux elements clos du decoupage: chaque composante de ˜t est de profondeur k et donc appartient
n ( ), a3 ♥. Donc, par le Lemme 3, Cn (t 0 ; ˜t ) est veri;ee: on peut trouver v0 dans CT et ˜v dans T ( )n tels que v0 :˜v appartienne a3 K et (v0 ) = t 0 ; (˜v) =˜t. On va montrer qu’on peut trouver un decoupage de v0 :˜v qui corresponde exactement par  au decoupage dec(t) et tel que chaque element du decoupage de v appartienne a3 P. Remarquons d’abord que par construction les composantes de ˜v sont dans P puisqu’elles ont pour image des terme clos de hauteur k et que (; K) est p-;d3ele. Il suVt donc maintenant de trouver une decomposition de v0 correspondant par  a3 dec(t). Raisonnons en fonction de n(t), le nombre d’elements non clos dec(t). Si n(t) = 1, le decoupage sera simplement v0 . Comme Cn (t 0 ; ˜t ) est veri;ee, (v0 ) = t 0 et est donc une quasi-lettre lineaire. De plus, comme (; K) est p-;d3ele, |v0 |6p× (1 + k).

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M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

Si n(t)¿1, on peut d’abord remarquer que puisque  est quasi-alphabetique et que les elements du decoupage sont des quasi-lettres ou des termes clos de hauteur superieure a3 ||, on peut decomposer v0 en 10 (w1 ; : : : ; wk ), (10 ) correspondant au premier element de dec(t) aux variables pr3es. De plus, comme Cn (t 0 ; ˜t ) est veri;ee, (10 ) est lineaire car chacune de ses variables est remplacee dans t par un terme qui contient une composante de ˜t. En iterant le processus sur les termes wi , on peut de;nir un decoupage de v0 tel que chaque element du decoupage ait pour image un element du decoupage de t (donc une quasi-lettre lineaire). De plus, comme (; K) est p-;d3ele, et que chaque element de dec(t) est un element de F(K), chaque element du decoupage a une hauteur au plus p × (1 + k) et donc est dans P. Donc v0 :˜v appartient a3 appartient a3 & ◦ &−1 (K) et t appartient a3  ◦ &(R). Remarque. Mˆeme si  est alphabetique, & peut ne pas eˆ tre alphabetique; par exemple, soit  : a(x) → b(x; x); c → c; K =a(c). Supposons qu’il existe R reconnaissable et lineaire, alphabetique, complet et strict tel que  ◦ soit lineaire, (R) ⊂ K et  ◦ (R) = (K); alors, soit t tel que  ◦ (t) = b(c; c); necessairement, comme  ◦ est complet, alphabetique, lineaire et strict, la taille de t est 3; mais alors, la taille de (t) est 3 egalement et (t) n’appartient pas a3 K. Nous conjecturons que cette proposition est vraie mˆeme si  n’est pas quasialphabetique (et alors bien sˆur  ◦ ne le serait pas non plus). La preuve donnee ici ne s’etend malheureusement pas, le
decoupage de l’image d’un arbre etant beaucoup plus complexe dans le cas general. Toutes les constructions ci-dessus sont e9ectives; en e9et, pour un morphisme quasialphabetique, on peut decider si l’image d’un reconnaissable est reconnaissable et si elle est co;nie (ce resultat est obtenu en utilisant une classe d’automates a3 tests [3]). On peut encore souligner le saut dans la diVculte quand on essaie de generaliser aux morphismes non-quasi-alphabetiques: Le probl3eme de la decidabilite de la reconnaissabilite de l’image d’un reconnaissable par un morphisme quelconque est ouvert. Cette proposition nous permet d’etendre le th eoreme de transversale rationnelle (ou cross-section) dans les arbres. Rappelons le theor3eme de cross-section: si  est un morphisme lineaire de T ( ) dans T (), toute forˆet reconnaissable F contient une partie reconnaissable qui est en bijection par  avec (F). A. Arnold a montre que l’hypoth3ese  lineaire ne pouvait eˆ tre supprimee [1]. Nous conjecturons qu’elle peut eˆ tre remplacee par
(F) reconnaissable, i.e.: Conjecture 1. Soit  un morphisme, F une forˆet reconnaissable telle que (F) soit reconnaissable. alors F contient une forˆet reconnaissable qui est en bijection par  avec (F). Nous montrons ici cette conjecture dans le cas o3u  est quasi-alphabetique: Th eoreme 1 (Theor3eme de transversale rationnelle). Soit  un morphisme quasi-alphabetique, R une forˆet reconnaissable tels que (R) est reconnaissable. Alors, il existe une forˆet reconnaissable R
incluse dans R tel que  soit une bijection de R
sur (R).

M. Dauchet et al. / Theoretical Computer Science 281 (2002) 219 – 233

231

D emonstration. La preuve est simple et utilise le theor3eme de cross-section pour les morphismes lineaires; en e9et, soit  un morphisme quasi-alphabetique et R une forˆet reconnaissable telle que (R) soit reconnaissable. On peut donc trouver & et K satisfaisant aux conditions de la Proposition 2. Comme  ◦ & est lineaire, on peut appliquer le theor3eme de cross-section a3  ◦ & et K. Donc, il existe K
reconnaissable inclus dans K tel que  ◦ &(K
)=  ◦ &(K) = (R) et tel que  ◦ & soit injectif sur K
. Soit R
= &(K
); comme &(K) est inclus dans R, R
est bien inclus dans R; comme & est lineaire, R
est reconnaissable; comme  ◦ & est injectif sur K
,  est injectif sur R
. 6. Bimorphismes Dans ce paragraphe, nous appliquons les resultats precedents aux bimorphismes d’arbres. Un bimorphisme est un triplet  = (; K; 2), ; 2 etant des morphismes et K un reconnaissable, et la transformation associee est de;nie par (t) = 2(−1 (t) ∩ K). Dans le cas des mots, les transformations de;nies par bimorphismes sont exactement les transductions rationnelles [16], dans le cas des arbres la situation est plus complexe [2]. Dans ce paragraphe, nous montrons que sous certaines conditions, on peut supposer que les composantes d’un bimorphisme ont certaines proprietes (linearite, completude, alphabecite; : : :). En particulier, le paragraphe precedent nous permet de reduire la nonlinearite sous des hypoth3eses de reconnaissabilite: La Proposition 2 nous permet d’obtenir les resultats suivants concernant les bimorphismes: Lemme 4. Soit =(; K; 2) un bimorphisme fonctionnel avec  quasi-alphabetique. Si le domaine de  est reconnaissable, on peut construire 
= (
; K
; 2
) e quivalent a<  avec: • 
lin eaire; • 
quasi-alphabetique; (mais l’eventuelle alphabecite de  n’est pas conservee). • 
(resp. 2
) strict (resp. complet) si  (resp. 2) l’etait; • 2
quasi-alphabetique si 2 l’etait; (mais l’eventuelle alphabecite de 2 n’est pas conservee). D emonstration. La preuve est immediate; d’apr3es la Proposition 2, on peut construire un reconnaissable R et un morphisme & lineaire, quasi-alphabetique complet et strict tel que  ◦ & soit lineaire, &(R)⊂K et  ◦ &(R) = (K). Alors soit 
= (
=  ◦ &; K
= R; 2
= 2 ◦ &): vu les proprietes de & et de  ◦ &,
et 2
veri;ent les conditions requises; soit t dans le domaine de ; comme &(K
) ⊂K, il est immediat que 
⊂ (au sens de l’inclusion des relations); comme  ◦ &(K
) = (K),  et 
ont mˆeme domaine; comme de plus  est fonctionnel, on a bien 
equivalent a3 . Symetriquement, on a: Lemme 5. Soit  =(; K; 2) un bimorphisme
injectif—i.e.. (t) ∩ (t
) = ∅, si t = t
— avec 2 quasi-alphabetique. Si l’image de  est reconnaissable, on peut construire

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= (
; K
; 2
) e quivalent a<  avec: • 2
lin eaire; • 2
quasi-alphabetique; (mais l’eventuelle alphabecite de 2 n’est pas conservee). • 2
(resp. 
) strict (resp. complet) si 2 (resp. ) l’etait; • 
quasi-alphabetique si  l’etait; (mais l’eventuelle alphabecite de  n’est pas conservee). D emonstration. Il suVt d’appliquer la Proposition 2 a3 (2; K). La conjonction des deux lemmes precedents donne: Th eorem 2. Soit une bijection de domaine et d’image reconnaissables realisee par un bimorphisme =(; K; 2) avec  et 2 quasi-alphabetiques. On peut alors realiser cette bijection par un bimorphisme 
=(
; K
; 2
) avec: • 
et 2
lin eaires; • 
et 2
quasi-alphabetiques; (mais l’eventuelle alphabecite de  ou 2 n’est pas conservee). • 
(resp. 2
) strict (resp. complet) si  (resp. 2) l’etait. 7. Conclusion L’approche developpee ici face aux probl3emes de non-linearite est complementaire de celle developpee par exemple dans [4,5] o3u sont de;nies de nouvelles classes d’automates pouvant reconnaˆRtre certaines images homomorphes non-lineaires de reconnaissables. L’etude des homomorphismes ni lineaires ni quasi-alphabetiques semble plus diVcile; certaines proprietes obtenues ici dans le cas quasi-alphabetique sont sans doute generalisables mais les techniques de preuve utilisees ici ne semblent pas s’etendre. Pour ;nir signalons un autre probl3eme resolu dans le cas quasi-alphabetique [3] mais ouvert dans le cas general: Peut-on decider si l’image homomorphe d’une forˆet reconnaissable est reconnaissable? References [1] A. Arnold, Le theor3eme de transversale rationnelle dans les arbres, Tech. Report I.T.3-78, Laboratoire d’Informatique Fondamentale de Lille, 1978. [2] A. Arnold, M. Dauchet, Morphismes et bimorphismes d’arbres, Theoret. Comput. Sci. 20 (1982) 33–93. [3] B. Bogaert, F. Seynhaeve, S. Tison, The recognizability problem for tree automata with comparisons between brothers, in: Proc. ETAPS 99 (FOSSACS), Lecture Notes in Computer Science, Vol. 1578, Springer, Berlin, 1999, pp. 150 –164. [4] B. Bogaert, S. Tison, Equality and disequality constraints on direct subterms in tree automata, in: Proc. 9th Annual Symposium on Theoretical Aspects of Computer Science, Vol. 577, Springer, Berlin, 1992, pp. 161–172. [5] A.-C. Caron, H. Comon, J.-L. Coquide, M. Dauchet, F. Jacquemard, Pumping, cleaning and symbolic constraints solving, in: S. Abiteboul, E. Shamir (Eds.), Automata, Languages and Programming, 21st Internat. Colloq., Lecture Notes in Computer Science, Vol. 820, Springer, Berlin, Jerusalem, Israel, 11–14 July, 1994, pp. 436 – 449.

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[6] A.-C. Caron, J.-L. Coquide, M. Dauchet, Encompassment properties and automata with constraints, in: Proc. 5th Internat. Conf. on Rewriting Techniques and Applications, Vol. 690, 1993, pp. 328–342. [7] H. Comon, Equational formulas on order-sorted algebras, in: Proc. Internat. Colloq. on Automata Languages and Programming, Lecture Notes in Computer Science, Vol. 443, Springer, Berlin, 1990, pp. 674 – 688. [8] H. Comon, M. Dauchet, R. Gilleron, D. Lugiez, S. Tison, M. Tommasi, Tree automata techniques and applications, 1997, available on: http://www.grappa.univ-lille3.fr/tata [9] J. Engelfriet, Bottom-up and top-down tree transformations— A comparison, Math. System Theory 9 (1975) 198–231. [10] F. Gecseg, M. Steinby, Tree Automata, Akademiai Kiado, 1984. [11] D. Hofbauer, M. Huber, Linearizing term rewriting systems using test sets, J. Symbolic Comput. 17(1) (1994) 91–129, also in Conditional term rewriting systems (Pont-3a-Mousson, 1992). [12] G. Kucherov, On relationship between term rewriting systems and regular tree languages, in: Ron Book (Ed.), Proc. 4th Internat. Conf. on Rewriting Techniques and Applications, Como, Italy, Lecture Notes in Computer Science, Vol. 488, Springer, Berlin, April 1991, pp. 299 –311. [13] G. Kucherov, M. Tajine, Decidability of regularity and related properties of ground normal form languages, Inform. Comput. 118 (1) (1995) 91–100. [14] S. Limet, P. Rety, E-uni;cation by means of tree tuple synchronized grammars, in: Michel Bidoit, Max Dauchet (Eds.), TAPSOFT ’97: Theory and Practice of Software Development, Lecture Notes in Computer Science, Vol. 1214, Springer, Berlin, 1997, pp. 429–440. [15] H.A. Maurer, M. Nivat, Rational bijection of rational sets, Acta Inform. 13 (4) (1980) 365–378. [16] M. Nivat, Transductions des langages de Chomsky, Ann. de l’Inst. Fourier 18 (1968) 339–456 (in French). [17] S. VagvOolgyi, R. Gilleron, For a rewrite system it is decidable whether the set of irreducible ground terms is recognizable, Bull. Eur. Assoc. Theoret. Comput. Sci. 48 (1992) 197–209.

Theoretical Computer Science (2002) 235 – 249

www.elsevier.com/locate/tcs

Reconstructing permutation matrices from diagonal sums  Alberto Del Lungo Dipartimento di Matematica, Universita di Siena, Via del Capitano 15, 53100 Siena, Italy

Abstract In this paper, we present a result concerning the reconstruction of permutation matrices from their diagonal sums. The problem of reconstructing a sum of k permutation matrices from its diagonal sums is NP-complete. We prove that a simple variant of this problem in which the permutation matrices lie on a cylinder instead of on a plane can be solved in polynomial time. We give an exact, algebraic characterization of the diagonal sums that correspond to a sum of permutation matrices. Then, we derive an O(kn2 )-time algorithm for reconstructing the sum of k permutation matrices of order n from their diagonal sums. We obtain these results by means c 2002 Elsevier of a generalization of a classical theorem of Hall on the 0nite abelian groups.
Science B.V. All rights reserved. Keywords: Combinatorial problem; Discrete tomography; Permutation; Permutation matrix; Diagonal sum; Finite abelian group

1. Introduction A lattice set is a 0nite subset of the integer lattice Z2 . It can be represented by a (0; 1)-matrix 1 as shown in Fig. 1. A discrete X-ray of a lattice set in a direction u is the function giving the number of points on each line parallel to u (see Fig. 1). Discrete Tomography studies the inverse problem of reconstructing a lattice set from its discrete X-rays. This problem is of fundamental importance in 0elds such as image processing [15], statistical data security [10], biplane angiography [12], graph theory [1] and reconstructing crystalline structures from X-rays taken by an electron microscope [11]. For a survey on the state-of-the-art in discrete tomography we suggest the recent book [9]. 

This work is partially supported by Vigoni project and by PAR University Siena project. E-mail address: [email protected] (A. Del Lungo). 1 If k ∈ N, then a (0; 1; : : : ; k)-matrix is one whose entries all are from {0; 1; : : : ; k}.

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 5 - 4

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1 2

1

1

1

1

1

3

X(1,0)F

2

3

3

F

3

0

2

0

1 2

2

3

2

3

1

1

X(1,1)F

1

1

2

1

1

1 1

0

0

0

0

0

3 1

1

1

0

0

0

3 0

1

1

0

1

0

3 0

0

1

0

1

1

2 0

0

0

1

1

0

1 0 2

0 2

0 3

1 2

0 3

0 1

1 2 3 0 0

X(0,1)F Fig. 1. A lattice set F with the corresponding (0; 1)-matrix. X(1;0) F; X(0;1) F and X(1;1) F are the discrete X-rays in the directions (1; 0); (0; 1) and (1; 1).

A permutation matrix A of order n is a (0; 1)-matrix of size n by n such that AAT = AT A = In , where AT is the transpose of A and In denotes the identity matrix of order n. The n! permutation matrices of order n are obtained from In by permuting the columns of In . The de0nition implies that a permutation matrix has a single entry “1” in each row and column and all other entries “0”. Therefore, the value of its X-ray in horizontal or vertical direction is equal to 1. The values of the X-ray in horizontal and vertical directions are equal to the row and column sums of the matrix, respectively. A sum of k permutation matrices gives a (0; 1; : : : ; k)-matrix A such that all of the row and column sums of A are equal to k. This is a well-known class of matrices having some interesting combinatorial properties [2]. In [3] the authors study the reconstruction of permutation matrices from their X-rays in some prescribed 0nite set of directions. They present a 0rst approach to characterizing permutation matrices that are accessible only via their X-rays. The purpose of this paper is to study the reconstruction of sums of permutation matrices from their X-rays in the diagonal direction (1; 1). The X-ray in the direction (1; 1) is equal to the diagonal sums of the matrix. From the results proved in [3], it follows that the reconstruction of sums of k permutation matrices from their diagonal sums is NP-complete. We tackle a very simple variant of this problem in which the permutation matrices lie on a cylinder instead of a plane. In this case, quite surprisingly, the problem can be solved in polynomial time. We give an exact, algebraic characterization of the diagonal sums that correspond to a sum of permutation matrices. This means that we provide a necessary and suHcient consistency condition for an integral vector to be the diagonal sums of a sum of permutation matrices. We point out that this condition can be checked in linear time. Moreover, we derive an O(kn2 ) time algorithm for reconstructing a sum of k permutation matrices of order n from its diagonal sums. We obtain these results by generalizing a classical theorem of Hall on 0nite abelian groups [7]. The algorithm follows from the constructive proof of the theorem. We point out that the characterization of permutation matrices that are accessible only via their X-rays is a project of Maurice Nivat, and this paper arose from several discussions with Maurice on the topic.

A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

237

2. The planar case A sum of k permutation matrices of order n gives a matrix A = (ai; j ) such that 06ai; j 6k and all the row and column sums of A are equal to k (see Fig. 2). Conversely, if A is a (0; 1; : : : ; k)-matrix of order n having all its row and column sums equal to k, then there exist k permutation matrices P1 ; : : : ; Pk of order n such that A = P1 + · · · + Pk . This result is a corollary of the celebrated BirkhoL–von Neumann Theorem [2, pp. 9,10]. Let A = (ai; j ) be a sum of k permutation matrices of order n, and let D = (d1 ; d2 ; : : : ;
i be its diagonal sums. This means that, di = h=1 ah; i−h+1 , for i = 1; : : : ; n; and d2n−1 )
n dn+i = h=i+1 ah; n+i−h+1 , for i = 1; : : : ; n−1. Then D satis0es the following conditions: 0 6 di 6 ki

for i = 1; : : : ; n;

0 6 dn+i 6 k(n − i) 2n−1
i=1

i=1

for i = 1; : : : ; n − 1;

(2.2)

di = kn:

Moreover, since k 2n−1


(2.1)

(2.3)


n

i=1

i+k


n

j=1

j=




i+j=k+1 (i

+ j) dk , we have

idi = kn2 :

(2.4)

We can now introduce the following decision problem on the sum of permutation matrices. Consistency of the diagonal sums of sums of permutation matrices (DSSPM) Instance: A vector D = (d1 ; d2 ; : : : ; d2n−1 ) with integer entries and satisfying conditions (2.1) – (2.4), and a non-negative integer k. Question: Is there a (0; 1; : : : ; k)-matrix of order n that has all the row and column sums equal to k and diagonal sums equal to D? Conditions (2.1) – (2.4) are not suHcient conditions for the existence of a solution of the problem. For example, suppose that D = (1; 0; 1; 1; 0; 0; 0; 1; 1), with n = 5 and k = 1. The vector D satis0es 06di 6i, for i = 1; : : : ; 5, 06d5+i 65 − i, for i = 1; : : : ; 4,

D =0 1 1 1 1 2 1 1 2

1 2 5 2 2 1 1 3 1 1 1 1 0 2= 1 2 0

1

1 1

1 1

1 1

+

1 1

1

1 1

+

1 1 1

1 1

1

Fig. 2. A sum of three permutation matrices of order 6 and its diagonal sums.

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9


9 di = 5 and i=1 idi = 25 (i.e., D satis0es conditions (2.1) – (2.4)). But there is no a permutation matrix of order 5 that has diagonal sums equal to D. In [3] it is shown that the previous problem is NP-complete on the class of permutation matrices (i.e., when k = 1). From this result it follows that DSSPM is NPcomplete. i=1

3. The cylindrical case We now consider a simple variant of DSSPM in which the permutation matrices belong to a cylinder instead to a plane (see Fig. 3(b)). Let A = (ai; j ) be a sum of k permutation matrices of order n, and let D = (d1 ; d2 ; : : : ; d2n−1 ) be its diagonal sums. By connecting the two opposite vertical sides of A as shown in Fig. 3, we obtain a sum of k permutation matrices AO that lie on a cylinder. The diagonal sums of AO are given by a vector DO = (dO1 ; dO2 ; : : : dOn ) such that dOi = di +dn+i for 16i¡n, and dOn = dn . We also say that DO is the cylindrical diagonal sums of A. From conditions (2.1) – (2.3) we deduce that the cylindrical diagonal sums DO satisfy 0 6 dO i 6 kn n
i=1

for i = 1; 2; : : : ; n;

(3.1)

dO i = kn:

(3.2)


n
2n−1
n−1
n Moreover, i=1 idOi = i=1 idi − n i=1 dn+i ; and so by condition (2.4): i=1 idOi =
n−1
n
n−1 n(kn − i=1 dn+i ). From condition (2.3) it follows that kn − i=1 dn+i = i=1 di . Thus n
i=1

idO i = n;

(3.3)

_ D =1 3 0 2 0 1 0 1 1 1 1 1 1 1

(a)

1

1

1 1 1 1 1

(b)

Fig. 3. (a) A permutation matrix A and its cylindrical diagonal sums. (b) Its corresponding permutation matrix AO on a cylinder.

A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

239

where  is an integer number such
that k66kn, and it gives the sum of entries in n the 0rst n diagonals of A (i.e.,  = i=1 di ). We can now de0ne Consistency of the Diagonal Sums of Sums of Permutation Matrices (CDSSPM) assuming that the permutation matrices are on a cylinder. Consistency of the cylindrical diagonal sums of sums of permutation matrices (CDSSPM) Instance: A vector DO = (dO1 ; dO2 ; : : : ; dOn ) with integer entries and satisfying conditions (3.1) – (3.3), and a non-negative integer k. Question: Is there a (0; 1; : : : ; k)-matrix of order n that has all the row and column O sums equal to k and cylindrical diagonal sums equal to D? Quite surprisingly, conditions (3.1) – (3.3) imply the existence of a solution to this decision problem. Therefore, we have three necessary and suHcient consistency conditions for an integral vector to be the diagonal sums of a sum of permutation matrices, and we claim we can check these conditions in linear time. We prove this claim by proving a generalization of a classical theorem of Hall on the 0nite abelian groups [9]. We begin by reducing CDSSPM to the following decision numerical matching problem on the additive group of integers modulo n (i.e., Z n = {0; 1; : : : ; n − 1}). Numerical matching on Z n Instance: A non-negative integer k, and a multiset C = {c1 ; c2 : : : ; ckn } of Z n . Question: Are there k permutations 1 ; : : : ; k of the elements of Z n such that i − 1 + h (i − 1) ≡ ch;i (mod n)

for i = 1; : : : ; n; h = 1; : : : ; k;

(3.4)

where {ci; h : 16i6n; 16h6k} = C? Lemma 3.1. There is a linear transformation from CDSSPM to numerical matching on Z n . Proof. Given an instance DO = (dO1 ; : : : ; dOn ) of the 0rst problem, we determine a multiset C = {c1 ; : : : ; ckn } such that c1 = · · · = cdO1 = 0; cdO1 +1 = · · · = cdO1 +dO2 = 1; : : : ; cdO1 +···+dOn−1 +1 = · · · = ckn = n − 1:

(3.5)

By conditions (3.1) and (3.2), the multiset C = {c1 ; : : : ; ckn } is an instance of numerical matching on Z n . We prove that there exists a (0; 1; : : : ; k)-matrix A of order n that has all the row and column sums equal to k and cylindrical diagonal sums equal to DO if and only if there exist k permutations 1 ; : : : ; k of the elements of Z n satisfying condition (3.4). Let us suppose that there is a (0; 1; : : : ; k)-matrix A of order n that has all the row O There are k and column sums equal to k and cylindrical diagonal sums equal to D. (k) permutation matrices P1 = (pi;(1) ); : : : ; P = (p ) of order n such that A = P1 + · · · + Pk . k j i; j

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Let Bh = {(1; j1 ); (2; j2 ); : : : ; (n; jn )} be the set of pair such that pi;(h) ji = 1. By setting h (i − 1) = ji − 1, for i = 1; : : : ; n, we obtain that h is a permutation of {0; 1; : : : ; n − 1}. From A = P1 + · · · + Pk , it follows that DO = DO 1 + · · · + DO k , where DO h is the cylindrical diagonal sums of the permutation matrix Ph , for h = 1; : : : ; k. Since DO h = (dO1(h) ; : : : ; dOn(h) ) is the sequence of cylindrical diagonal sums of Ph , there are dOi(h) distinct pairs (l; jl ) of Bh such that l + jl = i + 1 or l + jl = n + i + 1; for each i = 1; : : : ; n. By the de0nition of h , l − 1 + h (l − 1) = i − 1 or l − 1 + h (l − 1) = n + i − 1; that is l − 1 + h (l − 1) ≡ i − 1 (mod n): Therefore, for each h = 1; : : : ; k, there is a set Lh ⊂ {1; : : : ; n} such that |Lh | = dOi(h) and l − 1 + h (l − 1) ≡ i − 1 (mod n);

∀l ∈ Lh :

(3.6)

By this relation, 1 ; : : : ; k is a solution for C of the matching problem. In fact, dOi = dOi(1) + · · · + dOi(k) and C contains dOi elements equal to i − 1. Conversely, let 1 ; : : : ; k be k permutations of the elements of Z n satisfying condition (3.4). By the de0nition of C, for each i = 1; : : : ; n, there are k subsets L1 ; : : : ; Lk of {1; : : : ; n} such that Lh satis0es condition (3.6), |Lh | = dOi(h) , for h = 1; : : : ; k, and dOi = dOi(1) + · · · + dOi(k) . Therefore, for each h = 1; : : : ; k, there are dOi(h) distinct pairs (l; h (l − 1) + 1) such that l + h (l − 1) + 1 = i + 1 or l + h (l − 1) + 1 = n + i + 1: (h) Consequently, the matrix Ph = (pi;(h) j ) having pl; h (l−1)+1 = 1, for l = 1; : : : ; n, and all other entries 0, is a permutation matrix of order n and its ith cylindrical diagonal sum is equal to dOi(h) . Since dOi = dOi(1) + · · · + dOi(k) , we have that the matrix A = P1 + · · · + Pk has all the row and column sums equal to k and ith cylindrical diagonal sums equal to dOi .

From this lemma it follows that, if Numerical Matching on Z n can be solved in polynomial time, then CDSSPM can be solved in the same computational time. Now, by using a classical theorem of Hall [7], we show that if the non-negative integer k of the instance is equal to 1, then this decision matching problem is solvable in linear time. Let G be a 0nite abelian group of order n. Let Mi denote the class of multisets of G of size i. Theorem 3.2 (Hall [7]). If G is a 4nite abelian group of order n, and C∈Mn , then there exists a permutation  of the elements of G such that C = {a + (a) : a ∈G} if

A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

and only if
c = 0: c∈C

241

(3.7)

Remark. This result was rediscovered by Salzborn and Szekeres [13], where they proved the following equivalent of Hall’s Theorem: • If C∈Mn−1 , then there exists a permutation  of the elements of G and an element g of G such that C = {a + (a) : a ∈G\{g}}: Their proof is diLerent from Hall’s, but it is at the same “level” and “diHculty”. These are constructive proofs of this result that give an algorithm to 0nd a permutation  satisfying the statement of the theorem in time O(n2 ). We point out that the theorem states that condition (3.7) is a necessary and suHcient consistency condition for the existence of a permutation  of the elements of G such that C = {a+(a) : a∈G}. Therefore, we can establish the existence of this permutation in linear time. However, if we want to construct such a permutation, then we have to perform the algorithm deduced by one of the constructive proofs of the theorem, and these algorithms have computational complexity O(n2 ). The additive group Z n is a 0nite abelian group of order n and so by Hall’s Theorem, if k = 1, there exists a solution of matching on Z n if and only if the instance
numerical n C satis0es condition (3.7) (i.e., i=1 ci = 0 on Z n ). Since we can check this condition in linear time, we obtain that, if k = 1, the decision matching problem can be solved in linear time, and so Lemma 3.1 establishes that CDSSPM can be also solvable in linear time. By proceeding in a similar fashion, we deduce that CDSSPM can be solved in linear time for each k¿1, if we are able to prove the following generalization of Hall’s Theorem. Theorem 3.3. If G is a 4nite abelian group of order n, and C∈Mkn , where k is a positive integer number, then there exist k permutations 1 ; : : : ; k of the elements of G such that C = {a + i (a) : a ∈ G; 1 6 i 6 k} if and only if
c = 0: c∈C

(3.8)

(3.9)

Condition (3.9) is necessary for the existence of k permutations 1 ; : : : ; k of G satisfying condition (3.8). The theorem can be proved by using a result from combinatorial group theory. We introduce a de0nition which allows us to present this theorem. The Davenport constant
D(G) of G is the least integer d such that for any C ∈Md there is K⊂C such that c∈K c = 0. Theorem 3.4. Let G be a 4nite abelian group of order
n. If C ∈Mm , with m¿n + D(G) − 1, then there is K ⊂C such that |K| = n and c∈K c = 0.

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Remark. This theorem has been proved by Gao [6] in 1996 and some generalizations can be found in [4,8]. This theorem belongs to “Zero-Sum Ramsey Theory” which is a newly established area of combinatorics. The paradigm of zero-sum problems is the following: • suppose the elements of a combinatorial structure are mapped into a 0nite group G. Does there exists a prescribed substructure such that the sum of the weights of its elements is 0 in G? A survey of Zero-Sum Ramsey Theory is given in [5], and a recent result can be found in [14]. Proof of Theorem 3.3. Since D(G)6n, it follows from Theorem 3.4 that for any C ∈
M2n−1
there is K ⊂C such that |K| = n and c∈K c = 0. Consequently, every C∈Mkn with c∈C c = 0 can be split into k subsets each of size n, and each of which also sums to 0. Theorem 3.3 follows from this result and Hall’s Theorem. By Theorem 3.3, there exists a solution of numerical matching on Z n if and only if its instance C satis0es condition (3.9). Consequently, the matching problem can be solved in linear time, and so CDSSPM can be also solved in linear time. Moreover, condition (3.9) for the instance C = {c1 ; : : : ; ckn } corresponds to condition (3.3) for the instance DO = (dO1 ; dO2 ; : : : ; dOn ). In fact, by the de0nition of C, kn
i=1

ci =

n
i=1

(i − 1)dO i ;


kn
n and by condition (3.2) we deduce that i=1 ci = 0 on Z n if and only if i=1 idOi = n, where  is an integer number such that k66kn. Therefore: Theorem 3.5. There is a (0; 1; : : : ; k)-matrix of order n that has all the row and column sums equal to k, and cylindrical diagonal sums equal to DO = (dO1 ; dO2 ; : : : ; dOn ) if and only if DO satis4es conditions (3.1) – (3.3). 3.1. The reconstruction problem Now we tackle the problem of reconstructing a (0; 1; : : : ; k)-matrix of order n from its cylindrical diagonal sums, given that its row and column sums all equal k. That is, given a vector DO = (dO1 ; dO2 ; : : : ; dOn ) that satis0es conditions (3.1) – (3.3), we want to reconstruct a (0; 1; : : : ; k)-matrix of order n that has all its row and column sums equal O to k, and its cylindrical diagonal sums equal to D. Suppose a multiset C of Mkn satisfying condition (3.9) is given. If we are able to provide a polynomial-time algorithm for determining k permutations 1 ; : : : ; k of G satisfying condition (3.8), then we can determine a solution of numerical matching on Z n in polynomial time. So, if we are able to devise this algorithm, then the proof of Lemma 3.1 gives a procedure for reconstructing a (0; 1; : : : ; k)-matrix from its cylindrical diagonal sums. Since the existing proofs of Theorem 3.4 are not constructive, we propose an alternative, direct and constructive proof of Theorem 3.3. This proof provides an eHcient

A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

243

algorithm for determining the k permutations 1 ; : : : ; k of G satisfying condition (3.8). An alternative and constructive proof of Theorem 3.3. We introduce the following notation and de0nitions: • Mk0 is the set of the multisets G of size k and sum 0; • S is the group of permutations of G; • T=S × G is the set of “tagged permutations” of G; • for  ∈S; g ∈G (i.e., (; g)∈T) and C ∈Mn write  C if C = {a + (a) : a ∈ G} (; g) C if C = {a + (a) : a ∈ G\{g}}: Using this notation, Hall’s Theorem can be formulated as follows: ∀C ∈ Mn : (∃ ∈ S :  C) ⇔ C ∈ Mn0 and the equivalent formulation of Salzborn and Szekeres (see the remark after Theorem 3.2) becomes ∀C ∈ Mn−1 ∃(; g) ∈ T : (; g) C: We now de0ne a transition system on T :

(c; d)

• for (; g); ( ; g )∈T and (c; d) ∈G × G write (; g) → ( ; g ) if  =  ◦ ((g); (g ));

g + (g ) = c;

g + (g ) = d;

(i.e.,  and  diLer just by exchanging the values taken for g and g (provided g = g )). • if g = g (or equivalently c = d or  =  ), we have a degenerate transition. Notice that, (c; c) • if (; g) C ∈Mn−1 and (; g) → (; g) (i.e., a degenerate transition), then  C ∪ 0 {c}∈Mn . The main tool of the proof is the following concept of a coupled system: Let A; B be two copies of T that will interact by sending each other “messages” (i.e., elements from G) in an alternating way: (c; d)

• if A is in state (; a) and receives c from B, it performs the transition (; a) → ( ; a ) and sends d to B;

(d; c )

• if B is in state (; b) and receives d from A, it performs the transition (; b) → ( ; b ) and sends c to A; • initially A is in state ( 0 ; a0 ) and B is in state (0 ; b0 ) and A receives c0 ∈G.

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This process generates transition sequences (c0 ;d0 )

(c1 ;d1 )

(c2 ;d2 )

in A: (0 ; a0 ) → (1 ; a1 ) → (2 ; a2 ) → · · · (d0 ;c1 )

(d1 ;c2 )

(d2 ;c3 )

in B: (0 ; b0 ) → (1 ; b1 ) → (2 ; b2 ) → · · · : We have the following properties. Let Aj ; Bj ∈Mn−1 with (j ; aj ) Aj and (j ; bj ) Bj . Then Aj+1 = (Aj \{dj }) ∪ {cj };

Bj+1 = (Bj \{cj+1 }) ∪ {dj }:

From these relations it is easy to deduce that aj+1 = aj + dj − cj ;

bj+1 = bj + cj+1 − dj :

(3.10) (c i ; c i )

The process ends when it produces a degenerate transition (i.e., (i ; ai ) → (i ; ai ) or (di ; di )

(i ; bi ) → (i ; bi )). Hence either the coupled system produces a degenerate transition or the process continues inde0nitely. We show that this second alternative cannot arise. Assume that the process continues inde0nitely. Since a0 ; : : : ; aj and b1 ; : : : ; bj are chosen from the 0nite group G, we have two possibilities: 1. there exist i and j such that j¿i, a0 ; : : : ; ai ; : : : ; aj are all distinct, b0 ; : : : ; bj are all distinct, but aj+1 = ai ; 2. there exist i and j such that j¿i, a0 ; : : : ; aj+1 are all distinct, b1 ; : : : ; bi ; : : : ; bj are all distinct, but bj+1 = bi ; In the 0rst case, by relation (3.10), aj+1 = ai + di − ci . Since aj+1 = ai we have di = ci , (c i ; c i )

and so by sending the (2i)th “message” we obtain (i ; ai ) → (i ; ai ) and the process ends. In the second one, by relation (3.10), bj+1 = bi + ci+1 − di . Since bj+1 = bi we have (di ; di )

ci+1 = di , and so by sending the (2i + 1)th “message” we obtain (i ; bi ) → (i ; bi ) and the process ends. Consequently, the coupled system produces a degenerate transition in i steps, with i¡n. More precisely: Lemma 3.6. Let ( 0 ; a0 ) A0 , (0 ; b0 ) B0 , c0 ∈G be a starting position of the coupled system. Then there exists an integer i¡n such that i Ai ∪ {ci } ∈Mn0 or i Bi ∪ {di } ∈Mn0 . This lemma allows us to prove Theorem 3.3: 0 Let M be the set of the 0rst 2n − 1 elements of a multiset C ∈Mkn (i.e., C satis0es condition (3.9)). We have that M = A0 ∪ B0 ∪ {c0 }, where A0 ; B0 ∈Mn−1 , c0 ∈G. By Hall’s Theorem, there exist ( 0 ; a0 ); (0 ; a0 ) ∈T such that ( 0 ; a0 ) A0 and (0 ; b0 ) B0 . By Lemma 3.6, there exists i¡n such that i Ai ∪ {ci } ∈Mn0 or i Bi ∪ {di } ∈ Mn0 . If i Ai ∪ {ci }, then set C1 = Ai ∪ {ci } and 1 = i , else set C1 = Bi ∪ {ci } and 0 . Therefore, 1 = i . Since 1 C1 ∈Mn0 and C ∈Mkn0 , we have that C  = C\C1 ∈M(k−1)n  by repeating the procedure recursively on C , we obtain k permutations 1 ; : : : ; k of

A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

245

G such that 1 C1 ∈Mn0 ; : : : ; k Ck ∈Mn0 , with C = C1 ∪ · · · ∪ Ck . This means that 1 ; : : : ; k satisfy condition (3.8) of Theorem 3.3. This direct proof of Theorem 3.3 provides an algorithm which 0nds k permutations 1 ; : : : ; k of G satisfying condition (3.8). The algorithm performs the procedure described in the proof of the theorem k − 1 times. At 0rst this procedure 0nds ( 0 ; a0 ); (0 ; a0 )∈T such that ( 0 ; a0 ) A0 and (0 ; b0 ) B0 in O(n2 ) time, by using the algorithm deduced by the proof of Hall’s Theorem. Then, by running the coupled system it achieves a degenerate transition 1 C1 ∈Mn0 in O(n) time. Therefore, the computational complexity of the procedure is O(n2 ). Consequently, we are able to determine a solution of numerical matching on Z n in time O(kn2 ). Example 3.7. Assume that the abelian group G = Z n , with n = 6. The multiset C = 0 {4; 3; 3; 0; 3; 5; 2; 5; 1; 4; 3; 4; 1; 1; 4; 1; 4; 0} ∈M18 . In fact, it is a sequence of 18 elements of Z 6 = {0; 1; 2; 3; 4; 5}, k = 3, and C satis0es condition (3.8) (the sum of its elements is 48 ≡ 0 (mod 6)). We take the 0rst 2n − 1 = 11 elements and we set A0 = {4; 3; 3; 0; 3}; B0 = {5; 2; 5; 1; 4} and c0 = 3. By performing the algorithm deduced by the proof of Hall’s Theorem on A0 and B0 , we 0nd ( 0 ; a0 ) A0 and (0 ; b0 ) B0 , where 0 = (4; 2; 1; 3; 5; 0);

a0 = 5;

0 = (5; 1; 3; 4; 0; 2);

b0 = 5:

Then we run the coupled system: (c ;d )

0 0 (0 ; a0 ) → (1 ; a1 ); where 1 = (0; 2; 1; 3; 5; 4); a1 = 0; d0 = 4; A1 = {3; 3; 0; 3; 3};

(d ;c )

0 1 (1 ; b1 ); where 1 = (2; 1; 3; 4; 0; 5); b1 = 0; c1 = 5; B1 = {2; 5; 1; 4; 4}; (0 ; b0 ) →

(c ;d )

1 1 (1 ; a1 ) → (2 ; a2 ); where 2 = (5; 2; 1; 3; 0; 4); a2 = 4; d1 = 3; A2 = {5; 3; 3; 0; 3};

(d ;c )

1 2 (1 ; b1 ) → (2 ; b2 ); where 2 = (3; 1; 2; 4; 0; 5); b2 = 2; c2 = 5; B2 = {3; 2; 1; 4; 4};

(c ;d )

2 2 (2 ; a2 ) → (3 ; a3 ); where 3 = (5; 2; 0; 3; 1; 4); a3 = 2; d2 = 3; A3 = {5; 3; 0; 5; 3};

(d ;c )

2 3 (3 ; b3 ); where 3 = (3; 2; 1; 4; 0; 5); b3 = 1; c3 = 2; B3 = {3; 3; 1; 4; 4}; (2 ; b2 ) →

(c ;d )

3 3 (4 ; a4 ); where 4 = 3 (3 ; a3 ) →

a4 = 2; d3 = 2; A4 = A3 : (c3 ; c3 )

The coupled system produces a degenerate transition (3 ; a3 ) → (3 ; a3 ) and the process ends. Therefore, 3 A3 ∪ {c3 } = {5; 3; 0; 5; 3; 2} ∈M60 and we set C1 = A3 ∪ 0 {c3 }, 1 = 3 , and C  = C\C1 = {3; 3; 1; 4; 4; 4; 1; 1; 4; 1; 4; 0} ∈M12 .  Now, we repeat the procedure on C . We take the 0rst 2n − 1 = 11 elements and we set A0 = {3; 3; 1; 4; 4}; B0 = {4; 1; 1; 4; 1} and c0 = 4. Since A0 is equal to the previous B3 , we have that ( 0 ; a0 ) A0 where 0 = (3; 2; 1; 4; 0; 5);

a0 = 1:

So, we perform Hall’s algorithm only on B0 . We 0nd (0 ; b0 ) B0 , where 0 = (4; 0; 5; 1; 3; 2);

b0 = 5:

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A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

Then we run the coupled system: (c ;d )

0 0 (0 ; a0 ) → (1 ; a1 ); where 1 = (2; 3; 1; 4; 0; 5); a1 = 0; d0 = 3; A1 = {4; 3; 1; 4; 4};

(d ;c )

0 1 (1 ; b1 ); where 1 = (2; 0; 5; 1; 3; 4); b1 = 0; c1 = 4; B1 = {1; 1; 4; 1; 3}; (0 ; b0 ) →

(c ;d )

1 1 (1 ; a1 ) → (2 ; a2 ); where 2 = (4; 3; 1; 2; 0; 5); a2 = 3; d1 = 1; A2 = {4; 4; 3; 4; 4};

(d ;c )

1 2 (1 ; b1 ) → (2 ; b2 ); where 2 = (1; 0; 5; 2; 3; 4); b2 = 3; c2 = 4; B2 = {1; 1; 1; 1; 3};

(c ;d )

2 2 (3 ; a3 ); where 3 = (4; 3; 2; 1; 0; 5); a3 = 2; d2 = 3; A3 = {4; 4; 4; 4; 4}; (2 ; a2 ) →

(d ;c )

2 3 (3 ; b3 ); where 3 = (1; 2; 5; 0; 3; 4); b3 = 1; c3 = 1; B3 = {1; 1; 3; 1; 3}; (2 ; b2 ) →

(c ;d )

3 3 (4 ; a4 ); where 4 = (4; 3; 5; 1; 0; 2); a4 = 5; d3 = 4; A4 = {4; 4; 1; 4; 4}; (3 ; a3 ) →

(d ;c )

3 4 (4 ; b4 ); where 4 = (1; 3; 5; 0; 2; 4); b4 = 4; c4 = 1; B4 = {1; 4; 1; 3; 3}; (3 ; b3 ) →

(c ;d )

4 4 ( 4 ; a4 ) → (5 ; a5 ); where 5 = 4

a5 = 5; d4 = 1; A5 = A4 : (c4 ; c4 )

The coupled system produces a degenerate transition ( 4 ; a4 ) → ( 4 ; a4 ) and the process ends. Therefore,  4 A4 ∪ {c4 } = {4; 4; 1; 4; 4; 1} ∈M60 and we set C2 = A3 ∪ {c3 }, 2 = 3 , and C  = C  \C2 = {1; 4; 1; 3; 3; 0} ∈M60 . Since C  = B4 ∪ {0}, we have that 4 C  . We set C3 = C  and 3 = 4 . Therefore, the algorithm provides three permutations 1 = (5; 2; 0; 3; 1; 4);

2 = (4; 3; 5; 1; 0; 2);

3 = (1; 3; 5; 0; 2; 4)

such that 1 C1 ; 2 C2 ; 3 C3 , with C = C1 ∪ C2 ∪ C3 . This means that 1 ; 2 ; 3 is a solution for the instance C of numerical matching on Z 6 . From the previous algorithm and the proof of Lemma 3.1, it follows that: Theorem 3.8. Let DO = (dO1 ; : : : ; dOn ) be a vector satisfying conditions (3.1), (3.2) and (3.3). We can reconstruct a (0; 1; : : : ; k)-matrix of order n that has all the row and column sums equal to k, and cylindrical diagonal sums equal to DO in time O(kn2 ). The basic steps of this reconstruction algorithm are the following: given an instance DO = (dO1 ; : : : ; dOn ) satisfying conditions (3.1) – (3.3), we determine the multiset C = {c1 ; : : : ; ckn } satisfying condition (3.8). Then, we perform the previous algorithm which 0nds a solution 1 ; : : : ; k for C of the matching problem. We de0ne the permutation (h) matrix Ph = (pi;(h) j ) that has pi; h (i−1)+1 = 1, for i = 1; : : : ; n, and all other entries 0, for each h = 1; : : : ; k. Finally, the matrix A = P1 + · · · + Pk is a (0; 1; : : : ; k)-matrix of order O n that has all the row and column sums equal to k and cylindrical diagonal sums D. For instance, if DO = (2; 4; 1; 4; 5; 2), we determine C = {0; 0; 1; 1; 1; 1; 2; 3; 3; 3; 3; 4; 4; 4; 4; 4; 5; 5}: The previous example gives the solution 1 = (5; 2; 0; 3; 1; 4); 2 = (4; 3; 5; 1; 0; 2); 3 = (1; 3; 5; 0; 2; 4) for C. Thus, the permutation matrices P1 = (pi;(1) j ); P2 = (2) (3) (pi; j ) and P3 = (pi; j ) are such that: (1) (1) (1) (1) (1) (1) = p2;3 = p3;1 = p4;4 = p5;2 = 1; p6;5 = 1; p1;6 (2) (2) (1) (2) (2) (2) = p2;4 = p3;6 = p4;2 = p5;1 = 1; p6;3 = 1; p1;5

A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

D =2 4 1 1 1 2 1 1 2

1 4 5 2 1 1 1 1 1 1 1 2

1

1 1

1

1

1

=

247

1

+

1 1

1

1

+

1 1 1

1

1

1

1

Fig. 4. A sum of three permutation matrices of order 6 and its cylindrical diagonal sums. (3) (3) (3) (3) (2) (3) p1;2 = p2;4 = p3;6 = p4;1 = p5;3 = 1; p6;5 = 1;

and all other entries 0. Therefore, the matrix A = P1 + P2 + P3 is a (0; 1; 2; 3)-matrix that has all the row and column sums equal to 3 and cylindrical diagonal sums DO = (2; 4; 1; 4; 5; 2) (see Fig. 4). 3.2. Conjecture for CDSSPM on the class of (0,1)-matrices In this subsection we tackle CDSSPM on the class of (0; 1)-matrices. Given a vector DO = (dO1 ; dO2 ; : : : ; dOn ) with integer entries, we have to establish the existence of a (0; 1)-matrix of order n that has all the row and column sums equal to k and cylindrical O diagonal sums equal to D. The BirkhoL–von Neumann Theorem [2, pp. 9 and 10] provides an interesting decomposition property for those (0; 1; : : : ; k)-matrices of order n having all the row and column sums equal to k (see Section 2), which implies that if A is a (0; 1)-matrix of order n all of whose row and column sums are equal to the positive integer k, then there exist k permutation matrices P1 ; : : : ; Pk of order n such that A = P1 + · · · + Pk . From this decomposition, we deduce that CDSSPM applied to the class of (0; 1)-matrices could give a result similar to Theorem 3.5. Let DO = (dO1 ; : : : ; dOn ) be the cylindrical diagonal sums of a (0; 1)-matrix of order n such that all row and column sums are equal to k. The vector DO satis0es conditions (3.2), (3.3), and 0 6 dO i 6 n;

for i = 1; : : : ; n;

(3.11)

instead of conditions (3.1). Therefore, condition (3.11) is necessary for the existence of a (0; 1)-matrix of order n having all the row and column sums equal to k, and O cylindrical diagonal sums D. Condition (3.11) implies conditions (3.1), and so by Theorem 3.5, if DO satis0es conditions (3.11), (3.2) and (3.3), there is at least one (0; 1; : : : ; k)-matrix of order n that has all the row and column sums equal to k, and cylindrical diagonal sums equal O Now, we have to answer the following question: to D. • does this family of (0; 1; : : : ; k)-matrices contain at least one (0; 1)-matrix?

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A. Del Lungo / Theoretical Computer Science (2002) 235 – 249

Maurice Nivat and I feel that conditions (3.11), (3.2) and (3.3), imply the existence of such (0; 1)-matrix. This means that Conjecture 3.9. There is a (0; 1)-matrix of order n that has all the row and column sums equal to k, and cylindrical diagonal sums equal to DO = (dO1 ; dO2 ; : : : ; dOn ) if and only if DO satis4es conditions (3.11), (3.2) and (3.3). The instance DO = (2; 4; 1; 4; 5; 2) of the previous example satis0es conditions (3.11), (3.2) and (3.3). There is at least one (0; 1; 2; 3)-matrix that has all the row and column sums equal to 3 and cylindrical diagonal sums DO (see Fig. 4). This family of (0; 1; 2; 3)matrices contains the following (0; 1)-matrix that has all the row and column sums O equal to 3, and cylindrical diagonal sums equal to D.

Acknowledgements The author wishes to thank Volker Strehl whose suggestions greatly improved the presentation of the proof of Theorem 3.3 and for pointing out the existence of Theorem 3.4. A un amico. Volevo ringraziare Maurice Nivat a cui sono profondamente debitore. I numerosi suggerimenti, le sue intuizioni ed il suo senso estetico sono stati, e tuttora sono per me una guida insostituibile nell’aLascinante mondo dei modelli discreti. References [1] R.P. Anstee, Invariant sets of arcs in network Tow problems, Discrete Appl. Math. 13 (1986) 1–7. [2] R.A. Brualdi, H. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, Cambridge, 1991. [3] S. Brunetti, A. Del Lungo, P. Gritzmann, S. de Vries, On the reconstruction of permutation and partition matrices under tomographic constraints, preprint.

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Y. Caro, Remarks on a Zero-Sum Theorem, J. Combin. Theory, Ser. A 76 (1996) 315–322. Y. Caro, Zero-sum problems—a survey, Discrete Math. 152 (1996) 93–113. W.D. Gao, A combinatorial problem on 0nite abelians groups, J. Number Theory 58 (1996) 100–103. M. Hall Jr., A combinatorial problem on abelian groups, Proc. Amer. Math. Soc. 3 (1952) 584–587. Y.O. Hamidoune, On weighted sums in abelian groups, Discrete Math. 162 (1996) 127–132. G.T. Herman, A. Kuba (Eds.), Discrete Tomography: Foundations, Algorithms and Applications, Birkhauser, Boston, Cambridge, MA, 1999. R.W. Irving, M.R. Jerrum, Three-dimensional statistical data security problems, SIAM J. Comput. 23 (1994) 170–184. C. Kisielowski, P. Schwander, F.H. Baumann, M. Seibt, Y. Kim, A. Ourmazd, An approach to quantitative high-resolution transmission electron microscopy of crystalline materials, Ultramicroscopy 58 (1995) 131–155. G.P.M. Prause, D.G.W. Onnasch, Binary reconstruction of the heart chambers from biplane angiographic image sequence, IEEE Trans. Med. Imaging 15 (1996) 532–559. F. Salzborn, G. Szekeres, A problem in combinatorial group theory, Ars Combin. 7 (1979) 3–5. W.A. Schmid, On zero-sum subsequences in 0nite abelian groups, Integers: Electron. J. Combin. Number Theory 1 (2001) #A01. A.R. Shliferstein, Y.T. Chien, Switching components and the ambiguity problem in the reconstruction of pictures from their projections, Pattern Recognition 10 (1978) 327–340.

Theoretical Computer Science 281 (2002) 251 – 289

www.elsevier.com/locate/tcs

Reconnaissance parallele des langages rationnels sur automates cellulaires plans Marianne Delorme ∗ , Jacques Mazoyer LIP ENS Lyon, 46 Allee d’Italie, 69-364 Lyon Cedex 07, France

Abstract Peu de r+esultats sont connus sur la puissance de calcul des automates cellulaires 2D. Dans l’approche “reconnaissance de langages”, la di/cult+e provient de la multiplicit+e des plongements possibles d’un mot sur le plan discret. Une question tres simple comme la reconnaisance des langages rationnels en temps r+eel devient vraiment d+elicate. Nous montrons ce r+esultat sur deux plongements particuliers (spirale d’Archimede et 5l de Hilbert). Cette premiere investigation met en e+ vidence que ce qui est di/cile est moins l’algorithmique que la nature du plan discret. c 2002 Elsevier Science B.V. All rights reserved.


1. Introduction 1.1. Problematique Les automates cellulaires constituent un modele du parall+elisme massif et savoir en quoi et comment ce parall+elisme acc+elere les calculs reste une question intrigante. Une maniere de l’aborder est d’+etudier et de comparer les classes de complexit+e qui peuvent eˆ tre d+e5nies par di=+erents types de machines, et en particulier pour ce modele, de saisir les e=ets de la dimension sur la puissance du modele. Les classes de complexit+e sont des classes de langages. Il y a, pour les automates cellulaires deux modes de reconnaissance de langages: le mode s+equentiel, pour lequel le mot test+e est donn+e a l’automate lettre a lettre sur une cellule 5x+ee, et le mode parallele ou le mot tout entier d+etermine la con5guration initiale. Dans le premier cas, il a e+ t+e prouv+e par Cole [2] que la puissance de reconnaissance augmente strictement avec la dimension. Nous ne nous int+eresserons donc ici qu’au cas de la reconnaissance parallele. ∗

Corresponding author. E-mail addresses: [email protected] (M. Delorme), [email protected] (J. Mazoyer). c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 6 - 6

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M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

Mais alors le passage de la dimension 1 a une dimension sup+erieure, en mode parallele, conduit a s’interroger sur la pertinence du choix des mots comme objet de base. Il semble plus naturel de consid+erer dans Z2 une “image plane” plutˆot qu’un mot qui est un objet totalement ordonn+e ? Cependant, bien que de tres nombreux travaux adoptent ce point de vue, il ne semble pas ais+e de d+e5nir une notion d’image qui repr+esenterait une sorte d’objet canonique pour le plan. Diverses propositions ont e+ t+e formul+ees [9,17], qui impliquent des d+emarches di=+erentes et ne permettent pas a un choix de s’imposer. Par ailleurs c’est la comparaison avec les r+esultats obtenus en dimension 1 qui devient alors plus probl+ematique. Ainsi, dans [9], un mot est “sectionn+e, puis pos+e” ligne apres ligne dans un rectangle donn+e alors qu’il est dans [17] “repli+e” dans ce mˆeme rectangle. Nous avons donc fait le choix des mots, et cherch+e, en cons+equence, commentil e+ tait possible de disposer de faKcon uniforme — et e/cace du point de vue du temps de reconnaissance — les lettres d’un mot sur un plan de cellules. Cela nous a conduits a introduire une notion de “5l”, puis d’automate cellulaire 2-dimensionnel avec 5l. La question de la complexit+e a e+ t+e tres e+ tudi+ee pour les automates cellulaires de dimension 1 (des 1969 [2]), et le probleme di/cile qui reste ouvert est celui de la comparaison entre temps lin+eaire et temps r+eel, plus pr+ecis+ement savoir si le temps lin+eaire est, en fait, le temps r+eel. La dimension 2 reste a explorer, et c’est dans ce cadre que se situe ce travail sur la reconnaissance en temps r+eel des langages rationnels sur automates cellulaires de dimension 2 avec 5l. 1.2. Pourquoi des 6ls Il s’agit en fait d’exhiber de “bonnes” e+ num+erations de Z2 , c’est-a-dire des applications injectives de N dans Z2 qui ordonnent les cellules de Z2 de telle sorte que les lettres des mots test+es puissent y eˆ tre assign+ees de faKcon uniforme, compacte et non ambiguMe. D’un point de vue plus g+eom+etrique, nous nous int+eressons a des applications qui d+eterminent des chemins connexes et sans cycles dans Z2 , reliant des sites a partir desquels toute l’information n+ecessaire a la d+ecision de l’automate reconnaisseur pourra eˆ tre trait+ee le plus rapidement possible. C’est ainsi que nous avons e+ t+e amen+es, a titre exp+erimental, a e+ tudier deux 5ls particuliers. 1.3. Pourquoi ces 6ls Le premier, le 5l (ou la spirale) d’Archimede s’est impos+e imm+ediatement. En e=et, comme la reconnaissance de mots par un automate cellulaire a lieu sur une cellule donn+ee et comme on cherche une r+epartition dense des lettres, il est assez naturel de penser a la cellule origine et a une courbe discrete qui s’enroule r+egulierement et sans laisser d’espace inutile autour d’elle. L’id+ee du second, le 5l que nous avons baptis+e 5l de Hilbert, procede e+ galement de la recherche d’une courbe discrete remplissant le (une partie du) plan d’une faKcon r+eguliere et sans recoupements, plus compliqu+ee a d+e5nir que la spirale et qui ne donne pas a l’origine cette position de “centre de sym+etrie”. Nous avons alors conKcu, a partir

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253

d’une version discrete de la courbe de Hilbert [7,15], une courbe qui s’+eloigne de l’origine en d+ecrivant de faKcon dense une partie seulement du quart nord-est du plan. Quelles sont les performances possibles des automates 2-dimensionnels munis de tels 5ls? 1.4. Rationalite et temps reel Les automates cellulaires en dimension 1 et 2 sont e+ quivalents en tant que reconnaisseurs de langages, par cons+equent les langages rationnels sont e+ videmment reconnus par nos automates. La question int+eressante est celle du temps de reconnaissance. Or les langages rationnels sont reconnus en temps r+eel par automates cellulaires en dimension 1. Par temps r+eel, on entend le temps minimum pour que l’automate puisse d+ecider, c’est-a-dire le temps minimum en lequel la cellule sur laquelle se fait la d+ecision connaˆPt toute l’information n+ecessaire. En dimension 1, le temps r+eel est, dans le cas g+en+eral, la longueur de l’entr+ee augment+ee de 1, ce 1 provenant de la n+ecessit+e de marquer la 5n du mot, et de la longueur du mot dans le cas de langages pr+e5xes. Dans le cas des automates cellulaires de dimension 2, ce temps peut eˆ tre strictement inf+erieur a la longueur du mot grˆace au 5l le long duquel est dispos+ee l’entr+ee, comme nous allons le voir par la suite. Mais la question de la comparaison du temps r+eel en dimension 1 (1D) et en dimension 2 (2D+F) est d+elicate puisqu’il ne s’agit pas, en fait, du mˆeme temps! La suite du texte est compos+ee de quatre parties. La premiere est consacr+ee a des d+e5nitions, la seconde a la preuve que tout langage rationnel est reconnu en temps r+eel par un automate cellulaire 2-dimensionnel avec spirale d’Archimede. La troisieme est consacr+ee a la d+e5nition du 5l de Hilbert et a l’+evaluation du temps r+eel sur ce 5l. En5n, la derniere est r+eserv+ee a la preuve, beaucoup plus d+elicate, du r+esultat analogue au pr+ec+edent sur le 5l de Hilbert.

2. Denitions Nous rappelons d’abord quelques d+e5nitions e+ l+ementaires. Puis nous pr+ecisons ce que nous entendons par 5l et par automate cellulaire avec 5l. En5n, nous formalisons la notion de temps r+eel. Denition 1. Un automate 5ni d+eterministe F est un quintuplet (Q; ; ; q0 ; Qf ) ou • Q est un ensemble 5ni, l’ensemble des e tats, •  est un alphabet 5ni, •  de Q ×  dans Q est la fonction de transition, • q0 ∈Q est l’etat initial et Qf ⊆Q est l’ensemble des e tats d’acceptation. Si l’on d+esigne encore par  la fonction de transition e+ tendue aux mots sur , le langage reconnu par F est l’ensemble {u=u ∈? et (q0 ; u) ∈Qf }. Un langage est rationnel (ou r+egulier) s’il est reconnu par un automate 5ni.

254

u

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

Dans la suite, si F=(Q; ; ; q0 ; Qf ) et si u est un mot de ? , les fonctions, not+ees u

 et d+e5nies par q ∈Q et (q)= (q; u), jouent un rˆole essentiel, comme le fait qu’elles sont en nombre 5ni. Les automates cellulaires, dans ce texte, sont des automates avec voisinage de Moore, c’est-a-dire que l’+etat d’une cellule a l’instant t est fonction de son propre e+ tat et de ceux de ses huit voisines les plus proches. Denition 2. (1) Un automate cellulaire A de dimension 2 (ou 2-automate cellulaire, ou 2-AC) est un systeme de transitions, (S; ), ou: • S est un ensemble 5ni dont les e+ l+ements sont les e tats de A, •  : S 9 → S est la fonction de transition locale de A. (2) Une con5guration C de A est une application de Z2 dans S. D’une con5guration C a l’instant t, l’automate cellulaire passe a l’instant t +1 a la con5guration C ? d+e5nie par: i; j ∈Z et C ? (i; j) = (C(i; j); C(i + 1; j); C(i + 1; j + 1); C(i; j + 1); C(i − 1; j + 1); C(i − 1; j); C(i − 1; j − 1); C(i; j − 1); C(i + 1; j − 1)): La fonction qui associe a toute con5guration C la con5guration C ? qui lui succede directement est la fonction globale de l’automate cellulaire. L’+evolution de l’automate a partir d’une con5guration initiale, con5guration du systeme au temps t = 0, est la suite (Ct )t∈N de ses con5gurations successives. Ct est appel+ee la con5guration au temps t et l’+etat Ct (i; j) de la cellule (i; j) au temps t est parfois not+e i; j t . On repr+esente, en g+en+eral, une telle suite de con5gurations par un diagramme, appel+e diagramme espace-temps. Ceux des 2D-automates cellulaires sont donc des empilements de plans et un calcul de l’automate a partir d’une con5guration initiale est un volume, pas facile a repr+esenter. On se contente le plus souvent de coupes (horizontales comme sur les Figs. 10–15 ou verticales sur la Fig. 13). Denition 3. Un 5l est une application injective (r+ecursive ou non) ! de N dans Z2 telle que, si pour n∈N on pose !(n)=(!x (n); !y (n)), alors: (i) !(0) =(0; 0), (ii) !x (n + 1)= !x (n) et !y (n + 1)= !y (n) + 1 ou !x (n + 1)= !x (n) et !y (n + 1)= !y (n) − 1 ou !y (n + 1)= !y (n) et !x (n + 1)= !x (n) + 1 ou !y (n + 1)= !y (n) et !x (n + 1)= !x (n) − 1, ce qui signi5e simplement que si l’image de n par ! est un point du plan, l’image de n+1 est l’un des points situ+e a l’est, au nord, a l’ouest ou au sud du pr+ec+edent. Nous voulons e+ quiper un 2-automate cellulaire d’un 5l qui indique comment disposer les lettres du mot d’entr+ee dans les cellules de l’automate en con5guration intiale, l’automate doit donc connaˆPtre ce 5l ind+ependamment de ce mot d’entr+ee. C’est pourquoi l’ensemble des e+ tats d’un 2-automate avec 5l a deux composantes: l’une

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255

relative au 5l, qui ne change pas lors de l’+evolution de l’automate, l’autre qui d+etermine cette e+ volution. Nous allons utiliser le symbolisme suivant pour la composante-5l des e+ tats. Il permet d’indiquer comment le 5l arrive et part de la cellule courante, ou qu’il n’y passe pas. • •

signi5e que le 5l n’atteint pas la cellule, , ✲ signi5ent que le 5l a pour origine cette cellule , ✻ , ✛ ❄ et part dans la direction indiqu+ee par la Ueche,

• ✛

✲ : le 5l arrive sur la cellule (i; j) depuis la cellule (i; j + 1) , ❄ situ+ee au nord et sort suivant la direction indiqu+ee,

•

,

, ✲ , ✻ , ❄ se d+e5nissent de mˆeme.

✻ ,

✛

,

,

✲

✻ ,

✛

,

❄

:

Nous sommes alors en mesure de poser la d+e5nition suivante: Denition 4. Un automate cellulaire 2-dimensionnel A est dit capable de supporter un 5l !, ou plus simplement eˆ tre un 2-AC avec 5l !, si son ensemble d’+etats est de la forme W × S ou: • W est l’ensemble des 17 symboles suivants: ,

✻ ,

✛

,

❄

,

✲

,

✛

,

❄

,

✲

,

✻

,

, , , , et ✲ , ✲ , ✻ , ✛ ✻ , ✛ ❄ ❄ • S est un ensemble 5ni dont les e+ l+ements, ind+ependants de ceux de W , sont encore appel+es les e tats de l’automate. et si sa fonction locale de transition laisse la composante dans W invariante. Notation 1. Un automate cellulaire 2-dimensionnel A avec 5l ! sera not+e A! . De plus, pour un tel automate, l’+etat de la cellule (i; j) au temps t sera not+e !(i; j) ; s(i; j) t , avec !(i; j) ∈W et s(i; j) ∈S. La reconnaissance d’un langage par un 2-AC avec 5l di=ere peu de la notion classique. Elle a e+ galement lieu sur une cellule distingu+ee que nous choisissons eˆ tre la cellule origine. Denition 5. Soient L un langage sur un alphabet , ! un 5l et A un 2-automate cellulaire. On dit que L est reconnu par A! ou qu’il est reconnu par A via ! lorsque: (i)  ⊆S, (ii) l’ensemble S des e+ tats de A! contient deux sous-ensembles d’+etats disjoints distingu+es, Saccepte et Srejette ,

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(iii) S a un e+ l+ement distingu+e, sq , appel+e e+ tat quiescent, qui satisfait ((!c ; sq ); : : : ; (!se ; sq ))=(!c ; sq ), ou !c d+esigne la composante-5l de l’+etat de la cellule courante c, !i et !ij les composantes-5l des e+ tats des cellules voisines de c, (iv) Pour tout mot a0 : : : a‘−1 de ? , l’automate partant de la con5guration initiale d+e5nie par: !!(i) ; s!(i) 0 =(!!(i) ; ai ) pour i ∈ {0; : : : ; ‘ − 1} avec !!(0) ∈ { ✻ ; ✛ ; ; ✲ } et !!(i) ∈W − { ✻ ; ✛ ; ; ✲ } pour i ∈ {1; : : : ; ‘ − 1}, ❄ !!(‘+1) , les autres cellules e+ tant dans l’+etat ( ; sq ), il existe un temps  tel que: • 0; 0  =(w; s!(0) ) avec w ∈W , s!(0) ∈Saccepte si et seulement si le mot a0 : : : a‘−1 appartient a L, s!(0) ∈Srejette sinon, et • pour tout temps t, t¡, s!(0) n’appartient ni a Saccepte ni a Srejette . ❄

Finalement pr+ecisons ce qu’est le temps r+eel. La cellule distingu+ee sur laquelle doit avoir lieu la reconnaissance est l’origine, le voisinage est le voisinage de Moore et la distance d’un point (x; y) de Z2 a l’origine est Max(|x|; |y|). Par d+e5nition le temps r+eel est le temps minimum en lequel la cellule (0; 0) reKcoit toute l’information n+ecessaire a sa d+ecision, c’est-a-dire connaˆPt tout le mot et sait en particulier qu’il a e+ t+e int+egralement lu. D’ou la d+e5nition suivante. Denition 6. Un langage L est reconnu en temps r+eel par un automate cellulaire 2dimensionnel A! avec 5l ! lorsque le temps de reconnaissance d’un mot a0 : : : a‘−1 est Maxh∈{0;:::; ‘−1} (!x (h); !y (h)) + 1. 3. Le l ou spirale d’Archimede 3.1. De6nition et representation graphique Notation 2. Nous d+esignerons par C (x; y) , x; y ∈Z, le carr+e de cˆot+e unit+e dont le coin sud-ouest est le point (x; y). Denition 7 (Courbe et 5l d’Archimede). Le 5l d’Archimede est d+e5ni a partir de la courbe C faite de segments du plan. (1) Construction de la courbe (voir Fig. 1a): Partant du centre du carr+e C (0; 0) , on trace un segment horizontal jusqu’au centre de C (1; 0) , un segment vertical jusqu’au centre de C (1; 1) , un segment horizontal jusqu’au centre de C (−1; 1) , un segment vertical jusqu’au centre de C (−1; −1) et en5n un segment horizontal jusqu’au centre de C (1; −1) . Le processus est ensuite it+er+e: a partir du point (h; −h), on trace des segments vers les points successifs (h+1; −h), (h+1; h+1), (−(h+1); h+1), (−(h+1); −(h+1)) et (h + 1; −(h + 1)).

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257

Fig. 1. Archimede: (a) La courbe d’Archimede. (b) Le 5l d’Archimede.

(2) Le 5l d’Archimede A est une application de N dans Z2 d+e5nie comme suit (voir Fig. 1b): • A(0)=(0; 0). • si pour tout j, j ∈ {0; : : : ; i}, les A(j) sont d+e5nis, alors, A(i + 1) est le coin sud-ouest de l’unique carr+e — dont le coin sud-ouest n’appartient pas a A({0; : : : ; i}), — et qui est connect+e par C au carr+e dont le coin sud-ouest est A(i). 3.2. Temps reel sur le 6l d’Archim:ede Proposition 1. Sur le 6l d’Archim:ede, le temps reel est RArch (n) = 1 + max{d | (2d + 1)2 6 n}: d∈N

Preuve. Pour la distance max(|x|; |y|), les boules B(0;0) (d) sont les carr+es dont le centre est (0; 0), num+erot+e A(0), de cˆot+e 2d + 1. Soit n un entier tel que A(n)∈B(0; 0) (d) et A(n) ∈ B(0; 0) (d − 1), d¿1. Le temps pour que l’information port+ee par le point num+erot+e A(n) parvienne a la cellule (0; 0) est d. Le temps pour que cette cellule sache qu’elle a reKcu toute l’information n+ecessaire a sa d+ecision est d + 1 car alors tout ce qui lui parviendra des composantes non-5l des e+ tats sera sq . Le nombre de points dans la boule B(0; 0) (d) de cˆot+e 2d + 1 est (2d + 1)2 , donc le premier point du 5l d’Archimede qui est hors de B(0; 0) (d) est son 1 + (2d + 1)2 -ieme point, image de (2d + 1)2 par A.

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Il en r+esulte que pour un mot de longueur n d+eploy+e sur le 5l d’Archimede, le temps r+eel est donn+e par la formule de l’+enonc+e. 3.3. Langages rationnels sur le 6l d’Archim:ede Theoreme 1. Tout langage rationnel est reconnu en temps reel par un automate cellulaire 2-dimensionnel avec le 6l d’Archim:ede. Preuve. On commence par observer que chaque cellule d’un 2-AC avec le 5l dArchimede connaˆPt parfaitement sa position par rapport a la cellule origine grˆace a la composante-5l (dans W ) de son e+ tat (D+e5nition 4). Ceci est r+esum+e dans le tableau suivant pour une cellule (i; j). Coordonn+ees

Etat-5l

Coordonn+ees

i¿0 et −i + 1¡j¡i

j¿0 et −j¡i¡j

i¡0 et i¡j¡ − i

j¡0 et j¡i¡1 − j

i¿0 et i=j

i¡0 et j= − i

i¡0 et i=j

i¿0 et j= − i

Etat-5l

i=0 j =0

L’id+ee de l’algorithme est alors que chaque cellule envoie le plus vite possible les lettres du mot d’entr+ee qu’elle connaˆPt vers la cellule origine du 5l. D’apres la quasisym+etrie du 5l, les diagonales vont jouer un rˆole sp+ecial, mais pas toutes le mˆeme du fait de la non-sym+etrie. Les cellules situ+ees sur diagonales nord-est (x =y; x; y¿0), nord-ouest (−x =y; x; y¿0) et sud-ouest (x =y; x; y60) vont avoir un comportement analogue consistant a recueillir sur la diagonale l’information de trois cellules formant un coin. Ce n’est pas la diagonale sud-est (x =y, x¿0, y60), mais la droite x =−y+1, x¿0; y60 qui va o/cier pour le quatrieme coin (Fig. 2). Les lettres reKcues sont regroup+ees sur les “diagonales” x = y, −x =y (avec x¡0) et u

x = −y + 1 (avec x¿0). L’amalgamation des lettres est r+ealis+ee grˆace aux fonctions  ou  est la fonction de transition l’automate 5ni F reconnaissant le langage. La cellule origine reKcoit alors (2t − 1)2 lettres amalgam+ees au temps t et est ainsi e+ ventuellement capable d’e=ectuer en une unit+e de temps (2t − 1)2 transitions d’+etats de F. Donnons maintenant une preuve plus formelle du r+esultat.

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

259

Fig. 2. Mouvement des donn+ees dans la preuve du Th+eoreme 1.

Soit L un langage rationnel reconnu par un automate 5ni d+eterministe F = (Q; ; F ; q0 ; Qf ). Il s’agit de d+e5nir le 2-AC AC!; F = (W × S; ) qui le reconnaˆPt en temps r+eel via le 5l d’Archimede. La composante S des e+ tats de AC!; F est QQ ∪Q ∪ {sq ; saccepte ; srejette } ou sq est un e+ tat quiescent indiquant l’absence de lettre. Ainsi, au temps t = 0, sur l’entr+ee a0 : : : a‘−1 ai

de longueur ‘, les S-composantes des e+ tats sont F pour i ∈ {0; : : : ; ‘ − 1} et sq pour i¿‘. La fonction de transition locale de AC!; F est d+e5nie par: • Si la composante-5l (ou W -composante) de la cellule est

✻ (resp.

✛

,

, ✲ ), sa nouvelle S-composante est la S-composante de sa voisine est ❄ (respectivement nord, ouest, sud). Les donn+ees sont ainsi pouss+ees vers le centre. • Si la W -composante de l’+etat de la cellule est devient

ue une un

, sa composante-non-5l ✛ si les composantes-non-5l de ses voisines est, nord-est et nord sont

 F ue une un respectivement  F ,  F et  F . Il en va de faKcon analogue pour les cellules dont ✲ et , ✻ . Pour ce type de cellules, les ❄ donn+ees sont en mˆeme temps compress+ees et pouss+ees vers le centre.

la W -composante est

• Si la W -composante de la cellule est ✲ , c’est-a-dire si elle est la cellule origine, a0 — au temps 0, la S-composante de son e+ tat est  F , — au temps 1, cette S-composante est transform+ee en F (q0 ; a0 : : : a8 ), qui est un e+ tat de l’automate 5ni, d+etermin+e par le fait qu’elle reKcoit comme donn+ees les lettres du mot situ+ees sur elle-mˆeme et ses premieres voisines. A partir de cet instant, le rˆole de la cellule (0; 0) est de simuler cet automate 5ni sur les donn+ees qui lui arrivent.

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— au temps t, 1¡t¡RArch (n), la composante de la cellule e+ tant qt , elle reKcoit de ses huit voisines une S-composante qui appartient a QQ ∪ {sq }. On identi5e sq #

u

a F . Alors la cellule entre dans l’+etat dont la S-composante est F (qt ) avec u =ue une un unw uw usw us use , qui vaut F (qt ; u). — au temps RArch (n), la cellule (0; 0) ne reKcoit plus que sq comme information ind+ependante du 5l. Alors elle entre dans l’+etat saccepte ou dans l’+etat srejette suivant que la composante-non-5l de la cellule est dans Qf ou non. 4. Le l de Hilbert Ce second 5l provient d’une courbe connue depuis la 5n du 19-ieme siecle [7]. C’est un exemple de courbe remplissant l’espace, fournissant des exemples de fonctions de R dans R 2 continues et jamais di=+erentiables. Un ouvrage [15] est consacr+e a ce sujet. En fait nous ne choisirons pas cette courbe historique, mais une variante apparue dans le domaine des automates cellulaires [10] et fortement li+ee aux pavages ap+eriodiques du plan. Si le r+esultat que nous d+emontrons dans ce paragraphe s’+enonce de maniere analogue au pr+ec+edent, le temps r+eel n’a pas de bonne repr+esentation et l’algorithme de reconnaissance est beaucoup plus 5n. 4.1. De6nition et representation graphique Pour d+e5nir le 5l de Hilbert, nous proc+edons comme dans le cas du 5l d’Archimede, en d+e5nissant d’abord g+eom+etriquement une courbe discrete qui permet d’obtenir le 5l. La courbe que nous obtenons est inspir+ee de celle de Hilbert (en fait, nous partons d’un premier motif qui est obtenu de celui de Hilbert par une rotation de −$=2 [7]). Nous l’appellerons malgr+e tout courbe de Hilbert. Notation 3. (1) Le carr+e dont le coin sud-ouest est le point (0; 0) et dont le cˆot+e est de taille 2n sera d+esign+e par Cn , n∈N − {0}. (2) Ce carr+e Cn est partitionn+e en quatre carr+es e+ gaux. Le carr+e sud-ouest ainsi obtenu est d+enot+e Cn; 1 , le carr+e nord-ouest Cn; 2 , le carr+e nord-est Cn; 3 et le carr+e sud-est Cn; 4 . Denition 8 (Courbe de Hilbert). Le 5l de Hilbert est d+e5ni a partir d’une courbe C, construite elle-mˆeme a partir d’une famille (Cn )n∈N−{0} de courbes de Hilbert, chacune faite de segments du plan. 1. La famille (Cn )n∈N−{0} est construite par induction comme suit (voir Fig. 3): • Initialement on part de la 5gure C1 , inscrite dans le carr+e C1 et constitu+ee de trois segments unit+e: deux horizontaux, joints en leurs extr+emit+es gauche par un segment vertical. • Le second pas consiste a construire la courbe dans le carr+e C2 . Cela se r+ealise de la faKcon suivante: C1 est recopi+e dans C2; 1 et C2; 2 , puis dans C2; 3 apres une rotation de 3$=2 et en5n dans C2; 4 apres une rotation de $=2. La 5gure est compl+et+ee en joignant par un segment vertical entre C2; 1 et C2; 2 les courbes qui

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261

Fig. 3. La famille (Cn )n∈N−{0} .

y sont inscrites, puis par des segments horizontaux entre C2; 1 et C2; 4 d’une part, C2; 2 et C2; 3 d’autre part les courbes qui y sont e+ galement inscrites, de maniere a obtenir une courbe sans chevauchements a deux extr+emit+es libres, C2 . • Le processus est it+er+e a5n d’obtenir la courbe Cn+1 dans Cn+1 a partir de sa partie Cn inscrite dans le carr+e Cn . 2. La courbe C qui nous int+eresse est en fait:
C = C0 ∪ (Cn ∩ (Cn;2 ∪ Cn;3 )); n¿1

ou C0 d+esigne le segment qui joint le centre de C1;1 au centre de son bord nord. Nous sommes maintenant en mesure de pr+eciser ce qu’est le 5l de Hilbert. Denition 9 (Fil de Hilbert). Le 5l de Hilbert H est l’application de N dans Z2 d+e5nie comme suit (voir Fig. 4b): • H(0) =(0; 0), • si H( j) est d+e5ni pour tout j, j ∈ {0; : : : ; i}, alors H(i + 1) est le coin sud-ouest de l’unique carr+e unit+e — dont le coin sud-ouest n’appartient pas a H({0; : : : ; i}) — qui est connect+e par C a H(i). Nous prouvons maintenant une propri+et+e du 5l de Hilbert qui joue un rˆole essentiel pour la suite. 4.2. Une propriete du 6l de Hilbert Nous aurons a distinguer plusieurs types de carr+es de taille 2n , n¿1, suivant la maniere dont le 5l de Hilbert les traverse, ce qui n+ecessite de pr+eciser des notations. Nous nous int+eressons d’abord aux carr+es de taille 2. 2 Notation 4. (1) Par Cen , nous d+esignons tout carr+e de taille 2 dans lequel le 5l entre par l’est pour en ressortir par le nord. Plus formellement, tout carr+e pour lequel il

262

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

Fig. 4. Hilbert: (a) La courbe C. (b) Le 5l de Hilbert H.

existe deux entiers distincts i1 et i2 tels que H(i1 + 1)= H(i1 ) + (−1; 0) et H(i2 + 1) =H(i2 ) + (0; 1): (2) On d+e5nira de maniere analogue les carr+es Cij2 , pour i; j ∈ {e; n; w; s} ou la signi5cation des lettres est e+ vidente. Nous sommes alors en mesure de poser la d+e5nition suivante: Denition 10 (Courbes de Hilbert 2-group+ees). Soit n ∈N, n¿1. On d+e5nit la courbe de Hilbert 2-group+ee, Cn(2) , en partant de Cn et en parcourant les di=+erents carr+es Cij2 successivement le long de Cn , a partir de l’origine, comme suit (voir Fig. 5): • Joindre le milieu du carr+e C1 au milieu de son bord sup+erieur, ou bord nord, puis, • Si le carr+e courant est Cij2 , joindre le centre de ce carr+e au milieu de son bord i et au milieu de son bord j, • Le dernier carr+e de taille 2 rencontr+e intersecte Cn en exactement un point sur son bord ouest, joindre alors le centre de ce carr+e au milieu de son bord nord (celui par lequel sort le 5l du carr+e Cn de taille 2n ). Proposition 2. Soit HO;1=2 l’homothetie de centre (0; 0) et de rapport tout entier n, n¿1, (2) Cn = HO;1=2 (Cn+1 ):

Preuve. La preuve se fait par induction sur n. • Le cas n =1 se v+eri5e facilement sur la Fig. 5.

1 2.

Alors, pour

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263

(2)

Fig. 5. La courbe C3 .

• Dans le cas g+en+eral, on considere les carr+es Cn+1; k , k ∈ {1; 2; 3; 4}. — Puisque, par construction de Cn+1 , l’intersection de Cn+1 avec Cn+1;1 est l’intersection of Cn avec Cn , par hypothese d’induction on a Cn ∩ Cn;1 = HO;1=2 (Cn+1 ∩ Cn+1;1 ): — Par construction de Cn+1 , l’intersection de Cn+1 avec le carr+e Cn+1; 2 est obtenue par translation de vecteur (0; 2n ) a partir de l’intersection de Cn+1 avec le carr+e Cn+1; 1 qui est elle-mˆeme l’intersection de Cn avec le carr+e Cn . Par hypothese, l’intersection de Cn avec le carr+e Cn; 1 est obtenue par HO;1=2 a partir de l’intersection de Cn avec la carr+e Cn . Mais l’intersection de Cn avec le carr+e Cn; 2 est obtenue a partir de l’intersection de Cn avec le carr+e Cn; 1 par la translation de vecteur (0; 2n−1 ). Aussi Cn ∩ Cn;2 = HO;1=2 (Cn+1 ∩ Cn+1;2 ): — Les autres cas sont semblables en faisant intervenir les rotations utilis+ees pour la d+e5nition de Cn . La preuve se r+esume en les quatre diagrammes commutatifs suivants ou T(';() est la translation de vecteur ('; () et R) est une rotation d’angle ) (centr+ee au centre d’un carr+e). T(0;0)

Cn+1 −−−→ HO; 1  2 Cn

T(0;0)

Cn HO; 1  2

−−−→ Cn−1

T(0;−2n )

Cn+1  −−−→ HO; 1  2 Cn

T(0;−2n−1 )

Cn

HO; 1  2



−−−−→ Cn−1

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M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

Cn+1  HO; 1  2 Cn

T(2n ;0) ◦R 3$

2 −−−−−→

T(2n−1 ;0) ◦R 3$ 2

Cn  HO; 1  2

Cn+1  HO; 1  2

−−−−−−−→ Cn−1 Cas Cn+1;3

Cn

T(2n ;−2n ) ◦R $ 2

−−−−−−→

T(2n−1 ;2n−1 ) ◦R $ 2

Cn

HO; 1  2



−−−−−−−−→ Cn−1 Cas Cn+1;4

Corollaire 1. Le 6l entre et sort une seule fois de tous les carres de taille 2n qu’il traverse. Preuve. La preuve, par r+ecurrence sur n, est triviale. L’initialisation pour n = 1 s’observe sur C2 et le pas g+en+eral provient de la proposition pr+ec+edente. Ce corollaire sera utilis+e sous la forme suivante: si on pave le plan par des carr+es de taille 2n alors si le 5l passe sur un tel carr+e, apres en eˆ tre sorti une fois, il n’y rentre plus. 4.3. Temps reel sur le 6l de Hilbert Le temps r+eel sur le 5l de Hilbert ne s’exprime pas ais+ement. Nous donnons quelques indications sur comment le calculer. Nous d+esignons ce temps r+eel par RH(‘), ou ‘, ‘¿1, est la longueur du mot d’entr+ee. Sur la Fig. 6, nous avons indiqu+e par un cercle vert la position des points H(‘ − 1), 5n des mots de longueur ‘ pour laquelle le temps r+eel augmente d’une unit+e. En observant cette Fig. 6, nous voyons comment calculer la valeur de RH(‘) quand H(‘ − 1) appartient au carr+e Cn en fonction de ses valeurs quand H(‘ − 1) appartient a Cn−1 . Nous r+esumons ces observations dans la Table 1 ou nous avons suppos+e n¿3. Les cas ou n¡3 se calculant directement. Il ne reste plus qu’a formaliser et d+emontrer ces observations. Pour cela, nous avons besoin d’un lemme technique qui servira a expliciter les coordonn+ees en jeu. Lemme 1. Le point le plus en haut a: droite, c’est-:a-dire le coin nord-est, du carre Cn est H(*n ) avec *n =

n−1  i=0

2:4n−1 = 23 (4n − 1):

De mˆeme, le point d’o:u sort le 6l du carre Cn; h est *n−1 +(h−1)4n−1 (h ∈ {1; 2; 3}). Preuve. On le montre par r+ecurrence sur n. Si n = 1, on obtient H(2). Puis pour n, il y a (2:2n−1 : 2n−1 ) points de H dans les carr+es Cn; 2 et Cn; 3 . Donc *n+1 = *n + (2:4n−1 ). D’ou la formule. Il su/t ensuite de revenir a la d+e5nition de H et d’observer qu’il y a 4n−1 points de H dans un carr+e Cn; h .

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

265

Fig. 6. Ou le temps r+eel sur le 5l de Hilbert augmente d’une unit+e.

Maintenant examinons les cas les uns apres les autres: Cas (P1 ): Si *n−1 + 4n−1 + 16‘ − 16*n , alors RH(‘) = 2n . Les coordonn+ees de ces points (dont le num+ero est ‘) sont inf+erieures ou e+ gales a l’ordonn+ee du point d’ou le 5l sort de Cn; 2 qui est e+ gale a celle d’ou le 5l sort de Cn , qui est pr+ecis+ement 2n − 1. Cas (P2 ): Si *n−1 + 4n−2 + 16‘ − 16*n−1 + 2:4n−2 , alors RH(‘) = 2n−1 + 2n−2 . Les coordonn+ees de ces points sont inf+erieures ou e+ gales a l’ordonn+ee du point en haut a droite de Cn; 2; 4 . Cas (P3 ): Si *n−1 + 2:4n−2 + 16‘ − 16*n−1 + 3:4n−2 , alors RH(‘) = RH(‘ − n−1 4 ) + 2n−1 . Il su/t d’observer que, par construction, le carr+e Cn; 2; 2 est le translat+e du carr+e Cn−1; 2 par le vecteur (0; 2n−1 ) et que le nombre de points de H entre un point et son translat+e est e+ gal a la “surface” de Cn; 2 . Cas (P4 ): Si *n−1 + 3:4n−2 + 16‘ − 16*n−1 + 4n−1 , alors RH(‘) = 2n . Les coordonn+ees de ces points sont inf+erieures ou e+ gales a l’ordonn+ee du point en haut a droite de Cn .

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Table 1 Etude g+eom+etrique de RH(‘) pour n¿3

Carr+e d’origine

Observation

Cn; 1

Par hypothese: Cn; 1 = Cn

Cn; 3

Temps constant: (x; y) ∈ Cn; 3 ⇒ RH(H−1 (x; y) + 1) = 2n

(P1 )

Cn; 2; 1

Temps constant: (x; y) ∈ Cn; 2; 1 ⇒ RH(H−1 (x; y) + 1) = 2n−1 + 2n−2

(P2 )

Cn; 2; 2

Reli+e a Cn−1; 2 : (x; y) ∈ Cn; 2; 2 ⇒ RH(H−1 (x; y) + 1) = 2n−1 + RH(H−1 ((x; y) − (0; 2n−1 )) + 1)

(P3 )

Cn; 2; 3

Temps constant: (x; y) ∈ Cn; 2; 3 ⇒ RH(H−1 (x; y) + 1) = 2n

(P4 )

Cn; 2; 4; 1

Temps constant: (x; y) ∈ Cn; 2; 4; 1 ⇒ RH(H−1 (x; y) + 1) = 2n−1 + 2n−3

(P5 )

Cn; 2; 4; 2

Temps constant: (x; y) ∈ Cn; 2; 4; 2 ⇒ RH(H−1 (x; y) + 1) = 2n−1 + 2n−2

(P6 )

Cn; 2; 4; 3

Reli+e a Cn−1; 2; 4 : (x; y) ∈ Cn; 2; 3 ⇒ RH(H−1 (x; y) + 1) = 3 2n−3 + RH(H−1 ((x; y) − (2n−2 ; 3 2n−3 )) + 1)

(P7 )

Cn; 2; 4; 4

Reli+e a Cn−1; 2; 2 : (x; y) ∈ Cn; 2; 4 ⇒ RH(H−1 (x; y) + 1) = 2n−3 + RH(H−1 ((x; y) − (3 2n−3 ; 2n−3 )) + 1)

(P8 )

Cn; 2

Cn; 2; 4

Cas (P5 ): Si *n−1 + 3:4n−3 + 16‘ − 16*n−1 + 4n−2 , alors RH(‘) = 2n−1 + 2n−3 . Les coordonn+ees de ces points sont inf+erieures ou e+ gales a l’ordonn+ee du point en haut a droite de Cn; 2; 4; 3 . Cas (P6 ): Si *n−1 + 2:4n−3 + 16‘ − 16*n−1 + 3:4n−3 , alors RH(‘) = 2n−1 + 2n−2 . Les coordonn+ees de ces points sont inf+erieures ou e+ gales a l’ordonn+ee du point en haut a droite de Cn; 2; 4 . Cas (P7 ): Si *n−1 +4n−3 6‘−16*n−1 +2:4n−3 +1, alors RH(‘) =RH(‘−9:4n−3 )+ 3:2n−3 . Par construction, le carr+e Cn; 2; 4 est obtenu par T(2n−2 ; 2n−2 ) ◦R$=2 a partir du carr+e Cn−1; 2 . Il en est donc de mˆeme pour le carr+e Cn; 2; 4; 3 a partir du carr+e Cn−1; 2; 2 . Mais ce carr+e Cn−1; 2; 2 est aussi obtenu a partir du carr+e Cn−1; 2; 4 par T(−2n−3 ; 2n−3 ) ◦ R−$=2 . Il s’en suit qu’on obtient Cn; 2; 4; 3 a partir de Cn−1; 2; 2 par translation de vecteur (2n−2 ; 3:2n−3 ). Il su/t alors d’observer qu’entre deux points translat+es, le 5l H remplit la surface de 9 carr+es de taille 2n−3 .

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267

Fig. 7. Temps r+eel sur le 5l de Hilbert.

Cas (P8 ): Si *n−1 + 16‘6 − 1*n−1 + 4n−3 , alors RH(‘) = RH(‘ − 6:4n−3 ) + 2n−3 . Par construction, le carr+e Cn; 2; 4 est obtenu par T(2n−2 ; 2n−2 ) ◦R$=2 a partir du carr+e Cn−1; 2 . Il en est donc de mˆeme pour le carr+e Cn; 2; 4; 4 a partir du carr+e Cn−1; 2; 3 . Mais ce carr+e Cn−1; 2; 3 est aussi obtenu a partir du carr+e Cn−1; 2; 2 par T(2n−3 ;0) ◦ R−$=2 . Il s’en suit qu’on obtient Cn; 2; 4; 3 a partir de Cn−1; 2; 2 par translation de vecteur (3:2n−3 ; 2n−2 ). Il su/t alors d’observer qu’entre deux points translat+es, le 5l H remplit la surface de 6 carr+es de taille 2n−3 . Toutes ces observations donnent une faKcon r+ecursive de calculer RH(n) (voir la Fig. 7). L’aspect fractal de la courbe se comprend facilement car un calcul direct du temps r+eel par l’arithm+etisation du 5l de Hilbert donn+ee en [15] ferait intervenir r+ecursivement les chi=res de l’+ecriture en base 4 de ‘−‘0 (ou ‘0 = 4n −1−*n = 13 (4n −1) est la longueur de la courbe Cn n’appartenant pas a H). 5. Langages rationnels sur le l de Hilbert Ce paragraphe est constitu+e par la preuve, divis+ee en plusieurs sous-sections, du th+eoreme que nous allons e+ noncer. Dans la premiere, nous ferons quelques remarques g+en+erales justi5ant en particulier la pr+esentation de l’algorithme en trois parties: a l’ext+erieur d’une boule B(0; 0) (t), puis a l’int+erieur d’une telle boule pour 5nir par ce qui releve plus sp+eci5quement du temps r+eel. Theoreme 2. Tout langage rationnel est reconnu en temps reel par un 2D automate cellulaire avec le 6l de Hilbert. Preuve. La construction d’un 2-automate cellulaire avec 5l de Hilbert qui reconnaisse, en temps r+eel, un langage rationnel est assez longue et technique. 5.1. Remarques generales Nous reprenons les notations de la preuve du Th+eoreme 1. On part d’un langage rationnel L, suppos+e reconnu par un automate 5ni d+eterministe F= (Q; ; F ; q0 ; Qf ). Le 2D-AC que nous nous proposons de construire est d+esign+e par AAC = (QAC ; ). u

u

Nous associons a F l’ensemble 5ni des applications {}u ∈ ? ou (q) = (q; u) pour tout e+ tat q de Q.

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M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

Fig. 8. Mouvement des donn+ees dans la preuve du Th+eoreme 2 (Remarques g+en+erales). (a) Sch+ema. (b) Exemple.

Il s’agit, comme dans le cas du Th+eoreme 1 d’envoyer les donn+ees le plus vite possible sur la cellule d’origine. Mais la forme du 5l induit des di=+erences importantes. Une premiere est que, contrairement au cas de la spirale d’Archimede, la composante5l (W -+etat) d’une cellule ne renseigne pas n+ecessairement cette cellule sur sa position par rapport a l’origine. Une autre di=+erence essentielle est justement qu’une partie du 5l de Hilbert est au-dessous de la premiere diagonale et que le 5l n’est pas sym+etrique par rapport a cette diagonale, donc en particulier qu’une cellule ne sait pas quelle est sa situation par rapport a l’axe des abscisses. Cependant les cellules de l’axe des ordonn+ees savent qu’elles sont sur cet axe. Elles vont donc envoyer leur information vers le sud et permettre de r+eorienter vers le sud, en la 2-groupant avec leur information propre, l’information qui leur parvient du nordest de cellules au-dessus de la diagonale. Par ailleurs, les voisines sud des cellules qui contiennent le bord sud du 5l dans les carr+es Cn; 3 ont cette connaissance. Elles vont servir a indiquer a l’information qui leur parvient du nord-est de cellules au-dessous de la diagonale, en la 2-groupant avec la leur, de se porter vers l’ouest. Ceci est illustr+e sur la Fig. 8. La diagonale joue un rˆole bien particulier et essentiel qui exploite les “unit+es” du 5l de Hilbert que sont les carr+es Cij2 . En e=et, lorsque sur les cellules (t; t); (t + 1; t); ,1

,2

,3

,4

(t + 1; t + 1); (t; t + 1) apparaissent les donn+ees-lettres-5l (  ; w1 ); (  ; w2 ); (  ; w3 ); (  ; w4 ) u d’un tel motif de H, l’automate “absorbe” cette information sous la forme (; w) ou u est le mot form+e par les quatre lettres ,i dans l’ordre qui est indiqu+e par les composantes-5l correspondantes qui d+e5nissent le type w du carr+e Cij2 concern+e. Nous dirons que l’automate procede a une amalgamation ou que les donn+ees ont e+ t+e amalgamees. A ce mˆeme temps t, les donn+ees des cellules (i; t); (i; t + 1), 06i6t − 1, sont amalgam+ees sur la cellule (i; t), celles des cellules (t; j); (t + 1; j) sont amalgam+ees sur

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269

la cellule (t; j). Ainsi pour les temps t  , t  ¿t, l’information qui circule dans la boule B(0; 0) (t) est une information qui a d+eja e+ t+e pr+e-trait+ee en fonction de l’automate 5ni reconnaisseur de L, via au moins une amalgamation. Nous verrons que si les donn+ees se d+eplacent aussi vite que possible a l’ext+erieur d’une boule B(0; 0) (t), il pourrait y avoir perte de temps a l’int+erieur de la boule: chaque fois que des donn+ees sont amalgam+ees. Cela nous incite a initier alors en parallele un second calcul qui permettra de contrˆoler le temps r+eel. En cons+equence, nous concevrons l’algorithme r+ealis+e par l’automate cellulaire reconnaisseur comme la superposition de deux “feuilles”: une feuille Freguli  : et une feuille Fanticipation . ere L’+evolution de l’automate partiel courant sera e+ ventuellement illustr+ee par des 5gures. La 5gure au temps t repr+esente la con5guration obtenue apres l’ex+ecution de la t-ieme transition d’+etat de l’automate. 5.2. L’algorithme sur la feuille Freguli  : ere D’apres ce qui pr+ecede, l’ensemble des e+ tats QAC de A va eˆ tre construit comme produit de plusieurs composantes que nous pr+ecisons au fur et a mesure de la construction. Pour cette feuille, nous aurons besoin des composantes d+ecrites ci-dessous. • Une “composante-5l”, d+enot+ee W , qui donne la forme du 5l dans chaque cellule ou il passe. • Une composante D pour les d+eplacements de l’information, qui d+epend du mot test+e, de la forme du 5l, de l’automate 5ni, et qui reUete la dissym+etrie du 5l par rapport a la premiere diagonale. D ={sq } ∪ Q ∪QQ × W ∪(QQ × W × {0; 1}) ∪(QQ × W )2 . La composante sq signi5e l’absence, sur la cellule, d’information-lettre (elle sera e+ ventuellement assimil+ee a #

). Dans un e+ l+ement de QQ × W × {0; 1}, la composante de {0; 1} servira a indiquer que la donn+ee exprim+ee par les autres composantes a franchi ou non un certain seuil pr+ecis+e plus bas. • une composante E qui contient des e+ tats de l’automate ind+ependants des lettres du mot d’entr+ee, dont la signi5cation intuitive est indiqu+ee dans la Table 2. Ainsi E = {I; G; H; V; B; S; Ui) ; Ri) (i ∈ {0; 1})}. CommenKcons par d+e5nir le fonctionnement de l’automate (partiel relatif a cette feuille) a l’ext+erieur d’une boule B(0; 0) (t). La Fig. 9 illustre ce point. 5.2.1. Algorithme pour l’exterieur d’une boule B(0;0) (t) (1) Temps t =0, Con6guration initiale La con5guration initiale est d+e5nie par la r+epartition des lettres du mot d’entr+ee sur le 5l de Hilbert, comme dans le cas du 5l d’Archimede. C’est dire que la E-composante ,i

des e+ tats des cellules est I . Leur D-composante est ((F) ; wi ; 0) pour toutes les cellules qui portent une lettre ,i du mot d’entr+ee non vide ,0 : : : ,n−1 , et elle est sq pour toutes les autres. A partir de cette con5guration, un certain nombre de cellules connaissent, grˆace aux e+ tats-5l, et particulierement a l’+etat , quelle position sp+eciale elles

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M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289 Table 2 Signi5cation des e+ tats de E

E-Etat

Signi5cation

I

Cellules incapables de situer (0; 0).

G

Etat marquant des cellules sur lesquelles quatre donn+ees sont amalgam+ees, sur le bord interne d’une boule B0;0 (t), t ¿0.

V

Etat marquant des cellules situ+ees sur une verticale indiquant un groupement par deux de donn+ees.

B

Etat indiquant certaines cellules de la diagonale.

S

Etat marquant des cellules situ+ees imm+ediatement sous la partie du 5l contenue dans les carr+es Cn; 3 , n¿2, permettant de d+e5nir un seuil horizontal pour le d+eplacement nord-est/sud-ouest des donn+ees des cellules situ+ees sous la diagonale, utiles pour le temps r+eel.

H

Cellules situ+ees sur une horizontale indiquant un groupement par deux de donn+ees.

U0)

Etat indiquant les cellules au-dessus de la diagonale apres un nombre pair d’amalgamations des donn+ees d’origine. ) ∈ {0; 1} indique la parit+e de l’abscisse.

U1)

Etat indiquant les cellules au-dessus de la diagonale apres un nombre impair d’amalgamations des donn+ees d’origine. ) ∈ {0; 1} indique la parit+e de l’abscisse.

R)0

Etat indiquant les cellules au-dessous de la diagonale apres un nombre pair d’amalgamations des donn+ees d’origine. ) ∈ {0; 1} indique la parit+e de l’ordonn+ee.

R)1

Etat indiquant les cellules au-dessous de la diagonale apres un nombre impair d’amalgamations des donn+ees d’origine. ) ∈ {0; 1} indique la parit+e de l’ordonn+ee.

ont sur le plan et relativement au 5l, ce qui se marque, des le temps 1, par un E-+etat correspondant. Ceci est r+esum+e dans la Table 3. ,c

Remarquons que la composante i, i ∈ {0; 1}, des D-+etats de la forme (  ; wc ; i) ne sera utilis+ee que pour signi5er qu’une information provenant d’une cellule situ+ee au-dessous de la diagonale aura franchi ou non le premier “mur” d’+etats S qu’elle doit rencontrer, comme nous le verrons plus bas. Elle pourra alors e+ ventuellement eˆ tre 2-group+ee avec sa voisine nord et cette composante 1 n’aura plus de raison d’ˆetre. (2) Temps t¿1, e volution des composantes E et D des e tats Au temps 1, les cellules qui 5gurent dans la Table 3 entrent dans le E-+etat indiqu+e. La cellule c =(0; 0) entre dans le D-+etat

,ne ;,n ;,c



(q0 ), appartenant a Q. Les cellules de

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271

Fig. 9. Description g+en+erale de l’algorithme avant la premiere amalgamation.

,ne

,n

l’axe des ordonn+ees entrent dans le D-+etat ((  ; wne ); (  ; wn )), appartenant a (QQ × W )2 (groupage de deux donn+ees d’origine). Toutes les autres cellules prennent pour D-+etat ,ne

celui de leur voisine nord-est, (  ; wne ; 0), appartenant a QQ × W × {0; 1}. Puis les transitions suivantes sont e=ectu+ees: • (Pt1 ) Si une cellule c est dans le E-+etat I , V , B ou H , et si l’une de ses voisines est dans l’+etat G, alors, elle entre dans le E-+etat G et dans le D-+etat d+e5ni comme suit: ,c; ne ,c; n — (Pt1a ) si son E-+etat est V , c’est que son D-+etat est de la forme ((  ; wc;ne ); (  ; ,c ; ne

,c  ; n

wc; n )), si sa voisine nord c est e+ galement dans un e+ tat ((  ; wc ; ne ); (  ; wc ; n )), u alors la cellule c entre dans le D-+etat (; w) ou w est le type du carr+e du 5l

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Table 3 Marquage au temps 1 de cellules strat+egiques

Position de la (ou des) cellule(s)

Voisins dans le W -+etat

E-+etat au temps 1

Cellule (0; 0)

est, nord-ouest, ouest, sud-ouest, sud et sud-est

G

Cellule (0; i) (i¿1), sur le bord-ouest du 5l

nord-ouest, ouest et sud-ouest

V

Cellule (2n−1 ; 2n−1 ), n¿1, sud-ouest, sud et sud-est point du 5l dans le coin sud-ouest de Cn; 3 , sur la diagonale

B

Cellules (2n−1 ; 2n−2 ), (2n−1 + 1; 2n−2 ); : : : ; (2n ; 2n−2 ), n¿1, bordant inf+erieurement le 5l a l’ext+erieur du carr+e Cn;3

S

sud-ouest, sud, sud-est et est pour les cellules (h; 2n−2 ), 2n−1 ¡h¡2n sud, sud-est et est pour (2n−1 ; 2n−2 ), sud-ouest, sud, sud-est, est et nord-est pour (2n ; 2n−2 )

d+e5ni par les quatre donn+ees wc; ne ; wc; n ; wc ; ne et wc ; n , et u le mot obtenu a partir des lettres correspondantes dans l’ordre originel du 5l. u

— (Pt1b ) si son E-+etat est B, elle entre dans (; w) ou, en supposant que elle-mˆeme et ses voisines est, nord-est et nord possedent des donn+ees d’origine correspondant a un carr+e d’origine, w est le type de ce carr+e et u le mot obtenu a partir des lettres ,c , ,n , ,ne , et ,e , dans l’ordre originel du 5l. u

— (Pt1c ) si son E-+etat est H , elle entre dans (; w) ou, en supposant que ellemˆeme c et sa voisine est c possedent des donn+ees d’origine 2-group+ees du type ,c; e

,c  ; e

,c; ne

,c ; ne

((  ; wc; e ); (  ; wc; ne )) et ((  ; wc ; e ); (  ; wc ; ne )), alors w d+esigne le type de carr+e d+e5ni par les wc; e ; wc; ne ; wc ; e et wc ; ne et u le mot form+e par les lettres correspondantes. #

— (Pt1d ) (;

) sinon.

• (Pt2 ) Une cellule dont le E-+etat est I et dont le E-+etat de la voisine ouest est V ,ne

,n

entre dans le E-+etat V . Elle prend alors pour D-+etat ((  ; wne ); (  ; wn )) (2-groupage) ,c

en supposant que les D-+etats de ces deux voisines sont de la forme (  ; wc ; 0). • (Pt3 ) Une cellule dont le E-+etat est I et dont la voisine sud a pour E-+etat H entre dans ,ne

,e

le E-+etat H et prend comme D-+etat ((  ; wne ); (  ; we )) (2-groupage) en supposant ,i que les D-+etats de ses deux voisines est et nord-est sont de la forme (  ; wi ; 1), i ∈{e; ne}. • (Pt4 ) Une cellule dont le E-+etat est V reste dans V et prend pour D-+etat le D-+etat de sa voisine nord. • (Pt5 ) Une cellule dont le E-+etat est H reste dans H et prend pour D-+etat le D-+etat de sa voisine est.

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273

• (Pt6 ) Une cellule dont le E-+etat est B reste dans B et prend pour D-+etat le D-+etat de sa voisine nord-est. • (Pt7 ) Une cellule dont le E-+etat est I et dont la voisine sud-ouest a B pour E-+etat entre dans le E-+etat B et prend pour D-+etat le D-+etat de sa voisine nord-est. • (Pt8 ) Une cellule dont le E-+etat est S reste dans le E-+etat S et prend pour D-+etat le ,

,

D-+etat (; w; 1) si le D-+etat de sa voisine nord-est est (; w; 0). • (Pt9 ) Une cellule dont le E-+etat est I et dont la voisine ouest est dans le E-+etat S ,

,

prend le E-+etat S et le D-+etat (; w; 1) ou (; w; i) ou i ∈ {0; 1} est celui de sa voisine nord-est. , • (Pt10 ) Une cellule dont le E-+etat est I et le D-+etat est de la forme (; w; 1) et dont la voisine sud a pour E-+etat S prend pour E-+etat H . Elle prend alors pour D-+etat ,e

,ne

((  ; we ); (  ; wne )) (2-groupage) en supposant que les D-+etats de ces deux voisines ,c

sont de la forme (  ; wc ; 1). • (Pt11 ) Dans les autres cas, une cellule dans le E-+etat I reste dans le E-+etat I et prend pour D-+etat le D-+etat de sa voisine nord-est. Ainsi, les points (Pt1 ) et (Pt2 ) impliquent qu’au temps t, les cellules (x; y) avec x¡t sont dans le E-+etat G si y =t et dans le E-+etat V si y¿t. Les points (Pt6 ) et (Pt7 ) montrent que le E-+etat B ne peut apparaˆPtre que sur la diagonale. Les points (Pt3 ), (Pt5 ), (Pt8 ) et (Pt10 ) montrent que l’+etat H ne peut apparaˆPtre qu’au-dessous de la diagonale, et qu’il joue alors un rˆole analogue a celui de V . (3) Correction de l’automate partiel AC Il reste maintenant a montrer que l’automate partiel qui vient d’ˆetre d+ecrit r+ealise bien la strat+egie annonc+ee. D’apres la Proposition 2 et son corollaire, c’est le mouvement des donn+ees d’origine des cellules (2i; 2j), (2i + 1; 2j), (2i; 2j + 1) et (2i + 1; 2j + 1), qui forment les carr+es Cij2 , que nous devons e+ tudier. C’est ce que nous allons expliciter, en fonction de la position de la cellule (2i; 2j) par rapport a la diagonale. • Supposons que (2i; 2j) est au-dessus de la diagonale Les donn+ees de (2i; 2j) et de (2i; 2j + 1) se dirigent vers le sud-ouest par (Pt11 ) tant que le E-+etat de la cellule portant leur information d’origine est I . Ces donn+ees rencontreront un e+ tat V sur les cellules (i; 2j − i) et (i; 2j + 1 − i). Puisqu’elles sont situ+ees au-dessus de la diagonale, elles ne rencontrerons pas de E-+etat G avant ce temps. Donc, au temps i + 1, sur la cellule (i; 2j − i − 1) (resp. (i; 2j − i)), par (Pt2 ), les valeurs d’origine de (2i; 2j) et de (2i + 1; 2j) (resp. (2i; 2j + 1) et de (2i + 1; 2j + 1)) seront 2-group+ees. Ensuite ces 2-groupes se dirigeront vers le sud (grˆace aux E-+etats V et a (Pt4 )) jusqu’a la cellule (i; j) dont la voisine sud est dans le E-+etat G. Au temps suivant, donc au temps j − 1, les donn+ees 2-group+ees des cellules (i; j) et (i; j + 1) seront amalgam+ees sur cette mˆeme cellule (i; j) ((Pt1a )) qui passera alors dans le E-+etat G. Les hypotheses de (Pt1a ) et de (Pt2 ) sont ainsi v+eri5+ees en ce qui concerne la nature des D-+etats. Par le corollaire 1, les donn+ees d’origine peuvent eˆ tre ordonn+ees suivant l’ordre du 5l et la transition de (Pt1a ) est bien d+e5nie. Notons que la troisieme composante du D-+etat des cellules consid+er+ees ne change pas jusqu’au 2-groupage quand elle disparaˆPt naturellement. En fait, elle est inutile pour les cellules de cette partie du plan.

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• Supposons que (2i; 2j) est au-dessous de la diagonale et appartient a: H Observons tout d’abord que ce cas ne peut se produire que si (2i; 2j) est dans le carr+e Cn; 3 (avec n¿2) et qu’alors i ∈ {2n−2 +1; : : : ; 2n−1 −1}, j ∈ {2n−2 ; : : : ; 2n−2 −1} et j¡i. A un temps t (t ∈ {1; : : : ; 2n−1 }), la situation est la suivante: — Les cellules (2n−1 + t − 1; 2n−1 + t − 1) sont dans le E-+etat B et contiennent les donn+ees d’origine des cellules (2n−1 + 2t − 1; 2n−1 + 2t − 1) d’apres les points (Pt6 ) et (Pt7 ). — Les cellules (2i; 2j)=(2n−1 +k; 2n−1 +‘) avec 26k¡2n−1 −2 et 06‘¡2n−1 −2 et ‘¡k ont I pour E-+etat et contiennent la donn+ee d’origine de (2n − k + t; 2n − ‘ + t) ((Pt9 )). — Les cellules (2n−2 + k; 2n−2 − 1) avec k ∈ {t; : : : ; 2n−1 − 1 + t} sont dans le E-+etat S ((Pt8 )). Prenons le quadruplet “t+emoin” correspondant a (2i; 2n−1 ), 2n−1 + 26i62n − 2, (bord inf+erieur du carr+e Cn; 3 ). Il se d+eplace vers le sud-ouest ((Pt9 )) jusqu’a ce que la donn+ee d’origine de (2i; 2n−1 ) rencontre le E-+etat S au temps 1 sur la cellule ,

,

(2i − 1; 2n−1 − 1). Le D-+etat de cette cellule devient alors (; w; 1) ou (; w; 0) est le D-+etat de la cellule (2i; 2n−1 ) a l’origine. L’information ainsi modi5+ee continue de se d+eplacer par des cellules dont le E-+etat est I dans la direction sud-ouest jusqu’a ce qu’elle arrive dans une cellule dont la voisine sud est dans le E-+etat S. Ceci se produit au temps 2n−2 , sur la cellule (2i − 2n−2 ; 2n−1 − 2n−2 ). Alors, au temps suivant, 2n−2 + 1, les cellules (2i − 2n−2 − 1; 2n−1 − 2n−2 ) prennent le E-+etat H et ,e

,ne

,

,ne

le D-+etat 2-group+e ((  ; we ); (  ; wne )) si (e ; we ; 1) et (  ; wne ; 1) sont les D-+etats au temps pr+ec+edent de leurs voisines est et nord-est respectivement par (Pt10 ). Ainsi, a la 5n de la 2n−2 + 1-ieme transition, c’est-a-dire au temps 2n−2 + 1, les cellules (2n−2 + 1; 2n−2 ); : : : ; (2n−1 − 2; 2n−2 ) sont-elles dans le E-+etat H et avec des D-+etats 2-group+es. Finalement, a ce temps, les cellules (x; y) ou x ∈ {2n−2 +2; : : : ; 2n−1 +2}, y ∈ {2n−2 + ,

,

1; : : : ; 2n−1 − 1}, y¡x, sont dans le E-+etat I et le D-+etat (; w; 1) si (; w; 0) est le D-+etat de la cellule (x + 2n−2 + 1; y + 2n−2 + 1). Quant aux cellules (x; y) ou x ∈{2n−1 + 1; : : : ; 2n − 2}, y ∈ {2n−1 ; : : : ; 2n − 2}, y¡x, elles portent encore le D-+etat ,

(; w; 0) de la cellule (x + 2n−2 + 1; y + 2n−2 + 1). Le mouvement des donn+ees continue suivant les regles d+e5nies plus haut. Jusqu’au temps 2n−1 − 1 ou toutes les donn+ees contenues sur le 5l dans le carr+e Cn;3 sont soit 2-group+ees sur les demi-droites horizontales d’origine (x; y) avec x = 2n−1 − 1 et 2n−2 6y62n−1 −2 soit contenues par les cellules (2n−2 ; 2n−2 ); (2n−2 +1; 2n−2 ); (2n−2 + 1; 2n−2 + 1); (2n−2 ; 2n−2 + 1). Notons encore qu’aux temps 2n−1 , les cellules (2n−1 ; 2n−1 ); : : : ; (2n − 2; 2n−1 ) sont dans le E-+etat I et ne portent aucune information relative au mot car leur est arriv+ee l’information contenue exactement sur le bord externe du 5l dans le carr+e # Cn;3 . Elles sont donc dans le D-+etat (; ) et ce fait est encore vrai au temps 2n−1 +1 puisqu’un 2-groupage de donn+ees a lieu dans leurs voisines nord d’abscisses 2n−2 + 1; 2n−1 + 2.

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

• Supposons que (2i; 2j) est sur la diagonale Une telle cellule (2i; 2i) est dans un carr+e Cn;3 , donc 2n−2 6i62n−1 . D’apres le point (Pt7 ), la cellule (2i; 2i) = (2n−1 + j; 2n−1 + j) avec 16j62i;

275

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Fig. 10. R+epartition des E-+etats dans la preuve du Th+eoreme 2 (Algorithme pour l’int+erieur d’une boule).

Fig. 11. Repr+esentation des donn+ees au temps 27 dans la preuve du Th+eoreme 2 (Algorithme pour l’int+erieur d’une boule).

t doivent eˆ tre envoy+es imm+ediatement vers l’origine. Donc au temps t + 1 elles seront sur le bord int+erieur de B(0; 0) (t − 1) alors que la nouvelle amalgamation aura eu lieu sur le bord int+erieur de B(0; 0) (t + 1). Le mouvement des donn+ees est alors g+er+e a l’int+erieur de B(0; 0) (t) par les regles suivantes: • (Pt1? ) Une cellule dans le E-+etat U8) ou V considere ses voisines nord et nord-est: ? ◦ (Pt1A ) Si l’indice de leur parit+e d’amalgamation ainsi que celui de sa voisine sud sont le mˆeme que le sien: ? — (Pt1Aa ) Si ces deux cellules contiennent chacune une donn+ee non group+ee, elle les prend en les 2-groupant si son E-+etat est V ou si sa valeur pour ) est 0. Sinon, elle ne conserve aucune donn+ee. ? — (Pt1Ab ) Si la cellule est dans le E-+etat U81 et sa voisine-est est dans le E-+etat B, elle prend la donn+ee de sa voisine nord-est si cette derniere en a une.

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

•

•

• •

277

? — (Pt1Ac ) Si une, au moins, de ces voisines contient une donn+ee non group+ee ou 2-group+ee, elle la prend en choisissant sa voisine nord en cas de conUit. ? — (Pt1Ad ) Sinon, elle ne prend aucune donn+ee. ? ) Sinon, ◦ (Pt1B ? — (Pt1Ba ) Si sa voisine nord ne contient pas une donn+ee 2-group+ee, elle conserve sa propre donn+ee. ? — (Pt1Bb ) Si elle-mˆeme et sa voisine nord contiennent des donn+ees 2-group+ees, elle les amalgame et son indice d’amalgamation change au temps suivant. (Pt2? ) Une cellule dans le E-+etat R)8 ou H considere ses voisines est et nord-est: ? ◦ (Pt2A ) Si l’indice de leur parit+e d’amalgamation ainsi que celui de sa voisine ouest sont le mˆeme que le sien: ? — (Pt2Aa ) Si ces deux cellules contiennent chacune une donn+ee non group+ee, elle les prend en les 2-groupant si son E-+etat est H ou si sa valeur pour ) est 0. Sinon, elle ne conserve aucune donn+ee. ? — (Pt2Ab ) Si la cellule est dans le E-+etat R18 et sa voisine nord dans le E-+etat B, elle prend la donn+ee de sa voisine nord-est si cette derniere en a une. ? — (Pt2Ac ) Si une, au moins, de ces voisines contient une donn+ee non group+ee ou 2-group+ee, elle la prend en choississant sa voisine-est en cas de conUit. ? — (Pt2Ad ) Sinon, elle ne prend aucune donn+ee. ? ◦ (Pt2B ) Sinon, ? — (Pt2Ba ) Si sa voisine est ne contient pas une donn+ee 2-group+ee, elle conserve sa propre donn+ee. ? — (Pt2Bb ) Si elle-mˆeme et sa voisine est contiennent des donn+ees 2-group+ees, elle les amalgame. (Pt3? ) Une cellule dans le E-+etat B considere ses voisines est, nord-est et nord: ? ◦ (Pt1A ) Si l’indice de leur parit+e d’amalgamation ainsi que celui de sa voisine sud-ouest sont le mˆeme que le sien: ? — (Pt3Aa ) Si sa voisine nord-est contient une donn+ee non group+ee, elle la prend. ? — (Pt3Ab ) Sinon, elle ne prend aucune donn+ee. ? ) Sinon, ◦ (Pt3B ? — (Pt3Ba ) Si sa voisine nord-est ne contient pas de donn+ee, elle conserve sa propre donn+ee. ? — (Pt3Bb ) Si ces trois voisines et elle-mˆeme contiennent une donn+ee non group+ee, elle les amalgame. (Pt4? ) Une cellule dans le E-+etat G entre dans un des E-+etats V , U8) , B, R)8 ou B sans conserver aucune donn+ee. (Pt5? ) La cellule origine, dans le E-+etat O et le D-+etat q de Q, considere ses voisines nord et nord-est: ? ◦ (Pt4A ) Si elles contiennent toutes les deux des donn+ees non group+ees de QQ , elle une un

reste dans le E-+etat O et entre dans le D-+etat  (  (q)). ? ◦ (Pt4B ) Sinon, conserve ses E et D-+etats. Etudions maintenant les con5gurations auxquelles conduisent ces regles d’+evolution (voir les Figs. 11 et 12). PlaKcons-nous d’abord apres la premiere amalgamation sur le

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Fig. 12. Mouvement des donn+ees du temps 0 au temps 16 dans la preuve du Th+eoreme 2 (Algorithme pour l’int+erieur d’une boule, coupes horizontales du diagramme espace-temps).

bord interne de la boule B(0; 0) (t) lorsqu’une 8-ieme amalgamation y a e+ t+e r+ealis+ee au temps  que l’on supposera assez grand pour d+evelopper tous les cas. • Au temps  + 1, le bord interne de B(0; 0) (t − 1) ne contient pas de donn+ees par (Pt4? ).

M. Delorme, J. Mazoyer / Theoretical Computer Science 281 (2002) 251 – 289

279

• Au temps  + 1, les donn+ees du bord interne de B(0; 0) (t − 2) d+ependent de la parit+e de t: — Dans tous les cas, les cellules d’abscisse paire et d’ordonn+ee t − 2 de E-+etat U18 ou V contiennent des donn+ees (provenant du bord de B(0; 0) (t − 1) ou a eu lieu la 8-ieme amalgamation au temps ) qui sont 2-group+ees par ? (Pt1Aa ). — De mˆeme, les cellules d’ordonn+ee paire et d’abcisse t − 2 de E-+etat R81 ou H ? contiennent des donn+ees qui sont 2-group+ees par (Pt2Aa ). — Dans tous les cas, la cellule de ce bord, de E-+etat B, contient une donn+ee simple ? par (Pt3Aa ). — Si t est pair, alors il existe, sur ce bord, une cellule dans le E-+etat U81 et une autre de E-+etat R18 dont les voisines respectives est et nord ont pour E-+etat B. ? ? Alors, par (Pt1Ab ) ((Pt2Ab )), elles prennent la donn+ee de leur voisine nord-est. Si t est impair, seule la cellule de la diagonale est libre pour recevoir la donn+ee de sa voisine nord-est. On obtient ainsi, au temps  + 1, sur le bord interne de B(0; 0) (t − 2), des donn+ees 2-group+ees une cellule sur deux avec une (cas t pair) ou trois (cas t impair) cellules du coin nord-est avec une seule donn+ee (voir les temps 13 - t = 3 et  = 11 - et 16 - t = 4 et  = 14 - pour 8 = 2 sur la Fig. 12). • Au temps  + 2, les donn+ees du bord de B(0; 0) (t − 3) sont — Pour les cellules de E-+etat V (H ) celles de leur voisine nord (est) au temps ? ?  + 1 par (Pt1Ac ) et (Pt2Ac ). ? — Toutes les autres cellules prennent la donn+ee de leur voisine nord-est par (Pt1Ac ), ? ? (Pt2Ac ) et (Pt3Aa ). A ce moment-la, les deux cellules extrˆemes ont des donn+ees 2-group+ees puis ces donn+ees 2-group+ees apparaissent une cellule sur deux avec la disposition pr+ec+edente (t-parit+e d+ependante pour les cellules du coin). Par contre celles du bord de B(0;0)(t−2) ? ? ? n’ont pas de donn+ee par (Pt1Ad ), (Pt2Ad ) et (Pt3Ab ). • Au temps  + &, les donn+ees du bord de B(0; 0) (t − & − 1) sont du mˆeme type par ? ? ? (Pt1Ac ), (Pt2Ac ) et (Pt3Aa ): les &-cellules des extr+emit+es ont des donn+ees 2-group+ees puis de telles donn+ees apparaissent une cellule sur deux. Ceci donne la disposition en “damiers” avec “compression hyperbolique” observable sur la Fig. 11 entre la premiere et la seconde amalgamation. • Ainsi, si t est impair, pour & = t=2 on a: Au temps  + t=2, toutes les cellules du bord de B(0; 0) (t=2), a l’exception de la cellule du coin, contiennent des donn+ees 2-group+ees. Si on suppose que, a ce temps la, les cellules du bord de B(0; 0) (t=2 − 1) ont un indice 8 + 1 de parit+e ? ? ? ), (Pt2Ba ) et (Pt3Ba ) les cellules du bord de B(0; 0) (t=2 + d’amalgamation, par (Pt1Ba 1) auront les mˆemes donn+ees au temps  + t=2 + 1 puisque, par l’+etude pr+ec+edente, un bord de carr+e sur deux n’a pas de donn+ees. Mais au temps  + t=2 + 1, les cellules du bord de B(0; 0) (t=2 + 1) et de B(0; 0) (t=2) auront toutes, sauf les trois cellules du coin, des donn+ees 2-group+ees et l’amalgamation pourra avoir lieu sur le ? ? bord de B(0; 0) (t=2) lors de la  + t=2 + 2-ieme transition par (Pt1Bc ), (Pt2Bc ) et ? (Pt3Bb ). La encore, les transitions d’+etats li+ees a l’amalgamation sont bien d+e5nies

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car les fonctions a amalgamer repr+esentent des donn+ees se suivant dans l’ordre du 5l par le Corollaire 1. On observe que le cas t pair a aussi e+ t+e consid+er+e: ses donn+ees s’amalgament a celles produites par la boule B(0; 0) (t − 1) sur la boule B(0;0) ((t − 1)=2) par ce qui pr+ecede. Ainsi, sur la Fig. 12, au temps 14, sont amalgam+ees sur la boule B(0; 0) (2), les donn+ees provenant de la boule B(0; 0) (3) au temps 9 (3 est impair) et de B(0; 0) (4) au temps 12. Il est alors clair que l’+evolution que nous venons de d+ecrire se reproduit. L’amalgamation sur la boule B(0; 0) (t) au temps t donne lieu a un Uot de donn+ees toutes les deux unit+es de temps (voir la Fig. 11). Alors ce Uot de donn+es sera amalgam+e pour la seconde fois au temps 2 =t + (t − 1)=2 sur le bord de B(0; 0) ((t − 1)=2). Puis, en g+en+eral, il sera amalgam+e pour la j-i+eme fois, au temps     t−1 t−1 + t+ − 1 + ··· 2 2 sur la cellule       t−1 − 1 ::: − 1 : ::: 2  

 

j−1 fois

j−1 fois

Il ne reste plus qu’a examiner si l’hypothese faite est v+eri5+ee, soit a e+ tudier ou et quand ont lieu les amalgamations. Pour la seconde amalgamation, la Fig. 12 montre qu’elle est v+eri5+ee pour t assez petit. Ensuite, une preuve par r+ecurrence sur t est ais+ee: il su/t d’observer que la zone ou ) = 0 avance d’une cellule toutes les deux unit+es de temps car les donn+ees 2-group+ees restent un temps immobiles avant d’ˆetre amalgam+ees. Pour les amalgamations suivantes la situation est identique en changeant la valeur de ). Dans le diagramme espace-temps (dimension 3) du 2-automate cellulaire que nous venons de construire apparaˆPt une famille de cˆones emboˆPt+es de sommet l’origine, d’axe celui du temps, dont la “taille” du k + 1-ieme est diminu+ee de 2 par rapport a celle du k-ieme. La Fig. 13a repr+esente l’intersection de cette famille avec le plan perpendiculaire au plan de cellules suivant sa premiere diagonale. C’est dire que chaque droite y = t de ce diagramme montre, entre l’axe des ordonn+ees et la diagonale, l’+etat des cellules de l’automate situ+ees sur la diagonale au temps t. Les lignes bris+ees de ce mˆeme diagramme, qui sont les traces de ces cˆones, constituent e=ectivement une famille de signaux de pente 1=2k , ce qui explique la similitude entre ce diagramme espace-temps et celui d’une solution au Firing Squad Synchronization Problem de [1] utilisant pr+ecis+ement cette famille de signaux. L’engendrement de cette famille de cˆones est en fait le correspondant en dimension 2 de l’engendrement de la famille de signaux de vitesse 1=2k en dimension 1. Cependant l’int+erˆet principal de ce diagramme dans ce cadre n’est pas dans cette derniere remarque, mais bien plutˆot dans la suivante qui a trait au temps r+eel. Prenons l’exemple d’un mot de longueur 99. Sa derniere lettre va

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281

Fig. 13. Diagramme espace-temps suivant la diagonale (coupe verticale). (a) Diagramme espace-temps suivant la diagonale (mouvement d’une donn+ee sur Freguli  : ). (b) Diagramme espace-temps suivant la diagonale ere (mouvement d’une donn+ee sur Fanticipation ).

se trouver sur le point du 5l de Hilbert num+erot+e 98. L’information contenue dans la cellule (5; 13) parvient au temps 7 sur la cellule (2; 6) ou elle est amalgam+ee sur le bord interne de la boule B(0; 0) (7). Dans cette boule, si l’information se d+eplaKcait en temps r+eel, elle devrait parvenir a la cellule d’origine au temps 13, or, elle ne lui parvient suivant l’algorithme d+evelopp+e jusque-la qu’au temps 15 comme on peut le constater sur la Fig. 13a. L’automate cellulaire partiel dont nous disposons ne reconnaˆPt pas le langage en temps r+eel! Nous le modi5ons alors en superposant a la feuille pr+ec+edente une nouvelle feuille, dite d’anticipation (voir Fig. 13(b)). 5.3. Obtention du temps reel Cette feuille Fanticipation va servir, comme son nom l’indique, a anticiper les calculs aux moments ou des amalgamations sont r+ealis+ees sur la feuille Freguli  : qui font ere perdre du temps. Elle concerne donc, pour cette fonction, l’int+erieur des boules Bt . L’anticipation a lieu, de fait, avec des donn+ees qui sont de deux types: les donn+ees 2-group+ees qui arrivent sur le bord de la boule sur la feuille Freguli  : , qui se composent ere ui

alors de donn+ees-mot () et de donn+ees-5l (wi ). Le 2-groupage va donner lieu a une

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nouvelle donn+ee du type (; w) qui aura exploit+e de faKcon optimale les pr+ec+edentes, pour produire une nouvelle information e+ ventuellement partielle, en attente d’ˆetre compl+et+ee ult+erieurement. Par contre la gestion du temps r+eel doit aussi avoir lieu sur une partie de la zone de calcul au-dela des boules Bt . Soyons plus pr+ecis, et introduisons les nouveaux e+ tats n+ecessaires et les regles qui les gouvernent. En con5guration initiale sur Freguli  : , une cellule apprend de son voisinage un cerere tain nombre d’informations sur l’existence du 5l ou d’une lettre dans chacune de ses voisines. Ce fait donne lieu a des R-+etats pour la feuille Fanticipation . Le voisinage de l’automate e+ tant le voisinage de Moore, ce sont des e+ l+ements de V ={0; 1; 2}9 explicitant une connaissance sur la cellule courante, c, dans l’ordre “cellule, voisine est,. . . , voisine sud-est”. La signi5cation des composantes est la suivante: 1 marque l’existence u

d’une lettre ou d’une fonction , 2 marque le fait que le 5l traverse la cellule mais qu’elle ne contient plus d’information-mot, 0 dit que la cellule ne porte aucune indication ni mot, ni 5l. Un 9-uplet comme (1; 1; 0; 0; 0; 1; 1; 1; 1) indique que la cellule qui l’a peut e+ ventuellement d+e5nir le temps r+eel pour le mot d’entr+ee. Par contre, (1; 0; 0; 1; 1; 1; 0; 0; 0) ne le peut puisque la voisine nord de la cellule courante contient une donn+ee-mot non vide. En fait, seuls les bords de la zone sur laquelle le 5l contient un mot nous int+eresseront. Par ailleurs, a l’arriv+ee sur la boule B(0; 0) (t), cette information peut eˆ tre r+eduite, hors la diagonale, a une seule donn+ee 1 ou 2. En e=et, les cellules qui sont en-dessous de la diagonale ne peuvent certainement pas marquer le temps r+eel et pour celles qui sont au-dessus, sur le bord, seules les informations des trois cellules nord d’un e+ l+ement de V seront signi5catives. Nous ajoutons donc comme R-+etat possibles, 1 et 2. De faKcon plus g+en+erale, les regles d’+evolution de ce type d’information sont les suivantes: 1. A l’ext+erieur d’une boule B(0; 0) (t), leur mouvement est semblable a celui des informations sur la feuille Freguli ees envoie son  : : une cellule de l’axe des ordonn+ ere information a sa voisine sud, les autres envoient leur R-+etat a leur voisine sudouest, jusqu’a un 2-groupage n+ecessaire sur une verticale. 2. Sur le bord d’une boule B(0; 0) (t), ou les donn+ees de la feuille Freguli ˆ tre  : vont e ere amalgam+ees, plusieurs cas se produisent suivant que la cellule courante est situ+ee sur le bord nord hors la diagonale, sur la diagonale ou le bord est hors la diagonale. (a) Dans le premier cas, si les deux R-+etats des cellules 2-group+ees d’ordonn+ees t + 1 avaient leurs composantes 3, 4 et 5 de valeur 2, alors le R-+etat de la cellule sud d’ordonn+ee t − 1, est 2. Par contre si l’une de ces composantes a la valeur 1 pour l’une des deux cellules 2-group+ees, alors le R-+etat de la cellule sud d’ordonn+ee t − 1, est 1. (b) Dans le second cas, la cellule (t; t) prendra le R- e+ tat de sa voisine nord-est qui est alors le seul signi5catif. (c) Dans le troisieme cas, la cellule perd toute indication sur le temps r+eel d’apres l’+etude du temps r+eel sur le 5l de Hilbert (voir 4.3). De fait, on pourrait supprimer cette information temps-r+eel plus tˆot, au moins des qu’un E-+etat S est

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283

Fig. 14. Mouvement des donn+ees dans la preuve du Th+eoreme 2 au temps 27 sur les feuilles Freguli  : et ere Fanticipation .

rencontr+e sur la feuille Fr eguli e + ere , mais l’algorithme deviendrait plus compliqu+ sans gagner en e/cacit+e. 3. A l’int+erieur d’une boule B(0; 0) (t), les R-+etats possibles sont donc 1 ou 2. L’information se d+eplace toujours le plus vite possible. Lorsqu’une cellule doit recevoir simultan+ement une information de sa voisine nord et de sa voisine nord-est, elle prend le R-+etat 1 (2) si ses deux voisines sont dans le R-+etat 1 (2) ou si sa voisine nord-est ne contient pas de R-information et le R-+etat 1 si sa voisine nord-est est dans le R-+etat 1. 5.3.1. Le calcul sur la feuille Fanticipation Un tel calcul commence sur le bord interne des boules B(0;0) (t), t¿1. Le mouvement des donn+ees des feuilles Fr eguli etaill+e au temps 27 sur la Fig. 14;  : et Fanticipation est d+ ere en5n la Fig. 15 illustre le proc+ed+e entre les temps 0 et 16. Nous allons encore distinguer ce qui se passe sur le bord de la boule B(0; 0) (t). Ensuite, nous verrons comment le calcul progresse a l’int+erieur de la boule. Le calcul sur le bord de B(0; 0) (t) Nous nous souvenons que sur la feuille Freguli  : , au temps t, une amalgamation a ere lieu sur les cellules du bord interne (ordonn+ee t − 1 pour les cellules nord) de la boule, ce qui engendre une perte de temps d’une unit+e puisque au temps t + 1 les cellules du bord de B(0; 0) (t) n’ont pas reKcu de nouvelles donn+ees. A ce temps-la est alors initi+e sur la feuille Fanticipation un calcul qui consiste a di=user “aussi vite que possible le plus d’informations possibles”. Cela signi5e qu’au temps t + 1, sur les cellules du bord interne de la boule B(0; 0) (t) 5gure une information issue d’une amalgamation partielle des donn+ees provenant, pour les cellules nord, hors diagonale, de la boule, de leurs voisines nord et nord-est ou nordest (les autres cas e+ tant analogues), contenant chacune des donn+ees d+eja 2-group+ees du bord externe de B(0; 0) (t) au temps t comme on peut le voir sur la Fig. 15. Il s’agit

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Fig. 15. Mouvement des donn+ees dans la preuve du Th+eoreme 2 du temps 0 au temps 16 sur les feuilles Freguli  : et Fanticipation . ere

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285

donc d’amalgamer partiellement des donn+ees du type   u u v v   (; w); (  ; w ) et (; z); (  ; z ) : Ces donn+ees 5gurent donc au temps t sur des cellules (x; t); (x + 1; t), ce qui implique que ces informations peuvent ne pas correspondre a une donn+ee globale coh+erente de la con5guration initiale. L’amalgamation partielle consiste a ne conserver que la partie u

coh+erente des donn+ees-mot et 5l. On obtient ainsi des donn+ees de type (; w), mais w, qui indique alors comment le 5l entre et/ou sort de l’espace cod+e par u peut tres bien eˆ tre ind+etermin+e. Le calcul se poursuit alors a vitesse maximale sur Fanticipation jusqu’a ce qu’il coMPncide avec des donn+ees sur la feuille Freguli ees de  : . Alors les donn+ ere Fanticipation sont mises a jour et le processus pr+ec+edent continue. Fin du calcul Lors de l’arriv+ee de la marque du temps r+eel, la cellule origine prend en compte le calcul de Fanticipation qui coMPncide e+ ventuellement avec celui de Freguli  : . ere 5.3.2. Un exemple d’obtention du temps reel1 Consid+erons un mot d’entr+ee de longeur 51. Sa derniere lettre est sur H(50) qui est la cellule (6; 10) (Fig. 4b)). D’apres la d+e5nition 6, le temps r+eel est obtenu au temps 12 (voir la Section 4.3) et la derniere donn+ee signi5cative est celle de H(49) = (7; 11) qui doit arriver sur l’origine au temps 11. Sur la feuille Freguli ee de (7; 11)  : , la donn+ ere est amalgam+ee aux mˆemes temps que celle de (11; 11) qui se trouve sur la diagonale et dont le trajet dans le temps se lit sur la Fig. 13: • sur Freguli ee de (7; 11) est amalgam+ee avec celles de (7; 10), (6; 10) et de  : , la donn+ ere (6; 11) sur la cellule (3; 5) au temps 6. • La donn+ee, r+esultat de l’amalgamation pr+ec+edente en (3; 5), est a nouveau amalgam+ee en (1; 2) au temps 9 sur Freguli ee reste sur (0; 1) jusqu’a  : . Puis, la nouvelle donn+ ere l’amalgamation qui a lieu au temps 14 et arrive a l’origine au temps 15. • Au temps 6, il y a cr+eation sur la cellule (2; 4) d’une amalgamation sur Freguli  : ere (voir la Fig. 15). La donn+ee partielle repr+esente celles de (6; 10), (7; 10), (8; 10) et (9; 10). Elle s’+ecrit donc  47 (  ; ws? ); (sq ) : Cette nouvelle donn+ee atteint la cellule (1; 2) au temps 8. • Au temps 8, sur Fanticipation , il y a une nouvelle amalgamation partielle sur la cellule (1; 2) qui prend en compte  47 (  ; ws? ); (sq ) 1

Cet exemple est illustr+e dans [6].

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et les donn+ees de Freguli eja pr+esentes, a savoir  : d+ ere 

43−46



(sq ; (  ; wsn ) : • Au temps 9, la donn+ee, obtenue pr+ec+edemment et qui est 43−47

(  ; ws? ); arrive sur la cellule (0; 1). A ce mˆeme temps 9, il y a amalgamation des donn+ees de Freguli ee obtenue est  : sur la cellule (1; 2) (voir la Fig. 12). La donn+ ere 43−50

(  ; wsw ): • Sur Freguli  : , au temps 10, ere 

#

43−50



(; wen )(  ; wsw ) se trouve donc sur la cellule (0; 1). Par contre, sur Fanticipation , a ce mˆeme temps, la donn+ee 47

(  ; ws? ) a rejoint l’origine. Sur Fanticipation , la donn+ee de l’origine devient alors 0−47

 (q0 ):

A ce temps 10, il n’apparaˆPt pas de nouvelles anticipations sur les cellules (0; 1) et (1; 1) (voir la Fig. 15). • Au temps 11, la donn+ee  # 43−50 (; wen ); (  ; wsw ) de Freguli  : reste sur la cellule (0; 1). Mais, sur la feuille Fanticipation , la mise a jour ere donne comme seule donn+ee 43−50

(  ; wsw ) qui est sur la cellule origine au temps 11. Ainsi, sur Fanticipation , la donn+ee de l’origine 0−50

devient, au temps 11,  (q0 ). • A l’arriv+ee de la marque “temps r+eel”, cette donn+ee, pr+esente sur l’origine, est prise en compte pour donner le r+esultat au temps 12.

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6. Conclusion Nous avons pr+esent+e deux algorithmes paralleles en dimension 2 e+ tablissant que tout langage rationnel est reconnu en temps r+eel par un automate cellulaire 2-dimensionnel muni du 5l d’Archimede d’une part, du 5l de Hilbert d’autre part, qui exploitent tous deux la r+egularit+e de ces 5ls, forte dans le cas d’Archimede, plus distendue dans le cas de Hilbert. Ces di=+erences se reUetent bien dans la di/cult+e des algorithmes: si le premier est simple, le second ne l’est pas vraiment. u La rationalit+e du langage est impliqu+ee par l’usage des fonctions , qui assurent la 5nitude de l’ensemble des e+ tats de l’automate cellulaire, et par le rˆole de la cellule origine qui acheve le calcul de l’automate 5ni reconnaisseur du langage. Certains choix au niveau de l’automate cellulaire amenent des questions. Ainsi, par exemple, que se passe-t-il si on substitue au voisinage de Moore celui de Von Neumann? Il est facile de voir que, dans le cas d’Archimede, le mˆeme r+esultat vaut, mais qu’en est-il du cas Hilbert? La conception purement g+eom+etrique des algorithmes permet d’+echapper completement a l’arithm+etisation des courbes sous-jacentes, qui en explique pourtant bien l’aspect fractal, comme on peut le voir dans [15]. Cependant elle permet de g+en+eraliser nos r+esultats a d’autres 5ls, le 5l de Hilbert en 5gurant d’une certaine maniere le plus simple. Il a la propri+et+e remarquable de produire la seule faKcon de remplir un carr+e de taille 2n par rotation d’un motif lorsque le point d’entr+ee est le coin inf+erieur gauche et le point de sortie le coin sup+erieur droit. Les courbes dites de Peano dans [15], obtenues grˆace aux mˆemes transformations g+eom+etriques, remplissent des carr+es de taille 3n avec point d’entr+ee au coin inf+erieur gauche et point de sortie au coin sup+erieur droit. Mais plusieurs possibilit+es de remplissage existent alors. La Fig. 16a en illustre une pour le carr+e de taille 9, alors que la partie b en montre une autre. Il est alors facile d’obtenir des courbes non r+ecursives remplissant le quart de plan: il su/t de choisir, a chaque e+ tape et de faKcon non r+ecursive, un des motifs a ou b comme sur la Fig. 16c. La preuve du Th+eoreme 2 devrait s’adapter sans di/cult+e aux courbes de Peano. Le point d+elicat, sur le 5l de Hilbert, de la premiere compression des donn+ees sur les segments horizontaux devient sans objet. La proposition 2 ne s’+etend e+ videmment pas directement: la non r+ecursivit+e de la construction ne permet pas de d+e5nir une homoth+etie H(O; 13 ). Mais le corollaire 1 reste vrai en amalgamant maintenant 32i cellules. A chaque amalgamation de 81 cellules, l’observation des 9 cellules en bas a gauche permet d’induire la disposition du 5l dans un bloc de 81 cellules. Le 2-groupage devient un 9-groupage. On obtiendrait ainsi un exemple de reconnaissance en temps r+eel par 2D-CA des langages rationnels sur une courbe non r+ecursive. Cela justi5e que, dans la d+e5nition de la notion de 5l, nous n’ayons pas suppos+e que la fonction ! soit r+ecursive. Tous les exemples de 5ls que nous avons imagin+es conduisent aux mˆemes r+esultats. La di/cult+e est en fait de trouver des 5ls vraiment complexes qui ne se construisent pas a partir d’une partition en carr+es du plan tels que le 5l n’entre et ne sorte de ces motifs que par un nombre 5ni born+e de points. Il apparaˆPt 5nalement que les vraies questions ne proviennent, malgr+e le titre de l’article, ni de la rationalit+e, ni du modele

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Fig. 16. Courbes de Peano. (a) Un motif d’une courbe de Peano. (b) Un autre motif d’une courbe de Peano. (c) Un exemple de courbe de Peano non r+ecursive.

de machine, mais bien de l’ensemble des plongements de N dans Z2 et de leurs liens avec les pavages du plan discret. Le point de vue des classes de complexit+e mene bien au-dela des langages rationnels. L’appartenance a une telle classe signi5e une relation tout a fait particuliere entre la complexit+e du langage lui-mˆeme, celle du 5l et les possibilit+es que donne l’espace de calcul 2-dimensionnel. Si l’on d+esigne par RArch , RHilb les classes de langages reconnus en temps r+eel par automate cellulaire de dimension 2 avec 5l d’Archimede et de Hilbert, et par R1D la classe des langages reconnus en temps r+eel par automate cellulaire de dimension 1, nous avons e+ tabli [5] que RArch ⊂ R1D et RHilb ⊆R1D . Cela augure-t-il de l’existence d’une hi+erarchie (RF> )> dans R1D tenue par une famille de 5ls? En particulier le temps r+eel 2D d’une demi-droite est-il le temps r+eel 1D [9]? Par ailleurs, savoir si tout langage reconnu en temps lin+eaire par un automate cellulaire de dimension 1 l’est en temps r+eel, soit L1D ⊆R1D ?, est une question toujours ouverte. Il est facile de voir que R1D ⊆L2D , qu’en est-il de L2D ⊆R1D ? References [1] R. Balzer, A 8 states minimal time solution to the 5ring squad synchronization problem, Inform. and Control 10 (1967) 22–42. [2] S. Cole, Real time computation by n-dimensional arrays, IEEE Trans. Comput. 4 (1969) 349–365. [3] M. Delorme, J. Mazoyer, An overview on language recognition on one dimensional cellular automata, in: J. Almeida (Ed.), Semigroups, Automata and Languages, 1996, pp. 85 –100. [4] M. Delorme, J. Mazoyer (Eds.), Cellular Automata: a Parallel Model, Mathematics and its Applications, Kluwer, Dordrecht, 1999. [5] M. Delorme, J. Mazoyer, Real time recognition by 2D cellular automata with threads, 7th IFIP WG 1.5 Workshop, Osaka, 2000. [6] M. Delorme, J. Mazoyer, Reconnaissance parallele des langages rationnels sur automates cellulaires plans, RR 2001-20, LIP-ENSLyon, 2001.

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M [7] D. Hilbert, Uber die stetige Abbildung einer Linie auf ein FlMachenstMuck, Annalen 38 (1891) 559–560. [8] O.H. Ibarra, T. Jiang, On one-way cellular arrays, SIAM J. Comput. 16-6 (1987) 1135–1154. [9] O.H. Ibarra, Computational complexity of cellular automata: an overview, in: M. Delorme, J. Mazoyer (Eds.), Cellular Automata: a Parallel Model, Mathematics and its Applications, Kluwer, 1999, pp. 181– 199. [10] J. Kari, Reversibility of 2D Cellular Automata is undecidable, J. Assoc. Comput. Mach. 35 (3) (1990) 279–285. [11] J. Mazoyer, Entr+ees et sorties sur lignes d’automates, in: M. Nivat, C. Cosnard, Y. Robert (Eds.), Algorithmique Parallele, 1992, pp. 47– 62. [12] J. Mazoyer, R. Reimen, A linear speed-up theorem for cellular automata, Theoret. Comput. Sci. 101 (1992) 59–98. [13] J. Mazoyer, V. Terrier, Signals in one-dimensional cellular automata, Theoret. Comput. Sci. 217 (1999) 53–80. [14] M. Minsky, Finite and In5nite Machines, Prentice-Hall, Englewood Cli=s, NJ, 1967, pp. 28–29. [15] H. Sagan, Space-Filling Curves, Springer, Berlin, 1991. [16] A.R. Smith, Real time language recognition by one-dimensional cellular automata, Complex Systems 5 (1971) 19–30. [17] V. Terrier, Two dimensional cellular automata recognizers, Theoret. Comput. Sci. 218 (2) (1999) 325–346.

Theoretical Computer Science 281 (2002) 291 – 309

www.elsevier.com/locate/tcs

Counting H -colorings of partial k-trees  Josep D$%az ∗ , Maria Serna, Dimitrios M. Thilikos Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord-Modul C6, c=Jordi Girona Salgado 1-3, 08034, Barcelona, Spain Happy birthday Maurice!

Abstract The problem of counting all H -colorings of a graph G with n vertices is considered. While the problem is, in general, #P-complete, we give linear time algorithms that solve the main variants of this problem when the input graph G is a k-tree or, in the case where G is directed, when the underlying graph of G is a k-tree. Our algorithms remain polynomial even in the case where k = O(log n) or in the case where the size of H is O(n). Our results are easy to implement and imply the existence of polynomial time algorithms for a series of problems on partial k-trees c 2002 Elsevier Science B.V. such as core checking and chromatic polynomial computation.
All rights reserved. Keywords: Graph homomorphism; Counting problems; Treewidth

1. Introduction In this paper we consider a series of counting problems associated with various versions of homomorphisms from a graph G to a <xed graph H . Given a graph H = (G(H ); E(V )), with loops but without multiple edges, for any graph G = (V (G); E(G)) an homomorphism of G to H is a map : V (G) → V (H ) with the property that {v; w}∈E(G) ⇒ { (v); (w)} ∈E(H ). We will call such a homomorphism a H-coloring of G (for an example, see Fig. 1). For H <xed, the H -coloring problem asks if there is 

The work of all the authors was supported by the IST Program of the EU under contract number IST-99-14186 (ALCOM-FT) and by the Spanish CICYT TIC-2000-1970-CE. The work of the last author was partially supported by the Ministry of Education and Culture of Spain, Grant number MEC-DGES SB98 0K148809. ∗ Corresponding author. E-mail addresses: [email protected] (J. Diocute;az), [email protected] (M. Serna), [email protected] (D. M. Thilikos). c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 1 7 - 8

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Fig. 1. An example of an H -coloring of G is the mapping (a) = (d) = (f) = 1 and (b) = (c) = (e) = 2.

an H -coloring of G. If H is bipartite or it has a loop, then the H -coloring problem can be trivially solved in polynomial time, but in the case that H is not-bipartite and loopless the problem is known to be NP-complete [24]. See also [22] for the complexity of the same problem for bounded degree graphs. An application of the H -coloring problem is illustrated by the following example taken from [20]. Suppose that we have a graph whose vertices represent jobs to be processed and whose edges represent communication demands between two jobs. Let also H be a computer network where its vertices represent the processors and its edges correspond to communication links between them. We would like to assign the jobs to the processors so that all the communication demands between jobs are satis<ed. This question can be modeled by the problem of asking whether there exists a homomorphism from G to H , the H-coloring problem. An earlier version of the H -coloring problem is the exact H-coloring problem: given a graph G, decide if there is an H -coloring of G such that (V (G)) = V (H ) and (E(G)) =E(H ). This problem is known to be NP-complete, even for the case where H is a triangle [30]. We also consider the intermediate problem of asking whether there is an H -coloring of G such that (V (G)) = V (H ) and we call it vertex exact H-coloring. Given a graph G, let H(G; H ) denote the set of all H -colorings of G, and let E(G; H ) denote the set of all exact H -colorings of G. Also, we denote as V(G; H ) the set of all vertex H -colorings of G. For a <xed graph H , denote by #H -coloring the problem of, given G as input, computing |H(G; H )|. Also, denote by #EH -coloring the problem of, given as input G, computing |E(G; H )| and by #VEH -coloring the problem of, given as input G, computing |V(G; H )|. Dyer and Greehill [18] have proved that unless every connected component of H is an isolated vertex without a loop, a complete graph with all loops present or a complete unlooped bipartite graph, the #H -coloring problem is #P-complete, even when G has bounded degree. It is easy to verify that the same result hold for the #EH -coloring (#VEH -coloring) as the knowledge of E(G; H
) (V(G; H
)) for any (induced) subgraph H
of H implies the knowledge of H(G; H ). For further negative results on the approximability of the #H -coloring see [12] and [17]. For positive and negative complexity results concerning special cases of the #H -coloring problem, depending on restrictions on both H and G, see [19,16].

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The notion of treewidth appears to play a central role in many areas of graph theory. Roughly, a graph has small treewidth if it can be constructed by assembling small graphs together in a tree structure, namely a tree decomposition with small width (see Section 2 for the formal de

C


C


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405

• Finally, assume p = rec y:p1 with y = x. Since q is closed, we have p[q=x] = rec x:(p1 [q=x]). We ,rst show by induction on n that yn (
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Then, x0 = x0 (
P. Gastin, M. Mislove / Theoretical Computer Science 281 (2002) 369 – 421

407

xn+1 = xn+1 (
Finally, by taking the least upper bound of the sequence (xn )n¿0 , we get

This example illustrates how the resource x reveals the potential divergence of q. This is the case because {x } ∩ a = ∅ and weak sequential composition allows the unwinding of recursion to continue without the action a being executed. In fact, contrary to the previous example, this recursion is unguarded, causing the possible divergence. Note that it is not safe to hide a variable resource before checking the process for divergence. Indeed, if we hide x we cannot observe the unwindings of recursion and therefore, we cannot observe divergence anymore. Again, applying the de,nition for hiding 4.20, we see that,
with B1 (a; c) = (rec x:(a ; c ; x))\x and B1 (c; b) = (rec x:(c ; b ; x))\x . As with the previous examples, we get the following semantics:
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Finally, hiding c from this complex pomset gives the denotational semantics of B2 (a; b):

We can see that this ful,lls the property of being a 2-place buEer. The buEer has to receive a’s before sending the corresponding b’s. The buEer may also receive two a’s (but not three) before sending the ,rst b. The semantics of a 3-place buEer would be

5. Relating semantics: the congruence theorem In this section we complete the picture by showing that the operational behavior of a process de,ned in Section 3 is essentially the same as the --algebra map we de,ned for the denotational model in the last section. We ,rst show that the denotational semantics is adequate with respect to the behavior function which associates to a process p the set of strings of actions that p can execute. Then, we prove a congruence theorem for our semantics, and we conclude by showing that the denotational semantics is fully abstract with respect to the behavior function de,ned above. Our proof relies on two crucial propositions: the ,rst states that any action the process p can execute is a pre,x of its denotational semantics
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409

• If u ∈∗ and a∈, then ua / s if u / s and a6u−1 s; in this case, (ua)−1 s = a−1 (u−1 s). We can then extend this de,nition to u ∈∗ and x ∈C by u / x if and only if u / Re(x), and in this case, u−1 x = (u−1 Re(x); Im(x)). Remark 5.2. (i) We note that for s ∈R and u ∈∗ , we have u / s if and only if there exists a ,nite pre,x r ∈Rf of s such that u ∈Lin(r), where Lin (r) denotes the set of linearizations of r. Moreover in this case, u−1 s as de,ned above is equal to r −1 s as de,ned in De,nition 2.3. (ii) Any word u ∈∗ can be regarded as a real pomset when endowed with the linear order. Note that, if a ∈ and s ∈R then a6s if and only if a / s. This equivalence does not hold if we replace a by u ∈∗ . If u6s, then u / s, but u / s does not imply u is a pre,x of s in general. For example,   a a (a → b) / ; but clearly (a → b)  : b b (iii) We have extended the operation r −1 s given in De,nition 2.3 to those pairs (r; s) ∈∗ × R with r / s. And, if r6s, then the two de,nitions coincide. Therefore, we can use the same notation without causing confusion. (iv) Throughout this section, whenever we write u−1 x = y, we implicitly mean that u / x also holds and u−1 x is the complex pomset de,ned in De,nition 5.1. a

Recall that p →p denotes a transition from our operational semantics. If u ∈∗ u then we use p ⇒p to denote the fact that we can write u = a1 · · · an with n¿0, an a1  a1 ; : : : ; an ∈ ∪ {1} and p = p0 →p 1 · · · pn−1 →pn = p . Note that we always have 1 p ⇒p. Proposition 5.3. Let p ∈Lc be a closed term. a (i) If p →p with a ∈ ∪ {1}, then a /
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• If p = p1 ◦ p2 , then there are also two cases: a . p1 →p1 . Same as above using Lemma 4.13. a . p2 →p2 and a I p1 . Then p =p1 ◦p2 and we obtain by induction a−1
a

. p1 →p1 and a I p2 ; C. Then p =p1  p2 and we obtain by induction a−1
By Lemma 4.30 we deduce that a I
C

C

. p2 →p2 and a I p1 ; C is similar. a a . p1 →p1 , p2 →p2 , and a∈C. Then, p =p1  p2 and we obtain by induction a−1
=
C

b

• If p = p1 \R, then p1 →p1 for some b and p1 with b\R = a and p1 \R = p . By induction, b−1
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The nice property is a syntactic restriction that is preserved by the operational transitions. This is important because it ensures that if we start with a nice process, we will never leave the class of nice processes. Lemma 5.6. (i) If p ∈Lc; n and q ∈Ln , then q[p=x] ∈Ln . a (ii) If p ∈Lc; n , a∈ ∪ {1} and p →p , then p ∈Lc; n . Proof. For part (i), we ,rst note that there is nothing to prove if x is not free in q. On the other hand, if x is free in q we proceed by induction on q. • If q = x, then q[p=x] = p ∈Ln , and the result is clear. • If q = q1 op q2 , then q[p=x] = q1 [p=x] op q2 [p=x]. Now, if p = p1 \R 4 q[p=x] then p 4 q1 [p=x] or p 4 q2 [p=x], and so the result is clear by induction. • If q = q1 \R, then q ∈Ln implies that either R ∩ RV = ∅ or q1 ∈Lc is closed. But x is free in q, so the ,rst case must hold. Now, q[p=x] = q1 [p=x]\R, and so any subterm p = p1 \R 4 q[p=x] is either equal to q[p=x] or it is a subterm of q1 [p=x]. The result follows from the inductive hypothesis and R ∩ RV = ∅. • Assume q = rec y:q1 . Since p is closed we have q[p=x] = rec y:(q1 [p=x]). Again, if p = p1 \R is a subterm of q[p=x] then it is a subterm of q1 [p=x]. The result follows by induction. This completes the proof of part (i). For part (ii), we proceed by structural induction on p. • The cases p = SKIP or p = x are not possible. a • If p = b, then p →p implies a = b and p = SKIP, which is nice. a a • If p = p1 op p2 , then p = p1 op p2 →p1 op p2 = p with p1 →p1 and p2 = p2 or a a a p1 = p1 and p2 →p2 or p1 →p1 and p2 →p2 . By induction, p1 and p2 are nice and therefore, p is nice. a b • If p = p1 \R, then p →p implies p1 →p1 for some p1 and b with b\R = a and    p = p1 \R. By induction, p1 is nice. Since p is closed, so is p1 and p1 and we deduce that p = p1 \R is nice. a • If p = rec x:p1 , then p →p implies a = x and p = p1 [p=x]. The ,rst part of this lemma then implies p is nice. We can now state the converse of Proposition 5.3. Proposition 5.7. Let p ∈Lc; n be a closed, nice process. a (i) If a∈ and a /
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5.2. The rank function The rank function we need has two components, the ,rst of which measures the hiding depth of variable resources. The second component essentially measures the structural depth of a term except that it is reset to 0 when we hide some variable resources. This will not be a problem since in this case the ,rst component of the rank function increases, and we endow N × N with the lexicographic order. We denote by n ∨ m the maximum of n; m ∈N. Denition 5.8. De,ne l : L → N × N by l(p) = (l1 (p); l2 (p)) with • l(SKIP) = l(a) = l(x) = (0; 0), for all a ∈Act and all x ∈V; • l(p1 op p2 ) = (l1 (p1 ) ∨ l1 (p2 ); 1 + l2 (p1 ) + l2 (p2 )), for all processes p1 and p2 , and the operators op = ◦ or op = ; or op =  ; C

• l(rec x:p) =  (l1 (p); 1 + l2 (p)), for all p ∈L and x ∈V; and (l1 (p) + 1; 0) if R ∩ RV = ∅ • l(p\R) = for all p ∈L and R ⊆ R. (l1 (p); 1 + l2 (p)) otherwise A crucial property of this rank function is that it is invariant when performing a transition of the operational semantics whose label is not a variable resource. This fact may seem strange since unwinding a recursion usually increases the structural depth of a process as in x

rec x:(a ; x) → a ; rec x:(a ; x): Indeed, the rank function does increase with this transition and this is the reason for the restriction “whose label is not a variable resource”. Still, unwinding a recursion whose variable resource is then hidden is admissible, as in (rec x:(a ; x))\x → (a ; rec x:(a ; x))\x : But since l2 is reset to 0 when we hide a variable resource, the result also holds for recursion unwindings that are hidden. We start by proving the result for the second component of the rank function. a

Lemma 5.9. If p ∈Lc is a closed term and a ∈Act ∪ {1}, then p →p implies l2 (p) = l2 (p ). Proof. We proceed by structural induction on p. • p = SKIP or p = x are not possible. • If p = b, then b = a and p = SKIP. Then l2 (p) = l2 (SKIP) = 0: a a • If p = p1 op p2 , then p = p1 op p2 →p1 op p2 = p with p1 →p1 and p2 = p2 , or a a a p1 = p1 and p2 →p2 , or p1 →p1 and p2 →p2 . In any case, l2 (pi ) = l2 (pi ) for each i = 1; 2 by induction. We deduce that l2 (p) = l2 (p ). b • If p = p1 \R, then p1 →p1 for some p1 and b ∈Act ∪ RV ∪ {1} with p1 \R = p and b\R = a. We have two cases: . If R ∩ RV = ∅, then l2 (p) = 0 = l2 (p ).

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413

. If R ∩ RV = ∅, then a∈Act ∪ {1} implies b ∈Act ∪ {1} as well. By induction, l2 (p1 ) = l2 (p1 ), and we obtain l2 (p) = l2 (p ). • Finally, p = rec x:p1 is not possible, since this term can only make a x transition, which is not allowed since a ∈Act ∪ {1}. An important property of the rank function l1 is that it is invariant under all transitions of the operational semantics, even unwindings of recursions that are not hidden. This results holds for nice processes, which is precisely why we introduced this notion. Lemma 5.10. (i) If p ∈Lc and q ∈Ln , then l1 (q)6l1 (q[p=x])6l1 (q) ∨ l1 (p). a (ii) If p ∈Lc; n , a∈ ∪ {1} and p →p , then l1 (p) = l1 (p ). Proof. For part (i), we ,rst note that there is nothing to prove if x is not free in q. On the other hand, if x is free in q, then we proceed by induction on q. • If q = x, then q[p=x] = p, so 0 = l1 (q)6l1 q[p=x] = l1 (p) = l1 (q) ∨ l1 (p): • If q = q1 op q2 , then q[p=x] = q1 [p=x] op q2 [p=x]. By induction we have l1 (qi )6 l1 (qi [p=x])6l1 (qi ) ∨ l1 (p) for each i, and so l1 (q) = l1 (q1 op q2 ) = l1 (q1 ) ∨ l1 (q2 ) 6 l1 (q1 [p=x]) ∨ l1 (q2 [p=x]) = l1 (q1 [p=x] op q2 [p=x]) = l1 (q[p=x]) 6 (l1 (q1 ) ∨ l1 (p)) ∨ (l1 (q2 ) ∨ l1 (p)) = l1 (q) ∨ l1 (p): • If q = q1 \R, then q[p=x] = (q1 [p=x])\R. Since x is free in q and q is nice, we must have R ∩ RV = ∅. Using the induction hypothesis and the de,nition of l1 we obtain l1 (q) = l1 (q1 )6l1 (q1 [p=x]) = l1 (q[p=x])6l1 (q1 ) ∨ l1 (p) = l1 (q) ∨ l1 (p): • If q = rec y:q1 , then q[p=x] = rec y:(q1 [p=x]) since p is closed. Using the induction hypothesis and the de,nition of l1 we obtain l1 (q) = l1 (q1 ) 6 l1 (q1 [p=x]) = l1 (q[p=x]) 6 l1 (q1 ) ∨ l1 (p) = l1 (q) ∨ l1 (p): For part (ii), we proceed by structural induction on p. • If p = SKIP or p = x ∈V, then there is nothing to prove. • If p = b∈Act, then b = a, p = SKIP and we have l1 (p) = 0 = l1 (p ). a a • If p = p1 op p2 , then p = p1 op p2 →p1 op p2 = p with p1 →p1 and p2 = p2 , or a a a p1 = p1 and p2 →p2 , or p1 →p1 and p2 →p2 and we obtain by induction l1 (p) =    l1 (p1 ) ∨ l1 (p2 ) = l1 (p1 ) ∨ l1 (p2 ) = l1 (p ). • If p = rec x:p1 , then a = x and p = p1 [p=x]. We can apply (i), and conclude that l1 (p) = l1 (p1 )6l1 (p1 [p=x]) = l1 (p )6l1 (p1 ) ∨ l1 (p) = l1 (p). b • If p = p1 \R, then p1 →p1 for some p1 and b ∈ ∪ {1} with p1 \R =p and b\R =a. Then l1 (p1 ) = l1 (p1 ) by induction, and we obtain l1 (p) = l1 (p ).

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Now, we summarize in a proposition the results concerning the rank function that will be needed for the proof of Proposition 5.7. u

Proposition 5.11. (i) Let p ∈Lc; n and u ∈∗ . If p ⇒p , then p ∈Lc; n is closed and nice. u (ii) Let p ∈Lc; n and u ∈∗ . If p ⇒p , then l1 (p) = l1 (p ). u ∗ (iii) Let p ∈Lc and u ∈Act . If p ⇒p , then l2 (p) = l2 (p ). (iv) Let p; q ∈L. If q is a proper subterm of p, then l(q)¡l(p). Proof. (i), (ii) and (iii) follow directly by induction on the derivation length from Lemmas 5.6 and 5.10 and 5.9. The proof of (iv) is by structural induction on p. • p = SKIP, or p = a ∈Act, or p = x ∈V are not possible. • If p = p1 op p2 , then q 4 pi for some i = 1 or 2. By induction we get l(q)6l(pi ) and we have l(pi )¡l(p). • If p = rec x:p1 or p = p1 \R, then q 4p1 . By induction we get l(q)6l(p1 ) and we have l(p1 )¡l(p). We are now ready to prove the converse of Proposition 5.3. Proof of Proposition 5.7. (ii) follows directly by induction on the length of u using (i). In order to prove (i) we use induction on l(p). Let p ∈Lc; n and assume that for all q ∈Lc; n with l(q)¡l(p) property (i) holds for q. We consider the possible structure for p. • p = SKIP, or p = x ∈V are not possible. a • If p = b ∈Act, then a /
C

. a /
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415

. a /
X∗ (p) = {u ∈ ∗ | p ⇒}: From Propositions 5.3 and 5.7, we deduce immediately the following characterization of the observable behavior of a process.

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Proposition 5.13. If p ∈Lc; n is closed and nice, then X∗ (p) = {u ∈ ∗ | u /
Denition 5.15. Let p ∈Lc be a closed process. We de,ne u

XC (p) = {(∩[u]p ; res(p )) | p ⇒ p }; v

where [u]p = {v ∈∗ | p ⇒ and Vu = Vv }. Recall (De,nition 2.14) that ((x) = {(s; res(s−1 x)) | s6x and s ∈Rf }, for each x ∈C. We are now able to state the congruence theorem. In particular, the following result implies that XC (p) consists of complex pomsets, something that is not clear from its de,nition.

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417

Theorem 5.16 (Congruence). For all closed, nice processes p ∈Lc; n , we have XC (p) = ((
and

 XC (p) =
Proof. We ,rst claim that for s ∈Rf , if s6
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all a ∈. The canonical representations for s = (Vs ; 6s ) and t = (Vt ; 6t ) have the same vertex set Vs = Vt = {(a; i) | 06i¡|s|a = |t|a }. Since s = t we must have 6s = 6t hence we ,nd p; q ∈Vs = Vt with for instance p 6t q and p s q. Let r = (↓s q; 6s|↓s q × ↓s q )6s. Since p ∈= ↓s q and p6t q, ↓s q is not a lower set of (Vt ; 6t ) hence r is not a pre,x of t. Now, taking any linearization u of r we get u / s and u /= t. Proof of Theorem 5.17. (i) implies (ii) is just Theorem 5.14. Conversely, let p; q ∈ Lc; n with
u

implies p ⇒, while Proposition 5.3 implies q ⇒ = . Hence, the identity context serves to distinguish p and q in this case. On the other hand, if Re(
u

of Re(
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419

We claim that for all i; j, we have vi ¡vj if and only if vi ¡1 vj . One direction is easy. Indeed, assume that vi ¡vj . Then, ci ∩ aj = {6i; j } and therefore vi ¡1 vi ¡1 vj ¡1 vj . Conversely, we use an induction on j − i. If vi ¡1 vk ¡1 vj for some k, then i¡k¡j and we can conclude by induction. If bi ∩ bj = ∅ then, the over-synchronization condition for s gives vi ¡vj . Hence, we assume that bi ∩ bj = ∅ and that there is no k with vi ¡1 vk ¡1 vj . Using ak ; ck I bl for all k; l and the de,nition of weak sequential composition, we deduce from vi ¡1 vj that vi ¡1 vi ¡1 vj ¡1 vj . Using in addition that for k¡l we have ak I al , ak I cl and ck I cl , we can deduce that there is no vertex between vi and vj in t. Then, the de,nition of weak sequential composition implies ci ∩ aj = ∅. We must have ci ∩ aj = {6i; j } which implies vi ¡vj and we have proved the claim. Finally, let R = {71 ; : : : ; 7n } ∪ {6i; j | vi ¡vj }. Using the claim and the de,nition of hiding, we immediately get that
6. Summary In this paper we have presented a truly concurrent semantics for a process algebra. We have used resource pomsets to de,ne the denotational semantic model. We have presented an operational semantics for the language, which was shown to be congruent to the denotational semantics, and we also proved that the denotational model is adequate and fully abstract with respect to the behavior that extracts from a process’s operational semantics only the set of strings the process can execute. Notable in our semantics is the treatment of hiding—we support the hiding of resources as well as of actions, which generalizes somewhat the original notion of hiding from CSP [18]. Our semantics also uses special variable resources that make the unwinding of recursion observable. In particular, this allows us to observe divergence. If we do not want to observe these unwindings of recursion, we can hide the variable resources. It should be noted that our congruence and full abstraction theorems only apply to the processes we called nice (see De,nition 5.4), that is processes that do not hide variable resources over a process containing some free variables. We believe this is not a serious restriction to the expressiveness of our language and all natural processes

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we could think of can be written as nice processes. Still, we have this restriction and we leave as an open problem whether this restriction can be overcome. The work reported here is part of an on-going program the authors have embarked upon whose goal is to understand better the diEerence between concurrency and nondeterminism. Our earlier work [5,6] considered a language similar to the one treated here: the hiding operator was not present in that language, nor was strict sequential composition, but, on the other hand, the earlier language included a restriction operator p|R that con,nes process p to the resource set R. The operational rule for restriction is simply a

p → p ;

a

a⊆R

p|R → p |R

:

We have not considered that operator here because it can be derived from our parallel composition and hiding. One can show that p|R = (p  p)\RT where C = {a ∈ | a ⊆ R} C

and RT = R\R. For the future, we intend to add nondeterminism to our language. It is clear that the language we have studied here has many interesting properties, and that there are interesting processes that can be written in the language. Nonetheless, we believe that a full understanding of the diEerence between the “true concurrency” approach to modelling concurrency and the more standard interleaving approach will be possible only in a setting where both are supported. We also are looking at potential applications for the language we are developing. We believe there will be many such, but to name some areas in particular at this point is somewhat premature. The approach we take to de,ning the semantics—using resources—has obvious appeal, and suggests many possible applications. But until these appealing potential areas of application have actually been explored, we prefer not to predict what utility the language may have. In any case, we believe the results presented here shed light on how a semantics depending on pomsets and using resources can be crafted. Other authors have considered event structures [19] and pomsets [14,15] as models of concurrency, but none that we know of has actually written a speci,c language and attempted the sort of analysis—operational and denotational—presented here. Most of this work—ours included—has its origins in Mazurkiewicz trace theory [10] and Petri nets [17], and it would be interesting to see how expressive the language we have presented is for modelling these constructs. Nonetheless, we believe that, along with [5,6], this represents the ,rst presentation of a truly concurrent denotational semantics for a language including parallel composition and weak sequential composition. Acknowledgements The authors would like to take this opportunity to thank Maurice Nivat for providing us with the opportunity to discuss our mutual research interests. This opportunity has led to a fruitful collaboration, part of which is reLected in this work. We also wish

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to express our gratitude to the members of LIAFA and of the math department of Tulane University for their kind hospitality and for the stimulating environment they provided while this work was being carried out. Thanks are also due to the US OKce of Naval Research and the National Science Foundation, whose support also helped to allow this work to take place. References [1] S. Abramsky, A. Jung, Domain Theory, Handbook of Computer Science and Logic, Vol. 3, Clarendon Press, Oxford, 1995. [2] S.D. Brookes, C.A.R. Hoare, A.W. Roscoe, A theory of communicating sequential processes, JACM 31 (1984) 560–599. [3] V. Diekert, P. Gastin, Approximating traces, Acta Informatica 35 (1998) 567–593. [4] V. Diekert, G. Rozenberg (Eds.), The Book of Traces, World Scienti,c, Singapore, 1995. [5] P. Gastin, M.W. Mislove, A truly concurrent semantics for a simple parallel programming language, in: J. Flum, M. RodrUVguez-Artalejo (Eds.), Proceedings of the Annual Conference of the European Association for Computer Science Logic (CSL’99), Lecture Notes in Computer Science, Vol. 1683, 1999, pp. 515 –529. [6] P. Gastin, M.W. Mislove, A truly concurrent semantics for a simple parallel programming language, (journal version), Mathematical Structures for Computer Science, to appear. [7] P. Gastin, D. Teodesiu, Resource traces: a domain for processes sharing exclusive resources, Theoret. Comput. Sci., to appear. [8] G. Gierz, K.H. Hofmann, K. Keimel, J. Lawson, M. Mislove, D. Scott, A Compendium of Continuous Lattices, Springer, Berlin, Heidelberg, New York, 1980. [9] M. Hennessy, G.D. Plotkin, Full abstraction for a simple parallel programming language, Lecture Notes in Computer Science, Vol. 74, Springer, Berlin, 1979. [10] A. Mazurkiewicz, Trace theory, in: M.P. Chytil et al. (Eds.), Proceedings of the 11th Symposium on Mathematical Foundations of Computer Science (MFCS’84), Lecture Notes in Computer Science, Vol. 255, 1987, pp. 279 –324. [11] R. Milner, A Calculus for Communicating Systems, Lecture Notes in Computer Science, Vol. 92, 1980. [12] M.W. Mislove, F.J. Oles, Full abstraction and recursion, Theoret. Comput. Sci. 151 (1995) 207–256. [13] M.W. Mislove, Algebraic posets, algebraic cpos and models of concurrency, in: G.M. Reed, A.W. Roscoe, R.F. Wachter (Eds.), Topology and Computer Science, Oxford University Press, Oxford, 1990. [14] V.R. Pratt, On the composition of processes, Proceedings of the 9th ACM POPL, 1982, pp. 213–223. [15] V.R. Pratt, Modelling concurrency with partial orders, J. Parallel Programming 15 (1987) 33–71. [16] G.D. Plotkin, A structural approach to operational semantics, Report DAIMI-FN-19, 1981. [17] W. Reisig, Petri Nets, EATCS Monographs in Theoretical Computer Science, Vol. 4, Springer, Berlin, 1985. [18] A.W. Roscoe, CSP and determinism in security modelling, Proceedings of the 1995 IEEE Symposium on Security and Privacy, IEEE Computer Society Press, 1995. [19] G. Winskel, Events in Computation, Ph.D. Thesis, University of Cambridge, 1980.

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www.elsevier.com/locate/tcs

Modelization of deterministic rational relations Serge Grigorie& LIAFA, Universite Paris 7, 2 Pl. Jussieu, 75251 Paris Cedex 05, France

Abstract The de'nition of the class of deterministic rational relations is fundamentally based on the Read-only One-way Turing machine approach. The notion of deterministic automata developed up to now is too strong and asks for an unnatural detour via end-markers to give all deterministic rational relations (cf. Section 3:1). We stress that several conditions usually considered as related to determinism are mere normalizations of determinism and are not inherent to the notion (cf. Section 3:2). In this paper, we introduce pertinent notions of deterministic labelled graph automata (cf. Section 3:3) which avoid any use of end-markers: strong deterministic, n-deterministic automata for n ∈ N. These notions form an increasing in'nite hierarchy of classes of automata which all lead to the same usual class of deterministic rational relations. Moreover, the class corresponding to the natural extension to the case n = ∞ is exactly the class of unambiguous automata. We also consider Nivat’s characterization via multimorphisms applied to rational languages and introduce a hierarchy of deterministic versions of multimorphisms. Properties of determinism and unambiguity are compared. The decision problems for ambiguity or determinism relative to automata and multimorphisms are settled. Roughly, all problems are undecidable in case of arity ¿ 2 with at least two non-binary alphabets, else they are decidable, c 2002 Elsevier Science B.V. All rights reserved. most being even polynomial time decidable. 

Contents 1. Modelization of rational relations over words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 1.1. The product monoid 1∗ × · · · ×k∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 1.2. Rational relations de'ned via set theoretical operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 1.3. Multitape automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 1.4. Multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 1.5. Normalizing automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 1.6. Multimorphisms and Nivat’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 1.7. Read-only One-way Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 1.8. In-between automata and ROTM’s: ROTM transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 1.9. Tally rational relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 E-mail address: [email protected] (S. Grigorie&). c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 2 - 1

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2. Modelization of unambiguous rational relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 2.1. Unambiguous automata and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 2.2. Decidability of ambiguity for automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 2.3. Unambiguous multimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 3. Modelization of deterministic rational relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 3.1. End-markers and super-deterministic automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 3.2. What is a deterministic automaton? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 3.3. Strongly deterministic and n-deterministic automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 3.4. Ambiguity and ∞-determinism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 3.5. Deterministic automata compute what is expected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 3.6. Decidability of determinism for automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 3.7. Deterministic multimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

1. Modelization of rational relations over words 1.1. The product monoid 1∗ × · · · ×k∗ As usual, ∗ denotes the set of 'nite words in a 'nite alphabet . The empty word is  and |u| is the length of u. We identify u with an application from {0; 1; : : : ; |u| − 1} into . We shall denote words by u; v; w and tuples of words by u; D vD; v. D Let
1 ; : : : ; k be 'nite alphabets, k¿2. The product monoid 1∗ × · · · ×k∗ , also denoted i∗ , consists of k-tuples of words with componentwise concatenation product. Its neutral element is D = (; : : : ; ). Denition 1.1. (1) The length and depth of a multiword uD = (u1 ; : : : ; uk ) are |u| D = |u1 | + · · · + |um |;

(1)

depth(u) D = sup(|u1 |; : : : ; |um |):

(2)

(2) The support of a k-tuple uD is Supp(u) D = {i ∈ {1; : : : ; k} | ui = }. Remark 1.2. The set of multiwords with length 1 (i.e. multiwords all components
with equal to  except one which is a letter) generates the monoid i∗ . In fact, it is the smallest (relative to set inclusion) set of generators.
The set-theoretical de'nition of rational relations included in i∗ and the associated machine-like models (namely automata) are mere particular cases of notions de'ned in the general context of monoids. We recall these notions in the next subsections and refer to the literature for proofs of Theorems (cf. [4,1,28, II.1.3 Theorem 1.1]). 1.2. Rational relations de4ned via set theoretical operations Denition 1.3. (1) The product operation on elements of a monoid M (with neutral element 1M ) induces operations on subsets of M . If R and S are subsets of M then

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the concatenation product of R; S and the plus and star of R are the relations RS = {xy | x ∈ R and y ∈ S}; R+ =

 n¿1

Rn ;

R∗ = {1M } ∪ R+ :

(2) The family Rat(M ) of rational subsets of M is the closure of the family of 'nite relations by the operations of union, product and star. 1.3. Multitape automata There are several machine counterparts for Rat(M ). Though the historical machine is the Read-only One-way multitape Turing Machine (cf. Section 1.7), the reference machine model is the labelled graph automaton introduced by Myhill [18]. Denition 1.4. (1) A 'nite automaton A over a monoid M (with neutral element 1M ) is a 'nite directed graph A = Q; M; ; I; F labelled by elements of M \{1M }. Nodes and edges form the respective sets Q and  ⊆ Q×M ×Q and are also called states and transitions. I and F are distinguished subsets of Q called initial states and 'nal states. (2) An A-path c is a 'nite sequence of transitions a

a

a

p0 →1 p1 →2 p2 : : : pn−1 →n pn : The origin and end of the path c are p0 and pn . The label of c is the element a1 a2 : : : an of the monoid M . Case n = 0 corresponds to an empty path, with label 1M and p0 as both origin and end. Path c is successful if its origin is in I and its end is in F. (3) The behaviour of A is the set of labels of successful paths. (4) States which belong to some path with origin in I (resp. last state in F) are called accessible (resp. coaccessible). Automaton A is trim if all states are accessible and coaccessible. Notation 1.5. (1) In case M is a free monoid ∗ (with  a 'nite alphabet), the behaviour of A is also called the language associated to A and denoted L(A). (2) In case M is a product of free monoids 1∗ × · · · ×k∗ (where all i ’s are 'nite alphabets), A is called a 'nite multitape (or k-tape) automaton and the behaviour of A is also called the relation associated to A and denoted Rel(A). The basic case M = ∗ of the following fundamental theorem is due to Kleene [15]. The general for arbitrary monoids is due to Elgot and Mezei [6], who detailed
statement the case i∗ in [6, Proposition 3.5, and mentioned the general result in a footnote p. 50]. Theorem 1.6 (Kleene’s theorem). A subset of M is rational if and only if it is the behaviour of some 4nite automaton (resp. trim automaton).

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1.4. Multiplicities
As in the case of languages, the set-theoretical operations de'ning Rat( i∗ ) can be augmented with a notion of multiplicity so as to allow deep algebraic considerations. This is done by replacing
relations (which may be viewed as
functions
R : i∗ → {0; 1}) by N-series over i∗ , i.e. multiplicity functions s : i∗ → N. The three set operations of union, product and star on relations, respectively, correspond to the following algebraic operations on series: sum s + t, Cauchy product st and Cauchy star s∗ , where st(u) D = s∗ (u) D =

 

{s(v)t( D w): D vw = u}; D

(3)

{sn (u): D n ∈ N} de'ned if s() D =0

(4)

with the convention that s0 is the neutral element for Cauchy product, i.e. such that s0 () D = 1 and s0 (u) D = 0 for uD = . D That there are only 'nitely many non-zero terms in the above sums is insured by easy considerations on the lengths of k-tuples of words.
Denition 1.7. The family of rational series over i∗ is the closure of the family of 'nite relations by the operations of sum, Cauchy product and Cauchy star.
∗ Denition 1.8. Let A be a 'nite automaton over i . We allow several edges with the same label between any two states (in other words, edges have multiplicities). The
A-multiplicity series A-mult : i∗ → N associates to any multiword uD the number of accepting paths with label uD (this number is 'nite due to the fact that no edge is labelled ). D The counterpart to Kleene’s 1.6 is due to SchKutzenberger [30] for the basic
theorem case ∗ . The extension to i∗ is easy and folklore. Theorem 1.9 (Kleene–SchKutzenberger’s theorem). Rational series are exactly the multiplicity series of 4nite automata (resp. trim and normal automata).
Remark 1.10. In particular, a relation R ⊆ i∗ is rational if and only if it is the support supp(s) of some rational series s, where supp(s) = {u: D s(u) D = 0}: 1.5. Normalizing automata For future reference (cf. proof of Proposition 1.21, Remarks
3.18, 3.19) we make some simple observations about normalization of automata over i∗ . Denition 1.11. Let A be an automaton over




i∗ .

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(1) A is quasi-normal if all labels of transitions have depth 1 (cf. De'nition 1.1) (i.e. if uD = (u1 ; : : : ; uk ) is a label then sup{|u1 |; : : : ; |uk |} = 1, which means that each ui is a letter or is  and that some ui is a letter). (2) A is normal if all labels of transitions have length 1 (cf. De'nition 1.1). (i.e. if uD = (u1 ; : : : ; uk ) is a label then |u1 | + · · · + |uk | = 1, which means that among the ui ’s exactly one is a letter and the other ones are ). D

The idea to normalize an automaton is to split a transition p → q into a sequence of transitions 1

2

m−1

m

    p →(p; D1 ) →(p; 1 2 ) · · · → (p; D1 : : : m−1 ) → q

(5)

following a decomposition D = 1 : : : m . Notation 1.12. If u is a word and uD = (u1 ; : : : ; uk ) a multiword, we let lettert (u) = IF 1 6 j 6 |u| THEN the tth letter of u ELSE ; lettert (u) D = (lettert (u1 ); : : : ; lettert (uk )); letterit (u) D = (; : : : ; ; lettert (ui ); ; : : : ; ): We consider two decompositions of u, D where m = depth(u): D uD = letter1 (u) D : : : letterm (u); D

(6)

uD = letter11 (u1 ) : : : letter1|u1 | (u1 ) : : : letterk1 (uk ) : : : letterk|uk | (uk ):

(7)

Denition 1.13. (1) We let QuasiNormal(A) be the automaton obtained from A by D splitting any A-transition p → q as shown in Eq. (5) above following the decomposition of D given by Eq. (6) above. This introduces new states (t; letter1 () D : : : lettert ()) D for 16t¡depth(). D Notice that some of these states may come from several transitions with origin p. (2) We let Normal(A) be the automaton obtained from A by splitting any A-transition as shown in Eq. (5) above following the decomposition of D given by Eq. (7) above. This introduces new states (p; (1 ; : : : ; j ; ; : : : ; )) (p; (1 ; : : : ; j ; letter1 (j+1 ) : : : lettert (j+1 ); ; : : : ; )) for 16j¡k and 16t¡|j+1 |. Notice that some of these states may come from several transitions with origin p.

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Proposition 1.14. Normal(A) (resp. QuasiNormal(A)) is a normal (resp. quasinormal) automaton which has the same behaviour and multiplicity series (cf. Section 1:4) as A. 1.6. Multimorphisms and Nivat’s theorem
∗ Rational relations over i can be given still another characterization
using usual ∗ rational languages and morphisms from some monoid %
to the monoid i∗ . This is due to Nivat [19]. Observe that a morphism & : %∗ → i∗ is a tuple of morphisms (’i : %∗ → i∗ )i=1; :::; k . Whence the name multimorphism. Denition 1.15. Let ’ : %∗ → ∗ be a morphism and & = (’i : %∗ → i∗ )i=1; :::; k be a multimorphism. (1) ’ is alphabetical (resp. strict alphabetical) if |’(a)|61 (resp. |’(a)| = 1) for all a ∈%. & is alphabetical (resp. strict alphabetical) if so are all ’i ’s. (2) ’ (resp. &) is proper if ’(a) =  (resp. &(a) = ) D for all a ∈%.
Theorem 1.16 (Nivat’s theorem [19]). A relation
R ⊆ i∗ is rational if and only if R = &(L) where L ⊆ %∗ is rational and & : %∗ → i∗ is a multimorphism (resp. strict alphabetical multimorphism). 1.7. Read-only One-way Turing Machines The following material will be needed for the modelization of deterministic rational relations in Section 3.3. As said in Section 1.3, the reference model for multitape automaton is the labelled graph model of De'nition 1.4. However, as pointed by Fischer and Rosenberg [7, pp. 90, 91], the basic intuition for multitape automaton remains that of non-deterministic Read-only One-way multitape Turing machine (ROTM) with exactly one head per tape. Denition 1.17. An ROTM is a non-deterministic Turing machine such that the symbols on the tapes are not modi'ed and there is no backward move and every transition moves at least one head. Thus (i) the machine starts in any initial state, (ii) a transition step depends (non-deterministically) on the state of the machine and on the letters (or the blank symbol B lying on the right of the inputs) read by the k heads on the k tapes, (iii) a transition step changes state and moves forward some heads (possibly none), (iv) the machine stops when it enters a halting state, (v) halting states are of two types: accepting or rejecting, (vi) a computation is accepting if it enters an accepting halting state. It is convenient to introduce a variant of the ROTM which we shall call modi4ed ROTMs.

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Denition 1.18. A modi4ed ROTM behaves according to De'nition 1.17(i,ii) and to (iiibis) a transition step changes state and moves forward at least one head (possibly several heads), (ivbis) a head which reads the blank symbol B does not move and the machine stops when all heads read the blank symbol B, (vbis) states are of two types: 'nal and non-'nal, (vibis) a computation is accepting if its last state is 'nal. Formally, a modi'ed ROTM is a tuple T = QT ; 1 ; : : : ; k ; T ; IT ; FT  with T ⊆ Q ×




(i ∪ {B}) × Q × {0; 1}k :

A tuple (q; a; D r; m) D in T is interpreted as follows: q; r are the states before and after the transition, the k heads read aD and move according to m. D The following is folklore. Proposition 1.19. A relation is computable by an ROTM (resp. deterministic, resp. unambiguous ROTM) if and only if it is computable by a modi4ed ROTM (resp. deterministic, resp. unambiguous modi4ed ROTM). Proof. (1) ROTM ⇒ modi4ed ROTM. Get rid of -transitions D (i.e. transitions which move no head) in the standard way: D (q; a; () First, iteratively add new transitions as follows: if (p; a; D q; 0); D r; m) D are transitions then add transition (p; a; D r; m), D () then suppress all -transitions. D Now, force the machine to read entirely its input: if q is a halting state then add all transitions (q; a; D q; m) D where mi = IF ai = B THEN 0 ELSE 1: Declare as 'nal states all accepting halting states and all states from which one can access an accepting halting state by a sequence of transitions reading (B; : : : ; B). It is easy to check that these modi'cations transform an ROTM into a modi'ed ROTM computing the same relation. Moreover, this transformation preserves determinism and unambiguity. But it cannot preserve multiplicities: • all multiplicities are necessarily 'nite with modi'ed ROTMs, • in'nite multiplicities are possible with ROTMs since -transitions D and transitions reading (B; : : : ; B) can create loops. (2) Modi4ed ROTM ⇒ ROTM. To every state p associate a new state phalt and add transitions D (p; (B; : : : ; B); phalt ; 0):

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Declare as accepting (resp. rejecting) halting states those states phalt such that p is 'nal (resp. non-'nal). It is also well-known that the ROTM and modi'ed ROTM models are equivalent to the labelled graph automaton model [7]. Remark 1.20. For an ROTM (as for any modi'ed ROTM or Turing machine), an occurrence of a symbol in an input on some tape is read again and again through successive transitions while there is no move of the head of that tape. The labelled graph automaton model can thus be viewed as a very smooth normalization of the ROTM model in which each occurrence of a symbol of each input is read only once. Proposition 1.21. A relation is rational if and only if it is computed by a modi4ed ROTM. This is also true when multiplicities are considered. Proof. (1) automaton ⇒ modi4ed ROTM Going from an automaton A to a modi'ed ROTM is quite easy and can be done as follows: (i) Quasi-normalize A as in De'nition 1.13. (ii) A quasi-normal automaton can directly be interpreted as a modi'ed ROTM. (2) Modi4ed ROTM ⇒ automaton The passage from a modi'ed ROTM to an automaton requires some care. As we shall need it for the proof of Theorem 3.17, we detail the construction in De'nition 1.22. It is easy to check that this construction leads to a quasi-normal automaton A which computes the same relation as T and also the same multiplicity series. Denition 1.22. Let
T = QT ; 1 ; : : : ; k ; T ; IT ; FT  be a modi'ed ROTM. We de'ne an automaton A = Q ; i∗ ; A ; IA ; FA  as follows:
A (i) QA = QT
× (i ∪ {B; new}),
× {new}, FA = FT × {B; new}, (ii) IA = I (iii) A = t∈T Trans(t), where if t = (q; a; D r; m)∈ D T then Trans(t) is the following family of transitions of A: D ; {((q; +); D (r; ,)) D : for some i = 1; : : : ; k +i = new and for all i = 1; : : : ; k +i ∈ {ai ; new} and (ai = B ⇒ +i = B) and i = IF (+i = new and ai = B) THEN ai ELSE  and ,i = IF mi = 1 THEN new ELSE ai }: Thus, Trans(t) contains 2l −1 transitions where l is the number of i’s such that ai = B. Observe that this number is positive since T halts when it reads (B; : : : ; B) (so that l¿1). Intuition: A letter which has just been read in a T -computation is retained in the state of the emulating A-computation. This allows A to read each letter only once and

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not several times as T possibly does. Such an emulation of T by A cannot be static (state to state and transition to transition), it has to be somewhat dynamic. This is why to a single T -transition t we associate a family Trans(t) of A-transitions which may emulate t. The ith component of the label of an A transition is non-empty if and only if this transition emulates a T transition in which the symbol on tape i is read for the 'rst time and is not the blank end-marker (i.e., either this is the 'rst transition or in D of the preceding transition, there has been a move on tape i). Thus, in a state (q; +) A we have +i = new if this state is initial or if it emulates a T -state obtained after a T -transition which makes a move on tape i. Else, +i is the symbol which was read and will again be read by T . Remark 1.23. If the modi'ed ROTM T moves exactly one head per transition then all labels of A-transitions have length 1 and A is a normal automaton. 1.8. In-between automata and ROTMs: ROTM transducers The notion of (k; h)-ROTM transducer is obtained by adding to the k input tapes of an ROTM a family of h output tapes with alphabets -1 ; : : : ; -h : T = QT ; 1 ; : : : ; k ; T ; IT ; FT ; -1 ; : : : ; -h ; .T  where .T ⊆ Q×((1 ∪ {B})× · · · ×(k ∪ {B}))×(-1 × · · · ×-h ). Now, k-tape ROTMs can be viewed as (k; 0)-ROTM transducers while labelled graph k-tape automata can be viewed as (0; k)-ROTM transducers. From this point of view, automata appear as pure output machines with no input. 1.9. Tally rational relations Tally relations are relations over unary alphabets. Via an obvious bijective map, they correspond to relations over N. A characterization as lattices in Nk was obtained by Parikh [21], reprinted in Parikh [22] (see also [10]). Another characterization via Presburger arithmetic was developed by Ginsburg and Spanier [9]. Denition 1.24. A relation R ⊆ Nk is linear if there is a 'nite sequence u0 ; : : : ; un of elements of Nk such that R = u0 + u1 N + · · · + un N = {u0 + x1 u1 + · · · + xn un : x1 ; : : : ; xn ∈ N}:

(8)

A relation is semi-linear if it is the union of 'nitely many linear relations. Theorem 1.25. (1) (Parikh [21]) Tally rational relations are exactly the semilinear relations and [9] are closed under boolean operations and projections. (2) (Ginsburg and Spanier [9]) Let R be a tally relation. The following conditions are equivalent: (i) R is rational.

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(ii) R is de4nable in the structure N; =; + (i.e. Presburger arithmetic). (iii) R is de4nable in the structure N; =; + by a 10 formula. Since Presburger arithmetic is decidable [25], this gives a tool to get decidability results about tally rational relations. For instance, Corollary 1.26. There is an algorithm to decide whether two tally rational relations are disjoint. A simple argument allows to extend this last result to the case of relations with at most one non-unary component. Corollary 1.27. There is an algorithm to decide whether two rational relations with at most one non-unary component are disjoint. Proof. Let R1 ; R2 be rational relations included in ∗ ×({0}∗ ) k and let R = {(v; D w) D ∈ ({0}∗ )2k : ∃u ∈ ∗ ((u; v) D ∈ R1 ∧ (u; w) D ∈ R2 )} be the composition of R1 ; R2 along their ∗ component. Clearly, R1 ; R2 are disjoint if and only if R and the diagonal relation Diagk = {(v; D v) D ∈ ({0}∗ )2k : v) D ∈ ({0}∗ )k } are disjoint. Now, R and Diagk are tally rational relations and automata to compute these relations can be polynomial time de'ned from automata for R1 ; R2 . This gives a polynomial time reduction of the disjointness problem for rational (k + 1)-ary relations with at most one non-unary component to the disjointness problem for tally 2k-ary rational relations. Remark 1.28. (1) The above result is also an easy application of the analog result (due to [13, Theorem 3.1, p. 124]) about 'nite-turn multicounter machines, i.e. onetape automata (over some non-necessarily unary alphabet) with k counters making at most r alternations between push and pop modes for some 'xed r (all counters being initially empty). Cf. the proof of Corollary 1.31. (2) In case at least two components are non-unary, the disjointness problem is undecidable ([26], cf. also the proof of Theorem 2.4 below). Using Eq. (8), the disjointness problem for k-ary tally rational relations can be related to linear programming with non-negative integers involving systems of k linear equations. Such systems with a 'xed number of equations are polynomial time solvable ([16], cf. also [29, Corollary 18.7, p. 260]). Unfortunately, the passage from the automaton to a semi-linear representation is a priori exponentially complex. Thus, it is not possible to directly apply Lenstra’s theorem in order to get a polynomial time algorithm in Corollary 1.26. Nevertheless, as proved by Gurari and Ibarra [11], this

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can be achieved via an argument which uses known sharp bounds to integral solutions of linear systems [3,31]. As in Ibarra [13], the result proved by Gurari and Ibarra [11, Corollary 1, p. 224], deals with 'nite-turn multicounter machines (cf. Remark 1.28). Theorem 1.29 (Gurari and Ibarra [11]). For 4xed k, there is a polynomial time algorithm to decide whether two 4nite-turn multicounter machines compute disjoint languages. Remark 1.30. However, the degree of the polynomial bound increases with k. A simple polynomial time reduction leads to the following improvement of Corollary 1.27. Corollary 1.31. For 4xed k, there is a polynomial time algorithm to decide whether two (k + 1)-tapes automata over k unary alphabets and one possibly non-unary alphabet compute disjoint rational relations. Proof. A (k + 1)-tape automaton A over k unary alphabets and one possibly non-unary alphabet can be emulated by a 'nite-turn 2k-counter machine as follows: (i) Code the k + 1 inputs 0 n1 ; 0 n2 ; : : : ; 0 nk ; u as a binary input 0 n1 10 n2 1 : : : 10 nk 1u. (ii) consider the 1-turn k-counter machine M which 'rst pushes the 'rst k blocks of zeros 0 n1 ; 0 n2 ; : : : ; 0 nk of its input into the k counters of M , then emulates A on inputs 0 n1 ; 0 n2 ; : : : ; 0 nk ; u so that a move on the ith tape of A becomes a pop on the ith counter of M . This emulation is clearly polynomial time computable and gives a polynomial time reduction of the associated disjointness problems.

2. Modelization of unambiguous rational relations This section reviews known notions and facts and proves new polynomial time decidability results for diverse problems about ambiguity in case there is at most one non-unary alphabet (cf. Corollary 2.6). It also stresses analogies with the material we are going to develop for the modelization of determinism in Section 3. 2.1. Unambiguous automata and relations Denition 2.1. (1) An automaton is unambiguous if for every input there exists at most one accepting run (but there can be many non-accepting runs), i.e. the associated multiplicity series (cf. De'nition 1.8) is {0; 1}-valued. (2) A rational relation is unambiguous if it is accepted by some unambiguous automaton.

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Example 2.2. Two unambiguous (but non-deterministic) relations: (1) The suOx relation for words on a non-unary alphabet .

(2) R = {(0 m ; 0 n ): m6n62m}. To get an unambiguous automaton for R, observe that every pair in R can be written in a unique way in the form (0p+q ; 0p+2q ).

Rational relations which cannot be accepted by unambiguous automata are called inherently ambiguous. The standard example is analog to the standard inherently ambiguous context-free language: {(0m 1n ; 0p ): p = m or p = n}: However, Eilenberg and SchKutzenberger [5] proved the following theorem. Theorem 2.3 (Eilenberg and SchKutzenberger [5]). Every tally rational relation is unambiguous. 2.2. Decidability of ambiguity for automata Ambiguity does not correspond to a simple property of machines. In the general case, this is an undecidable property. Theorem 2.4 (Rabin and Scott [26]). Let k¿2 and suppose that at least two among the k alphabets 1 ; : : : ; k are non-unary. Then the class of unambiguous automata is not recursive. It is, in fact, 110 -complete. Proof. Rabin and Scott [26] really proved that the disjointness problem is undecidable, but their proof applies with no change to the ambiguity problem. We briePy recall their argument. The Post Correspondence Problem for m sequences (PCPm ) is the class of

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pairs of homomorphisms ’; : {1; : : : ; m}∗ → {0; 1}∗ such that there exists u =  satisfying ’(u) = (u). Now, let A’ = {q0 ; q1 }; ’ ; {q0 }; {q1 } where  = {(q; i; ’(i); q1 ): q ∈ {q0 ; q1 }; i ∈ {1; : : : ; m}} be the obvious 2-tape automaton which computes the relation R’ = {(u; ’(u)): u ∈ {1; : : : ; m}∗ ; u = }: Observe that A’ is unambiguous (even super-deterministic automata, cf. De'nition 3.1). It is clear that the union automaton A’ ∪ A is ambiguous if and only if (’; ) ∈PCPm . This gives a recursive reduction (in fact polynomial time reduction) of PCPm into the class of ambiguous automata on alphabets {1; : : : ; m}; {0; 1}. It is known that PCPm is not recursive and is 10 -complete for large m [24], even for m = 7 [17]. This proves the theorem for the case of alphabets {1; : : : ; m}; {0; 1}. Standard coding transfers the result to any 'nite sequence of alphabets with at least two non-unary alphabets. However, when at most one alphabet is non-unary, unambiguous automata form a recursive class. First, recall the notion of truth-table reducibility (cf. standard textbooks on recursion theory, [27, pp. 109–110; 20, p. 268]). A class X ⊆ ∗ is truth-table reducible to Y ⊆ -∗ if there exists a recursive function & : ∗ → P4n (-∗ ) (where P4n (-∗ ) is the set of 'nite subsets of -∗ ) such that ∀u

(u ∈ X ⇔ &(u) ⊆ Y ):

X is polynomially truth-table reducible to Y if the above & is polynomial time computable (in particular, the number of words in &(u) and their lengths are bounded by a polynomial in |u|). Proposition 2.5. Fix alphabets 1 ; : : : ; k . The class of unambiguous automata is polynomially truth-table reducible to the class of pairs of automata computing disjoint relations. Proof. Suppose that A is a trim automaton (cf. De'nition 1.4, point 4). Then A is ambiguous if and only if there exists two distinct paths having the same label, starting in the same state but having di&erent 'rst transitions. Let us denote by Aq the automaton A with q as (unique) initial state. If (p; ; D q) is a transition of A, lets denote A(p; ;D q) the automaton obtained from A by adding a new D q). It is clear that state pbis as the (unique) initial state and a new transition (pbis ; ; Rel(A(p; ;D q) ) = D Rel(Aq ). Using these notations, A is ambiguous if and only if there exists two distinct tranD r) with the same origin such that D Rel(Aq ) ∩ D Rel(Ar ) = ∅. sitions (p; ; D q); (p; ; Thus, A is unambiguous if and only if Rel(A(p; ;D q) ) ∩ Rel(A(p; ;D r) ) = ∅ for all pairs D r) of distinct transitions with the same origin. (p; ; D q); (p; ;

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This gives a polynomial time truth-table reduction of the class of unambiguous automata to the class of pairs of automata computing disjoint relations. From Proposition 2.5 and Corollary 1.31 we get Corollary 2.6. If there is at most one non-unary alphabet then the class of unambiguous automata is polynomial time computable. 2.3. Unambiguous multimorphisms The analog of Nivat’s theorem 1.16 relative to unambiguous relations holds.
Theorem 2.7. A relation R ⊆ i∗ is unambiguous rational if and only if R = &(L)
∗ where L ⊆ % is rational and & : %∗ → i∗ is a multimorphism (resp. strict alphabetical multimorphism) which is injective on L.
Proof. (1) Let L ⊆ %∗ be computed by AL and & : %∗ → i∗ be a multimorphism. A simple non-deterministic automaton RA; & which computes &(L) acts as follows: it guesses the successive letters of a word w ∈%∗ , simulates AL to check that w ∈L and compares &(w) to the input. If the restriction of & to L is injective and if AL is unambiguous (which we may suppose without loss of generality) then the above automaton RA; & is clearly unambiguous. This proves that &(L) is unambiguous. (2) Now,
let R be a rational relation computed by a k-tape automaton A. Let A ⊆ QA × i∗ ×QA be the 'nite set of transitions of A. Considering A as an alphabet %, automaton A can be viewed as a 1-tape automaton
∗ computing a language L ⊆ %∗ . Also, the label function (q; u; D r) →  u D from % into i has a unique extension
to a multimorphism & : %∗ → i∗ . It is clear that &(L) = R. Clearly, A is unambiguous if and only if the restriction of & to L is injective. Lastly, observe that if A is normal (which can be supposed without loss of generality) then & is strictly alphabetical. As can be expected, the answer to the decision problem relative to the above characterization is much the same as in Corollary 2.6. However, this answer also depends on the cardinality of the source alphabet % of L and &. Proposition 2.8. (1) If at least two of the alphabets i ’s are non-unary then ∗ (i) There is a 4nite alphabet % and
∗a rational language L ⊂ % such0 that the family ∗ of multimorphisms & : % → i which are injective on L is 11 -complete hence undecidable.
(ii) There is a strict alphabetical multimorphism & : {0; 1; 2; 3}∗ → i∗ such that the family of rational languages on which & injective on L is 110 -complete hence undecidable. (2) If % has at most 3 letters then there is a polynomial time algorithm to de
cide whether an alphabetical multimorphism & : %∗ → i∗ is injective on a rational language L ⊆ %∗ .

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(3) If there is at most one non-unary alphabet then
there is a polynomial time algorithm to decide whether a multimorphism & : %∗ → i∗ is injective on a rational language L ⊆ %∗ . Remark 2.9. (1) Point (1i) cannot be improved with a restriction to alphabetical multimorphisms
since for a 'xed 'nite % there are 'nitely many alphabetical multimorphisms & : %∗ → i∗ . (2) The proof of point (1i) (together with the best known bound m = 7 for the undecidability of PCPm , [17]) leads to an alphabet % with 14 symbols. We do not know what is the least possible cardinality of %. Proof. (1) It is suOcient to consider the case k = 2 and 1 = 2 = {0; 1}. We keep the notations of the proof of Theorem 2.4. (i) Set % = {1; : : : ; 2m} and L = {1; : : : ; m}∗ ∪ {m+1; : : : ; 2m}∗ . Let &’; : %∗ → {0; 1}∗ ×{0; 1}∗ be the multimorphism such that &’; (i) = IF 1 6 i 6 m THEN (0i 1; ’(i)) ELSE (0i−m 1; (i)): It is clear that &’; is injective on L if and only if (’; ) ∈PCP = m . This last property is 110 -complete for m¿7 ([17], cf. proof of Theorem 2.4).
(ii) Let & : {0; 1; 2; 3}∗ → i∗ be the strict alphabetical multimorphism such that &(0) = (; 0);

&(1) = (; 1);

&(2) = (0; );

&(3) = (1; ):

Set L’; = {2i 3’(i): i = 1; : : : ; m} ∪ {2i 3 (i): i = 1; : : : ; m}. It is clear that &(2i 3’(i)) = (0i 1; ’(i)) and &(2i 3 (i)) = (0i 1; (i)). So that & is injective on L’; if and only if (’; ) ∈PCP = m. (2) We 'rst reduce the problem of injectivity of multimorphisms on rational languages to the ambiguity problem of multitape automata.
Let & : %∗ → i∗ be a multimorphism, L ⊆ %∗ be a rational language and A be a deterministic (one-tape) automaton computing L. De'ne A& as the non-deterministic k-tape automaton computing &(L) as follows: ∗ • A& (non-deterministically) guesses a word
∗ u ∈% , & • A compares &(u) with its input 8D ∈ i , • A& emulates A to check if u ∈L, • A& accepts if u ∈L and &(u) = 8. D
It is clear that the accepting runs of A& on an input 8D ∈ i∗ are in a 1–1 correD Thus, & is injective on L if and only if A& is spondence with the words u ∈&−1 (8). unambiguous. A priori, the alphabets of the k tapes of A& are the i ’s. Of course, one can reduce the ith alphabet i to the subalphabet Xi formed by the letters of the ith components of the multiwords in &(%).
Now, suppose & : %∗ → i∗ is alphabetical and let %i = {9 ∈ %: &(9) has a non-empty ith component}: Clearly, Xi is the set of ith components of &(%i ).

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The %i ’s form a partition of % and (denoting ](X ) the cardinality of a set X ) we have ](Xi )6](%i ). Now, if % has at most 3 letters then there is at most one component i for which %i has more than one element. A fortiori, there is at most one component i for which Xi has more than one element. But this means that A& has at most one non-unary alphabet. So that Corollary 1.31 allows to decide in polynomial time if A& is unambiguous. (3) Suppose now there is one non-unary alphabet  and k unary alphabets. Let L ⊆ %∗ be a rational language and & = (’; ;) a multimorphism with ’ : %∗ → ∗ , ; : %∗ → ({0}∗ ) k . Set T = {(’(); ;(); ;()): ;  ∈ L ∧ (’() = ’() ∧  = )}: Clearly, & is injective on L if and only if the projection of T on ({0}∗ )2k is disjoint from the diagonal of ({0}∗ ) k ×({0}∗ ) k . To conclude via Corollary 1.31, it suOces to prove that T is rational and construct in polynomial time some automaton for T . Now, the condition  =  can be expressed as the disjunction of conditions: (i)  is a strict pre'x of , (ii)  is a strict pre'x of , (iii)  = +a, and  = +b= for some +; ,; = ∈%∗ and a; b ∈% and a = b so that, with obvious notations  T = T¡ ∪ T¿ ∪ Ta;b : a;b∈;a=b

Let A be a one-tape deterministic automaton computing L and denote Ap and Lp (resp. Ap and Lp ) the automaton A with p as the unique initial (resp. 'nal) state and the language it computes. We then have  T¡ = (’; ;; ;)(Lp )({} × {} D × ;(Lp ∩ (’−1 ()\{}))); p∈QA

T¿ = Ta;b =



p∈QA

(’; ;; ;)(Lp )({} × ;(Lp ∩ (’−1 ()\{})) × {}); D



p∈QA

(’; ;; ;)(Lp )Ep;a;b ;

where Ep;a;b = {(’(au); ;(au); ;(bv)): au; bv ∈ Lp ∧ ’(au) = ’(bv)}: All these sets are obviously rational, hence also T , and associated automata can easily be constructed in polynomial time. 3. Modelization of deterministic rational relations For a detailed study of deterministic rational relations, we refer to Pelletier and Sakarovitch [23] and Sakarovitch [28]. Here, we shall be concerned with the problem of modelization of determinism along the di&erent approaches to rational relations described in Section 1.

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3.1. End-markers and super-deterministic automata The obvious notion of deterministic ROTM leads to the natural class of deterministic rational relations. Hence, we would like to de'ne a reasonable notion of deterministic labelled graph automata leading to the same class. However, from the very start of the theory [26, pp. 8, 86; 6, p. 48; 7, p. 89, through its development 2, 14, 12, : : :] up to the most recent papers [23, Section 2.2], the notion of deterministic automata which has been considered does not lead directly to the class of deterministic rational relations. A detour is made via auxiliary relations obtained by adding end-markers. We recall this classical de'nition (together with a variant in the vein of De'nition 3.8 below).
Denition 3.1. Let A = Q; i∗ ; E; I; F be a multitape automaton. (1) A is super-deterministic if it has a unique initial state and if the labels of two distinct edges with a common origin are pre'x incompatible (i.e. have no common extension). (2) (Rabin and Scott [26]) A is normal super-deterministic if it has a unique initial state and if there is a partition (Q1 ; : : : ; Qk ) of Q such that: (i) For all i ∈ {1; : : : ; k} and for all p ∈Qi the label of any edge with origin p has all components empty but the ith which is a letter. In particular, the support of the label is {i} (cf. De'nition 3.8). (ii) Di&erent edges with origin p have di&erent labels. These notions are e&ective. Proposition 3.2. One can decide in polynomial time whether an automaton is superdeterministic or normal super-deterministic. Remark 3.3. Super-deterministic automata are called deterministic in the literature. The reason why we depart from the standard terminology will be clear from De'nition 3.8 and Theorem 3.17 below. Remark 3.4. Exact polynomial complexity in Proposition 3.2 depends on the parameters taken into account for A (number of edges, number of nodes, outer degree, i.e. maximum number of edges having a common origin) and the presentation of the automaton (as a list of labelled edges or as a list of pairs consisting of a node and the list of labelled edges coming out of that node). This will apply as well to Proposition 3.20 below. The following simple fact insures that De'nition 3.1 is sound. Proposition 3.5. Let R ⊆ 1∗ × · · · ×k∗ and let R$ be obtained by adding an endmarker $ to each word in tuples of R. R is computed by a deterministic ROTM if and only if R$ is the behaviour of a super-deterministic (resp. normal super-deterministic) multitape automaton.

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The reason for such a detour is due to a diOculty arising from computation of multiwords having disjoint supports (cf. De'nition 1.12). This is illustrated by the following simple example. Let a; b be letters and consider the relation R = {(a; ); (; b)}. This relation is obviously ROTM deterministic. However, it is easy to see that it is not the behaviour of any super-deterministic automaton. Whereas, R$ = {(a$; $); ($; b$)} is easy to compute by a normal super-deterministic automaton:

Thus, despite the fact that multitape automata are always identi'ed with labelled graph automata, the modelization of determinism is still very much reminiscent of the ROTM model with the blank symbol appearing at the right of each input. 3.2. What is a deterministic automaton? Nevertheless, there does exist some reasonable notion of deterministic labelled graph automata which directly compute deterministic rational relations. As far as we know (and surprisingly as it may be), such notions seem to be original. Before entering the drier stu& of formal de'nitions, let us illustrate the intuition on some example. Example 3.6. The graph of the concatenation function, {(u; v; w): uv = w} is obviously ROTM deterministic: 'rst read (u; ; u) and then (; v; v). It is easy to see that it is not the behaviour of a super-deterministic automaton (argue as above with (a; ; a) and (; a; a) instead of R = {(a; ); (; b)}). Let us look at diverse automata which compute it. (1) First, two automata which can in no way be deterministic.

The 'rst one is clearly ambiguous on every triple (u; ; u) such that u = , hence surely non-deterministic. The second one is unambiguous. However, to decide which transition is the right one to go from state 0 to state 1, one has to know whether |v| is even or odd, and this is known only when v is completely read. So, there is no way for determinism.

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(2) Now, an automaton D1 which is not super-deterministic because there are tran(a; ; a)

(; a; a)

sitions 0 → 0 and 0 → 1 with pre'x compatible labels.

However, the 'rst transition reads an a on tape 1 whereas the second does not and leads to a state from which it is no more possible to read anything on tape 1 (we shall call such a state an 1-end). Thus, there is no problem to decide which transition is the right one. So, we shall consider this automaton as deterministic. The same with the following normalized versions D2 ; D3 (which have 2 + 3|| states):

(3) Now, automata D4 ; D5 which are not super-deterministic for two reasons: (i) There are transitions with pre'x compatible labels. (ii) There are several initial states.

Let us consider the 'rst automaton. (a; ; a) (a; ; a) As for (i), transitions 0 → 0 and 0 → 1 have the same label. However, the second one leads to a state (namely 1) from which it is no more possible to read anything

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on tape 1. Whereas, from the 'rst one, one is forced to read something on tape 1 in order to go to a 'nal state. Thus, there is no problem to decide which transition is the right one. As for (ii), a similar argument does work. If we start at state 0, we are forced to read something on tape 1 in order to go to a 'nal state. Whereas, if we start at state 1, then nothing can be read on tape 1. Thus, there is no problem to decide which initial state is the right one to start with. Let us now consider the second automaton. As for (i), argue as with automaton of point 2 above. As for (ii), if we start from state 2 then, nothing can be read on tape 2. Whereas, from state 0 we are forced to read tape 2 in order to go to a 'nal state. So, again, there is no problem to decide which initial state to start with. Thus, we shall also consider these automata as deterministic. Conclusion. From the above examples, we see that the pre4x incompatibility of the labels of di*erent transitions from a given state is not a condition inherent to determinism. Also, the unicity of the initial state is no more inherent to determinism, it is a mere normalization condition. Remark 3.7. There is still another reason which could be considered to get determinism. Let say that a state q is i-consistent if the label of every path from q to a 'nal state has a non-empty ith component. D

D

Suppose q is i-consistent and r is an i-end. If p → q and p → s are two transitions then there is no problem to choose between these two transitions: just look ahead at the ith component of the input. However, we shall not retain this type of deterministic character. The reason is that it does not carry to subautomata, contrarily to all above deterministic characters (cf. Proposition 3.13 below). 3.3. Strongly deterministic and n-deterministic automata Now, we come to the desired de'nitions and introduce two types of deterministic multitape automata. Denition 3.8 (Strong determinism). (1) A state p is an i-end (16i6k) for a k-tape automaton A if any path from p to a state in FA has an empty ith component. (2) A is strong deterministic if it has a unique initial state and if for every pair of D q), (p; ,; distinct transitions (p; +; D r) with the same origin, at least one of the following conditions hold: (i) +D and ,D are pre'x incompatible. D is a strict pre'x of ,(i) (ii) q is an i-end and +(i) D for some i ∈ {1; : : : ; k}. D for some i ∈ {1; : : : ; k}. (iii) r is an i-end and ,(i) D is a strict pre'x of +(i)

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(3) A is normal strong deterministic if it is strong deterministic and normal (cf. De'nition 1.11). Remark 3.9. A super-deterministic automaton (cf. De'nition 3.1) is obviously strong deterministic. Notation 3.10. (1) If n∈N ∪ {∞}, u is a word, uD = (u1 ; : : : ; uk ) is a multiword, we let max(n; u) = max(n; |u|);

max(n; u) D = (max(n; |u1 |); : : : ; max(n; |uk |)):

(2) If pD = (p1 ; : : : ; pk )∈(N ∪ {∞}) k and R a relation on words, we let n-Pre'x(u) = the pre'x of u with length min(n; |u|); n-Pre'x(u) D = (n-Pre'x(u1 ); : : : ; n-Pre'x(uk )); n-Pre'x(R) = {n-Pre'x(u): D uD ∈ R}; p-Pre'x( D u) D = (p1 -Pre'x(u1 ); : : : ; pk -Pre'x(uk )); p-Pre'x(R) D = {p-Pre'x( D u): D uD ∈ R}: Denition 3.11 (n-determinism). (1) A is n-deterministic if the following two conditions are satis'ed. (i) If p; q are distinct initial states then n-Pre'x(Rel(Ap )) ∩ n-Pre'x(Rel(Aq )) = ∅: D q), (p; ,; (ii) If (p; +; D r) are distinct transitions with the same origin then D ,)-Pre'x( D D D D +Rel(A max(n; +; ,Rel(A D q )) ∩ max(n; +; ,)-Pre'x( r )) = ∅: In case all labels of transitions of A have length6n (in particular if n¿1 and A is normal or quasi-normal, cf. De'nition 1.11) then condition (ii) can be expressed in a simpler form: D (iibis) n-Pre'x(+Rel(A D q )) ∩ n-Pre'x(,Rel(A r )) = ∅. (2) A is normal n-deterministic if it is n-deterministic and normal. Example 3.12. Let us review the deterministic character of the automata introduced in Example 3.6. Automaton D1 is strong deterministic: (; a; a)1 = ¡pre4x a = (a; ; a)1 and state 1 is an 1-end. Similarly, automaton D3 is normal strong deterministic: (; ; a)1 = ¡pre4x a = (a; ; )1 and state 2a is an 1-end. Automata D2 ; D4 ; D5 are not strong deterministic: • there are two D2 -transitions from state 0 with the same label (; ; a), • there are two D4 -transitions from state 0 with the same label (a; ; ), • D5 has two initial states 0 and 2.

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However, D2 ; D5 are 1-deterministic and D5 is 4-deterministic. (since these automata are quasi-normal, condition (ibis) is to be checked): (; ; a)Rel((D2 )0a )) = {(a1+m ; a1+n ; a2+m+n ): m; n ∈ N}; (; ; a)Rel((D2 )2a )) = {(; a1+n ; a1+n ): n ∈N} and the associated 1-Pre'x relations are disjoint. (a; ; a)Rel((D4 )0 )) = {(a2+m ; an ; a2+m+n ): m; n ∈ N}; (a; ; a)Rel((D4 )1 )) = {(a; an ; a1+n ): n ∈ N} and the associated 2-Pre'x relations are disjoint. Rel((D5 )0 )) = {(am ; a1+n ; a1+m+n ): m; n ∈ N}; Rel((D5 )2 )) = {(a; ; a)} and the associated 1-Pre'x relations are disjoint. Also, (a; ; a)Rel((D5 )0 )) = {(a1+m ; a1+n ; a2+m+n ): m; n ∈N}; (; a; a)Rel((D5 )1 )) = {(; a1+n ; a1+n ): n ∈ N} and the associated 1-Pre'x relations are disjoint. It is easy to check that D4 is not 1-deterministic. As announced in Remark 3.7, we have the following property (which is obvious from the de'nitions). Proposition 3.13. If an automaton is n-deterministic (resp. strong deterministic) then so is the automaton obtained by suppressing any collection of nodes or edges. The following result is easy. Proposition 3.14. (1) For all n∈N A is strong deterministic ⇒ A is 0-deterministic A is n-deterministic ⇒ A is (n + 1)-deterministic (2) The above implications cannot be reversed. Proof. Point 1 is straightforward. As for point 2, the 'rst implication cannot be reversed since a 0-deterministic automaton with two distinct initial states cannot be strong deterministic. We now deal with the second implication. For n = 0, observe that the following automaton is 1-deterministic but not 0-deterministic: the relations computed from the two initial states are Rel(A0 ) = {(a; )}; Rel(A2 ) = {(; b)}

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so that their 1-Pre'xes are distinct but their 0-Pre'xes are equal.

For n¿1, consider the following automaton A:

We have (a; )Rel(A1 ) = {(an c; b)}; (; b)Rel(A3 ) = {(an d; b)} so that if c = d then A is (n + 1)-deterministic but not n-deterministic. 3.4. Ambiguity and ∞-determinism Though straightforward, the following result is worth noticing. Proposition 3.15. (1) For all n∈N, every n-deterministic automaton is ∞-deterministic. (2) An automaton A is ∞-deterministic if and only if it is unambiguous. Remark 3.16. However, there are unambiguous automata which are not n-deterministic for any n. For instance, the second automaton given in point 1 of Example 3.6. 3.5. Deterministic automata compute what is expected Theorem 3.17. Let R be a relation. The following conditions are equivalent: (i) For some n ∈N the relation R is the behaviour of some n-deterministic automaton. (ii) For all n ∈N the relation R is the behaviour of some normal n-deterministic automaton. (iii) R is the behaviour of some strong deterministic automaton. (iv) R is the behaviour of some normal strong deterministic automaton. (v) R is computed by some deterministic ROTM. (vi) R is computed by some modi4ed deterministic ROTM. Before coming to the proof, we observe the following fact. Remark 3.18. Normalization and quasi-normalization of automata (cf. De'nition 1.13) do not preserve neither strong determinism nor n-determinism. A counterexample is

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obtained by considering the following strongly deterministic automaton A which computes {(a; b); (a; bc)}.

The associated normalized and quasi-normalized automata Anl and Aqnl are as follows (where A; B stand for states (0; (a; )); (0; (; b))):

Anl is not strongly deterministic: the pair of transitions starting at state 0 violates the condition for strong determinism since A is not a 2-end and B is not a 1-end. Anl is not even 1-deterministic since 1-Pre'x((a; )Rel(Anormal )) = 1-Pre'x((; b)Rel(Anormal )) = {(a; b)}: A B However, Anl is 2-deterministic. All the same properties hold with Aqnl . Proof of Theorem 3.17. We shall prove implications (i) ⇒ (vi), (vi) ⇒ (iii), (vi) ⇒ (iv). All other implications follow from these and Propositions 3.14, 1.19. (i) ⇒ (vi) Given an n-deterministic automaton A, we describe a deterministic ROTM T which has the same behaviour. Let m be the maximum width of labels of transitions of A. Then T acts as follows: (
) Before emulating any A-transition, T reads its tapes so as to memorize up to max(n; m) letters of each one of the k inputs (an information it retains in its state). () When T has completed this memorization, it emulates a transition of A (which is necessarily unique, since A is n-deterministic), changes state accordingly and forgets the portion of the memorized input corresponding to the label of the simulated transition. (vi) ⇒ (iii) De'nition 1.22 associates to a modi'ed ROTM T a quasi-normal automaton A which computes the same relation as T does (cf. Proposition 1.21). We show that if T is a deterministic modi'ed ROTM then A is strong deterministic. Since T has a unique initial state so does A. Thus, the 'rst condition for strong determinism is satis'ed. (A) A is functional D and label D are equal. i.e. two edges with the same origin (q; +)

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D ; In fact, suppose ((q; +); D (r; ,)) D is an A-transition. Observe that there is a unique D ; T -transition t = (q; a; D r; m) D such that ((q; +); D (r; ,)) D ∈Trans(t). The reason is that aD and mD are determined as follows: ai = IF +i = new THEN +i ELSE (IF i =  THEN B ELSE i ); mi = IF ,i = new THEN 1 ELSE 0:

(9) (10)

Since T is deterministic, from q and aD we get r and m. D Combined with Eq. (9) and D D we get r and ,, the de'nition of ,D from m, D this proves that from q; +; D i.e. A is functional. D such that Bi = B is clearly an i-end (B) Any A-state (s; B) D ; D ; D (s; B)) D out of some Consider now two distinct edges ((q; +); D (r; ,)) D and ((q; +); D A-state (q; +). D Suppose these distinct labels are Due to (A), they must have distinct labels: D = . pre'x compatible and let i be such that |i |¿|i |. Since A is quasi-normal, this implies that i =  and that i ∈i is a letter. Recall i = IF (+i = new and ai = B) THEN ai ELSE  so that, from i ∈i we get +i = new. D ; D (s; B)) D ∈Trans(t). From +i = new and i =  D s; m) Let t = (q; b; D be such that ((q; +); we get bi = B. Therefore T makes no move on tape i and mi = 0, whence Bi = bi = B. D is an i-end. Which proves the condition for strong deterUsing (B) we see that (s; B) minism of A. (vi) ⇒ (iv) The above automaton A is quasi-normal. To get a normal automaton, we argue as follows: • Normalize the ROTM so that it moves exactly one head per transition. Observe that the obvious way to do that does preserve determinism. • Use Remark 1.23 to conclude that A is then normal. Remark 3.19. The direct way of normalizing a multitape automaton does not preserve neither n-determinism nor strong determinism (cf. Remark 3.18). We can use the above construction to get a (rather tortuous) method to normalize a deterministic automaton: (1) Go from A to a deterministic modi'ed ROTM TM . (2) Transform TM to Atomic(TM ) so as there is exactly one move per transition. This does preserve ROTM determinism. (3) From Atomic(TM ) get A normal strong deterministic using the construction given in the proof of Theorem 3.17. 3.6. Decidability of determinism for automata Proposition 3.20. (1) The class of strong deterministic automata is polynomial time decidable (cf. Remark 3:4). (2) Let expl-DET = {(n; A): A is n-deterministic} (where “expl” stands for “explicit”). Caution: n is to be considered as an object of length n, i.e. it is written in unary.

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(i) If k¿2 and there are at least two non-unary alphabets then the class expl-DET is co-NP-complete. (ii) If there is at most one non-unary alphabet then the class expl-DET is polynomial time decidable. Proof. (1) It is easy to devise a polynomial time algorithms to decide if a state is an i-end of A. (2i) Deciding if the relations computed by two automata have a common n-Pre'x is clearly in NP. Hence the disjointness conditions of De'nition 3.11(i,ii) for n-determinism lead to an obvious co-NP algorithm. Conversely, a straightforward adaptation of the proof of Theorem 2.4 leads to a polynomial time reduction of the bounded Post Correspondence Problem (which is PCP in which we want a solution u with length6n) to the non-disjointness problem of the n-Pre'xes of relations computed by 'nite super-deterministic automata. This last problem reduces easily to the complement of the class expl-DET . We conclude using the well-known NP-completeness of the bounded-PCP (cf. [8, p. 228]). (2ii) An automaton A[n] computing n-Pre'x(Rel(A)) is as follows: • QA[n] = QA ×{0; 1; : : : ; n}k , • A[n] counts the number of letters read on each of the k components, • A[n] emulates A while all counts are 6n. Such an automaton can be constructed in polynomial time. Construct (as above) automata computing • relations n-Pre'x(Rel(Ap )) where p is an initial state of A, D ,)-Pre'x( D D • relations max(n; +; D +Rel(A p )) where +; ,D are labels of transitions starting at p. Using these automata (which can be constructed in polynomial time) and the fact that at most one alphabet is non-unary, we can apply Corollary 1.31 to check in polynomial time the disjointness of relations occuring in condition (ii) of De'nition 3.11 for n-determinism. However, for existentially quanti'ed n the problem is in general undecidable. Proposition 3.21. Let DET1 ; :::; k = {A: ∃ n (A is n-deterministic)}. (1) Suppose k¿2 and there are at least two non-binary alphabets. Then DET1 ; :::; k is 10 -complete hence undecidable. (2) If there is at most one non-unary alphabet then DET1 ; :::; k is decidable. Remark 3.22. We do not know the exact complexity of the class DET1 ; :::; k in case there is at most one non-unary alphabet. It is bounded by that of Presburger arithmetic. Proof of Proposition 3.21. (1) Let M be a deterministic Turing machine with input alphabet , set of states Q and initial state q0 . Sequences of instantaneous descriptions (i.d.) of M can be coded as words in alphabet - =  ∪ Q ∪ {$}, where $ serves as a Pag separating successive i.d.’s. If I is an i.d. with non-'nal state, we denote I + the i.d. obtained from I with one M -transition.

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For each u ∈∗ we de'ne 2-tape automata Au ; Bu as follows: (i) Au computes the relation {(q0 u$I1 $I2 $ : : : $It ; I1 $I2 $ : : : $It ): the Ii  s are i:d: s}

(The q’s vary over QA , the a’s vary over ). (ii) Bu computes the relation {(I0 $I1 $ : : : $It ; I0+ $I1+ $ : : : $It+ ): the Ii  s are non-'nal i:d: s}

Labels in T are (bqa; rbc); (qa; rc); (qa; cr) according to the emulated M -transition (q; a) = (r; c; −1) or (r; c; 0) or (r; c; 1); (where −1; 0; 1 mean “left move”, “no move”, “right move” and c is what M writes in place of a). It is clear that Au is super-deterministic. Also, Bu is 0-deterministic: • transitions starting at state 1 have incompatible labels, • transitions in T starting at state 0 have incompatible labels, • for transitions (0; (a; a); 0) and (0; +; 1) ∈T , the value max(0; (a; a); +) is 2 or 3 and max(0; (a; a); +)-Pre'x((a; a)Rel(B0u ) ∩ max(0; (a; a); +)-Pre'x(+Rel(B1u ) = ∅ Since Rel(Au ) ∪ Rel(Bu ) = ∅ the union automaton Au ∪ Bu is always unambiguous. We now look under which condition Au ∪ Bu is n-deterministic for some n. First, observe that condition (ii) in De'nition 3.11 is automatically satis'ed with n = 0. Condition (i) in De'nition 3.11 is satis'ed for some n if and only there is a bound to the depths of common pre'xes to a pair in Rel(Au ) and a pair in Rel(Bu ).

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Now, such common pre'xes are exactly the pre'xes of the pairs (q0 u$I1 $I2 $ : : : $It ; I1 $I2 $ : : : $It ); where the Ii ’s are the successive instantaneous con'gurations of the computation of M on input u. In particular, there is a bound to their depths if and only if M halts on input u. Thus, automaton Au ∪ Bu is n-deterministic for some n if and only if M halts on input u (and the smallest such n is then the sum of the lengths of the successive i.d.’s of the 'nite M -computation on input u). Considering a universal Turing machine M , we get a recursive reduction of the halting problem for M to the problem of determinism for automata. Hence the wanted undecidability and also the 10 -completeness. (2) In case there is one non-unary alphabet  and k unary alphabets, we reduce to Presburger arithmetic as in the proof of Corollary 1.27. Let us add an n-component to the sets considered in the conditions of De'nition 3.11(i,ii). For initial states q; r this leads to de'ne I q;r = {(w; s;D n): w ∈ ∗ ∧ sD ∈ (N)k ∧ n ∈ N ∧ (w; s) D ∈ n-Pre'x(Rel(Aq )) ∧ (w; s) D ∈ n-Pre'x(Rel(Aq ))}: D q), (p; (v; ,); For transitions (p; (u; +); D r) the sole interesting case is when p; q are pre'x comparable (otherwise condition (ii) is trivial). Thus, we restrict to the case u is a pre'x of v, i.e. v is of the form v = uu . This leads to de'ne q;r Su;u D n): w ∈ ∗ ∧ sD ∈ (N)k ∧ n ∈ N  ;+; D ,D = {(w; s;

D ∧ (w; s) D ∈ n-Pre'x((u; +)Rel(A q )) ∧ (w; s) D ∈ n-Pre'x((uu ; ,)Rel(A D r ))}: However, these relations I q; r ; Su;q;ur ; +;D ,D are not rational. So we introduce variants J q; r ; Tu;q;ur ; +;D ,D which are rational:

J q;r = {(w; s;D tD; m; n): w ∈ ∗ ∧ s;D tD ∈ (N)k ∧ m; n ∈ N D DD ∈ Nk ∧ ∃;  ∈ ∗ ∃.; D ∈ Rel(Aq )) ∧ (; .) ∧ (; D) D ∈ Rel(Ar ) D ∧ (w; s) D = m-Pre'x(; .) ∧ (w; tD) = n-Pre'x(; D)}: D

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q;r Tu;u D tD; m; n): w ∈ ∗ ∧ s;D tD ∈ (N)k ∧ m; n ∈ N  ;+; D ,D = {(w; s;

D DD ∈ Nk ∧ ∃;  ∈ ∗ ∃.; D ∈ Rel(Aq )) ∧ (; .) ∧ (; D) D ∈ Rel(Ar ) D ∧ (w; s) D = m-Pre'x(u; +D.) ∧ (w; tD) = n-Pre'x(uu ; ,DD)}: D Clearly, I q;r = {(w; s;D n): (w; s;D s;D n; n) ∈ J q;r } q;r q;r Su;u D n): (w; s;D s;D n; n) ∈ Tu;u  ;+;  ;+; D ,D = {(w; s; D ,D}:

Let K q; r ; Uu;q;ur ; +;D ,D be the projections of J q; r ; Tu;q;ur ; +;D ,D parallel to the ∗ component. Now, A is deterministic if and only if there exists n such that for all initial states D q), (p; (uu ; ,); p; q and all transitions (p; (u; +); D r) q;r {(w; s): D (w; s;D n) ∈ I q;r } = ∅; {(w; s): D (w; s;D n) ∈ Su;u  ;+; D ,D} = ∅;

i.e. q;r {(w; s): D (w; s;D s;D n; n) ∈ J q;r } = ∅; {(w; s): D (w; s;D s;D n; n) ∈Tu;u  ;+; D ,D} = ∅:

This amounts to say ∃n ∀sD (s;D s;D n; n) ∈= K q;r ;

q;r ∃n ∀sD (s;D s;D n; n) ∈= Uu;u  ;+; D ,D:

(11)

Since relations K q; r ; U u;q; ur  ; +;D ,D are rational, they can be expressed in Presburger arithmetic (cf. Theorem 1.25). Hence assertions (11) can be expressed in Presburger arithmetic and we get the wanted decidability. 3.7. Deterministic multimorphisms The tight relation between k-tape automata and multimorphisms (cf. point 3 of the proof of Theorem 2.7) leads to a natural notion of n-determinism and strong determinism.
Denition 3.23. Let & : %∗ → i∗ be a multimorphism and L ⊆ %∗ be a rational language. (1) & is an i-end for a language X ⊆ %∗ if &(a)i =  for all letters a occuring in some word in X (where &(x)i denotes the ith component of &(x)). (2) & is n-deterministic on L if for all words u ∈%∗ , for all distinct letters a; b ∈% if ua; ub are pre'xes of words in L then n-Pre'x({&(ax): uax ∈ L}) ∩ n-Pre'x({&(by): uby ∈ L}) = ∅:

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(3) & is strong deterministic on L if for all words u ∈%∗ , for all distinct letters a; b ∈%, if ua; ub are pre'xes of words in L then at least one of the following conditions holds:
∗ (i) &(a); &(b) are pre'x-incompatible in i . (ii) &(a)i is a strict pre'x of &(b)i and & is an i-end for (ua)−1 L. (iii) &(b)i is a strict pre'x of &(a)i and & is an i-end for (ub)−1 L. We can now state the deterministic version of Nivat’s Multimorphism Theorem, the proof of which is straightforward from the de'nitions.
Theorem 3.24. A rational relation R ⊆ i∗ is n-deterministic (resp. strong deterministic, resp. normal strong deterministic) if and only if R = &(L) where L ⊆ %∗
is a rational language and & : %∗ → i∗ is a proper multimorphism which is ndeterministic on L (resp. strong deterministic on L, resp. strong deterministic on L and alphabetical multimorphism). As for the e&ectiveness of the notion, results are analog to those of Proposition 3.21, 3.20 (with similar proofs). Proposition 3.25. (1) Consider the class of triples
(n; A; &) such that A is a onetape automaton on alphabet % and & : %∗ → i∗ is a multimorphism which is n-deterministic on L(A). (Caution: n is to be considered as an object of length n, i.e. it is written in unary). (i) If k¿2 and there are at least two non-unary alphabets then this class is co-NPcomplete. (ii) If there is at most one non-unary alphabet then this class is polynomial time decidable. (2) Consider the class
of pairs (A; & such that A is a one-tape automaton on alphabet % and & : %∗ → i∗ is a multimorphism which is n-deterministic on L(A) for some n. (i) If k¿2 and there are at least two non-unary alphabets then this class is 10 complete hence not recursive. (ii) If there is at most one non-unary alphabet then this class is recursive Remark 3.26. As for the notion of deterministic automaton (cf. Remark 3.22), we do not know the exact complexity in point (2ii). It is bounded by that of Presburger arithmetic.

Acknowledgements Many thanks to Christian Cho&rut and to an anonymous referee for comments and corrections.

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References [1] J. Berstel, Transactions and Context-Free Languages, B. G. Teubner, Stuttgart, 1979. [2] M.R. Bird, The equivalence problem for deterministic two-tape automata, J. Comput. System Sci. 7 (1973) 218–236. [3] I. Borosh, L. Treybig, Bounds on positive integral solutions of linear diophantine equations, Proc. Amer. Math. Soc. 55 (2) (1976) 299–304. [4] S. Eilenberg, Automata, Languages and Machines, vol. A, Academic Press, New York, 1974. [5] S. Eilenberg, M.P. SchKutzenberger, Rational sets in commutative monoids, J. Algebra 13 (1969) 173–191. [6] C.C. Elgot, J.E. Mezei, On relations de'ned by 'nite automata, IBM J. Res. Develop. 9 (1965) 47–68. [7] P.C. Fischer, A. Rosenberg, Multitape one-way nonwriting automata, J. Comput. System Sci. 2 (1968) 88–101. [8] M. Garey, D. Johnson, Computers and intractability. A guide to the theory of NP-completeness, Freeman, New York, 1979. [9] S. Ginsburg, E.H. Spanier, Semigroups, presburger formulas, and languages, Paci'c J. Math. 16 (1966) 285–296. [10] J. Goldstine, A simpli'ed proof of parikh’s theorem, Discrete Math. 19 (1977) 235–240. [11] E. Gurari, O. Ibarra, The complexity of decision problems for 'nite-turn multicounter machines, J. Comput. System Sci. 22 (1981) 220–229. [12] T. Harju, J. KarhumKaki, The equivalence problem of multitape 'nite automata, Theoret. Comput. Sci. 78 (1991) 347–355. [13] O. Ibarra, Reversal-bounded multicounter machines and their decision problems, J. Assoc. Comput. Mach. 25 (1) (1978) 116–133. [14] E.B. Kinber, The inclusion problem for some classes of deterministic automata, Theoret. Comput. Sci. 26 (1983) 1–24. [15] S.C. Kleene, Representations of events in nerve nets and 'nite automata, in: C. Shannan, J. McCarthy (Eds.), Automata Studies, Princeton Univ. Press, Princeton, NJ, 1956, pp. 3– 41. [16] H.W. Lenstra, Integer programming with a 'xed number of variables, Math. Oper. Res. 8 (4) (1983) 538–548. [17] Y. Matiyasevich, G. Senizergues, Decision problems for semi-Thue systems with a few rules, in: Proc. 11th Symp. on Logic in Computer Science, IEEE Computer Society Press, 1996, pp. 523–531. [18] J. Myhill, Finite automata and the representation of events, WADD Technical Report, Wright-Paterson Air Force Base, 57-624, November, 1957. [19] M. Nivat, Transactions des langages de Chomsky, Ann. Inst. Fourier 108 (1968) 339–456. [20] P. Odifreddi, Classical Recursion Theory, Studies in Logic and the Foundations of Mathematics, vol. 25, North-Holland, Amsterdam, 1989. [21] R.J. Parikh, Language-generating devices, Quart. Prog. Rep. 60 (1961) 199–212. [22] R.J. Parikh, On context-free languages, J. Assoc. Comput. Mach. 13 (1966) 570–581. [23] M. Pelletier, J. Sakarovitch, On the representation of 'nite deterministic 2-tape automata, Theoret. Comput. Sci. 225 (1999) 1–63. [24] E. Post, A variant of a recursively unsolvable problem, Bull. Amer. Math. Soc. 52 (1946) 264–268. [25] M. Presburger, Ueber die VollstKandigkeit eines gewissen Systems ganzer Zahlen, in welchen die Addition als einzige Operation hervortritt, in: 1er CongrWes des Math. des Pays Slaves, Varsovie, 1930, pp. 92– 101. (English translation: on the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation, Hist. Philos. Logic 12 (1991) 225 –233. [26] M.O. Rabin, D. Scott, Finite automata and their decision problems. IBM J. Res. Develop. 3 (1959) 63–90. (Reprinted in Sequential Machines, Moore (Ed.), Addison-Wesley, Reading, MA, 1964. [27] H. Rogers, Theory of Recursive Functions and E&ective Computability, McGraw-Hill, New York, 1967. [28] J. Sakarovitch, ElXements de thXeorie des automates, 2001, to appear. [29] A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1987. [30] M.P. SchKutzenberger, On the de'nition of a family of automata, Inform. and Control 4 (1961) 245–270. [31] J. Von zur Gathen, M. Sieveking, A bound on solutions of linear integer equalities and inequalities, Proc. Amer. Math. Soc. 72 (1) (1978) 155–158.

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www.elsevier.com/locate/tcs

On the algorithmic inversion of the discrete Radon transform  Peter Gritzmann, Sven de Vries Zentrum Mathematik, TU Munchen, D-80333 Munchen, Germany

Abstract The present paper deals with the computational complexity of the discrete inverse problem of reconstructing .nite point sets and more general functionals with .nite support that are accessible only through some of the values of their discrete Radon transform. It turns out that this task behaves quite di3erently from its well-studied companion problem involving 1-dimensional X-rays. Concentrating on the case of coordinate hyperplanes in Rd and on functionals : Zd → D with D ∈ {{0; 1; : : : ; r}; N0 } for some arbitrary but .xed r, we show in particular that the problem can be solved in polynomial time if information is available for m such hyperplanes when m 6 d − 1 but is NP-hard for m = d and D = {0; 1; : : : ; r}. However, for D = N0 , a case that is relevant in the context of contingency tables, the problem is still in P. Similar results are given for the task of determining the uniqueness of a given solution and for a related counting problem. c 2002 Elsevier Science B.V. All rights reserved.  MSC: 68R05; 68U10; 90C10 Keywords: Radon transform; Discrete inverse problem; Discrete tomography; Contingency table; Computational complexity; Polynomial-time algorithm; NP-hard

1. Introduction The problem of reconstructing .nite point sets from the values of their discrete X-ray transform in a few directions has attracted much attention; see the surveys [10], [11] and the monograph [12]. In particular, the computational complexity of the basic  The research of both authors was supported in part by the German Federal Ministry of Education, Science, Research and Technology Grant 03-GR7TM1 and through the German Academic Exchange Service Grant 314-vigoni-dr. E-mail addresses: [email protected] (P. Gritzmann), [email protected] (S. de Vries).

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter
PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 3 - 3

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underlying tasks has been determined completely, showing that the problem can be solved in polynomial time when m, the number of lines parallel to which the images are taken, is 2 but is NP-complete for any set of at least 3 lines in any dimension d at least 2, [8]. There is much less known about the ‘companion’ problem that involves the discrete Radon transform. Some uniqueness theorems are due to [5] and [19], some tractability results for contingency tables are given in [13] and [20], and some .rst intractability results for general k-dimensional X-rays are discussed in [6]. The starting point for the present paper is a recent result of [1] that solves a problem raised by the .rst named author in 1996 showing that checking the consistency of data that encode the values of the Radon transform for the 3 coordinate planes in R3 is NP-hard. The present paper extends this result to arbitrary dimensions, determines the computational complexity of related uniqueness and counting problems, and also treats the tasks for the more general objects of weighted sets or, equivalently, for functionals : Zd → D of .nite support. The most relevant domains D for the weights are {0; 1}, the standard case of .nite point sets, and, more generally, {0; 1; : : : ; r} for some .xed positive integer r, N0 , the case corresponding to contingency tables, and [0; 1], [0; r] and [0; ∞[, the associated linear programming relaxations. (Of course, in order to deal with these problems in the binary Turing machine model we will actually restrict all input and output data to rationals.) Our results show that the case of Radon transforms is much more involved than that of the standard X-ray problem in that • It is not only the parameters d and m that determine the computational complexity but also the intersection pattern of the hyperplanes. • Even in R4 there are hyperplanes S1 ; S 2 ; S 3 for which the consistency and uniqueness problems are NP-hard over {0; 1; : : : ; r} and N0 , but there are other hyperplanes S1 ; S 2 ; S 3 for which the problems can be solved in polynomial time over these domains. • The problems can be solved in polynomial time for m6d−1 coordinate hyperplanes over all relevant domains D, are NP-hard for m = d over {0; 1; : : : ; r} but polynomialtime solvable over N0 , [0; r] and [0; ∞[. Section 2 gives all relevant de.nitions, a brief overview of the known complexity results, and statements of our main results. Section 3 contains proofs for the new intractability results while some tractability results are proved in Section 4. 2. Denitions, preliminaries and main results Typically, in discrete tomography the objects that have to be reconstructed are .nite lattice sets F ⊆ Zd , see [12]. This is appropriate for most of the applications, particularly for those involving crystalline structures in semiconductor physics that have to be reconstructed from few of their images under high resolution transmission electron microscopy; see [11]. Of course, crystal lattices are more general than Z3 but due to the aIne invariance of the basic tasks the concentration on the standard lattice is no real restriction of generality. Another line of research focuses on the reconstruction of contingency tables from some of their marginal sums, see [13] and [20]. Here the

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entries of the tables are nonnegative integers. Also linear programming relaxations to ‘fuzzy sets’ with weights in [0; 1] have been studied in the literature; see [4]. Here we utilize a uni.ed approach by considering functions : Zd → D, with .nite support supp( ) = {x ∈ Zd : (x) = 0}, where D is some speci.c set of nonnegative numbers. The sets D that will be considered are {0; 1}, or more generally R = {0; 1; 2; : : : ; r} for some .xed r ∈ N, N0 and the corresponding relaxations [0; 1], [0; 1] ∩ Q, [0; r], [0; r] ∩ Q, R+ and Q+ , where Q+ and R+ denote the nonnegative rationals and reals, respectively. (We will sometimes express results explicitly for D = {0; 1} and D = R even though the latter case contains the former since it is only required that r is an arbitrary but .xed positive integer allowing the choice r = 1. This is done to emphasize that the corresponding result holds for the ‘classical case’ of .nite subsets of Zd .) The family of these sets D will be denoted by D. For any such set D let FDd denote the class of all functions : Zd → D with .nite d support. In order to emphasize the interpretation of a function ∈ F{0; 1} as the .nite d d set supp( ) we will sometimes abbreviate F{0; 1} by F , the notation introduced in [8]. The elements of Fd are then called lattice sets. The values (x) of a function ∈ FDd will in the following be denoted by x . For k; d ∈ N with k6d − 1, let Sk; d be the set of all k-dimensional subspaces in d-dimensional Euclidean space R d . Let Lk; d denote the subset of Sk; d of those spaces spanned by vectors v1 ; : : : ; vk ∈ Zd \{0}. The elements of Lk; d will be called lattice spaces. For S ∈ Sk; d let A(S) denote the set of all k-dimensional aIne subspaces of R d that are parallel to S. Let ∈ FDd and S ∈ Sk; d . The (discrete) k-dimensional X-ray of parallel to S is the function XS : A(S) → [0; ∞[ de.ned by
XS (T ) = x x∈T

for T ∈ A(S). The mapping that associates with every S ∈ Sk; d the X-ray XS is called the discrete k-dimensional X-ray transform of . For k = 1 it is the standard discrete X-ray transform while for k = d − 1 it is called discrete Radon transform of . Given a functional ∈ FDd its ‘1 -norm
| x| 1 = x∈Zd

will be called total weight. The present paper deals with algorithmic issues related to discrete inverse problems associated with the Radon transform. The natural model of computation is the usual binary Turing machine model; see [9] and [14]. Of course, all explicitly encoded data must have .nite bit length. Hence we will employ rational points only. So, for most purposes we will restrict the domains D to the subset DQ of elements of D that contain  d−1 only rational points. Also, all spaces S will be restricted to k=1 Lk; d . Let S ∈ Ld−1; d . In order to de.ne our basic algorithmic problems we need to parameterize A(S) appropriately. This is most easily done with the aid of a vector s ∈ S ⊥

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that has minimum Euclidean distance from the origin among all such nonzero vectors t with (t + S) ∩ Zd = ∅. Then, of course, A(S) = {s + S:  ∈ Z}: Hence we can regard XS as function on Z, and accordingly represent data for the reconstruction task by means of functions fS : Z → Q+ with .nite support, referred to as data functions. The graph {(; f()) ∈ Z × Q+ : f() = 0} of fS is called an X-set. In the following we will switch freely between the di3erent representations. Adequate data structures can also be de.ned for general k-dimensional X-rays. This has been done explicitly in [8] for 1-dimensional X-rays and can be extended analogously to full generality. Since some more technical details are involved and since we want to concentrate here on the case of the Radon transform anyway we refrain from going into details. However, in order to set our results into perspective we introduce the basic algorithmic tasks for general k-dimensional X-rays. In the following let always d; k; m ∈ N with 26d; m and k6d − 1. Further, D ∈ DQ , unless stated otherwise. Also let S1 ; : : : ; Sm be m di3erent elements of Lk; d . CONSISTENCYFDd (S1 ; : : : ; Sm ). Instance: Data functions fS1 ; : : : ; fSm . Question: Does there exist a ∈ FDd such that XSi = fSi for i = 1; : : : ; m? From a practical point of view, it is more relevant to actually reconstruct a solution. R ECONSTRUCTIONFDd (S1 ; : : : ; Sm ) is de.ned similarly, the input being the same but the question is replaced by the task of constructing a solution if one exists. Clearly, the reconstruction problem cannot be easier than the consistency problem. Of course, if is a solution for a given instance I of CONSISTENCYFDd (S1 ; : : : ; Sm ) then its total weight 1 equals fSi 1 for any i = 1; : : : ; m. Hence fS1 1 = · · · = fSm 1 is a necessary condition for consistency, a condition that can be checked eIciently. In the following we may assume that this condition is satis.ed; n = n(I) denotes the corresponding cardinality. Another important algorithmic task involves checking the uniqueness of a solution. UNIQUENESSFDd (S1 ; : : : ; Sm ) Instance: A functional ∈ FDd . Question: Does there exist  ∈ FDd \{ } such that XSi = XSi for i = 1; : : : ; m?



Note that UNIQUENESSFDd (S1 ; : : : ; Sm ) checks, in e3ect, nonuniqueness; but this way of de.ning the problem puts it into the class NP for all our relevant domains. UNIQUENESSFDd (S1 ; : : : ; Sm ) is a special case of #(CONSISTENCYFDd (S1 ; : : : ; Sm )), the counting problem that asks for the number of solutions.

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Here is an overview of the results about the computational complexity of the above tasks that are most relevant for our purposes. As pointed out before the best studied case is that of 1-dimensional X-rays. We begin with (some slight extensions) of results of [8]. Theorem 2.1. Let S1 ; : : : ; Sm ∈ L1; d be m di:erent lattice lines. If m¿3 then the problems CONSISTENCYFDd (S1 ; : : : ; Sm ) and UNIQUENESSFDd (S1 ; : : : ; Sm ) are NP-complete while R ECONSTRUCTIONFDd (S1 ; : : : ; Sm ) is NP-hard in the strong sense for D = {0; 1}; D = R and D = N0 . All three problems are solvable in strongly polynomial time for D = Q ∩ [0; 1]; D = Q ∩ [0; r] and D = Q+ . Further, #(CONSISTENCYFDd (S1 ; : : : ; Sm )) is #Pcomplete for D = {0; 1}; D = R and D = N0 . Proof. The results were proved for D = {0; 1} already in [8, Theorem 3.7] for consistency, in [8, Theorem 4.3] for uniqueness and in [8, Corollary 3.8] for the counting problem. In fact, in [8] it was actually shown, that the statements remain true, even for instances where two data functions fi take values only in {0; 1}. This observation immediately implies the intractability results for D = R and D = N0 because such instances permit only solutions with values in {0; 1}. In the other cases consistency, reconstruction and uniqueness can be reduced to linear programming. In fact, the linear feasibility problem modeling CONSISTENCYFDd (S1 ; : : : ; Sm ) and R ECONSTRUCTIONFDd (S1 ; : : : ; Sm ) is of the form
i = 1; : : : ; m; T ∈ A(Si ); x = fSi (T ); x∈T ∩Zd

x

∈ D;

x ∈ Zd :

Note that x is zero for all but polynomially many x ∈ Zd , hence the coeIcient matrix is a 0-1-matrix of size bounded by a polynomial in the input. Given ∈ FDd specifying an instance of UNIQUENESSFDd (S1 ; : : : ; Sm ) one can decide for each element x0 of supp( ) whether there exists another solution  with x0 = x0 by solving the linear programs
min x∈T ∩Zd

 x0

x

= fSi (T );

i = 1; : : : ; m; T ∈ A(Si );

 x

∈ D;

x ∈ Zd

and
max  x∈T ∩Zd

 x0

x

= fSi (T );

i = 1; : : : ; m; T ∈ A(Si );

 x

∈ D;

x ∈ Zd :

The tractability assertion follows now with the aid of [18]. Of course, the problem #(CONSISTENCYFDd (S1 ; : : : ; Sm )) is essentially meaningless for D ∈ {Q ∩ [0; r]; Q+ } because a given instance permits either no, one or in.nitely many solutions, since the set of solutions over the closure cl(FD ) is convex.

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The case of higher dimensional X-rays has not been studied as comprehensively as that of 1-dimensional X-rays. Of course, some additional features occur that make the problems more diIcult to handle. For instance, not even the case m = 2 or the membership in NP extends without further assumptions. For instance, since the intersection of two 2-dimensional lattice subspaces in R3 is a 1-dimensional lattice subspace it contains an in.nite number of lattice points. So, the number of points that belong to a solution need not necessarily be bounded by a polynomial in the bit length of the data functions fSi . This is no problem for D = N0 but for bounded D the following general assumption is needed. Assumption 2.2. Let D be R or [0; r] ∩ Q. Then for any instance I the cardinality n(I) is bounded by a polynomial in the bit length of the data functions. Remark 2.3. The problem CONSISTENCYFDd (S1 ; : : : ; Sm ) belongs to NP. Proof. Checking whether a guessed solution is actually a solution for a given instance can easily be done in time that is polynomial in the input size of I and the size of . Hence it suIces to show that in case of consistency there  m exists a solution whose size is polynomial in the size of I. This is clear if S = i=1 Si = 0. So, suppose that S contains a line. If D = N and D = Q+ then in case of consistency there always exists a solution such that supp( ) ∩ T is empty or a singleton for every translate T of S. For D = R and D = [0; r] ∩ Q the existence of a polynomial witness follows from Assumption 2.2, in the latter case with the aid of the fact that in case of consistency the corresponding linear program admits a basic feasible solution whose size is bounded by a polynomial in the input length. It is not too diIcult to see that under the Assumption 2.2 the case m = 2 is always simple. Theorem 2.4. Let S1 ; S 2 ∈ Lk; d . Then all three problems CONSISTENCYFDd (S1 ; S 2 ), R ECONSTRUCTIONFDd (S1 ; S 2 ) and UNIQUENESSFDd (S1 ; S 2 ) can be solved in polynomial time for D ∈ DQ . Proof. Proofs for the planar case can be found in [3], [17], and [16], and the general result for r = 1 is contained in [7]. For r¿1 it follows by a simple network-Row argument similar to that of [17]. It is also clear that the hardness results of Theorem 2.1 for the case of 1-dimensional X-rays can be used to obtain some hardness results for larger k. The following theorem is taken from [6]. Theorem 2.5 (Gardner and Gritzmann [6, Theorem 4.5.2]). Let S1 ; : : :  ; Sm ∈ Lk; d be m m di:erent spaces, and let D = R or D = N0 . Further, let m¿3 and dim( i=1 Si ) = k −1 or let m¿4 and k = 2. Then CONSISTENCYFDd (S1 ; : : : ; Sm ) is NP-complete.

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The .rst more intrinsically 2-dimensional transformation is constructed in [1] to determine the computational complexity of CONSISTENCYF3 (S13 ; S 23 ; S 33 ) for the standard coordinate planes S13 ; S 23 ; S 33 in R 3 . Theorem 2.6 (Brunetti et al. [1, Theorem 3.3]). CONSISTENCYF 3 (S13 ; S 23 ; S 33 ) is NPcomplete. This theorem was the starting point for the present paper. We will largely generalize and extend it and give results that show that even though there is an obvious geometric duality between lines and hyperplanes the case of Radon transforms is much more involved than that of the standard X-ray problems in that various phenomena occur for k = d − 1 that do not occur for k = 1. To be able to state our results precisely we need some additional notation. In order not to overload the paper with too many technicalities we will state our main results for coordinate hyperplanes and one additional ‘diagonal’ hyperplane. It should of course be clear that they extend to much more generality.
d d Let e1d ; : : : ; edd be the standard unit vectors of R d , let ed+1 = (1=d) i=1 eid , and let 1 d . The coordinate hyperplanes are denoted by S1d ; : : : ; Sdd be the ‘all-ones’ vector ded+1 i.e., Sid = {x ∈ Rd : x eid = 0};

i = 1; : : : ; d:

Further let d d Sd+1 = {x ∈ Rd : x ed+1 = 0}:

The subfamily of Sd−1; d that is most relevant for the purpose of the present paper is then Hd = {Sid : i = 1; : : : ; d + 1}: Since our problems will from now on be restricted to subsets of Hd we will use the abbreviation CONSISTENCYFDd (d; m) for CONSISTENCYFDd (S1d ; : : : ; Smd ); the problems UNIQUENESSFDd (d; m) and #(CONSISTENCYFDd (d; m)) are de.ned accordingly. Note that for any given instance I = (f1 ; : : : ; fm ) of CONSISTENCYFDd (d; m) all solutions are contained in the (generally in.nite) lattice set G = G(I) = Zd ∩

m  i=1

(supp(fi )eid + Si );

called the grid associated with I. Of course, the grid G can be .nitely represented and computed in polynomial time from the sets f1 ; : : : ; fm by solving systems of linear equations. Now we are ready to state our main results. The .rst is a far reaching intractability result for the algorithmic inversion of the discrete Radon transformation that generalizes Theorem 2.6.

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P. Gritzmann, S. de Vries / Theoretical Computer Science 281 (2002) 455 – 469

Theorem 2.7. Let d = m¿3 and D = {0; 1} or, more generally, D = R. Then CONSISTENCYFDd (d; m) and UNIQUENESSFDd (d; m) are NP-complete in the strong sense while #(CONSISTENCYFDd (d; m)) is #P-complete. For the domains D ∈ {[0; r] ∩ Q; Q+ }, the problems CONSISTENCYFDd (d; m) and UNIQUENESSFDd (d; m) can be solved in strongly polynomial time. Let us note in passing that the part of Theorem 2.7 dealing with uniqueness shows in particular that, unless P = NP, the uniqueness criterion for d = m = 3 of [19] cannot be checked eIciently. On the other hand, Wiegelmann [20] has given the following tractability result that is based on an algebraic study of the underlying toric ideals; it generalizes a result of [13]. Theorem 2.8 (Wiegelmann [20, Section 4.2.2]). Let d = m¿2. Then CONSISTENCYFNd 0 (d; m) and UNIQUENESSFNd (d; m) can be solved in polynomial time. 0

In Section 4 we will give an elementary proof and simple and fast algorithm for a slight generalization of the former tractability result that shows that whenever d¿m¿2 there is a solution for a given instance I = (f1 ; : : : ; fm ) of CONSISTENCYFNd (d; m) if 0 and only if f1 1 = · · · = fm 1 . This simple condition is in striking contrast to the more involved characterization of [15] and [16] of the feasibility of instances of CONSISTENCYF{0;2 1} (2; 2). As another consequence of Theorems 2.7 and 2.8 we see that the complexities for problems over {0; 1} and N0 may di3er dramatically, another feature that does not occur in the case of 1-dimensional X-rays. On the positive side, we obtain a simple tractability result for D = R. Theorem 2.9. Let d ∈ N with d¿m¿2. Then there is a polynomial time algorithm for CONSISTENCYFdR (d; m) and R ECONSTRUCTIONFdR (d; m). In particular Theorem 2.9 shows that there is no .xed number m0 , independent of d, such that our basic problem CONSISTENCYFd (d; m) becomes intractable whenever m¿m0 . In conjunction with Theorem 2.5 this shows that unlike in the case of 1-dimensional X-rays, the computational complexity of problems involving the Radon transform does not depend on the parameters d and m alone but also on the intersection pattern of the hyperplanes that are involved. 3. Intractability results By a modi.cation of the approach of [1] for CONSISTENCYF3 (3; 3), we will show in this section that for D = R and d¿3 the problems CONSISTENCYFDd (d; d) and UNIQUENESSFDd (d; d) are NP-complete and #(CONSISTENCYFDd (d; d)) is #P-complete. In [1] a reduction is given from a particularly restricted consistency problem, that was shown to be NP-complete in [2]. The construction fails, however, to show the NP-

P. Gritzmann, S. de Vries / Theoretical Computer Science 281 (2002) 455 – 469

463

hardness of the uniqueness problem and the #P-hardness of the counting problem. Here we construct a transformation from the unrestricted 1-dimensional X-ray problem that allows us to invoke Theorem 2.1 and determine the complexity not only in the general case of CONSISTENCYFDd (d; d) but also for UNIQUENESSFDd (d; d) and #(CONSISTENCYFDd (d; d)) at the same time. Together with the results of the next section this will in particular fully characterize the computational complexity for planes orthogonal to the coordinate axes in arbitrary dimension d¿3. The classical counting problem #(CONSISTENCYF2 (2; 2)) for two 1-dimensional X-rays remains, however, open. Let d¿3. For i = 1; : : : ; d let Si = Sid−1 , and Si = Sid . Of course, S1 ; : : : ; Sd are hyperplanes of R d while S1 ; : : : ; Sd will be regarded as hyperplanes of R d−1 . Whenever useful we will identify R d−1 with the hyperplane of R d of those vectors whose last coordinate is 0. For our complexity results we will use transformations whose validity can be most easily seen with the aid of a geometric argument which relies on the notion of the barycenter c of . As usual, the barycenter of a function ∈ FRd+ that is not identically 0 is de.ned by c =




1 1

x∈supp( )

x x:

Next we see how information about the barycenter of X-rays parallel to a hyperplane S.

can be derived from its

Lemma 3.1. Let S be a hyperplane and let v be a normal to S. If
 y∈S ⊥ XS (y + S) v y 
: v c = T ∈A(S) XS (T ) Proof. Let U = supp( ). First, note that


1 = x = x = x∈U

Second, v c = v =



1 1

T ∈A(S) x∈T ∩U

1
1 x∈U


 xx




T ∈A(S) x∈T ∩U

= xv


T ∈A(S)

1




1

x∈U



x=

∈ FRd+ then

XS (T ):

xv



x


1 XS (y + S) v y; 1 y∈S ⊥

which concludes the proof. Lemma 3.1 implies that the barycenter c of a function ∈ FRd+ is uniquely determined by the X-rays parallel to S1 ; : : : ; Sd . In the sequel we only need the weaker statements that 1 c can be computed in polynomial time from XSi , i = 1; : : : ; d and also from XSd+1 alone.

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In the following let Z always be a .nite subset of Nd0 , let n ∈ N, let r ∈ N, and denote by F(Z; n; r) the set of all ∈ F[0;d r] with supp( ) ⊂ Z and of total weight n. We will now characterize those elements ∗ of F(Z; n; r) which minimize the functional  : F(Z; n; r) → R de.ned by ( ) = 1 c . For ∈ N set PZ; = {x ∈ Z: 1 x ¡ };

HZ; = {x ∈ Z: 1 x = };

and F (Z; n; r) = { ∈ F(Z; n; r): PZ; ⊂ supp( ) ⊂ PZ; ∪ HZ; ∧ |PZ; ≡ r}; where

|PZ; denotes the restriction of

to PZ; .

Lemma 3.2. Let ; n ∈ N be such that F (Z; n; r) = ∅. Then a function minimizes the functional  if and only if ∈ F (Z; n; r). Proof. Let

∗

∈ F(Z; n; r);

1 c = 1 c

∗

+

∈ F (Z; n; r), U = supp( ) and U ∗ = supp(  

∗   x1 x − x1 x ;

1 ∗



∈ F(Z; n; r)

1

∗

). Note that

x∈U ∗

x∈U

hence for the ‘if’-assertion it suIces to show that  

∗   1 x − 1 x ¿ 0: x x x∈U ∗

x∈U

With the abbreviation P for PZ; we have
x∈U

x1

 =

Since

x

x−


x∈U ∗




x1

x∈U ∩P

 =




x∈U ∩P

(

x



∗  x1 x

x−


x∈P





r · 1 x +



− r)1 x −


x∈P\U




x1

x∈U \P

 

r·1 x

 +



x−


x∈U \P




x∈U ∗ \P

x1



∗ x




x−

x∈U ∗ \P

− r 60, 1 x6 − 1 for x ∈ P and 1 x¿ for x ∈ U \P we have


x∈U

 x1 x

−


x∈U ∗

 ¿ ( − 1)

∗  x 1 x


x∈U ∩P

 x

 
− r|P| +

x∈U \P

x

−


x∈U ∗ \P

 ∗ x

:

 ∗ x

:

P. Gritzmann, S. de Vries / Theoretical Computer Science 281 (2002) 455 – 469

Now, r|P| +
x∈U




x1

x∈U ∗ \P 

x−


x∈U ∗

∗ x

=

465

∗

1 . Hence 
∗  x1 x ¿

x∈U

x

∗

−

 1

= 0;

concluding the proof of the ‘if ’-part of the assertion. As to the ‘only if ’-part observe that equality holds in the above computation only if


 r · |P| = and x x1 x = x; x∈U ∩P

which implies

x∈U \P

x∈U \P

∈ F (Z; n; r).

In the following we describe a lifting procedure that constructs from a feasible instance of CONSISTENCYFd−1 (S1 ; S2 ; : : : ; Sd ) an equivalent instance of CONSISTENCYF[0;d r] [0; r] (S1 ; S 2 ; : : : ; Sd ). So let I = (f1 ; : : : ; fd ) be an instance of CONSISTENCYFd−1 (S1 ; S2 ; : : : ; Sd ). The con[0; r] struction will be described in terms of the corresponding X-sets; let them be of the form Xi = {( j; %i;j ): j ∈ Ui }

for i = 1; : : : ; d;

where Ui = supp(fi  ) for i = 1; : : : ; d. As the algorithmic problems are invariant under translation, we may assume that 0 = min Ui for i = 1; : : : ; d − 1. This is to say that the , and 0 ∈ G(I ). Let & = max Ud , set grid G(I ) is contained in Nd−1 0  × Z: 1 x ∈ Ud \{&}}: P = {x = ('1 ; : : : ; 'd ) ∈ U1 × U2 × · · · × Ud−1

Now we de.ne an instance I = (f1 ; : : : ; fd ) of CONSISTENCYF[0;d r] (S1 ; S 2 ; : : : ; Sd ) by specifying its X-sets as follows: Xi = {(j; %i;j + r · |P ∩ (jeid + Sid )|): j ∈ Ui }

for i = 1; : : : ; d − 1;

Xd = {((& − j); %d;j + r · |P ∩ ((& − j)edd + Sdd )|): j ∈ Ud }: Observe that the grids of I and I are related by G(I) = {x + jedd : x ∈ G(I ) ∧ (& − j) ∈ Ud }: Further, with the notation introduced before Lemma 3.2, P = PG(I);&

and

 (Sd HG(I);& = G(I );

where (Sd denotes the orthogonal projection on Sd . Theorem 3.3. Let I be a feasible instance of CONSISTENCYFd−1 (S1 ; S2 ; : : : ; Sd ), and let [0; r] I be the corresponding instance of CONSISTENCYF[0;d r] (S1 ; S 2 ; : : : ; Sd ) constructed above.  Then the solutions in F[0;d−1 r] of I are in one-to-one correspondence with the solutions

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in F[0;d r] of I. The bijectivity persists even when the ranges of the functionals are restricted to [0; r] ∩ Q or R.  Proof. Let  ∈ F[0;d−1 r] be a solution for I . We construct a solution as follows. Let

∈ F[0;d r] for I

F = {x + (& − 1 x)edd : x ∈ supp(  )} and

x

=

 r    

for x ∈ PG(I);& ;  (Sd (x)

0

for x ∈ F; else:

This construction can easily be reversed for solutions ∈ F& (G(I); 1 ; r); simply by taking  x

=

for all x ∈ G(I ):

x+(&−1 x)edd

 It follows from the explicit construction above that the solutions  ∈ F[0;d−1 r] of I are in one-to-one correspondence with those solutions ∈ F[0;d r] for I that belong to F& (G(I); 1 ; r). Hence all that remains to be shown is that all solutions of I are of this kind. So, let ∈ F[0;d r] be a solution for I. By our assumption there exists a solution   ∗ ∈ F[0;d−1 ∈ F& (G(I); 1 ; r) be the corresponding solution for I, in r] for I . Let F[0;d r] . By Lemma 3.1,

1 c = 1 c ∗ ; thus Lemma 3.2 implies ∈ F& (G(I);

∗

1 ; r);

concluding the proof of the .rst assertion. The last two claims of the theorem follow by observing, that under the bijection the values of the functions are not changed. Therefore, integers correspond to integers and rationals to rationals. The next result is needed for an induction step in the proof of Theorem 2.7. Remark 3.4. Using the notation introduced before Theorem 3.3, let I be a feasible instance of CONSISTENCYFd−1 (S1 ; S2 ; : : : ; Sd ) and let I = (f1 ; : : : ; fd ) be the correspond[0; r] ing instance of CONSISTENCYF[0;d r] (S1 ; S 2 ; : : : ; Sd ). Further, we de.ne an additional data

P. Gritzmann, S. de Vries / Theoretical Computer Science 281 (2002) 455 – 469

467

d d function fd+1 corresponding to the hyperplane Sd+1 orthogonal to ed+1 by d d + Sd+1 )| if  = &; r · |P ∩ (ed+1 fd+1 () = n(I ) if  = &: d Then (f1 ; : : : ; fd ; fd+1 ) is an instance Iˆ of CONSISTENCYF[0;d r] (S1 ; S 2 ; : : : ; Sd ; Sd+1 ). The sets ˆ of solutions of I and I are identical. This fact persists even when the ranges of the functionals are restricted to [0; r] ∩ Q or R.

Now we can prove Theorem 2.7, our main intractability result. Note that for D ∈ {[0; r] ∩ Q; Q+ } consistency, reconstruction and uniqueness can be reduced to linear programming with matrices and objective function with values only 0 and 1. From [18] it follows then, that they are solvable in strongly polynomial time. So let D = R now. Membership of the problems in the relevant complexity classes is trivial. The hardness assertions for d = 3 follow from the corresponding results for (S12 ; S 22 ; S 32 ) of Theorem 2.1. In fact, since the feasibility of a given instance over [0; r] can be veri.ed in strongly polynomial time the intractability results of Theorems 2.1 persists if we restrict the set of instances to those that are feasible over [0; r]. Hence the assertion follows directly from Theorem 3.3. For d¿3 we proceed by induction. Suppose we know already that CONSISTENCYFd−1 R (d − 1; d − 1) and UNIQUENESSFd−1 (d − 1; d − 1) are NP-complete in the strong R sense, and that #(CONSISTENCYFd−1 (d − 1; d − 1)) is #P-complete. By Remark 3.4 R CONSISTENCYFd−1 (d − 1; d) and UNIQUENESSFd−1 (d − 1; d) are NP-complete in the strong R R sense, and the counting problem #(CONSISTENCYFd−1 (d − 1; d)) is #P-complete. The asR sertion follows then again with the aid of Theorem 3.3. 4. Tractability results We begin by proving a lemma characterizing consistency. Lemma 4.1. Let d¿m¿2 then there is a solution for a given instance I = (f1 ; : : : ; fm ) of CONSISTENCYFNd (d; m) if and only if f1 1 = · · · = fm 1 . If m6R, the assertion 0 persists even for D = {0; 1}. Proof. Let I = (f1 ; : : : ; fm ) be an instance of CONSISTENCYFNd (d; m). All we need to 0 show is that the condition f1 1 = · · · = fm 1 is suIcient. This can be done by induction over n = n(I). Of course, for n = 0 there is nothing to show. So suppose n¿1. For every index i = 1; : : : ; m we choose a point i ∈ supp(fi ) and set fi () − 1 for  = i ;  fi () = fi () for  = i :

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Then I = (f1 ; : : : ; fm ) is an instance of CONSISTENCYFNd (d; m) with n(I ) = n−1. Since 0 f1 1 = · · · = fm 1 , the instance I is feasible by the induction hypothesis; let  be m a solution. Further, T = i=1 (i eid + Sid ) = ∅, hence we can raise the value of  on one point in T to obtain a solution for I. This implies the .rst assertion. Turning to the second, let I be an instance of CONSISTENCYF{0;d 1} (S1 ; : : : ; Sm ) satisfying the condition. We regard I as an instance of CONSISTENCYFNd (S1 ; : : : ; Sm ), and apply the 0 .rst part of Lemma 4.1 to obtain a solution . Note that since now m6d − 1, S=

m  i=1

Si

is at least 1-dimensional. Let T be any translate of S that meets supp( ). If we replace |T by an arbitrary functional )T : T ∩ Zd → {0; 1} with support of cardinality
 by x = )T (x) for x ∈ T and T ∈ A(S) x∈T ∩Zd x and de.ne a new functional then we obtain a solution for I as an instance of CONSISTENCYF{0;d 1} (S1 ; : : : ; Sm ). Next, Algorithm 1 is a polynomial-time method proving Theorem 2.8 for the general case of d¿m¿2. Algorithm 1. Reconstruction

Initialize: For i = 1; : : : ; m, let supp (fi ) = {i( j) : j = 1; : : : ; ni } and *i( ji ) ← fi (i( j) ) (ni )  while (*(1) i ; : : : ; *i ) = 0 for all i = 1; : : : ; m do for i = 1 to m do ji ← arg min{*(i j) : j = 1; : : : ; ni ∧ *(i j) = 0} end for i0 ← arg min{*i( ji ) : i = 1; : : : ; m}  y0 ← (1( j1 ) ; 2( j2 ) ; : : : ; m( jm ) ; 0 d−m ) ( j0 ) y0 ← *i0 for i = 1 to m do *i( ji ) ← *i( ji ) − y0 end for end while if *(i j) = 0 for some index i then report ‘infeasible’ else output end if

Note, that each iteration step can be performed in polynomial time. Further, in each step at least one of the *(i j) turns 0, hence the algorithm runs in strongly polynomial time. This implies Theorem 2.8. Also, Theorem 2.9 is now a consequence of the second part of Lemma 4.1. In fact, due to Assumption 2.2, the construction in its proof can be speci.ed in such a way, so as to run in polynomial time.

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References [1] S. Brunetti, A. Del Lungo, Y. Gerard, On the computational complexity of reconstructing three-dimensional lattice sets from their two-dimensional X-rays, Linear Algebra Appl. 339 (2001) 59–73. [2] S. Brunetti, A. Del Lungo, P. Gritzmann, S. de Vries, On the reconstruction of permutation and partition sets under tomographic constraints, Manuscript (2001) 13. [3] S.K. Chang, The reconstruction of binary patterns from their projections, Commun. ACM 14 (1) (1971) 21–25. [4] P. Fishburn, P. Schwander, L. Shepp, R.J. Vanderbei, The discrete Radon transform and its approximate inversion via linear programming, Discrete Appl. Math. 75 (1) (1997) 39–61. [5] P.C. Fishburn, J.C. Lagarias, J.A. Reeds, L.A. Shepp, Sets uniquely determined by projections on axes II: Discrete case, Discrete Math. 91 (2) (1991) 149–159. [6] R. Gardner, P. Gritzmann, Uniqueness and complexity in discrete tomography, in: G. Herman, A. Kuba (Eds.), Discrete Tomography: Foundations, Algorithms, Applications, BirkhVauser, Boston, 1999, pp. 85–113. [7] R.J. Gardner, P. Gritzmann, D. Prangenberg, in: R.A. Melter, A.Y. Wu, L. Latecki (Eds.), On the Reconstruction of Binary Images from their Discrete Radon Transforms, Vision Geometry V, Society of Photo-Optical Instrumentation Engineers Proceedings, Vol. 2826, 1996, pp. 121–132. [8] R.J. Gardner, P. Gritzmann, D. Prangenberg, On the computational complexity of reconstructing lattice sets from their X-rays, Discrete Math. 202 (1999) 45–71. [9] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, 1979. [10] P. Gritzmann, On the reconstruction of .nite lattice sets from their X-rays, in: E. Ahronovitz, C. Fiorio (Eds.), Proc. 7th Internat. Workshop, DGCI ’97, Montpellier, France, Lecture Notes in Computer Science, Vol. 1347, Springer, 1997, pp. 19 –32. [11] P. Gritzmann, S. de Vries, Reconstructing crystalline structures from few images under high resolution transmission electron microscopy, in: W. JVager (Ed.), Mathematics: Key Technology for the Future, Springer, 2002, in print. [12] G.T. Herman, A. Kuba (Eds.), Discrete Tomography: Foundations, Algorithms, and Applications, BirkhVauser, Boston, 1999. [13] R.W. Irving, M.R. Jerrum, Three-dimensional statistical data security problems, SIAM J. Comput. 23 (1994) 170–184. [14] D.S. Johnson, A catalog of complexity classes, in: J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, Vol. A, Elsevier Science Publishers B.V., Amsterdam, 1990, pp. 69 –161. [15] H.J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957) 371–377. [16] H.J. Ryser, Combinatorial Mathematics, Mathematical Association of America and Quinn & Boden, Rahway, NJ, 1963. [17] C.H. Slump, J.J. Gerbrands, A network Row approach to reconstruction of the left ventricle from two projections, Comput. Graphics Image Process. 18 (1982) 18–36. X Tardos, A strongly polynomial algorithm to solve combinatorial linear programs, Operations Research [18] E. 34 (2) (1986) 250–256. [19] A. Torres-ChXazaro, E. Vallejo, Sets of uniqueness and minimal matrices, J. Algebra 208 (1998) 444–451. [20] M. Wiegelmann, GrVobner bases and primal algorithms in discrete tomography, Ph.D. thesis, Technische UniversitVat MVunchen, 1998.

Theoretical Computer Science 281 (2002) 471 – 498

www.elsevier.com/locate/tcs

A uniform semantic proof for cut-elimination and completeness of various $rst and higher order logics
Mitsuhiro Okadaa; b; c; ∗; 1 a Department

of Philosophy, Keio University, 2-15-H5 Mita, Minatoku, 108-8345, Tokyo Japan b Department of Philosophy, Universit# e Paris-I (Sorbonne), France c Coq-Group, INRIA-Rocquencourt, Computer Science Laboratories (LRI and LIX) Universit#e Paris-Sud and Ecole Polytechnique, France

Abstract We present a natural generalization of Girard’s ($rst order) phase semantics of linear logic (Theoret. Comput. Sci. 50 (1987)) to intuitionistic and higher-order phase semantics. Then we show that this semantic framework allows us to derive a uniform semantic proof of the ($rst order and) higher order cut-elimination theorem (as well as a ($rst order and) higher order phase-semantic completeness theorem) for various di/erent logical systems at the same time. Our semantic proof works for various di/erent logical systems uniformly in a strong sense (without any change of the argument of proof): it works for both $rst order and higher order versions and for linear, substructural, and standard logics uniformly, and for both their intuitionistic and c 2002 Published by Elsevier Science B.V. classical versions uniformly.


1. Introduction In a preceding paper [21] we gave a phase semantic framework for a $rst and higher order cut-elimination proof for classical linear logic, by extending Girard’s ($rst order)
This paper is the full and revised version of the $rst part of Okada [20] (the extended abstract “Phase Semantics for Higher Order Completeness, Cut-Elimination and Normalization Proofs” appeared in Electronic Notes of Theoretical Computer Science 3 (1996) 22). ∗ Corresponding author. Department of Philosophy, Keio University, 2-15-45 Mita, Minatoku, Tokyo 108-8345, Japan. E-mail address: [email protected] (M. Okada). 1 This work was partly done when the author was invited to Univ. Paris-I, INRIA, Univ. Paris-Sud and Ecole Polytechnique for the year 2000 –2001. The author would like to express his thanks for their support on this work.

c 2002 Published by Elsevier Science B.V. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 4 - 5

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phase semantics [12] to the higher order setting. Our phase semantic proof was presented as a stronger version of the completeness proof of Girard [12] (for the $rst order case), that not only implies completeness, but also implies cut-eliminability (namely, the normal form theorem in the sequent calculus formulation). The purpose of this paper is to present a general framework for higher order phase semantics, namely the intuitionistic version of higher order phase semantics. The intuitionistic phase semantics presented in this paper can be viewed as a very general framework for providing a $rst and higher order completeness and cut-elimination proof in a uniform way in the strong sense: our proposed ($rst order and higher order) semantic proof works not only for intuitionistic linear logic, but also for various, di/erent logical systems including linear versions, various substructural versions and traditional (standard) versions of both intuitionistic and classical logics (line by line without any change of the proof argument). The di/erence between the linear logic versions, various substructural logic versions, and the standard logic versions can be given simply by using di/erent monoid structures underlying the phase semantics domains; namely, di/erent choices for these underlying monoid structures induce di/erent variants of the phase spaces. Then we show that the ($rst and higher order) cut-elimination (at the same time phase-semantic completeness) proof itself can be described independently of the choice of the variant of the phase spaces, which provides a uniform argument for the ($rst and higher order) cut elimination (and completeness) proofs for various di/erent logical systems at the same time. For the substructural logics, we consider the most typical two cases as examples, namely the case with the weakening rules and without the contraction rules (this type of logic is often referred to as a=ne logic), and the case without the weakening rules and with the contraction rules (which is typical for relevance logics). However, one can also extend our argument to the cyclic logic versions and the non-commutative versions. Note that standard (traditional) logics (either classical or intuitionistic) are obtained as the special case where both the weakening rules and the contraction rules are included, while linear logics (either classical or intuitionistic) are obtained as the special case where both the weakening rules and the contraction rules are omitted. As the uniform phase semantic cut-elimination and completeness proof we take the intuitionistic phase semantics for linear logic as an explicit example $rst, since in this setting all above variants of logics can be obtained by imposing some additional speci$cations on this basic monoid structure of the phase space domains. We only outline how to modify the main setting (for the intuitionistic linear logic) for the above-mentioned variants of logical systems. We also should remark that the higherorder phase semantic completeness theorems have a certain limited meaning; our higher order completeness theorems (for di/erent variants of logics mentioned above) hold only with respect to the class of non-standard higher order phase semantics. (For this purpose we introduce higher order non-standard phase models.) This situation is very natural when compared with the well-known situation of second-order completeness of standard (classical) predicate logic; it is well-known that the second-order completeness theorem ($rst shown by Henkin) for the (traditional) classical predicate logic holds only with respect to the non-standard second-order Tarskian models.

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The fact that our phase semantic cut-elimination and completeness proof works for a very wide range of logical systems was $rst noticed in Okada [20,23]. For simplicity of the argument we mainly consider the framework of $rst order and higher order propositional logics, as is customary in the linear logic literature. But our method can be generalized naturally with $rst order quanti$ers to the case of full $rst order and higher order predicate logics. Semantic proofs of cut-elimination and normalization have been studied by many authors mainly in the traditional logical framework, such as for intuitionistic logic and classical logic (for example, cf. [8,10,5]). On the other hand, our approach shows that the phase semantic framework provides a uniform cut-elimination proof for a very wide range of logics, including the traditional logics. Our uniform phase semantic setting is closely related to the uniform setting of pretopological semantics of Sambin [32]. On the other hand, our interest is not only to provide a uniform framework for completeness, but also to show the same, single proof can be used both for completeness and for cut-elimination uniformly for a wide range of logics. Prior work on semantic cut-elimination proofs usually relies on a (bottom-up) proofsearch based model construction (or equivalently, a Beth Tableaux based model construction). This traditional model construction depends heavily on the underlying definition of inference rules of the logical system under consideration. Applicability of this method is also limited by the fact that the method repeatedly uses the structural (contraction and weakening) rules in an essential way to construct the model, and it is, therefore, not applicable to linear logic or to substructural logics. On the other hand, our method works for a wide range of logical systems uniformly. (See [25,38] for the speci$c techniques required for the proof search-based model construction of linear logic and substructural logics.) Moreover, our proposed uniform single proof works for a wide range of logical systems independently of the choice of logical syntax. Our phase semantic uniform proof also can be seen as a uniform semantic proof for proof-normalization. As mentioned in Section 1 in [21] it was shown how to interpret our phase semantic completeness=cut-elimination proof as various proof-normalization proofs for classical ($rst and higher order) linear logic, as one example. We present this in detail elsewhere for a more general case than is treated in this paper. This research direction seems to be closely related to the work of Altenkirch, Hofmann and Streicher on a reduction-free normalization proof [3,4], and to related work. The author would like to point out that although those works depend heavily on the choice of syntax of the logical system/typed-lambda calculus, our phase-semantic approach is so constructed that it can provide a strongly uniform framework for a wide range of syntax choices. In the appendix, we list the syntax of the two-sided sequent calculus for the various logical systems used in this paper. 2. First and higher order intuitionistic phase space In this section, we $rst give a general framework for phase semantics based on a commutative monoid, essentially due to Girard, in a generalized setting for the intuitionistic case. In particular, we present the second-order and higher order-phase semantics.

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We $rst give the intuitionistic phase space, using the notion of closure properties. Then we give another, additional requirement on this space in order to obtain a higher order phase space. In later sections, we use these phase spaces for a cut-elimination proof. First order intuitionistic phase space has been introduced by various authors in the literature (cf. [1,27,31,39]). Our $rst de$nition is essentially the same as those, except for the introduction of higher order space. We also introduce a more concrete, but essentially equivalent, version (the second de$nition) for intuitionistic phase space. Denition 2.1 (The $rst de$nition of intuitionistic phase space). An intuitionistic phase space is a tuple (M; cl; J ) where M is a commutative monoid, cl is a map from P(M ) 2 to P(M ), J ⊆ M is a submonoid satisfying the weak idempotent property: (∀a ∈ J ) cl({a}) ⊆ cl({a · a}) (J can be {a : aa = a}), and cl satis$es the following properties: For any  ⊆ M and ⊆ M , C1.  ⊆ cl(). C2. cl(cl()) = cl(). C3. If  ⊆ then cl() ⊆ cl( ). C4. cl() · ⊆ cl( · ). On a phase space, the interpretation of the linear operators & (with), ⊕ (and), ⊗ (tensor), ( (linear implication) and constants are de$ned as follows: L1. & =  ∩ 3 L2.  ⊕ = cl( ∪ ). L3.  ⊗ = cl( · ). L4.  ( = {d: d ·  ⊆ } = {d: for all a ∈  da ∈ }. 4 L5. 1 = cl({1}). 5 L6.  = P; 0 = cl( ). 6 L7. ! = cl( ∩ 1 ∩ J ). More concrete formulations of the intuitionistic phase spaces can be obtained by specifying the closure operator. The following is one such. Denition 2.2 (The second de$nition of intuitionistic phase space). An intuitionistic phase space is a tuple (M; DM ; J ) where M is a commutative monoid, J is a submonoid, and DM is a subset of P(M ), called the set of closed sets. We impose the following conditions on DM ;

P1. DM is closed under arbitrary ; namely if i ∈ DM for any i ∈ , then i∈ i ∈ DM , where  is an arbitrary index set. P2. For any ∈ DM , linear implication  ( = {b: for all a ∈  ab ∈ } ∈ DM . 2

P(M ) stands for the powerset of the monoid M . Note that cl() ∩ cl( ) = cl(cl() ∩ cl( )). Hence & is closed under cl. 4 In this paper, we use d · ;  · d,  · to denote {d · a : a ∈ }, {a · d : a ∈ }, {a · b : a ∈ ; b ∈ }, respectively. 5 This 1 represents the unit of the underlying monoid M . 6 This represents the empty set. 3

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Then we can de$ne the linear logical operators L1–L7 as above. (Note that  ( is already introduced in P2 above.) It is easy to see that the above two de$nitions of the intuitionistic phase space are in fact equivalent: To see that the second de$nition implies the $rst, one can de$ne cl() for any  ⊆ M in the second de$nition by cl() is the
smallest closed set (namely the smallest ∈ DM ) such that  ⊆ , namely cl() = { ∈ DM :  ⊆ }. Then it is easily shown that the closure conditions C1– C5 are satis$ed with this operator cl. In particular, P2 implies C4 as follows:  · ⊆ cl( · ), which implies  ⊆ ( cl( · ). Thus, cl() ⊆ ( cl( · ) (by P2), and so cl() · ⊆ cl( · ). Conversely, to see that the $rst de$nition implies the second, one can de$ne DM in the $rst de$nition as DM = {cl() :  ⊆ M }. Then P1 is obvious. P2 can be shown as follows: Assume that is closed (hence, ∈ DM ). Then  · ( ( ) ⊆ , hence by C4 (and C1– C3),  · cl( ( ) ⊆ cl( ) = . Hence, cl( ( ) ⊆  ( , and so  ( ∈ DM . It is easily seen that cl() is the smallest closed set that includes , for any  ⊆ M . A classical phase space (M; ⊥; J ) of Girard [12] is a special case of intuitionistic phase space (in the sense of the $rst de$nition) in which the closure condition is speci$ed as follows: cl() = ⊥⊥ where ⊥ is an arbitrarily $xed subset of M and ⊥ is  ( ⊥. 7 However, in this paper we introduce a slightly generalized version of classical phase space (M; cl; ⊥; J ): Denition 2.3 (Classical phase space). A classical phase space (M; cl; ⊥; J ) is an intuitionistic phase space (M; cl; J ) where • a special closed set ⊥ is speci$ed, and •  = ⊥⊥ for any closed set . We denote the classical phase space (induced from M; ⊥ and J ) by (M; cl; ⊥; J ). We also often omit ⊥ and write (M; cl; J ) for a classical phase space when ⊥ is implicitly assumed. It is easily seen that the original phase spaces of Girard [12] (which is the standard reference for the notion of a classical phase space) are the classical phase spaces in our sense. We use this slightly generalized de$nition of classical phase space in order to treat both the intuitionistic and the classical logical systems uniformly; in particular, it is used for presenting a single proof (of the main lemma) which works for both the intuitionistic versions and the classical versions of various logical systems uniformly. 2.1. Phase models On an intuitionistic phase space, as in [12], the interpretation A∗ of a formula A is de$ned to be a closed set. Assuming we are given an assignment of closed 7

A closed set of a classical phase space is called a “fact” in Girard [12].

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sets for the (propositional) variables occurring in A, we can extend the de$nition as follows: • R∗ is a closed set (i.e., R∗ ∈ DM ) for atomic R. • (A ( B)∗ = A∗ ( B∗ . • (A&B)∗ = A∗ &B∗ . • (A ⊕ B)∗ = A∗ ⊕ B∗ . • (A ⊗ B)∗ = A∗ ⊗ B∗ . • (!A)∗ = !(A∗ ). • 1∗ = cl({1}). • ∗ = M . • 0∗ = cl( ). We call A∗ the inner value of A. We call (M; DM ; cl; J ), which we also denote (M; cl; J ), a (standard) phase space for any commutative monoid M . Given a proper subset D ⊂ DM that is closed under all the above operators, we call (M; D; cl; J ) a non-standard phase space. A non-standard phase model is a (non-standard) phase space (M; D; cl; J ) with assignments R∗ ∈ D for each propositional variable R. One can easily extend these interpretations to the case of $rst-order quanti$ers (by interpreting these as additive operators), as usual. Although we do not treat $rst-order quanti$ers explicitly in this paper, we give a short sketch of this extension. 8 We consider (M; S; cl; J ) instead of (M; cl; J ) for a standard phase space, where S is a set representing the $rst-order individual domain. For a standard phase model (M; S; cl; J ), the interpretation of $rst-order quanti$ers is added in the same way as the additive propositional connectives, as follows. For any : S → D,
L1 : ∀x:(x) = ∈S (),  L2 : ∃ x:(x) = cl( ∈S ()). A $rst-order predicate, say, P(∗) is interpreted as P ∗ : S → D. The notion of valuation (assignment) of $rst order variables by the elements of S is introduced in the natural way. Then, the inner value A∗ [x := a] for a formula A and a valuation a from the elements of S for (a vector of) variables x can be easily de$ned by extending the de$nition of the inner value of the propositional case. This extension can be generalized to non-standard phase spaces in the same way. A formula A is said to be true if 1 ∈ A∗ (or equivalently, 1 ⊆ A∗ ), where 1 is the unit element of underlying monoid M . We can de$ne the classical phase models based on the classical phase spaces in the same way. Note that, as usual, the inner value of the negation A⊥ can be de$ned as (A ( ⊥)∗ . Now we extend this idea to higher order phase models. By higher order phase models we mean phase models extended by the interpretation of higher order quanti$ers. We introduce higher order operators as additive operators in the syntax, in the obvious manner. (See the appendix for the list of syntactic rules.) We $rst extend the notion of (propositional) phase space to that of second-order phase space. Here, A[X ] means that X is the list of the free propositional variables 8

An explicit, detailed exposition of this will be presented elsewhere.

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occurring in A; A[B=X ] or A[B] denotes the formula obtained from A[X ] by substituting a vector of formulas B for X ; A∗ [=X ] or A∗ [] denotes the inner value of A assigning a vector of closed sets  for X . In this section we use Form for the set of (second order) formulas. The interpretation of second-order operators is added to a non-standard phase model (M; D; cl; J ) as follows. For any  : D → D,
L8. ∀X:(X ) =  (), ∈D L9. ∃X(X ) = cl( ∈D ()). Denition 2.4 (Second-order phase space). A (non-standard) phase space (M; D; cl; J ) is called a (non-standard) second-order phase space if D is closed under the operators ∀ and ∃ above; namely if ∀X:(X ) ∈ D and ∃X:(X ) ∈ D for  : D → D of the form (X ) ≡ A∗ [X; 1 =Y1 ; : : : ; n =Yn ] for all A ∈ Form and all i ∈ D. The above condition can be expressed by the following equivalent condition. (∗) For any (second order) formula A (possibly with free second-order variables X ≡ X1 ; : : : ; Xn ), if i ∈ D then A∗ [=X ] ∈ D, where  ≡ 1 ; : : : ; n . Next we give an alternative de$nition of a second-order phase space which is actually a special case of the above notion of non-standard second-order phase space. Denition 2.5 (Alternative de$nition of second-order phase space). An intuitionistic phase space is called a second-order phase space if the following additional conditions are satis$ed: P4. For any (second order) formula A (possibly with free second order variables), a set A of closed sets (i.e., A ⊆ DM ) are associated. We call  ∈ A a candidate for the value of A. P5. For any formula A[X ], where X is a vector of second order variables, for any formula B, and for any closed sets  ∈ B, A∗ [=X ] ∈ A[B=X ]. In particular, A can be a singleton: we can take A as a singleton for all secondorder formulas A for the phase-semantic second-order completeness proof of logics. However, we shall actually construct a slightly generalized form of the second-order phase model for our main lemma by taking A as a set which is not necessarily a singleton, in order to make our “uniform proof” work both for a version of the completeness theorem and for the cut-elimination theorem (of various versions of second order logics) in a later section. It is easy to see that a second-order phase space in  the latter sense is a second-order phase space in the former sense, by de$ning D = A∈Form A ⊆ DM . In particular, with these alternative (more speci$c) de$nitions of second-order quanti$ers, the above interpretations L8
and L9 could be expressed by,
for any  : D → D, L8 : ∀X:(X ) = {() :  ∈ B; B ∈ Form} = ∈B; B∈Form (),  L9 : ∃X:(X ) = cl( ∈B; B∈Form ()), where Form is the set of the second order formulas. A classical second-order phase spaces can be de$ned in the same way from the classical ($rst order) phase space. One can easily extend our de$nition and arguments

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for second order phase semantics to higher order logic (of $nite types). We now give a brief sketch of the de$nition. According to the usual $nite type structure of the higher-order syntax, 9 we consider the functional space based on DM . More precisely, for each type  we de$ne D as follows: D0 = DM , D × ··· × Dn

D1 × ··· × n → 0 = D01

.

We can generalize the notion of non-standard domain D ⊆ DM of the non-standard ($rst order) phase space to the functional space above. The interpretation of higher order operators (in general, de$ned on a non-standard functional space)
is added as follows. For any  : D → D, L10. ∀X:(X ) = ∈D   (), L11. ∃X:(X ) = cl( ∈D ()). One can de$ne the notion of a non-standard higher order phase space in the same way as that of a non-standard second-order phase space. The closure condition for the non-standard functional space domain can be expressed in the same way as for the second order case, as follows: (∗) For any higher order formula A of type 1 × · · · × n → 0 (with free higher order variables X ≡ X1 ; : : : ; Xn of types 1 ; : : : ; n ), if i ∈ Di then A∗ [=X ] ∈ D, where  ≡ 1 ; : : : ; n . As in the case of a second-order phase space, we can give an alternative de$nition of a higher order phase space which is actually a special case of the above notion of non-standard higher order phase space. Alternative de?nition of non-standard higher order phase space: The functional space domain of non-standard higher order phase space is a subspace of the functional space based on D that satis$es the following; P4. For any (higher order) abstract A of type , A ⊆ D is de$ned. Here, an abstract (or semi-formula) is an expression obtained from a (higher order) formula A by bounding some (higher order) variables X with the -symbol, (see e.g. [37]). P5. Given an abstract A[X ], where X is a vector of free (higher order) variables, given abstracts B of the same types as X , and given closed sets ∈ B, A∗ [ =X ] ∈ A [B=X ]. As before, A can be a singleton—we can take A as a singleton for all higher order formulas A for the phase-semantic completeness proof of logics of $nite orders. However, constructing a slightly generalized form of a second order phase model leads to our main lemma for the higher order cases. Namely, we take A to be a set which is not necessarily a singleton, in order to make our “uniform proof” work both for a version of completeness theorem and for the cut-elimination theorem (for various versions of higher order logics).

9

See Takeuti [37] for the type structure of the simple type theory. Note that our higher order syntax is based on propositional logic, instead of predicate logic (which is the typical case of the simple type theory), as the $rst-order level.

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As before, it is easy to see that a higher order phase space in  the latter sense is a higher order phase space in the former sense, by de$ning D = A∈Form A ⊆ D , where Form means the set of formulas of type  (in the sense of Takeuti [37]). In particular, with these alternative (more speci$c) de$nitions of the higher order quanti$ers, the above interpretations L10 and L11 could be expressed by, for any  : D → D,
L10 : ∀X:(X ) = ∈B; B∈Form (), where Form stands for the set of abstracts of type , i.e., the  types of X . L11 : ∃X:(X ) = cl( ∈B; B∈Form ()) A classical higher order phase spaces (of $nite orders) can be de$ned in the same way from the classical ($rst order) phase space. In this paper, we present a uniform proof of the higher order cut-elimination theorem for variants of logics by explicitly using the second order case. However, the entire argument can be shifted easily to the higher order case (of $nite orders), by changing the framework for second-order phase semantics to the higher order phase semantics presented above. Our choice of the second-order case (for demonstrating the uniform proof) is mainly for readability of the paper (by avoiding inessential formal complications of the higher order case in general). Although the notions of phase space and of phase semantics were $rst introduced for the semantics of linear logic, the same notions work well for the standard (intuitionistic and classical) logics and for various variants of substructural logics with no essential changes, but only by changing the underlying monoid structure M . We outline how to modify the above de$nitions to accomodate phase space and phase semantics for the standard (i.e., non-linear) intuitionistic and classical logics, and for phase space and phase semantics of some typical substructural logics. [I] The modi$cation depends on the structural rules permitted in the syntax of the corresponding logical system. (1) If both the contraction rules and the weakening rules are permitted in the syntax (in this case, the logical systems are standard intuitionistic logic or standard classical logic), the underlying commutative monoid M satis$es (a) M = 1, namely, for any a ∈ M , a ∈ cl({1}). (b) M = J , namely, for any a ∈ M , a · a = a. Here we do not consider the modality !. (2) If the contraction rules are not permitted but the weakening rules are permitted in the syntax (in this case, the logical systems are often called a=ne logics), the underlying commutative monoid requires only (a). (3) If the contraction rules are permitted but the weakening rules are not permitted in the syntax (in this case, the logical systems are the typical version of relevance logics), the underlying commutative monoid requires only (b). [II] Now, for each of the three modi$cations above, one can consider the intuitionistic version or the classical version in exactly the same way as we did above to distinguish between intuitionistic phase space and classical phase space, (i.e., the classical version is considered a special case with ⊥). [III] For each of the above modi$cations, one can extend the $rst-order phase spaces and the $rst-order phase semantics to the second order and higher order ones, in exactly the same way as above.

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3. The uniform cut-elimination and (phase-semantic) completeness proof for rst-order logics In this section, we demonstrate our paradigm in the simplest setting—the case of $rst-order logic. In the same way as in Girard [12], if we take the phase spaces to be the classical phase spaces and if we take the logical system to be that of classical linear logic (of the two-sided sequent calculus formulation), 10 then we have Theorem 3.1 (Soundness theorem). If a formula A is provable, then A is true for any phase space and for any assignment. More generally, if B1 ; : : : ; Bn  A is provable, then B1∗ · · · · · Bn∗ ⊆ A∗ . This theorem holds as is for all the variants of logics mentioned in the previous section (with the suitable variants of phase space). The proofs of these theorems are performed uniformly by induction on the length of proof of each logical system. Since this is routine and inessential for the purposes of our paper, we omit the proof. The following completeness theorem was presented by Girard [12] for classical phase semantics for the case of classical linear logic. Theorem 3.2 (Phase-semantic completeness theorem). If a formula A is true for any phase space and for any assignment, A is provable in classical linear logic. The above remark on the soundness theorem also can be applied to this phasesemantic completeness theorem: one can prove the phase semantic completeness theorem for all the above logical systems. (The case of intuitionistic linear logic is well known. The substructural logic cases are also implicit in the literature.) On the other hand, the paradigm throughout this paper is to consider the following slightly re$ned form of the above completeness theorem; Theorem 3.3 (Strong completeness theorem). If a formula A is true for any phase space and for any assignment, A is provable without the cut-rule. The phase-semantic proof of this strong completeness theorem was $rst given in [21] (cf. also [22]). An argument similar to the proof of this result for classical linear logic will be used repeatedly in later sections. We construct a speci$c (canonical) phase model (uniformly in a strong sense) for which the truth of a formula A implies its cut-free provability. The construction of the phase space is essentially the same as in [12] in his completeness proof, except that we use cut-free provability instead of provability in the de$nition of
We use the two sided formulation of sequent calculus for various variants of logical systems in this paper. This is because the two sided sequents formulation provides a uniform presentation of various logical systems and $ts our proposed uniform cut-elimination and phase semantic completeness proof. We shall present the two sided formulation of sequent calculus for various variants of logical systems (which we use in this paper) in the appendix.

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For any formula A (of linear logic), we de$ne

j∈





{
Proof of Sublemma. For any formula E,
E ∈ ( ⇔ · E⊆
⇔ A; E ∈
(1) (2)

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The only non-trivial part of the above equivalence is the direction from (2) to (1). Here, for the case of the strong outer-value
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483

(2.1) A ( B ∈ A∗ ( B∗ : Assume  ∈ A∗ (and
we shall show A ( B;  ∈ B∗ ). By the induction
hypothesis, A∗ ⊆
B  Bj A&B  Bj



hold for any i ∈ 1 and any j ∈ 2 .] Hence A&B ∈ i∈1

holds.] Therefore, A ⊕ B ∈ {
B  A⊕B

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hold.] Hence, A∗ ⊕ B∗ ⊆

holds.] Therefore, A ⊗ B ∈ {

holds.] Hence, !A ∈ j∈
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the above single proof is valid line by line for all variants of logics mentioned without any change. We also note that we do not need some of the above cases for some variants of logics; for example, since we can take the additive connectives forms of inference rules for traditional (intuitionistic and classical) logics—namely the forms of &-inference rules for ∧ (the traditional conjunction) and the forms of ⊕ -inference rules for ∨ (the traditional disjunction)—we do not have to consider the multiplicative connectives cases when we apply this proof to traditional logics. We also note that we need to treat the negation-rules explicitly for the two-sided sequent calculus formulation of the classical version of these variants of logics, which was not explicitly mentioned in the above argument. However, as we mentioned above, negation is identi$ed with a special case of implication (A⊥ is A ( ⊥). With this assumption, our arguments for the implications are valid for negation as they are, even for the classical versions of the logics. 4. The uniform cut-elimination and (phase-semantic non-standard) completeness proof for higher order logics In this section, we extend the semantical cut-elimination proof of the previous section to the higher order case, using higher order phase space. In particular, we introduce a notion analogous to Girard’s “candidates of reducibility” in his well-known syntactical higher order normalization proof [11]. We shall discuss the correspondence between our cut-elimination proof by phase semantics and Girard’s syntactical normalization proof later. To simplify the argument, we demonstrate the cut-elimination proof based on secondorder phase semantics in detail below. The higher order case can be obtained exactly in the same way, using higher order phase semantics, instead of the second-order ones. First we can extend the Soundness Theorem from Section 3 in the obvious way. Theorem 4.1 (Soundness theorem (second-order version)). For any second-order phase space, if A[X ] is provable (in second-order linear logic with cut), then A[X ] is true in any second-order phase space. Here, “A[X ] is true” means “1 ∈ A∗ [=X ] for any i ∈ D”. More generally, if B1 [X ]; : : : ; Bn [X ]  A[X ] is provable, then for any i ∈ D and ∈ D, 1 · B1∗ [1 =X ] · · · · · Bn [n =X ] ⊆ A[ =X ], where X and  denote vectors whose length are the same. The proof is carried out by induction on the length of proof, essentially in the same way as for the usual $rst-order case, except for the following second-order quanti$er cases. As emphasized in Section 3, the argument is valid for all variants of logical systems considered. Although we use the form of intuitionistic sequents below, the argument is valid for the (two-sided) classical sequent rules; This is because ∗ ⊆ ∗ oA∗ is equivalent to ∗ ⊥∗ ⊆ A∗ , hence the arguments on the classical sequents can be reduced to the arguments on the intuitionistic sequents presented below.

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Proof. (1) ∀-right rule: [X ]  A[X; Y ] ; [X ]  ∀Y [X; Y ] where Y does not appear as a free variable in . By the induction hypothesis, for any  ∈ D, and for any ∈ D, ∗ [] ⊆ A∗ [; ]. Since this holds for any ∈ D,
∗ ∗ [] ⊆ A [; ] = ∀YA∗ [; Y ]: ∈D

Hence, [] ⊆ ∀YA∗ [; Y ]. (2) ∀-left rule: A[B[X ]; X ]; [X ]  C[X ] : ∀YA[Y; X ]; [X ]  C[X ] By the induction hypothesis, for any  ∈ D, A∗ [B∗ []; ] · ∗ [] ⊆ C ∗ []. By the condition on the second-order phase space, B∗ [] ∈ D. Therefore,
∗ A [ ; ] · ∗ [] ⊆ A∗ [B∗ []; ] · ∗ [] ⊆ C ∗ []: ∈D

Hence, ∀YA∗ [Y; ] · ∗ [] ⊆ C ∗ []. (3) ∃-right rule: [X ]  A[B[X ]; X ] [X ]  ∃YA[Y; X ] By the induction hypothesis, for any  ∈ D, ∗ [] ⊆ A∗ [B∗ []; ]. By the condition on the second-order phase space, B∗ [] ∈ D. Therefore,  ∗ ∗ [] ⊆ A [ ; ] ⊆ ∃YA∗ [Y; ]: ∈D

(4) ∃-left rule: A[X; Y ]; [X ]  B[X ] ; ∃YA[X; Y ]; [X ]  B[X ] where Y does not appear as a free variable in . By the induction hypothesis, for any  ∈ D and for any ∈ D, A∗ [; ]·∗ [] ⊆ B∗ []. Hence, A∗ [; ] ⊆ ((B∗ [])⊥ · ∗ [])⊥ for any ∈ D. Hence,  ∗ A [; ] ⊆ (( ∗ [])⊥ · ∗ [])⊥ : ∈D

Since the right-hand side is a closed set,  
∗ ∗ ∃YA [; ] ≡ cl A [; ] ⊆ ((B∗ [])⊥ · ∗ []): ∈D

Hence, ∃YA∗ [; Y ] · ∗ [] ⊆ B∗ [].

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Now we prove the main lemma for the second-order case. As we did for the $rstorder case, we specify one canonical phase model (M; D; cl; J ). The de$nition of the canonical phase model is exactly the same as before except for the following changes; • For any second order formula A, we de$ne the outer-value of A as

holds.] Hence ∀Y A[C; Y ] ∈ A∗ [; ] for any D, ∈ D. Hence, ∀Y A[C; Y ] ∈ ∈D D∈Form ∗ ∗ A [; ] = ∀Y A [; Y ]. (8.2) A[X ] is of the form ∀Y B[X; Y ]: We prove ∀Y B∗ [=X; Y ] ⊆ <∀Y B[C=X; Y ]= for any i ∈ Ci  and any Ci ∈ Form. Assume that
B∗ [; ]:  ∈ ∀Y B∗ [; Y ] = ∈D D∈Form

By the induction hypothesis, B∗ [; ] ⊆
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does not appear in A[C; Z] and in E. Take this Y for D above. Then, A[C; Y ] ∈ <E=. On the other hand, if A[C; Y ] ∈ <E= then ∃Y A[C; Y ] ∈ <E=. [This is because A[C; Y ]  E ∃Y A[C; Y ]  E

 holds.] Hence, ∃Y A[C; Y ] ∈ ∈D A∗ [; ] = ∃Y A∗ [; Y ] D∈Form (9.2) ∃Y B∗ [; Y ] ⊆ <∃Y B[C; Y ]=: Take  ∈ ∃Y A∗ [; Y ]. Namely, for any E such that  ∗ ∗ ∈D A [; ] ⊆ <E=;  ∈ <E=. By the induction hypothesis, A [; ] ⊆
∈ D and for any D ∈ Form. On the other hand,
Hence,  ∈ <∃Y A[C; Y ]=. Note that, as we already remarked just after the proof of the main lemma for the $rst-order case at the end of Section 3, in the above proof, our explicit reference to (intuitionistic) linear logical rules for the explanations “[This is because : : : hold(s)]” for each case does not limit our proof to this case, since the rules used are cut-free admissible rules of all variants of logics which we consider. So the above single proof is valid line by line for all those variants of logics without any change. We also recall from the remark at the end of Section 3 that we do not need some of the above cases for some variants of logics; for example, since we can take the additive connectives forms of inference rules for the traditional (intuitionistic and classical) logics, we do not have to consider the multiplicative connectives cases when we apply this proof for the traditional logics. The classical negations can be treated as a special case of implications, as remarked at the end of Section 3. The above de$nition of A satis$es the $rst condition P4 of second-order phase space. On the other hand, this main lemma ensures the second condition P5. Hence our phase space D is actually a second-order phase space. By using the soundness theorem and main lemma together, we can obtain the cut-elimination theorem in exactly the same manner as before. On the other hand, the main lemma, of course, implies the strong completeness theorem, as before. Theorem 4.3 (Strong completeness(second-order version)). If A[X ] is true for any second-order phase model, A is provable without cut. Remark 4.4. Note again that this strong completeness theorem with respect to the second-order (non-standard) phase models is proved uniformly for all variants of logical systems in consideration: Namely, the second-order classical and intuitionistic linear

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489

logics, the second-order classical and intuitionistic traditional logics, the second-order classical and intuitionistic aRne logics and the second-order classical and intuitionistic relevance logics (logics without weakenings) are all complete with respect to the corresponding version of (non-standard) second-order phase models. As in the $rst-order case, by combining the soundness theorem and the above strong completeness theorem, we obtain the higher order cut-elimination theorem. Theorem 4.5 (The cut-elimination theorem (second-order version)). If A is provable with the cut rule in the second-order linear logic, it is also provable without the cut rule. Namely, the second-order classical and intuitionistic linear logics, the second-order classical and intuitionistic traditional logics, the second-order classical and intuitionistic a=ne logics and the second-order classical and intuitionistic relevance logics (logics without weakenings) are all cut-eliminable. Remark 4.6. We can extend this proof of the main lemma to higher order linear logic using the higher order phase space, exactly in the same way, as we remarked in Section 2. We also remark that a weak version of completeness theorem can be obtained in the similar way, but with constructing an even simpler second-order model (for the main lemma). As we remarked in Section 3, Girard [12] used a weaker version of our main lemma to show the completeness of the classical $rst-order linear logic. One can extend this original lemma of Girard in our general higher order setting as follows: We consider the weaker version of the outer value de$nition. We de$ne the outervalue of A as
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while the above weaker version uses the form A∗ [=X ] =
For example, with the original form of the !-contraction rule, !P; !Q  (P ⊗ Q) ⊗ (P ⊗ Q) does not have any non-commutative cut-free proof in the usual syntactic setting of non-commutative versions and cyclic versions, although with the modi$ed contraction rule it does.

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491

di/erent from such completeness theorems of certain logics that are not monoid-based. However, one can often easily $nd the relationship between the standard known completeness theorems and the phase-semantic completeness theorems obtained in this paper. For example, in the case of the traditional classical logic our phase models are easily shown to be Boolean valued models; in the case of the traditional intuitionistic logic our phase models are easily shown to be Heyting valued models. Acknowledgements The author would like to express his thanks to Dr. Kazushige Terui for his careful reading of the earlier draft and for his comments, and to Mr. Koji Hasebe for his assistance in careful typing the materials. The author also would like to express his sincere thanks to Drs. Jean-Yves Girard, Max Kanovitch, and Andre Scedrov for stimulating discussions and encouragements during the preparation of this paper. The author would also like to express his sincere thanks to the anonymous referees and the guest editor, Dr. Pierre-Louis Curien, of this special issue for invaluable comments to improve readability of this paper; without their comments publication of this work would be very much delayed. Appendix A. A.1. Inference rules for intuitionistic linear logic Below, A, B and C represent arbitrary formulas and  and  represent arbitrary ($nite) sequence of formulas, including the case of empty sequence. • Axiom sequent Logical axiom sequent AA Logical constants 1 ⊥

 0;   C

• Rules for constants C 1;   C • Exchange Exchange ; A; B;   C ; B; A;   C

 ⊥

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• Cut-rule   A A;   C ;   C • Multiplicative ⊗ − left

⊗ − right

A; B;   C A ⊗ B;   C

A B ;   A ⊗ B

• Additive & − left

& − right

A;   C A&B;   C

A B   A&B

B;   C A&B;   C ⊕ − left

⊗ − right

A;   C B;   C A ⊕ B;   C

A  A⊕B

B  A⊕B • Linear implication ( − left

( − right

  A B;   C A ( B; ;   C

A;   B A(B

• Linear negation ⊥

− left

A A;  

⊥

⊥

− right

A;     ⊥A

• Modality ! − left

! − right

(dereliction) A;   C !A;   C

!  C ! !C

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493

(weakening) C !A;   C (contraction) !A; !A;   C !A;   C A.2. Second-order intuitionistic linear logic The above $rst order system can be extended to a second order system by adding the following second order quanti$er rules. ∀ − left

∀ − right

A[X := B];   C ∀XA;   C

A   ∀XA

(X does not appear as a free variable in .) ∃ − left

∃ − right   A[X := B]   ∃XA

A;   C ∃XA;   C

(X does not appear as a free variable in ; C.) A.3. Classical linear logic The logical axioms and inference rules of classical linear logic can be obtained from the above intuitionistic case by allowing possibly multiple occurrences (namely an environment sequence) of formulas in the right hand of sequents. As usual, we take the negation-rules, instead of the implication rules, for the classical version. (Note that these classical rules for negation are considered special cases of the implication rules where A⊥ is identi$ed with A ( ⊥.) • Axiom sequent Logical axiom sequent AA Logical constants 1

  ; 

⊥

0;   

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• Rules for constants    ; ⊥

 1;    • Exchange Exchange-left

Exchange-right

; A; B;    ; B; A;   

  ; A; B;    ; B; A; 

• Cut-rule   ; A A;    ;   ;  • Multiplicative ⊗ − left A; B;    A ⊗ B;    o − left

⊗ − right   ; A    ; B ;   ;  ; A ⊗ B o − right

A;    B;    AoB; ;   ; 

  ; A; B   ; AoB

• Additive [& − left A;    A&B;   

& − right   ; A   ; B   ; A&B

B;    A&B;    ⊕ − left

⊕ − right

A;    B;    A ⊕ B;   

  ; A   ; A ⊕ B

• Linear implication ( −left

( −right

  ; A B;    A ( B; ;   ; 

A;   ; B   ; A ( B

  ; B   ; A ⊕ B

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495

• Linear negation ⊥

− left

⊥

  ; A A⊥ ;   

− right

A;      ; A⊥

• Modality ? − left(promotion)

! ?; A ! ?; !A

A! ? ?A; ! ? ! − left

! − right(promotion)

? − right

(dereliction − left) A;    !A;   

(dereliction − right)

  ; A   ; ?A

(contraction-left)

(contraction-right)

!A; !A;    !A;    (weakening-left)

  ; ?A; ?A   ; ?A

(weakening-right)

 !A;   

   ; ?A

A.4. Second-order classical linear logic ∀ − left

∀ − right

A[X := B];    ∀XA;   

  ; A   ; ∀XA

(X does not appear as a free variable in ; .) ∃ − left A;    ∃XA;   

∃ − right   ; A[X := B]   ; ∃XA

(X does not appear as a free variable in , .) A.4.1. Substructural systems We can consider the typical substructural systems, one obtained by adding the weakening rules, and the other by adding the contraction rules to the above linear logic systems. Here, one can get both intuitionistic version and classical version of these

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substructural systems. For the case in which the weakening rules appear, we do not need the rules for constants. Note also that due to the presence of the weakening rules (the contraction rules, respectively), one can omit the corresponding weakening rules (and the contraction rules, respectively) of the modalities. the intuitionistic weakening weakening-left C A;   C the intuitionistic contraction contraction-left A; A;   C A;   C the classical weakening weakening-left weakening-right  A;   

   ; A

the classical contraction contraction-left contraction-right A; A;    A;   

  ; A; A   ; A

A.4.2. Traditional systems When both weakening rules and contraction rules are added, one can obtain traditional classical logic and traditional intuitionistic logic, respectively, from classical linear logic and from intuitionistic linear logic, respectively. Since the distinctions between the additive logical connectives and constants and the multiplicative logical connectives and constants disappear, as is well known, in this case, one can take only one of the original two connectives for the traditional cases. In this paper, we take the additive rules and constant-axioms. Note that due to the presence of the weakening rules, we do not need the rules for constants. We also do not need modal operators or the modality rules. Following tradition, we use thw symbol → instead of (, ∧ instead of &, ∨ instead of ⊕ and ¬ A instead of A⊥ for the traditional cases (cf. [37] for the sequent calculus of traditional (classical and intuitionistic) $rst and higher order logics.) References [1] V.M. Abrusci, Sequent calculus for intuitionistic linear propositional logic, Mathematical Logic Plenum Press, 1990. [2] V.M. Abrusci, Non-commutative intuitionistic linear logic, Zeischrift fVur Math. Logik und Grundlagen d. Math. 36 (1990).

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[3] T. Altenkirch, M. Hofmann, T. Streicher, in: David Pitt, David E. Rydeheard, Peter Johnstone (Eds.), Categorical reconstruction of a reduction free normalization proof, Category Theory and Computer Science, Lecture Notes in Computer Science, Vol. 953, 1995, pp. 182–199. [4] T. Altenkirch, P. Dybjer, M. Hofmann, P. Scott, Normalization by evaluation for typed lambda calculus with coproducts, 16th Annual IEEE Symposium on Logic in Computer Science, IEEE Press, 2001, pp. 303–310. [5] W. Buchholz, Ein ausgezeichnetes Modell fVur die intuitionistische Typenlogik, Archive for Mathematical Logic 17 (1975) 55–60. [6] V. Danos, La Logique LinWeaire appliquWee aX l’Wetude de divers processus de normalisation, ThXese de doctorat, UniversitWe de Paris 7, 1990. [7] V. Danos, L. Regnier, The structure of multiplicatives, Archive for Mathematical Logic 28 (1989). [8] A. Dragalin, Mathematical Intuitionism: Introduction to Proof Theory, AMS, 1993. [9] A. Dragalin, A completeness theorem for higher-order intuitionistic logic: an intuitionistic proof, in: D. Skordev (Ed.), Mathematical Logic and its Applications, Plenum Press, New York, 1987. [10] A. Dragalin, Cut-elimination theorem for higher-order classical logic. An intuitionistic proof, in: D. Skordev (Ed.), Mathematical Logic and its Applications, Plenum Press, New York, 1987. [11] J.-Y. Girard, Doctoral Thesis, Univ. Paris VII, 1972. Also in the Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1973. [12] J.-Y. Girard, Linear logic, Theoretical Computer Science 50 (1987). [13] J.-Y. Girard, Linear logic: its syntax and semantics, in: Girard, Lafont, Regnier (Eds.), Advances in Linear Logic, Cambridge University Press, 1995. [14] J.-P. Jouannaud, M. Okada, A computation model for excecutable higher order algebraic speci$cation languages, in: Proceedings of LICS 91, IEEE International Symposium on Logic in Computer Science, IEEE Press, 1991. [15] M. Kanovitch, M. Okada, A. Scedrov, Phase semantics for light linear logic, Theoretical Computer Science, to appear. A prior version appeared in Proceedings of the International conference on the Mathematical Foundations of Programming Languages, Electronic Notes in Theoretical Computer Science 6 (1997). The full-version is to appear in Theoretical Computer Science, Special Issue on Linear Logic. (2002), to appear. [16] Y. Lafont, La logique lineaire du second ordre sans exponentielle est indecidable (version preliminaire), Dec. 1994. [17] Y. Lafont, From proof-nets to interaction nets, in: Girard, Lafont, Regnier (Eds.) Advances in Linear Logic, Cambridge University Press, 1995. [18] Y. Lafont, The $nite model property for various fragments of linear logic, Journal of Symbolic Logic (1997). [19] Y. Lafont, A. Scedrov, The undecidability of second order multiplicative linear logic, Information and Computation 125 (1996) 46–51. [20] M. Okada, Phase semantics for higher order completeness, cut-elimination and normalization proofs, (Extended Abstract), Electronic Notes in Theoretical Computer Science 3 (1996) 22. [21] M. Okada, Phase semantic cut-elimination and normalization proofs of $rst- and higher-order linear logic, Theoretical Computer Science 227 (1999) 333–396. [22] M. Okada, Introduction to linear logic: expressivenes and phase semantics, in: M. Takahashi, M. Okada, M. Dezani (Eds.), Theories of Types and Proofs, Memoir of Mathematical Society of Japan, 2, 1998 (1st Edition), 1999 (2nd Edition). [23] M. Okada, A uniform phase-semantic proof of higher order completeness and cut-elimination theorems (abstract), Proceedings of the International Congress on Logic, Methodology and Philosophy of Science, Poland, August 1999. [24] M. Okada, Intuitionistic logic and linear logic, Revue internationale de philosophie, to appear. Also available as a lecture at the Institute of History and Philosophy of Sciences and Technology of Paris, March 2001, also at the Logic, Mathematics and Philosophy Meeting, Aix-en-Provence, May 2001. [25] M. Okada, K. Terui, Completeness proofs for linear logic based on the proof search method (Preliminary Report), in: J. Garrigue (Ed.), Proceedings of the Workshop on Type Theory and its Applications to Computer Systems, Kyoto University, January 1997, pp. 57–75.

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[26] M. Okada, K. Terui, The $nite model property for various fragments of intuitionistic linear logic, Journal of Symbolic Logic 64 (1999) 790–802. [27] H. Ono, Semantics for substructural logics, K. Dosen, P. SchrVoder (Eds.), Oxford Univ. Press., 1994, pp. 259–291. [28] H. Ono, Y. Komori, Logics without the contraction rule, Journal of Symbolic Logic 50 (1) (1985). [29] D. Prawitz, Hauptsatz for higher order logic, Journal of Symbolic Logic (1968) 452–457. [30] L. Regnier, Lambda-calcul et rWeseaux, ThXese de doctorat, UniversitWe de Paris 7, 1992. [31] G. Sambin, Intuitionistic formal space, in: Logic Colloquium 88, North-Holland, 1989. [32] G. Sambin, Pretopologies and completeness proofs, Journal of Symbolic Logic 60 (3) (1995) 861–878. [33] K. SchVutte, Beweisstheorie, Springer, 1960. [34] W. Tait, A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic, Bulletin of the American Mathematical Society, Vol. 72, pp. 980 –983. [35] W. Tait, Intensional interpretation of functionals of $nite type I, Journal of Symbolic Logic 32 (1967) 198–212. [36] M. Takahashi, On Takeuti’s conjecture, Journal of Math. Soc. of Japan (1967). [37] Gaisi Takeuti, Proof Theory, North-Holland, 1985. [38] K. Terui, Proof Search in Linear Logic: Semantic View and Computational View, Master Thesis, Department of Philosophy, Keio University, March 1997, submitted. [39] A. Troelstra, Lectures on Linear Logic, CSLI Lecture Notes, 1991, Vol. 29, Stanford.

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www.elsevier.com/locate/tcs

A uni#ed language processing methodology  Teodor Rus ∗ Department of Computer Science, University of Iowa, Iowa City, IA 52242, USA

Abstract This paper discusses a mathematical concept of language that models both arti#cial and natural languages and thus provides a framework for a uni#ed language processing methodology. This concept of a language is regarded as a communication tool that allows language users to develop knowledges, while interacting with their universe of discourse, and to communicate with each other, while exchanging knowledges. Criteria for consistent usage of a language are established using a Galois connection between language syntax and language semantics. Solutions to ambiguity, paraphrase, attitude, and other problems concerning the relationship between syntax and semantics are addressed. A general schema for language speci#cation is introduced and algorithms that perform language generation and language analysis are discussed as universal tools de#ned by the speci#cation schema. Language transformations performed by various kinds of translators are examined and correctness criteria of these translators are de#ned using the language Galois connection. The paper is structured as follows: Section 1 introduces the framework and justi#es the necessity of a uni#ed methodology for language processing. Section 2 presents the mathematical concept of a language. Section 3 illustrates the mathematical concept of a language with three kinds of language structures: natural language, logical language, and programming language. Section 4 discusses the algebraic mechanism of language speci#cation that uni#es the methodology for language processing tool development. Section 5 formalizes the criterion for the consistency of the language usage, de#nes the architecture of a uni#ed language processing system, and shows how the consistency criteria for language usage can be employed as correctness criteria c 2002 Elsevier Science B.V. for the algorithms performing various language transformations.
All rights reserved. Keywords: Algebra; Algorithm; Galois connection; Language; Speci#cation

 ∗

This paper was supported in part by Roy J. Carver Charitable Trust through the Grant #01-162. Corresponding author. E-mail address: [email protected] (Teodor Rus). URL: http://www.cs.uiowa.edu/rus

c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 5 - 7

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1. Introduction By general agreement a language is considered to be a collection of well-formed strings over a given alphabet. Since languages are used as communication mechanisms, various people provide the rules of string well-formedness and associate well-formed strings with meanings in ways which depend upon their interests. Logicians consider languages as formal systems and language meanings as models that satisfy the construction and deduction rules of the formal system; computer scientists consider languages as collections of well-formed strings speci#ed by formal grammars and language meanings as machine computations; computational linguists consider languages as collections of “grammatical expressions” used through innate or learned convention and language meanings as entities of the universe of discourse. Logicians and computer scientists specify the rules for string well-formedness and the meanings of the strings these rules generate; computational linguists discover the rules for string well-formedness and the meanings of the strings they generate from observations of their usage in communication. The expression power of well-formedness rules is controlled by their speci#ers for the case of logic and programming languages; this is however a great challenge for computational linguists as they have to answer questions such as: are these rules mentally represented and if so, what kind of structure do these rules have [2]. Answers to such questions lead to approximations of natural languages by formal systems [25]. Logicians and computer scientists look at language lexical elements as monads while computational linguists observe that the form and meaning of lexical elements depend upon the phrase structure where they are used and the information content they express [21]. These observations challenge the grammar structure of well-formedness rules [5] and lead to constraint-based grammar formalisms [45]. Language processing implies the development of mathematical models of languages and the study of the transformations, that map well-formed texts of one language into well-formed texts of another language, while preserving text meaning (and style). Examples of algorithms that perform such transformations are compilers of programming languages and language-to-language translators of high-level programming languages or natural languages. Computational diCculties involved in the development of such translators are due to the implicit nature of the language universe of discourse and the lack of mathematical models of the language parlance. However, the ubiquity of computer applications proliferates computer languages and thus requires the development of system software that allows computer users to personalize their machines through language translation. Unfortunately, compiler constructors (in charge of programming language processing) and computational linguists (in charge of natural language processing) developed their language processing paradigms following methodologies that best #tted their domains of interest, without having relied on the same formal concept of a language. Consequently, they do not share their research results, though these results present solutions to the same problems. Attempts to develop mathematical models of natural languages and language translations have a long history [46]. The need for natural language modeling and translation is recently reiterated by the “global economy” and the requirement to develop “personalized” computer systems [50]. Earlier methods for the design of machine translation

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systems [16] are based on such paradigms as table lookup, rule-based, syntactic transfer, interlingua, statistical modeling, etc. None of these approaches has yet produced domain-independent, fully unambiguous translations. Experiments with compositional machine translation [35] suggest that both language modeling and language translation can be achieved within the framework of universal algebra. On the other hand, a methodology for compiler construction based on a language concept formally de#ned within the framework of the universal algebra is already available [41]. It seems that the glue that connects natural and arti#cial languages, leading to a uni#ed methodology for language processing, is a method of language modeling capable of de#ning formally a language as a communication mechanism. This method is used in this paper by looking at language elements as knowledges [18]. Knowledges are obtained by language users in the process of their interaction with the universe of discourse and are used by language users in their communication process. Thus, we consider a language to be a collection of symbolic expressions used through innate or learned convention by a group of communicators to construct knowledges, while interacting with their environment, and to communicate with each other. Hence, language de#nition needs to include the four fundamental components of a language: the universe of discourse (called semantics), the collection of symbolic expressions used to represent semantic objects (called syntax), the process of learning to generate knowledges by representing semantic objects using symbolic notations, and the process of using knowledges by mapping symbolic notations into the semantic objects they represent. This view of a language splits the language users into two classes: speakers, who are communicators producing language expressions, and auditors, who are communicators interpreting language expressions. The speaker’s universe of discourse may be diIerent from the auditor’s universe of discourse. However, the communication consistency is ensured by language processing tools that: (i) allow speakers to generate incrementally valid language expressions that represent objects of their universe of discourse; (ii) facilitate auditors to interpret language expressions produced by the speakers in the speaker universe of discourse. On the other hand, a speaker and an auditor using a language to communicate may be of diIerent nature. For example, a man-made robot and a human may communicate using an appropriate communication language. Hence, looking at language as a communication tool leads to the requirement of a language processing methodology that uni#es natural and arti#cial languages, and allows languages to be incrementally speci#ed and processed. The framework for a uni#ed language processing methodology discussed in this paper builds on the conventional mechanisms for language speci#cation (formal systems, used by logicians; automata theory, used by computer scientists; and formal grammars, used by computational linguists). Each of these conventional mechanisms regards a language as a collection of well-formed strings over a given alphabet, keeping syntax and semantics, the two major components of a language, in diIerent #elds of interest. Recent developments in computer applications make language design and implementation an essential ingredient of information technology. This, in turn, requires new methods in language processing that support language usage as a communication mechanism by systems of diIerent nature. The framework we propose in this paper responds to this requirement by providing a uni#ed methodology for language processing based on the

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following accomplishments: (1) integrate syntax and semantics into a mathematical concept of a language that models natural, logical, and programming language de#nition, (2) develop computational tools for the simulation of the cognition process, (3) provide formal mechanisms for incremental development of language syntax and language semantics integrated into a language object, (4) elaborate criteria for consistent language usage employed as mathematical characterizations of correctness of language transformations, and (5) discover algebraic methods as alternatives to automata for the automatic generation of language processing tools. Moreover, since repetition is diCcult to observe in man–machine interacting systems, computer education is a major limitation of conventional methods in language processing. Automatic generation of language processing tools supported by our framework reveals new principles in education where learning is based on the automatic generation of solutions rather than on repetition. The rest of the paper is structured as follows: Section 2 introduces the mathematical concept of language. Section 3 illustrates the mathematical concept of a language with three kinds of language structures: natural language, logical language, and programming language. Section 4 discusses the algebraic mechanism of language speci#cation that uni#es the methodology for language processing tool development. Section 5 formalizes the criterion for the consistency of the language usage, de#nes the architecture of a uni#ed language processing systems, and shows how the consistency criterion for language usage can be employed as a correctness criterion for the algorithms performing various language transformations. 2. Language as a communication tool A language in this paper is a mechanism of communication that allows its users to interact with their universe of discourse and with each other. The interaction with the universe of discourse is called cognition and the interaction with each other is called communication. Language elements are tuples of the form
symbol; entity called knowledges. The entity component of a knowledge is an abstract or concrete object of the environment called knowledge meaning. The symbol component of a knowledge is a sensory (acoustic, visual, etc.) representation of the object, called knowledge token. Fig. 1 provides an example of knowledge. The cognition process can be modeled in terms of the operations perception, recognition, and action [12], de#ned as follows: • perception: associates the environment’s objects with innate or learned representations called matching concepts (M-concepts). • recognition: instantiates perception patterns (i.e., M-concepts) as knowledge tokens, called internal concepts (I-concepts), by substituting appropriate values for M-concept parameters. • action: maps I-concepts into the environment’s objects associated with their M-concepts by the identi#cation of the appropriate parameter values.

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Fig. 1. Knowledge representation.

M-concepts, I-concepts, values, and parameter processing (evaluation, testing, substitution) in the cognition process are modeled here by computational objects called macro-operations and macro-processors. 2.1. Macro-operations A macro-operation is a tuple M =
name; body, where body designates a class of objects and name designates references to the objects of the body. For example, the lexicon of a language can be seen as a set of macro-operations such as noun =
N; set-of -nouns, identi4er =
Id; identi#er-speci#cation-rules, and number =
Nr; number-speci#cation-rules. Macro-operations can be primitive, where name and body are given constants, as seen in Fig. 1, or composed, where name and body are parameterized patterns. The name component of a composed macro-operation speci#es the component elements of the body; the body of a composed macro-operation shows how the elements designated by the name component are composed from the elements of its constituent bodies. The parameter values of a composed macro-operation are denoted by @, accompanied by indices if necessary, and the property p of the value @ is denoted by @ :p. For example, a class of phrases of English language is speci#ed by the macrooperation Phrase =
S → NP VP; @1 : agreement = @2 : agreement, @0 : surface = @1 : surface ◦ @2 : surface, where @0 ; @1 ; @2 stand for the values taken by the parameters S; NP; VP, respectively, and ◦ denotes string concatenation. A branching statement of C language is speci#ed by the macro-operation BranchC =
S → ifE{S}; @1 : type = boolean; @0 : type = @2 : type; @0 : surface = if ◦ @1 : surface ◦ {◦ @2 : surface ◦}. A stack-machine branch may be de#ned by the macro-operation BranchM =
Branch(@0 ; @1 ; @2 ); Push(@2 : label); Push(@1 : value); BT  where BT is the mnemonic for branch operation. A triangle can be speci#ed by the macro-operation Triangle =
graph(@1 ; @2 ; @3 ); @1 :type = @2 :type = @3 :type = edge, (@1 ; @2 ) =“1 , (@2 ; @3 ) =“2 , (@3 ; @1 ) = “3 , Constraint: “1 + “2 + “3 = 180◦ , ∀i; j; k ∈ {1; 2; 3}; i = j = k (@i : size + @j : size¿@k : size), if @1 : size = @2 : size = @3 : size then equilateral triangle else if (@1 : size = @2 : size ∨ @1 : size = @3 : size ∨ @2 : size = @3 : size) then isosceles triangle else triangle. The process of mapping a macro-name or a macro-body into the object it speci#es is called macro-expansion and is performed by a macro-processor, M. Hence, a macroprocessor takes macro-operation names and parameter values as arguments and expands the associated macro-bodies by checking constraints and substituting parameter values

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for parameters. That is, we use macro-processors provided with an appropriate logic that allows them to check properties and to perform conditional substitution of the parameters. For example: M(Triangle; (2; 3; 4)) = triangle, M(Triangle; (3; 3; 4)) = isosceles triangle, M(Triangle; (3; 3; 3)) = equilateral triangle, M(BranchC; (x¿y; x = x + y)) = if x¿y {x = x + y}. Both components of a macro-operation M , M: name and M: body, are parameterized patterns using the same parameters. Parameters @i , 06i6n, of M can also be macro-operations; macro-names @i : name, 06i6n, constrained by a relationship Ename (@0 : name; : : : ; @n : name), are used for the construction of the M: name; macrobodies @i : body, 06i6n, constrained by a relationship Ebody (@0 : body; : : : ; @n : body), are used for the construction of the M: body. The relationships Ename and Ebody are in general diIerent and can be provided as components of their macro-bodies, as seen in the macro-operations Phrase and BranchC, where @1 : agreement = @2 : agreement, @1 : type = boolean, are attached to the macro-body; Ename , Ebody can also be introduced as separate components of the macro-operation using appropriate keywords, as in the macro-operation Triangle where the keyword Constraint is used to introduce the graph-constraints that de#ne a triangle. Macro-names are mapped into knowledge tokens representing M-concepts and macro-bodies are mapped into knowledge meanings representing I-concepts. Thus, the macro-processor that expands M: name(@0 ; : : : ; @n ) into the object it speci#es interprets the parameters @i , 06i6n, diIerent from the interpretation of the parameters @i , 06i6n, by the macro-processor that expands M: body(@0 ; : : : ; @n ), into the object it speci#es. An example of a macrooperation in this framework is a branch operation whose name is a C language construct and whose body is the stack-machine operation performing it. Another example is an English language phrase speci#ed by the macro-operation Phrase whose body is the lambda-expression of its Montague semantics, as seen in [14], p. 16. 2.2. Communicators, languages, and language systems Using (parameterized) macros the operations perception, recognition, and action of a cognition process can be formally de#ned as follows: Perception: is de#ned by an open-ended list of speci#cation rules of the form M =
parameter-pattern: macro-name; macro-body , where parameter pattern is a relation of the form lhs → rhs (left-hand-side → right-hand-side) which speci#es the parameters, and macro-name and macro-body are macro-operations de#ning an M-concept and its associated I-concept, respectively, using the parameters in the parameter-pattern, as seen in Table 1. This list contains innate elements for natural systems and may be expanded (by learning) or shrunk (by forgetting). For an arti#cial system this list is provided by the system designer. Recognition: takes as input an (instantiated) macro-name, i.e., a knowledge token, and some parameters, and expands the macro-body associated with the macroname by processing the appropriate parameters to obtain an internal representation of the knowledge meaning, i.e., an I-concept. For example, make triangle (2; 3; 4) →

(2; 3; 4).

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Table 1 Example of a cognition environment speci#cation

1. Command→ VP: name: @0 : surface = @1 : surface; body: @0 : action = @1 : action 2. VP → V N: name: @0 : surface = @1 : surface ◦ @2 : surface; body: @1 : type = @2 : type → @0 : type; @0 : action = call(@1 : action(@2 : args)) 3. N → “triangle”: name: @0 : surface = triangle; body: @0 : args = int size1; size2; size3; (size1; size2; size3) 4. V → “search”: name: @0 : surface = search; body: @0 : type : int × int × int → ; @0 : action = SearchTriangle(int; int; int) 5. V → “make”: name: @0 : surface = make; body: @0 : type : int × int × int → ; @0 : action = MakeTriangle(int; int; int)

Action: takes as input an (instantiated) macro-body, i.e., a knowledge meaning, and some parameters, and uses the associated macro-name to deliver the object from the environment, i.e., the knowledge token, represented by the I-concept. For example, search (2; 3; 4) → triangle. By macro-processing, a macro-processor Mn expands macro-names into external representations of I-concepts (i.e., into knowledge tokens), while a macro-processor Mb expands the macro-bodies into internal representation of I-concepts (i.e., into knowledge meanings). Macro-names serve as abstract representations of classes of similar objects of the environment; the associated macro-bodies identify individual objects of the class and the operations de#ned on these objects. Using this representation of M-concepts and I-concepts, a knowledge can be de#ned as a tuple
symbol; entity, where symbol is a macro-name representing an M-concept and entity is a macro-body where some (may be none or all) parameters are replaced by parameter values. A cognition system, CS, can be modeled by a tuple CS =
Environment; Record; Action; Recognition; Control where Environment is a universe of discourse de#ned by a given list of knowledge speci4cation rules (as seen in Table 1), Record is a mechanism of knowledge representation, for example
triangle; , Action maps objects in the environment into knowledges in the Record, Recognition maps knowledges in the Record into the objects they represent, and Control is a logical expression in terms of Action and Recognition that controls the system behavior. For example, repeat forever (Recognition → Action) ∨ (Action → Recognition), is such a control expression. While the Environment and the Record of two diIerent cognition systems are in general diIerent, the operations Action, Recognition, and Control are the same. Therefore, we use the notation Envc , Recordc , as references to the environment and the knowledge representation of the cognition system c.

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Fig. 2. The structure of a communicator.

A language user (communicator) is a natural or arti#cial system consisting of (1) a cognition system c =
Envc ; Recordc ; Action; Recognition; Control (2) a collection of knowledges represented by a database, Dbasec , de#ned by Dbasec = {w | ∃o ∈Envc ∧
w; o ∈ Recordc }, such that Action(o) = w and Recognition(w) = o. Dbasec is also called the context of c. (3) a mechanism that consists of a learning action, denoted Learnc , that maps objects in Envc into knowledge representations in Dbasec , and an evaluation action, denoted Evalc , that maps knowledge representations in Dbasec into the objects of Envc , as shown in Fig. 2. The Envc is used twice in Fig. 2 to emphasize its double relation with the Dbasec : as the source of knowledges represented in the Dbasec , and as the target of the actions represented by the knowledges in the Dbasec . Hence, communicators can be modeled by tuples of the form
Env; Dbase; Learn; Eval, where Learn and Eval are binary relations, Learn ⊆ Env × Dbase, Eval ⊆ Dbase × Env. As usual, ∀m ∈Env, Learn(m) ={w ∈Dbase |(m; w) ∈ Learn} and ∀w ∈ Dbase, Eval(w) = {m ∈ Env | (w; m) ∈ Eval}. A language L is modeled here by a tuple L =
U; P; L ⊆ U × P; E ⊆ P × U , where U is a universe of discourse (i.e., a cognition environment), P is a collection of wordexpressions, called parlance, L ⊆ U × P is a learning relation that associates objects u ∈ U with word-expressions p ∈ P thus generating knowledges (u; p), and E ⊆ P × U is an evaluation relation that interprets word-expressions p ∈ P and objects u ∈ U to determine if (u; p) is a language knowledge. A linguistic system, LS, is a tuple LS = (L; C) where L is a language and C is a collection of communicators using L to interact with their environments and to communicate with each other. For LS = (L; C), L =
U; P; L ⊆ U × P; E ⊆ P × U  and c ∈ C, c =
Envc ; Dbasec ; Learnc ; Evalc  is provided with two mappings, Generatec : Dbasec →P and Interpretc : P → Dbasec , such that Learnc ◦ Generatec ⊆L, Interpretc ◦ Evalc ⊆E, where ◦ is the relation composition. In order to interact and communicate using L, the assumption is that c has learned L, i.e., Envc ⊆ U and ∀m ∈ Envc ∧ ∀w ∈ Learnc (m) : (m; Generate(w)) ∈L, and ∀w ∈ P ∧ ∀m ∈ Evalc (Interpret(w)) : (w; m ) ∈E. These relations are satis#ed if Dbasec ⊆ P. 2.3. Language use Language use by communicators consists of generating knowledges and exchanging them with other communicators. A communicator c generates knowledges (m; wmc ) ∈ Envc × Dbasec by its own cognition process (described by macro-expansion). When c uses a linguistic system to communicate, c embeds m in the language universe of discourse and maps wmc into a language well-formed word-expression wm . Hence, for each m ∈ U the word-expression wm ∈ P such that (m; wm ) ∈L ∧ (wm ; m) ∈E is the

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Fig. 3. The structure of a communication system.

literal meaning of (m; wm ). The linguistic phenomenon by which a word-expression, wm , means the object, m, of the universe of discourse represented by wm in a knowledge is called a reference relation of m. For a language L =
U; P; L ⊆ U × P; E ⊆ P × U , the reference modeling can also be described by a macro-expansion process. Literal meanings wm are language types (M-concepts) de#ned by the parameterized macro-operations of the cognition environment de#ning U . The contextual references wmc are language tokens (I-concepts) obtained by the expansion of macro-operations de#ning the cognition environment of the communicator c, i.e., wmc ∈ Dbasec . Hence, the reference in a linguistic system generates two notions of meaning: (i) literal meaning, also called speaker’s meaning, is a word-expression wms such that there is a knowledge meaning m expressed by it, i.e. (m; wms ) ∈L implies (wm ; m) ∈E, and (ii) utterance meaning, also called auditor’s meaning, is a word-expression wma such that there is a knowledge meaning m to which wma is evaluated, i.e. (wma ; m ) ∈E implies (m ; wma ) ∈L. The relationship between wms (which exists independent of its contextual use) and a wm (whose existence depends upon the existence of a language user) is described by the communication system in Fig. 3, organized on three levels: language level, where a language L is introduced as a universal communicator, speaker level, where a speaker S is shown as a particular communicator using L to generate knowledges, and auditor level, where an auditor A is seen as a particular communicator using L to interpret knowledges. The rules describing the language use by the speaker, S, are: (1) A knowledge (m; wms ) is generated by LearnS and wms is stored in DbaseS . (2) (m; wms ) is mapped into a linguistic knowledge, that is, m is embedded in U and GenerateS (wms ) is performed thus mapping wms into wm ∈ P ensuring that (m; GenerateS (wms )) ∈L. (3) The knowledge (m; wms ) is communicated using L, i.e., GenerateS (wms ) is issued as the reference to m. The rules describing the language use by the auditor, A, are: (1) Receives the word-expression wm , where (wm ; m) ∈E. (2) Maps wm into the contextual token wma = InterpretA (wm ) by its own interpretation procedure.

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(3) Evaluates wma , i.e., determine m = EvalA (wma ) ∈ EnvA . If m is identical to m then the auditor interpreted correctly the speaker’s message encoded in wm and a successful communication took place. Language L is consistently used if GenerateS and InterpretA ensure that every valid knowledge issued by speaker is interpreted correctly by auditor. Note: wma must be evaluated in the speaker’s context, i.e., Envs , in order for the communication to take place. That is, EnvS ⊆ U; EnvA ⊆ U , and E(wma ) must interpret wma over EnvS portion of U . The parameterized macro-operations representing M-concepts provide support for paraphrase and synonyms while parameterized macro-operations representing I-concepts provide support for a continuum of semantics where knowledges are built incrementally on top of other knowledges. This allows the auditor to add appropriate semantic properties to I -concepts, customizing them to the context, and explains the paraphrase phenomena. Therefore, when a speaker exclaims “look, a mushroom”, pointing to a rock, the auditor correctly interprets it as “look, a rock with the shape of a mushroom”. 2.4. Meaning of word-expressions For a natural language to function as a communication mechanism, the meanings of word-expressions are established by conventions which are maintained by their constant use within a linguistic system. Arti#cial languages function as communication mechanisms by their de#nitions. That is, the usage of well-formed expressions to denote objects of the environment is encapsulated in the de#nition of the well-formed expressions rather than in an innate or learned convention. However, the de#nitions of well-formed expressions in turn are provided by formal or informal conventions. That is, for arti#cial languages the meaning of a word-expression is “wired” in that expression by the process of language speci#cation. Conventions which associate wordexpressions with environment objects (provided by historical usage or by de#nition) thus generating knowledges, are learned (i.e., internally established) by each communicator member of the linguistic system. Hence, language meanings are mental objects in the communicator’s cognition. Let LS =
L; C be a linguistic system. In a communication using LS, the speaker’s word-expressions need to be evaluated by the auditor in the speaker’s context. L makes the evaluation process feasible by providing the universal relationship between wordexpressions and their meanings. The computational aspect of word-expression evaluation was envisioned by Gotlob Frege (1848 –1925), and is currently called Frege’s principle. Since Frege did not actually use this principle in the algebraic sense used here, we adopt the terminology introduced by Janssen [19], calling it the principle of compositionality which is stated as follows: The meaning of a complex word-expression is a function of the meaning of its component parts and their mode of composition. The principle of compositionality implies that the two main components of a language, the universe of discourse and the collection of word-expressions which represent its objects, have a similar structure. Consequently, in order to construct (or specify) a lan-

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guage as a mathematical object we should proceed as follows. Consider a speci#cation schema S =
N;
F, ,∈- ; Syn; Sem where: (i) N is a not-empty set of names; (ii) is a set of indices and ∀, ∈ -, F, is a scheme of operations, i.e., F, : n1 × · · · × nk → n, where k¿0 and n; n1 ; : : : ; nk ∈ N , not necessarily diIerent; (iii) Syn is a surface structure construction functor; and (iv) Sem is a semantic structure construction functor. Syn maps elements of N into syntax categories and operation schemes F, into rules for word-expression well-formedness; Sem maps elements of N into computation types and operation schemes F, into operations of the types constructed by Sem. A language speci#ed by S is a tuple L =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL , where SemL = Sem(N;
F, ,∈- ), SynL = Syn(N;
F, ,∈- ), and LL and EL are relations such that ∀(m; w) ∈ SemL × SynL if (m; w) is a language element then (m; w) ∈ LL and (w; m) ∈EL and vice versa, if (w; m) ∈EL and (m; w) ∈LL then (m; w) is a language element. That is, language elements are knowledges. As required by the principle of compositionality the language universe of discourse and the language parlance are modeled by similar universal algebras called semantics, SemL , and syntax, SynL , respectively. Word-expressions w ∈ SynL are constructed by composition operations ◦Syn from components w1 ; : : : ; wn ∈ SynL , i.e., w = ◦Syn (w1 ; : : : ; wn ). The relationship between w ∈ SynL and E(w) ∈ SemL is given by the identity: E(◦Syn (w1 ; : : : ; wn )) = ◦Sem (E(w1 ); : : : ; E(wn )); where ◦Sem =EL (◦Syn ). That is, this construction provides a mathematically well de#ned concept of a language where syntax and semantics are similar universal algebras. Examples of schema usage for language speci#cation are: A Montague language: Consider
N; F, ,∈- to be a signature [49] and let X be any N -sorted set (of variables and constants). Then SynL =
Syn(N; X ); Syn(F, ),∈-  may be constructed as a Montague syntax, i.e., a word-algebra generated by Syn(F, ),∈in terms of the elements in X . Similarly, SemL =
Sem(N ); Sem(F, ),∈-  may be constructed as a Montague model and EL : SynL → SemL as a Fregean interpretation [24]. The construction of LL : SemL → SynL provides the integration of the syntax and the semantics into a language. A programming language: Consider N to be the set of syntax categories used for the speci#cation of a programming language and (F, ),∈- to be the set of BNF rules used to specify the syntax of that programming language. Then SynL =
Syn(N ); Syn(F, ),∈-  is the algebra of the derivation trees speci#ed by the grammar GN =
N; T; (F, ),∈- ; Z ∈ N , where T is the set of lexical elements de#ned by Syn(F, ),∈- . Similarly, SemL =
Sem(N ); Sem(F, ),∈-  is an algebra of actions [27] and EL : SynL → SemL is an interpretation homomorphism and LL : SemL → SynL provides the integration mechanism. 2.5. Galois connection as a criterion for consistent language usage Let
P; 6P  and
Q; 6Q  be two preorders (i.e., 6P and 6Q are rePexive and transitive relations). Two mappings f : P → Q and g : Q → P form a Galois connection [1] if ∀p ∈ P and ∀q ∈ Q, f(p) 6Q q iI (if and only if ) p 6P g(q). We show here that for a language L =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL  speci#ed by a schema S, the two language components SynL and SemL are preorders. Then, the formal characterization of the consistent language usage relies on the con-

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struction of LL ⊆ SemL × SynL and EL ⊆ SynL × SemL such that they form a Galois connection. To show that SemL and SynL are preorders we de#ne their compositionality relations 4sem ⊆ SemL × SemL and 4syn ⊆ SynL × SynL as follows. For m1 ; m2 ∈ SemL we say that m1 is a component of m2 and denote it by m1 4sem m2 , iI m1 is identical to m2 or exists an operation ◦Sem such that m2 = ◦Sem (: : : ; m1 ; : : :). For w1 ; w2 ∈ SynL we say that w1 is a component of w2 and denote it by w1 4syn w2 , iI w1 is identical to w2 or exists an operation ◦Syn such that w2 = ◦Syn (: : : ; w1 ; : : :). Note that 4sem and 4syn are antisymmetric, i.e., ∀m1 ; m2 ∈ SemL , m1 4sem m2 and m2 4sem m2 implies m1 = m2 and ∀w1 ; w2 ∈ SynL , w1 4syn w2 and w2 4syn w1 implies w1 = w2 . Moreover, m1 4sem m1 and w1 4syn w1 (by de#nition), i.e., 4sem and 4syn are rePexive. If m1 4sem m2 and m2 4sem m3 then m2 = ◦sem (: : : ; m1 ; : : :) and m3 = ◦sem (: : : ; m2 ; : : :) by de#nition. Hence, m3 = ◦sem (: : : ; m2 ; : : :) = ◦sem (: : : ; ◦sem (: : : ; m1 ; : : :); : : :) and thus m1 4sem m3 , i.e., 4sem is also transitive. Similarly, 4syn is a transitive relation. That is, the tuples
SemL ; 4sem  and
SynL ; 4syn  are preorders. Fact 0. If LL ⊆ SemL × SynL and EL ⊆ SynL × SemL form a Galois connection, then for c; c two communicators using the language L, (m; w) ∈ Learnc ◦ Generatec and (w; m ) ∈ Interpretc
◦ Evalc
implies m = m . Proof. (m; w) ∈ Learnc ◦ Generatec implies (m; w) ∈LL and (w; m ) ∈Interpretc
◦ Evalc
implies (w; m ) ∈EL (see Section 2.2). That is w ∈LL (m) and m ∈EL (w). Since LL and EL form a Galois connection, w 4syn w implies m 4sem m. (m; w) ∈LL and (m; w) knowledge implies (w; m) ∈E, i.e., m ∈EL (w); similarly, w ∈LL (m ). Hence, m 4sem m , i.e., m = m . Thus, consistency of language usage is expressed by the following proposition: Language L =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL  can be consistently used if the relations LL and EL de4ne a Galois connection between the preorders
SemL ; 4sem  and
SynL ; 4syn , that is, ∀m ∈ SemL ; ∀w ∈ SynL , ∀w ∈LL (m) and ∀m ∈ EL (w) w 4syn w i; m 4sem m. In Section 3.3 we give examples of schema S usage for the speci#cation of SynL and SemL and show the construction of computable L and E that form a Galois connection and thus ensure consistency of language usage.

3. Example language structures The mathematical concept of a language is illustrated in this section by sketching the structure induced by a schema S on natural languages, logical languages, and programming languages. The goal of this illustration is to show the unifying properties of this structure rather then to construct eIectively these languages. Hence, each example is discussed by showing language components used in language de#nition: semantics, syntax, the learning, and the evaluation relationships.

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3.1. Natural language A natural language is a tuple Ln =
Un ; Pn ; Ln ⊆ Un × Pn ; En ⊆ Pn × Un  where Un is the universe of discourse, Pn is the collection of utterances which express entities of Un as knowledges, Ln allows language users to interact with their universe of discourse to generate knowledges, and En allows language users to interpret elements of Pn during their communication. Natural languages are social conventions which result from repeated usage of particular symbols to represent particular entities of the universe of discourse. The structure of the language components is provided by the universe of discourse whose entities are classi#ed as objects, actions and events, and properties. The fundamental relationship among the entities of Un is that of “an entity being a component (or a constituent) of another entity, including itself ”. Hence, a speci#cation schema for a natural language is a tuple S =
N;
F, ,∈- ; Syn; Sem where Syn(N;
F, ,∈- ) = Pn , Sem(N;
F, ,∈- ) = Un , Ln is the natural process of learning, and En is the identi#cation of a word-expression with the entity it denotes. Since Un , Pn , Ln , and En are given, the speci#cation schema S is rather a theory (called the grammar) explaining an approximation of a language taken as a snapshot in the in#nite process of language evolution. Montague [25] and Janssen [19] have constructed the algebras Syn(N;
F, ,∈- ) and Sem(N;
F, ,∈- ) for fragments of English language. The excessive complexity of these constructions prevents us from reproducing them here. This complexity measures the diCculty of #tting social conventions within the logical framework of algebra rather than the complexity of natural language. The conventional nature of the language elements allows the usage of language wordexpressions as references for the elements of the universe of discourse. The evolutionary nature of this referential relationship induces the lexical ambiguity in the language by which the same utterance may be used to represent diIerent entities and diIerent utterances to represent the same entity. Lexical ambiguity is inherited by valid language word-expressions and is further extended into syntactic and semantic ambiguities by the rules that build the natural language hierarchy. Informally these rules can be speci#ed as follows: Language lexicon: is a collection of words available at a give time as references for the entities of Un . Since entities of Un are classi#ed as objects, actions and events, and properties, the lexical elements are themselves classi#ed as nouns, used to represent objects, verbs used to represent actions and events, and modi4ers used to represent properties of nouns and verbs. However, due to lexical ambiguity, words of one category of lexical elements (walk, noun) may also be used to represent other categories of lexical entities, (walk verb). Consequently, only the concrete context of utterance usage may decide precisely what is the entity of Un referred to by a speci#c usage of a word. Basic propositions: are word-expressions used to represent the fundamental relationship of Un : the entity e1 is a component of the entity e2 where e1 and e2 may not be diIerent, denoted e1 4Un e2 . The structure of basic propositions is a free juxtaposition of nouns, verbs, and modi#ers, expressing the relationship 4Un . Though this juxtaposition may be #xed for a language over a given time period, one cannot conclude that it will not change during the language evolution process. We accept that a basic propo-

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sition has the standard structure functor argument modi4er. The components functor, argument, modi4er of a basic proposition are lexical elements which represent verbs, nouns, add-verbs (adverbs) and add-nouns (adjectives), respectively, and can be permuted in the context of the use according to given juxtaposition relations, which we consider #xed for a language approximation. Composed propositions: are word-expressions whose components are basic propositions, thus representing complex relationships in the universe of discourse. The representation of the composed propositions extends the language lexicon with words used to denote various composition rules such as conjunction, prepositions, determiners, etc. Again, the expression of complex propositions is a convention that evolves from the historical usage of the language. Consequently, there are no inherently wrong expressions in a natural language. There are only grammatical and ungrammatical propositions, where grammar is a set of rules, deduced from the historic usage of the language. Semantic and pragmatic functioning of word-expressions in natural languages is based on three diIerent reference mechanisms of the words composing word-expressions: symbol, index, name. A symbol refers to the entity it represents by matching its Iconcept; an index refers to the entity it represents by identifying a symbol reference anaphoric (i.e., pointing back in the sequence of word-expressions) or cataphoric (i.e., pointing ahead in the sequence of word-expressions); a name refers to the entity it represents by naming that entity and then matching this name. By language de#nition both the universe of discourse Un and the set of word-expressions Pn are #nitely speci#ed. By the rules for language usage the speaker cognition environment, EnvS , and the auditor cognition environment, EnvA , are embedded in Un . That is, the context of a communicator’s cognition environment can be de#ned as the portion of Un which coincides with the environment of the respective communicator. The fundamental laws for contextual-evaluation of word-expressions to their meanings were suggested by De Saussure [44] and are: (i) the linguistic words (signs) used as basic constructs of a language parlance are arbitrary (for example, go in English and aller in French represent the same action), and (ii), words (signs) composing a word-expression are time-linear ordered. That is, words composing phrases are sequentially ordered in space and phrase evaluation to its meaning is sequential in time following word ordering. Hausser [13] used these laws to design the database semantics for natural languages and to develop the context-based semantic evaluation mechanism. The compositionality relations 4Pn and 4Un and the consistency criteria for the language usage are as de#ned in Section 2.5. 3.2. Logical language A logical language LL =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL  is de#ned by a schema S =
N;
F, ,∈- ; Syn; Sem, where SynL = Syn(N;
F, ,∈- ) is the word algebra of formulas, SemL = Sem(N;
F, ,∈- ) is the algebra of an intensional logic, and LL and EL are homomorphisms between these algebras. A construction of these components of a logical language is shown in [19]. The three layers of the language

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513

Table 2 The speci#cation of SynL by the functor Syn

3∈4

Syn(3)

Syn(F; 3)

t f

true false

true ∈ Syn(F) false ∈ Syn(F) if f ∈ Syn(F) then ¬f ∈ Syn(F) if f1 ; f2 ∈ Syn(F) then f1 ∧ f2 ∈ Syn(F) if f1 ; f2 ∈ Syn(F) then f1 ∨ f2 ∈ Syn(F) if f ∈ Syn(F) then AX f ∈ Syn(F) if f ∈ Syn(F) then E X f ∈ Syn(F) if f1 ; f2 ∈ Syn(F) then A[f1 Uf2 ] ∈ Syn(F) if f1 ; f2 ∈ Syn(F) then E[f1 Uf2 ] ∈ Syn(F)

¬ ∧ ∨ ✷ ✸ ✷u ✸u

¬ ∧ ∨

AX EX AU EU

hierarchy are: (1) The lexicon consists of a collection of symbols denoting variables and constants. Though this collection may be in#nite it is #nitely speci#ed. (2) The basic propositions are obtained by using F, ; , ∈ -, as signature of operations over the lexicon. (3) The composed propositions are obtained by considering basic propositions as a family of constructions closed with respect to the action of Syn(F, ); Sem(F, ); , ∈-, on this family. The lexical ambiguity is avoided in logical languages by employing diIerent lexical elements to denote diIerent variables and constants. The ambiguity is further avoided by the functor Syn which maps names in N into syntax categories, and operators F, ; , ∈ -, into string constructors, ensuring that any well-formed word-expression has a unique representation of the form Syn(F, )(w1 ; : : : ; wk ) of category Syn(n) for some F, : n1 × · · · × nk → n, k¿0, for n; n1 ; : : : ; nk ∈ N , not necessarily diIerent, where wi are of category Syn(ni ), i = 1; : : : ; k, and Syn(n1 ) × · · · × Syn(nk ) → Syn(n) is the signature of Syn(F, ). Example of schema application for the speci#cation of a logical language is the speci#cation of CTL [4] by the schema S =
{F}; 4 = {t : ∅ → F; f : ∅ → F; ¬ : F → F; ∧ : F × F → F; ✷ : F × F → F; ✸ : F → F; ✷u : F × F → F; ✸u : F × F → F}; Syn; Sem. The set of CTL formulas are de#ned by Syn functor as seen in Table 2. Let M =
S; E; P : AP → 2S  be a directed graph where S is a #nite set of nodes, E is a #nite set of directed edges, and P is a function which labels each node with logical propositions. For each s ∈ S, succ(s) = {s ∈ S | (s; s ) ∈ E} and we assume succ(s) = ∅. A path in M is an in#nite sequence of nodes (s0 ; s1 ; s2 ; : : :) such that ∀i; i¿0; (si ; si+1 ) ∈ E. The labeling function P maps an atomic proposition given in AP to the set of nodes in S on which that proposition is true. A model M of CTL formulas is constructed by the functor Sem as an algebra of subsets of S as shown in Table 3, where LFP is a “least #xed point” operator. The satisfaction of CTL formulas is de#ned by the homomorphism ECTL : SynCTL → SemCTL where SynCTL is in Table 2 and SemCTL is in Table 3. The learning relation LCTL : SemCTL → SynCTL allows the user of CTL to express the model’s constraints using CTL formulas and is also a homomorphism [43].

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Teodor Rus / Theoretical Computer Science 281 (2002) 499 – 536 Table 3 The speci#cation of SemL by the functor Sem

3

Sem(3)

Ssem(F; 3)

t f

S

S ∈ Sem(F) ∅ ∈ Sem(F) if s ∈ Sem(F) then C(s) = Sem(F)\s if s1 ; s2 ∈ Sem(F) then s1 ∪ s2 ∈ Sem(F) if s1 ; s2 ∈ Sem(F) then s1 ∩ s2 ∈ Sem(F) Nextall (s) = {s ∈ S | succ(s ) ⊆ s} Nextsome (s) = {s ∈ S | succ(s ) ∩ s =  ∅} LFPall (s1 ; s2 ) = s2 ∪ {s ∈ s1 | succ(s) ⊆ LFPall (s1 ; s2 )} LFPsome (s1 ; s2 ) = s2 ∪ {s ∈ s1 | (succ(s) ∩ LFPsome (s1 ; s2 ) = ∅}

¬ ∧ ∨ ✷ ✸ ✷u ✸u

∅ C ∪ ∩

Nextall Nextsome LFPall LFPsome

The two algebras SynL and SemL may be freely generated as seen in the CTL example. In that case the mappings EL : SynL → SemL and LL : SemL → SynL with the properties required in the language de#nition exist by virtue of the universal construction, shown in Section 3.3, which ensures that any function de#ned on the generators of an algebra with values in the carrier of a similar algebra is uniquely extensible to a homomorphism [10]. Note: SemL and SynL are partially ordered by the compositionality relations 4sem and 4syn de#ned in Section 2.5. Since SemL and SynL are generated by #nite sets of generators the homomorphisms LL and EL can be constructed such that they are inverse to each other. Hence, they form a Galois connection and thus ensures the consistency of logical communication. 3.3. Programming language A programming language LP =
SemP ; SynP ; LP ⊆ SemP × SynP ; EP ⊆ SynP × SemP  is speci#ed by a schema S =
N;
F, ,∈- ; Syn; Sem where SemP = Sem(N;
F, ,∈- ) is an algebra of computations expressible in the programming language LP , SynP = Syn(N;
F, ,∈- ), is the algebra of well-formed expressions in the language LP , and LP and EP are relations such that ∀(m; w) ∈ SemP × SynP , (m; w) ∈LP implies that w is an expression of m in SynP and (w; m) ∈EP implies that w is evaluated by an abstract machine characterizing LP to m. Each F, ; , ∈ -, is speci#ed by an equation of the form n = t0 n1 t1 : : : tk−1 nk tk , for some n; ni ∈ N , 16i6k, and ti , 06i6n, some #xed strings (including 6) not in N . For a given speci#cation rule F, : n = t0 n1 t1 : : : tk−1 nk tk , n is the left hand side of the rule and is denoted by lhs(F, ) and t0 n1 t1 : : : tk−1 nk tk is the right hand side of the rule and is denoted by rhs(F, ). Let S be the set of #xed strings used by rules F, ; , ∈ -, and W =
N ∪ S; ◦; 6 be the semigroup of words with unity 6 generated by concatenation ◦ over the alphabet V = N ∪ S. The syntax algebra, SynP =
{Syn(n) | n ∈ N }; {Syn(F, ); , ∈ -}, is a context-free algebra [15] constructed by the rules: (1) For each n ∈ N , Syn(n) is a family of well-formed word-expressions called the syntactic category n, whose elements denote computations.

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(2) For each rule F, ; , ∈ -, F, : n = t0 n1 t1 : : : tk−1 nk tk , Syn(F, ) is an algebraic operation Syn(F, ) : Syn(n1 ) × · · · × Syn(nk ) → Syn(n) which constructs elements w0 ∈Syn(n) from elements wi ∈ Syn(ni ), 16i6k, by the rule: Syn(F, )(w1 ; : : : ; wk ) =t0 w1 t1 : : : tk−1 wk tk . Fact 1. SynP is embedded in W =
N ∪ S; ◦; 6 by derived operations. The compositionality relation 4syn on SynP de#ned in Section 2.5 becomes: for all w; w ∈SynP , w 4syn w iI w = w or there is F, ; , ∈-, such that w = Syn(F, )(w1 ; : : : ; wk ) and w 4syn wi for some i, 16i6k. Using this relation we can introduce the context of w ∈ SynP as the set w[ ] of strings over the alphabet of the language that contains a hole, denoted by [ ]. When this hole is replaced by w an element w [w] ∈ SynP is obtained, i.e., w[ ] = {w1 [ ]w2 | w1 ww2 ∈ SynP }. We assume that for w ∈ w[ ], w [w] = w1 ww2 such that w 4syn w1 . The semantics algebra SemP =
{Sem(n) | n ∈ N }; {Sem(F, ); , ∈ -} is a universal algebra chosen by the language designer by the following rules: (1) For each n ∈ N , Sem(n) is a family of computations, called the semantic domain n, that may be expressed in the language. (2) For each rule F, ; , ∈ -, F, : n = t0 n1 t1 : : : tk−1 nk tk , Sem(F, ) is an algebraic operation Sem(F, ) : Sem(n1 ) × · · · × Sem(nk ) → Sem(n) which constructs elements c ∈ Sem(n) from elements ci ∈ Sem(ni ), 16i6k, by the computation rule associated with Sem(F, ), denoted here by Sem(F, )(c1 ; c2 ; : : : ; cn ) = c. Computations SemP are speci#ed in terms of three generic elements [41]: the universe of discourse de#ned by a collection of types, T, the state, 4, de#ned as a collection of mappings assigning values of types t ∈ T to sets of names (variables and constants), and state transitions, T, that are operators mapping states before computations into states after computations. The collection of types T contains the type called computation process, 1 denoted P, whose values are processes performing given computations. We assume that a computation operates on a set D, of typed data variables and constants, that denote values in the universe of types, and a set C, of control variables and constants, that denote processes in P. To simplify presentation, we use here only two control variables: ↑ which identi#es a computation process before its execution, and ↓ which identi#es a computation process after its execution. Formally, the three elements used to specify computations are: (1) T is an algebra of types, T =
T; 8, where T is a given set of prede#ned types, which are available to every computation, and 8 is a set of type constructors that can be used by a computation to construct new types in T from the available types. The scope of a type thus constructed is determined by the computation which de#nes it. For t ∈ T, V(t) is the set of values of type t. We assume that there are two universal values, @, which denotes any good value of any type,

1

A process is a tuple P = Agent; Action; Status where Agent is speci#ed by the set of data types it recognizes, Action is an expression in the language of the Agent describing a computation activity, and Status is the status (active, passive, etc.) of the computation performed by Agent.

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and ?, which denotes any bad value of any type. Moreover, @t is the good value of type t and ?t is the bad value of type t. (2) 4 is a set of mappings 3 : D ∪ {↑; ↓} → V(t); t ∈ T, such that for x ∈ D and t ∈ T 3(x) ∈V(t), 3(↑) identi#es the process that performs next, and 3(↓) identi#es the process that performed previously. 9(3)

(3) T is a set of transitions
T; 3 : D ∪ {↑; ↓} → V(t); t ∈ T  −→
T  ; 3 : D → V(t); t ∈ T  , where T; T  ⊆ T. If ↑ and ↓ enclose the expression w of the transformation 9w   ; 3 ; w↓ where 9w performed by 9, the expression of 9 becomes
T; 3; ↑w −→
T denotes the process performing the transformation w with variables bound to values de#ned in the state 3. Abstracting the expression of the transformation performed 9 by a transition 9 we may represent it by
T; 3 →
T  ; 3 . A computation is an in#nite sequence of transitions [29], as seen below. 9

9

9i−1

9

9i+1

i 1 2
T1 ; 31  −→
T 2 ; 32  −→ · · · −→
Ti ; 3i  −→
Ti+1 ; 3i+1  −→ · · · :

The initial state of a computation is a state whose variables and constants satisfy an initial condition : : D ∪ {↑; ↓} → {true; false}; the #nal state of a computation is a state whose variables and constants satisfy a #nal condition ; : D ∪ {↑; ↓} → {true; false}. Fact 2. The set T of transitions forms a semigroup with unity. 9

9

Proof. We de#ne the composition of two transitions
T1 ; 31  →1
T2 ; 32  and
T2 ; 32  →2 91 ◦ 9 2 91 92
T3 ; 33 , by
T1 ; 31  −→
T2 ; 32  −→
T3 ; 33 . It is obvi
T3 ; 33 , denoted
T1 ; 31  −→ < ous that 91 ◦ (92 ◦ 92 ) = (91 ◦ 92 ) ◦ 93 . The identity transition, <, is:
T; 3 →
T; 3. Note that < is both a left and right unity for the transition composition operation ◦, i.e., ∀9 ∈ -: 9 ◦ < = < ◦ 9 = 9. The semantics algebra SemP of a particular language is obtained by: (a) letting Sem(n), for each n ∈ N , be a set of transitions of the semigroup Tn =
T(n); ◦; < where 9 ∈T(n) represents a type, a state, or a state mapping, and (b) letting Sem(F, ), for each , ∈ -, be an algebraic operation on transitions. The expression of Sem(n) for n ∈ N is given by the formula: Sem(n) =  <  n ∈ T; 
n; ∅; ∅ → V(n) →
n; ∅; ∅ → V(n); < n ∈ 4;
T; D; 3 : D → V(t); t ∈ T  →
T; D; 3 : D → V(t); t ∈ T ;  9 
T; D; 3 : D → V(t); t ∈ T  →
T  ; D ; 3 : D → V(t); t ∈ T  ; n ∈ T: Note: Types n ∈ T are transitions de#ned by the operator that maps the function ∅ → V(n), which de#nes the set of values of the type, into itself; states
T; D; 3 ∈ 4 are transitions de#ned by the operator that maps the function 3 into itself; transitions 9 =
T; D; 3 →
T  ; D ; 3  ∈ T are de#ned by the operator that maps the function 3 into the function 3 . For F, : n = t0 n1 t1 : : : tk−1 nk tk , Sem(F, ) is constructed as follows: ∀wi ∈ Syn(ni ), 16 i6k, and w = Syn(F, )(w1 ; : : : ; wk ) ∈ Syn(n), the transition Sem(w) performed by w is

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517

expressed in terms of transitions Sem(wi ), 16i6k, performed by the components wi of w, 16i6k. That is, Sem(F, ; w) = EF, (Sem(w1 ); : : : ; Sem(wk ); ◦; <), where EF, is a particular composition of transitions Sem(ni ), 16i6k, using the laws ◦ and <. Hence, SemP =
{Tn ; n ∈ N }; {EF, ; , ∈ -}; ◦; <. Examples are given in [42]. Fact 3. SemP is embedded in the semigroup T by derived operations. The compositionality relation 4sem on SemP de#ned in Section 2.5 becomes: for all 9w
9w
T1 ; D1 ; 31 ; w↓,
T2 ; D2 ; 32 ; ↑w  →
T2 ; D2 ; 32 ; w ↓, 9w 4sem transitions
T1 ; D1 ; 31 ; ↑w →  9w
iI 9w
= E(: : : ; 9w ; : : :). Note that if w ∈ w[ ] then 9w 4sem 9w
. If w(9) is the expression of the computation performed by 9 then one can see that w(91 ) 4syn w(92 ) implies 91 4sem 92 and vice versa. To construct the relations LP ⊆ SemP × SynP and EP ⊆ SynP × SemP used in the language de#nition we #rst observe that the following facts are true: Fact 4. SemP is freely generated by the signature {Sem(F, ); , ∈ -} whose generators are the set of transitions SemP0 = Con(SemP ) ∪ Var(SemP ), where Con(SemP ) = {
∅; t; ∅ 9F,

9F,

−→
n; t; t → tn  | F, : n = t ∈S; n ∈ N ; t ∈ S} and Var(SemP ) = {
∅; x; x → @ −→
n; x; x → @n  | F, : n = X ∈S; p; X ∈ N ; x ∈ Syn(X )}. Fact 5. SynP is the term algebra generated by the signature {Syn(F, ) | , ∈ -} whose generators are the lexical elements SynP0 = Con(SynP ) ∪ Var(SynP ) of SynP where Con(SynP ) = {t | F, : n = t ∈S; n ∈ N; t ∈ S} and Var(SynP ) = {x | F, : V = X ∈S; V; X ∈ N ; x ∈ Syn(X )}. The mapping LP is de#ned as follows: (1) First we de#ne the function LP0 : SemP0 → SynP by the equality: 9

w ∀9 ∈ Sem0P ∧ 9 =
∅; w; = →
n; w ; =  set L0P (9) = n

Note that by construction n is the token name of the class of lexical elements 9 speci#ed by F, ; , ∈ -. Hence, for 91 ; 92 ∈ SemP0 de#ned by
∅; w1 ; =1  →1
n1 ; w1 ; =1 , 92
∅; w2 ; =2  →
n2 ; w2 ; =2 , 91 = 92 implies n1 = n2 . Thus, LP0 is a surjection. (2) The mapping LP : SemP → SynP is now de#ned by the equality:  0 if 9 ∈ Sem0P ; LP (9) LP (9) = Syn(F, )(LP (91 ); : : : ; LP (9k )) if 9 = Sem(F, )(91 ; : : : ; 9k ): (3) Since SemP and SynP are similar and SemP is freely generated by SemP0 , the unique extension lemma [3] ensures that this function is the unique extension of LP0 to the homomorphism LP : SemP → SynP . Note that since LP0 is a surjection, LP is a surjection as well. The mapping EP is de#ned as follows: (1) First we de#ne the function EP0 : SynP0 → SemP by the following procedure: ∀w ∈ SynP0 , let Syn(F, ) be the rule that speci#es w, that is, lhs(F, ) = n, for some token n, and rhs(F, ) matches w. If rhs(F, ) is a constant then w = rhs(F, ) and there is a

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Teodor Rus / Theoretical Computer Science 281 (2002) 499 – 536 9rhs(F, )

transition 9 ∈ SemP0 , namely, 9 =
∅; rhs(F, ); ∅ −→
lhs(F, ); rhs(F, ); rhs(F, ) → c ∈ V(lhs(F, ) and EP0 (w) = 9; if rhs(F, ) is a variable then ∃9 ∈ SemP0 , namely 9 =
∅; 9lhs(F, )

{x; x → @ | x ∈ Syn(rhs(F, )} −→
lhs(F, ); {x; x → @lhs(F, ) | x ∈ Syn(rhs(r))} such that w ∈ Syn(rhs(r)) and EP0 (w) = 9. Note that EP0 maps classes of lexical elements into transitions, therefore it de#nes an equivalence relation on SynP0 where two lexical elements w; w ∈ SynP0 are equivalent, w ≡0 w , if EP0 (w) =EP0 (w ). Hence, for 9 ∈ SemP0 , SynP0 =9 = {w ∈ SynP0 | EP0 (w) = 9} is an equivalence class on SynP0 . Denote by SynP0 =≡0 the quotient of SynP0 with respect to ≡0 . Then EP0 : SynP0 =≡0 → SemP de#ned by EP0 (SynP0 =9 ) = 9 is an injection. (2) Now the mapping EP : SynP → SemP is de#ned by the equality:  0 if w ∈ Syn0P ; EP (w) EP (w) = Sem(F, )(EP (w1 ); : : : ; EP (wk )) if w = Syn(F, )(w1 ; : : : ; wk ): (3) Since SynP is freely generated by SynP0 and SynP is similar to SemP , the function EP0 : SynP0 → SemP has a unique extension to the homomorphism EP : SynP → SemP . The equivalence ≡0 on SynP0 is extended to the equivalence ≡ on SynP de#ned by: ∀w; w ∈ SynP :w ≡ w iI EP (w) = EP (w ). That is, EP : SynP =≡ → SemP is an injection. Fact 6. LP and EP de#ne a Galois connection. Proof. The relations 4sem on SemP and 4syn on SynP have been de#ned componentwise. Since 4sem and 4syn are rePexive and transitive, the tuples
SemP ; 4sem  and
SynP ; 4syn  are preorders. Since LP is a surjection and EP is an injection it results that LP ◦EP = 1SemP . Assume that LP (9) 4syn w, i.e., LP (9) = w and w ∈ w [ ]. This means that there is F, ; , ∈ - such that w = Syn(F, )(: : : ; w ; : : :). Hence, EP (w) = Sem(F, )(: : : ; EP (LP (w )) : : :). Since LP ◦ EP = 1SemP we have E(w) = Sem(F, )(: : : ; 9; : : :), i.e., 9 4sem EP (w). Now assume that 9 4sem EP (w) and let w = LP (9). Then, there is F, ; , ∈ -, such that w = Syn(F, )(: : : ; w ; : : :). That is, w ∈ w [ ] and thus w = LP (9) 4syn w. Hence, ∀9 ∈ SemP and ∀w ∈ SynP we have LP (9) 4syn w iI 9 4sem EP (w) and thus LP and EP de#ne a Galois connection. 4. Language speci%cation Here we analyze a language speci#cation schema S =
N;
F, ,∈- ; Syn; Sem from three viewpoints: (i) the structural properties of the operation schemes, (ii) the universal algorithms supplied by S in the languages it speci#es, and (iii) the support S provides for automatic generation of language processing tools. 4.1. Structural properties of a speci4cation schema Structural properties of a speci#cation schema S =
N;
F, ,∈- ; Syn; Sem are discussed here in view with the properties induced by S in the languages it speci#es. These properties are determined by the structure of the operation schemes F, ; , ∈ -,

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519

and the functors Syn and Sem mapping these schemes of operations into algebraic operations. For that we recall that each operation scheme F, , , ∈ -, is associated with three elements: (1) a symbol denoting the operation it speci#es, (2) the signature of this operations, and (3) the operation law performed by this operation. That is, each F, ; , ∈ - is associated with a triple
Sign(F, ); Sig(F, ); Op(F, ) where Sign(F, ) is a constant symbol chosen by the functors Syn and Sem, respectively, to denote the operation Op(F, ) performed by Syn(F, ), and Sem(F, ), respectively. The Sig(F, ) is however given in S as an operation scheme n1 × · · · × nk → n, for n; n1 ; : : : ; nk ∈ N , not necessarily distinct. For F, ; , ∈ -, of signature Sig(F, ) = n1 × · · · × nk → n we use the following notations: Dom(F, ) = (n1 ; : : : ; nk ), Ran(F, ) = n, domain selector set of F, , DS(F, ) = {n | n ∈ Dom(F, )}. Note that Dom(F, ) is a vector, DS(F, ) is a set, and Ran(F, ) is an element. An operation scheme F, is recursive if Ran(F, ) ∈ DS(F, ). A schema S is recursive if it contains at least one recursive operation scheme. To structure the set F = {F, | , ∈ -} of the operation schemes of a schema S, F can be sorted on the following hierarchy: F0 = {F, | DS(F, ) = ∅}; Dom(F0 ) = {Ran(F, ) | F, ∈ F0 }; F1 = {F, | DS(F, ) ⊆ Dom(F0 }; Dom(F1 ) = {Ran(F, ) | F, ∈ F1 };  F2 = F, | DS(F, ) ⊆

1  i=0

 Dom(Fi ) ; Dom(F2 ) = {Ran(F, ) | F, ∈ F2 };

.. .

  m−1  Dom(Fi ) ; Dom(Fm ) = {Ran(F, ) | F, ∈ Fm }: Fm = F, | DS(F, ) ⊆ i=0

Fact 7. For each schema S with a #nite set of operation schemes F, there is an m¿0 called the hierarchy degree of S such that F =F0 ∪F1 ∪ · · · ∪Fm where Fi ∩Fj = ∅, 0 6 i = j 6 m. Proof. Obvious from the construction of Fi , i = 0; 1; : : : ; m. This structuring allows us to de#ne an order relation on the set of operations schemes F, ∈F of a schema S by: F, ≺ F,
(read F, precedes F,
in F) iI F, ∈Fi , F,
∈Fj and i¡j. This order relation is preserved by the functors Syn and Sem and thus allows the automatic development of automata-based [33] and pattern-matching based [41] algorithms for generating and analyzing the languages speci#ed by the schema S. The structuring of the operations schemes F, ; , ∈ - of a schema S on layers and the ordering relations between the operation schemes depend only on the signature of these operations. Therefore, this structuring is independent of the functors Syn and Sem which construct the language algebras. Consequently, all algorithms that are based on this structuring are universal over the class of languages speci#ed by a schema

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S =
N;
F, ,∈- ; Syn; Sem. A few properties induced by this structuring in the languages speci#ed by Syn and Sem are: (1) The layer F0 of the hierarchy de#nes the free generators of the algebras Syn(N;
F, ,∈- ) and Sem(N;
F, ,∈- ). (2) The layering F0 ; F1 ; : : : ; Fm induces in the algebra Syn(N;
F, ,∈- ) the hierarchy Syn(F0 ), Syn(F1 ); : : : ; Syn(Fm ) where: (a) Syn(F0 ) is a primitive subalgebra of Syn(N;
F, ,∈- ) (see primitive subgrammar in [33], for an example of this concept). (b) The hierarchy Syn(F0 ), Syn(F1 ); : : : ; Syn(Fm ) corresponds to the decomposition of the algebra Syn(N;
F, ,∈- ) into a cascade of layers of generation Syn(N0 ;
F0 ), Syn(N1 ;
F0 ∪F1 ); : : : ; Syn(Nm ;
F0 ∪ · · · ∪Fm ). This allows the decomposition of the parsing algorithms on layer components that can be composed into a pipe-line parsing algorithm (the Algorithm 7 in [33] is an example). (3) Similar properties are induced by the layering F0 ; F1 ; : : : ; Fm in the algebra Sem(N;
F, ,∈- ). Note: The structuring of a speci#cation schema on layers of generation supports the design of tools for incremental schema development, and provides the theoretical basis for the design of incremental and interactive language processing tools. 4.2. Universal tools for language processing Language processing tools that are independent of the languages speci#ed by a schema S are determined by the structuring of the operation schemes F. This structuring is used by the functors Syn and Sem as the mechanism for the development of the processing tools that generate and analyze languages speci#ed by S. Such mechanisms employed so far are rewriting systems [6,8], 4nite automata and automata on in4nite objects [32,47,31], and universal algebras [24,37,11,19]. Rewriting systems are nondeterministic Markov algorithms [23]. The automata based tools, such as pushdown automata, #t the architecture of Von Neumann machine and are practically used to implement all other mechanisms, though they are naturally sequential. Algebraic language processing tools [28,3,38,7,40], are based on universal constructs in algebra, are compositional, and thus are naturally parallel. These properties are lost when the implementation of algebraic tools for language processing rely on automata based methods. Due to their generality, algebraic tools provide the model of a unifying methodology for language processing. We illustrate them here by pattern-matching algorithms (used to implement homomorphism based computations) and by semantic macroprocessing algorithms (used to implement isomorphic embedding of one algebra within another algebra by polynomial operations). For that, the operations schemes F, ; , ∈ -, Sig(F, ) = n1 × · · · × nk → n, n; ni ∈ N , 16i6k, are mapped by the functor Syn into equations of the form n = t0 n1 t1 : : : tk−1 nk tk , where the operation symbol Sign(F, ) = t0 t1 : : : tk is distributed on its operands thus de#ning a mix4x notation. The operation performed by Syn(F, ) is de#ned by the equality: ∀wi ∈ Syn(ni ), 16i6k, Syn(F, )(w1 ;

Teodor Rus / Theoretical Computer Science 281 (2002) 499 – 536

521

: : : ; wk ) = t0 w1 t1 : : : tk−1 wk tk ∈Syn(n0 ), and is performed by a syntax macro-processor, Msyn , that is, Msyn (Syn(F, )(w1 ; : : : ; wk )) = t0 w1 t1 : : : tk−1 wk tk . The operation scheme F, is mapped by the functor Sem into a semantic macro-operation, Sem(F, ), in a semantics expression language, SEL [34]. Sem(F, ) speci#es SEL constructs of type Sem(n) in terms of SEL constructs of type Sem(ni ), 16i6k. The SEL macro-operations that specify Sem(ni ), 16i6k, are taken by Sem(F, ) as parameters. If mi are SEL constructs of types Sem(ni ), 16i6k, then a semantics macro-processor, Msel , expands Sem(F, )(m1 ; : : : ; mk ) into a SEL construct of type Sem(n). Hence, the equation lhs(F, ) = rhs(F, ), where lhs(F, ) = n and rhs(F, ) = t0 n1 t1 : : : tk−1 nk tk , can be used by both Msyn and Msem as macro-operation name. The algorithm which generates wordexpressions in terms of their word-expression components interprets the symbols ni ∈ N as syntax categories while the algorithm which generate meanings of word-expressions as SEL constructs interpret ni ∈ N as semantics types called SEL images. That is, the speci#cation rule lhs(F, ) = rhs(F, ) is used by Syn as the speci#er of a class of wordexpressions of syntax category lhs(F, ) and by Sem as a speci#er of a class of SEL images of semantics type lhs(F, ). Furthermore, rhs(F, ) can be used as the representative structure of the class of constructs speci#ed by Syn(F, ) and Sem(F, ), respectively. Thus, a pattern-matching algorithm may use rhs(F, ) as a parameterized pattern in a process that determines if a construct w (or a SEL image m) is of syntax category (semantics type) lhs(F, ). The same algorithm, while using wi , 16i6k, to decide if w is of syntax category lhs(F, ), can generate the SEL image of w in terms of the SEL images of the components wi , 16i6k, of w, and vice versa. The information needed for the pattern-matching algorithm to perform this task is identi#ed as context and noncontext, is independent of the functors Syn and Sem, is associated with each operation scheme F, , and is de#ned as follows, where V = N ∪ S: • Context (F, : n = t0 n1 t1 : : : tk−1 nk ) is the set {(=; @) ∈ V ∗ × V ∗ } such that if a valid word (or meaning) expression w (respectively, m) has the form : : : =t0 w1 t1 : : : tk−1 wk tk @ : : : (: : : =t0 m1 t1 : : : tk−1 mk tk @ : : :), then the component t0 w1 t1 : : : tk−1 wk tk of w (t0 m1 t1 : : : tk−1 mk tk of m) is speci#ed by F, . • Noncontext (F, : n = t0 n1 t1 : : : tk−1 nk tk ) is the set {(=; @) ∈ V ∗ × V ∗ } such that if a valid word (or meaning) expression w (respectively, m) has the form : : : = t0 w1 t1 : : : tk−s wk tk @ : : : (: : : = t0 m1 t1 : : : tk−s mk tk @ : : :) then the component t0 w1 t1 : : : tk−1 wk tk of w (t0 m1 t1 : : : tk−1 mk tk of m) is not speci#ed by F, . An algorithm that computes context and noncontext for each operation scheme in F is given in [36]. Let C(F, ) and N(F, ) be the context and noncontext associated with F, and A(F, ) =C(F, ) ∩N(F, ) for each F, ∈F. If A(F, ) = ∅ for each F, ∈F then C(F, ) and N(F, ) are disjoint sets and one of these sets (usually the smaller) suCces as the decision mechanism used by the pattern-matching processor to reduce a word-expression (or to expand a macro-operation) to its equivalence class (or to the image it speci#es) while preserving the validity of the input. If A(F, ) = ∅, whenever the match of a portion of input with rhs(F, ) is discovered in a context in A(F, ) the algorithm postpones the reduction (expansion) decision. The hierarchy F0 ; F1 ; : : : ; Fm can be organized into a table of universal scheme of operations, TUSO[0..m], and the

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algorithm performed by a pattern matching parser, PMP, is: for j = 0; 1; : : : ; m for i = 0; 1; : : : ; j for each F, ∈Fi for each =; x; A; y; @ such that w = = x A y @ if rhs(F, ) matches A then if (x; y) ∈ A(F, ) and (x; y) ∈C(F, ) then w : = = x lhs(F, ) y @; else do nothing; else do nothing; if w ∈ N then Accept else Diagnose(w) This algorithm is naturally parallel in the sense that all patterns rhs(F, ), F, ∈F, can be searched in parallel in the input w. Its complexity is O(n), though depending upon F the constant of proportionality could be of the order of 106 [39]. The average time of the algorithm for an input string of n symbols is n ∗ log(n) and the worst case is n2 [22]. To improve PMP performance, relations constructive depth and recognition depth [39] between classes of hierarchy F0 ; F1 ; : : : ; Fm , that characterize the complexity of the languages speci#ed by Syn and Sem in terms of the complexity of the operation schemes in F, can be used. These relations are de#ned by: • The constructive depth of Fi is Fj , j¿i, for the smallest j, that satis#es the property: ∃F, ∈Fi , ∃F,
∈Fj , such that Ran(F, ) ∈ Dom(F,
). • The recognition depth of Fj is the class Fi , j¿i, for the smallest i, that satis#es the property: ∃F, ∈Fj , ∃F,
∈Fi such that Ran(F, ) ∈ Dom(F,
). The recognition depth allows us to control the pattern matching operation after a match using Syn(F, ); F, ∈Fj , to the recognition depth of Fj , rather than to F0 . To reduce further the amount of pattern matching performed by the parsers based on pattern-matching algorithms we rede#ne the relation ≺ on F introduced in Section 4.1 to assure that a construct speci#ed by Syn(F, ) (or Sem(F, )) can be a component of a construct speci#ed by Syn(F,
) (Sem(F,
)) only if F, ≺ F,
. For that ≺ is de#ned by: F, ≺ F,
iI Ran(F, ) ∈ Dom(F,
). An appropriate tool computes the set Next(F, ) = {(F,
; k) | F, ≺ F,
∧ Dom(F,
)[k] = Ran(F, )} for each F, ∈F. TUSO is now k organized on a hierarchy of two layers: F0 and the rest of the rules, that is i=1 Fi . m It is obvious that F0 ∩ ( i=1 Fi ) = ∅. With this decomposition, the parser based on a pattern matching algorithm has two components: a pattern matcher, PM, and a pattern jumper, PJ. PM performs pattern matching of the input using rules in F0 and records each F, which participated in a successful match together with the successful matching point p in the input. PJ tries to match only those rules which are in the Next(F, ) where F, was recorded as having had a successful match in a previous try. For that PJ positions rhs(F, ) right over the portion p of the input where a potential match may be found, ([41], 130–131). To describe the new pattern-matching algorithm, called the jumping parser, we use the following notation: w[p::q] is the substring w[p]w[p + 1] : : : w[q] of the string w, B(s) is the length of the string s, and B is a stack implementing a multiset which initially is empty. With this notation the

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523

Table 4 TUSO table of the speci#cation Fe

Rule

Context

Noncontext

Next

r4 : F = id

{∗; 6; +; 6; $; 6}

N (r4 ) = ∅

{(r3 ; 2); (r2 ; 0)}

r0 : E = T r1 : E = E + T r2 : T = F r3 : T = T ∗ F

{$; +; $; $} {6; +; 6; $} {+; 6; $; 6} {+; 6; $; 6}

{+; 6; 6; ∗} {6; ∗} {∗; 6} ∅

{(r1 ; 0)} {(r1 ; 0)} {(r1 ; 2); (r3 ; 0)} {(r3 ; 0); (r1 ; 2)}

algorithm performed by the jumping parser is: for each F, ∈F0 for each =; x; A; y; @ such that w = = x A y @ if rhs(F, ) matches A then if (x; y) ∈ A(F, ) and (x; y) ∈C(F, ) then begin w : = = x lhs(F, ) y @; Push((F, ; B(= x)); B) end else do nothing; else do nothing; while (B = ∅) let (F, ; p) : = Top(B); Pop(B); for each (F,
; k) ∈ Next(F, ) if rhs(F,
) matches w[p − k ::B(rhs(F,
))] ∧ w = = x rhs(F,
) y @ then if (x; y) ∈ A(F,
) ∧ (x; y) ∈C(F,
) then begin w : = = x lhs(F,
) y @; Push((F,
; B(= x)); B) end else do nothing; else do nothing; if w ∈ N then Accept else Diagnose(w) The #rst for loop of this algorithm is a parallel pattern-matching algorithm which updates the input as PMP did and collects information in B to be used by the second part of the algorithm. The while loop is a parallel pattern-matching algorithm that does not glide patterns over the string. Rather, it positions patterns over those portions of the input string w, where a match could potentially be found. Solutions to various implementation problems are discussed in [22,41]. We illustrate the action of this algorithm with the speci#cation Fe of a simple expression language, whose TUSO is in Table 4, where $ denotes begin and end of #le, and 6 is the empty string. The behavior of the jumping parser on id + id ∗ id is recorded in the Table 5 where 8 is used as a deleted symbol. 4.3. Properties of languages speci4ed by a schema A speci#cation schema allows us to study a language by studying its speci#cation which is a much simpler object. The fundamental problems raised by the study of

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Teodor Rus / Theoretical Computer Science 281 (2002) 499 – 536 Table 5 The behavior of jumping parser

Step

w

1. 2. 3. 4.

id + id ∗ id F + id ∗ id F + F ∗ id F +F ∗F

5. 6. 7. 8. 9. 10. 11.

T +F ∗F T +T ∗F T +T ∗F E+T ∗F E + T88 E8888 E8888

Tuple

B { } {r4 ; 1)} {r4 ; 1; r4 ; 3} {r4 ; 1; r4 ; 3; r4 ; 5}

 r 4 ; 1  r 4 ; 3  r 4 ; 5  r 2 ; 1  r 2 ; 3  r 0 ; 1  r 0 ; 3

{r4 ; 3; r4 ; 5; r2 ; 1} {r4 ; 5; r2 ; 1; r2 ; 3} {r2 ; 1; r2 ; 3} {r2 ; 3; r0 ; 1} {r0 ; 1; r3 ; 3} {r0 ; 3} { }

languages LS speci#ed by a schema S =
N;
F, ,∈- ; Syn; Sem are: (1) Is LS a #nite or an in#nite language? (2) Are the elements of LS ambiguous or not? (3) For m ∈ SemL is there an algorithm to determine w(m) ∈ SynL such that (m; w(m)) is a knowledge, i.e., (m; w(m)) ∈L and (w(m); m) ∈E? This is the language (learning) generation problem. (4) For w ∈ SynL is there an algorithm to determine m(w) ∈ SemL such that (m(w); w) is a knowledge, i.e., (m(w); w) ∈L and (w; m(w)) ∈E? This is the language (evaluation) interpretation problem. To discuss solution algorithms to these problems, independent of the arti#cial or natural character of the language, we assume that schema S is #nite (i.e., N and - are #nite sets) and the operations schemes F, ; , ∈ -, are split on the hierarchy F0 ; F1 ; : : : ; Fm discussed in Section 4.1. Finiteness of a language speci#ed by the schema S can be determined easily by an algorithm that searches the operation schema F for recursive elements, i.e., for operation schemes F, : n1 × · · · × nk → n where n ∈ {n1 ; : : : ; nk }. If no such operation scheme is discovered, then LS is #nite, otherwise it is in#nite. Since we assume that only a #nite set of operation schemes are in S an algorithm that performs this search is easily designed. However, as observed in Section 4.1, the set of operation schemes F0 is interpreted by the functors Syn and Sem as generator sets of the algebras SynL and SemL , which are closed with respect to the operations Syn(F, ), and Sem(F, ), k F, ∈ i=1 Fi . Though the set F0 is assumed #nite, it can specify a #nite or an in#nite set of generators using one of the following mechanisms: (1) Each F, ∈F0 , F, : ∅ → n; n ∈ N , speci#es a #nite (enumerated) set of words (constants and variables) {w1n ; w2n ; : : : ; wsnn } in SynL using the functor Syn and a #nite (enumerated) set of semantics objects (constants and variables) {m1n ; m2n ; : : : ; mtnn } using the functor Sem. These sets can be seen as mappings of the operation schemes F, : ∅ → n into the set of functions {∅ → w | w ∈ Syn(n)} and {∅ → m | m ∈ Sem(n)}, respectively. This is the case of a natural language dictionary.

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525

Table 6 Tabular de#nition of LL , EL on generators

F, ∈ F0

∅ → n1

∅ → n2

:::

∅ → nk

Sem

m11 ; : : : ; m1t1

m21 ; : : : ; m2t2

mk1 ; : : : ; mktk

Syn

w11 ; : : : ; ws11

w12 ; : : : ; ws22

::: ::: :::

L|E

↓ | ↑

↓ | ↑

↓ | ↑

w1k ; : : : ; wskk

(2) Each F, ∈F0 , F, : ∅ → n; n ∈ N , speci#es an indexed set of words (constants and variables) used as generators for both SynL and SemL which can be seen as mapping F, : ∅ → n into the set of functions {∅ → xi | i ∈ N}; where N is the set of natural numbers. This is the case of logical languages where F0 is mapped by the functors Syn and Sem into in#nite sets of constants and variables that can be used as generators. (3) Each F, ∈F0 , F, : ∅ → n; n ∈ N , is mapped into an equation of the form n = rhs, where rhs is a regular expression that speci#es a #nite or in#nite set of generators. This is the case of programming languages where F0 , though #nite, speci#es an in#nite set of lexical elements. The relations LL and EL can be constructed by universal constructs of algebra, de#ning them on the free generators of the algebras Syn(N;
F, ,∈- ) and Sem(N;
F, ,∈- ) and then extending them homomorphically to the entire algebras. For natural languages this is usually done using tables of the form of Table 6. For logical languages constants and variables are interpreted as denoting truth values. For programming languages, the interpretation of the generators is usually provided by attributes associated with speci#cation rules. With schema S macro-operations are used in all cases. Let LS =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL  be a language speci#ed by the schema S =
N;
F, ,∈- ; Syn; Sem, that is, SemL = Sem(N;
F, ,∈- ), and SynL = Syn(N;
F, ,∈- ). Following [24], we say that LS is syntactically unambiguous if the following two conditions hold: k (1) For every w = Syn(F, )(w1 ; : : : ; wk ) ∈ Syn(n), where F, : n1 × · · · ×nk → n ∈ i=1 Fi and wi ∈ Syn(ni ), 16i6k, w is not the value of Syn(F, ) for any F, ∈F0 . k (2) For every w = Syn(F, )(w1 ; : : : ; wk ) ∈ Syn(n), where F, : n1 × · · · × nk → n ∈ i=1 Fi and wi ∈ Syn(ni ), 16i6k, Syn(F, ) and the arguments wi , 16i6k, are unique in the syntax algebra SynL . A language LS speci#ed by schema S that does not satisfy these properties is syntactically ambiguous. If LS is an ambiguous language one can associate each w ∈ SynL with the set of analyses of w denoted Anal(w) and de#ned by Anal(w) = {Syn(F,i )(w1i ; : : : ; wni i ) | w = Syn(F,i )(w1i ; : : : ; wni i )}, where F,i : n1i × · · · × nki i → n is the signature of F,i , wji ∈ Syn(nji ), 16j6ki , and w ∈ Syn(n). Since every w ∈ Anal(w) is an element of SynL , the set Anal(w) establishes a binary relation on SynL de#ned by w ∼ = w if  ∼ w ∈ Anal(w). It is obvious that = thus de#ned is an equivalence relation on SynL and ∼ ∼ = = = =
Sem ; Syn =∼ ; L∼ consequently, the language LS L = ; EL : SynL =∼ =→ L = L : SemL → SynL =∼

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SemL  is a syntactically disambiguated language. Thus, for evaluation purposes the relation EL can always be constructed as a function. A word-expression w ∈ SynL is semantically ambiguous if there are at least two different semantic entities m1 ; m2 ∈ SemL such that (m1 ; w); (m2 ; w) ∈LL and (w; m1 ); (w; m2 ) ∈EL . The language LS is semantically ambiguous if it contains semantically ambiguous word-expressions. Semantic ambiguity can be approached similarly to syntax ambiguity. That is, we say that LS is semantically unambiguous if the following two conditions hold: k (1) For every m = Sem(F, )(m1 ; : : : ; mk ) ∈ Sem(n), where F, : n1 × · · · × nk → n ∈ i=1 Fi and mi ∈ Sem(ni ), 16i6k, m is not the value of Sem(F, ) for any F, ∈F0 . k (2) For every m =Sem(F, )(m1 ; : : : ; mk )∈Sem(n), where F, : n1 × · · · × nk → n ∈ i=1 Fi , and mi ∈ Sem(ni ), 16i6k, Sem(F, ) and the arguments mi , 16i6k, are unique in the semantics algebra SemL . A language LS speci#ed by schema S that does not satisfy these properties is semanti: ∼ cally ambiguous. However, using an equivalence : relation = on SemL ,=:similar with = de= #ned on SynL , we can consider the language LS =
SemL ==: ; SynL ; LL : SemL ==: → SynL ; : EL= : SynL → SemL ==:  which is a semantically disambiguated language, and consequently the relation LL can be constructed as a function as well. Combining the two results obtained above one can see that LS can be rede#ned : : as LS =
SemL ==: ; SynL =∼ = ; LL : SemL == → SynL =∼ = ; EL : SynL =∼ = → SemL ==  which is both syntactically and semantically disambiguated and thus LL and EL can be constructed as functions. Thus, algebraically speaking the ambiguity problem seems to be manageable from a practical point of view, though it is an unsolvable problem [17]. Practical solutions for language generation and evaluation problems of a language LS depend on the form of the speci#cation rules F, ; , ∈ -, and the functors Syn and Sem in the speci#cation schema S. The solution we advocate in this paper is based on a speci#cation schema S =
N;
F, ,∈- ; Syn; Sem, where the family of operation schemes F, ; , ∈ -, is a #nite list of rules of the form
SyntaxMacro; SemanticsMacro. For historical reasons SyntaxMacro is an equation of the form lhs = rhs, where lhs ∈ N and rhs is a pattern of the form t0 n1 t1 : : : tk−1 nk tk where n1 ; : : : ; nk ∈ N and t0 ; t1 ; : : : ; tk are #xed symbols, including the empty symbol, 6. SemanticsMacro is a macro-operation in a semantics expression language de#ning the semantic image of the word-expressions speci#ed by the SyntaxMacro. Note that t0 ; t1 ; : : : ; tn are used in the SyntaxMacro as a mix4x operation symbol, i.e., an operation symbol distributed on the operands taken by the operation it denotes. The rationale for this notation seems to be the Pavor of a natural language provided to the Algol-like programming languages originally speci#ed by such rules [30] and its potential to disambiguate word-expressions thus speci#ed. SemanticsMacro used so far as semantics speci#cation rules are denotations [26] used in the study of semantics of programming languages, attributes [9] used in compiler construction, semantics feature representations [18,12] used in natural language processing, and derived operations [19,40] used in algebraic treatment of languages. The language generation problem is solved by de#ning a process of valid wordexpression generation that starts with the identi#cation of a SyntaxMacro of the

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form Z = t0 n1 t1 : : : tk−1 nk tk , where Z ∈ N is a unique symbol called the start symbol, and continue by rewriting the symbols n1 ; : : : ; nk with the appropriate rhs-s of SyntaxMacro-s ni = rhsi , ni ∈ N , 16i6k, until no more variables are in the resulting string. The language evaluation problem is solved by a process of interpreting the steps of the language generation process. For that the derivation of a word-expression w using SyntaxMacro-s is represented by a tree called the derivation tree of w, DT (w). The root of DT (w) is labeled by the start symbol Z, the interior nodes of DT (w) are labeled by the variables rewritten at various generation steps, and the leaves of DT (w) are labeled by constant symbols used at various derivation steps (which are lexical elements of the language) [48]. We advocate the replacement of the derivation tree DT (w) by the abstract parsing tree, APT (w) [41]. The nodes of the APT (w) are labeled by the speci#cation rules used to rewrite variables at various steps of the generation process and the leaves of the APT (w) are labeled by the rules that specify lexical elements. If APT (w) (or DT (w) for that matter) is unique then w is an unambiguous element of the language and the evaluation of w can be performed by an algorithm that traverses APT (w) and evaluates SemanticsMacro-s at its nodes. Since each node of the APT (w) is labeled by a tuple (SyntaxMacro; SemanticsMacro) the process is well-de#ned and allows us to use APT as an abstract representation which describes both, the generation of w using SyntaxMacro-s and the evaluation of w using SemanticsMacro-s. In addition, there is no need for a single start symbol, that is, every valid word-expression of the language can be generated and evaluated, irrespective of its syntax category or semantic type. The algorithms performing language generation and evaluation using the APT are de#ned as follows: (1) The functor Syn is associated with a macro-processor MSyn that maps wordexpressions into abstract representations using SyntaxMacro-s as speci#cation rules, i.e., MSyn (w; S) = APT (w); (2) The functor Sem is associated with a macro-processor MSem that maps semantics entities into abstract representations using Semantics-Macro-s as speci#cation rules, i.e., MSem (m; S) = APT (m); (3) APT (w) and APT (m) have the same topological shape. The nodes of APT (w) are labeled by tuples
SyntaxMacro; SEL-image; the nodes of APT (m) are labeled by tuples
Word-expression; SemanticsMacro. (4) An algorithm that, while mapping a semantic entity m into the process of its derivation from SemanticsMacro-s, generates the syntax expression w(m) of m using SyntaxMacro-s associated with the SemanticsMacro-s provides a solution to the language generation problem (Fig. 4). (5) An algorithm that, while mapping a word-expression w into the process of its derivation from SyntaxsMacro-s, generates the semantics image m(w) of w using SemanticsMacro-s associated with the SyntaxMacro-s provides a solution to the language evaluation problem, (Fig. 5). The illustration of these solutions for language generation and evaluation problems are provided by interpreters, compilers, and translators of natural and arti#cial languages. We take a closer look at these solutions in the Section 5.2.

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Fig. 4. Language generation.

Fig. 5. Language evaluation.

5. The architecture of a language processing system This section gives a constructive de#nition of the criterion for the consistency of the usage of a language speci#ed by a schema S =
N;
F, ,∈- ; Syn; Sem, de#nes the architecture of a uni#ed language processing system, and shows how the consistency criterion for language usage can be employed as a correctness criterion for the algorithms performing various language transformations. 5.1. Consistency criterion for language usage The family of operation schemes F, ; , ∈ -, was organized in Section 4.1 into a hierarchy of layers F0 ; F1 ; : : : ; Fk which facilitates the structuring of SynL and SemL as two similar algebras, freely generated. To simplify presentation we rely here on two layers only, F0 = {F, : ∅ → n | , ∈ -; n ∈ N } and F1 = {F, : n1 × · · · × nk → n | , ∈ -; n; n1 ; : : : ; nk ∈ N; k¿1} and use the following notation: (1) The functors Syn and Sem map each F, : ∅ → n ∈F0 into a set of functions, Syn(F, )= {∅ → w | w ∈ Syn(n)}, and Sem(F, ) = {∅ → m |m ∈ Sem(n)}, respectively, which are used as free generators. We call the set of free generators a basis  and use the notation Bp = F, ∈F0 Syn(F, ) as the basis of the algebra SynL , and  Bu = F, ∈F0 Sem(F, ) as the basis of the algebra SemL . (2) The functors Syn and Sem map each F, : n1 × · · · × nk → n, k¿1, into the algebraic operations Syn(F, ) : Syn(n1 ) × · · · × Syn(nk ) → Syn(n), and Sem(F, ) : Sem(n   1) × · · · × Sem(nk ) → Sem(n), respectively. Let Rp = F, ∈F1 Syn(F, ) and Ru = F, ∈F1 Sem(F, ). Then SynL =
Bp ; Rp  and SemL =
Bu ; Ru  where Bp , Bu are the free generators and Rp , Ru are the algebraic operations generating SynL and SemL , respectively. The consistency of language usage for knowledge development and communication is formulated now by constructing LL ⊆ SemL × SynL and EL ⊆ SynL × SemL to observe the following properties: (1) ∀e ∈ Bp construct the set of meanings EL0 (e) ⊆ Bu , and ∀m ∈ Bu construct the set of expressions LL0 (m) ⊆ Be such that ∀m ∈ Bu ∃e ∈ Bp ∧ m ∈ EL0 (e) and ∀e ∈ Bp ∃m ∈ Bu ∧ e ∈LL0 (m). The feasibility of this construction is ensured by the #niteness of the bases Bp and Bu .

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Fig. 6. Semantic isomorphism of two languages.

(2) ∀rp ∈ Rp de#ne the meaning construction rules L(rp ) ⊆ Ru and ∀ru ∈ Ru de#ne the expression construction rules EL (ru ) ⊆ Rp such that ∀r ∈ Rp ∃rm ∈ LL (r) ∃rs ∈ EL (rm ) : EvalSyn (r) ∩ EvalSyn (rs ) = ∅ and ∀r ∈ Ru ∃rs ∈EL (r) ∃rm ∈LL (rs ) : EvalSem (r) ∩ EvalSem (rm ) = ∅ where EvalSyn and EvalSem are the macro-expansion processes performed by the algorithms using MSyn and MSem discussed in Section 4. During the process of developing knowledges and communicating, often communicators use languages where the same universe of discourse is paired with diIerent parlances. That is, SynL1 =
Bp1 ; Rp1 , SynL2 =
Bp2 ; Rp2 , and SemL1 = SemL2 =
Bu ; Ru . In this case, communication consistency relies on a meaning-preserving isomorphism between SynL1 and SynL2 , h : SynL1 → SynL2 . In the algebraic framework presented here, this isomorphism is shown in Fig. 6 and is de#ned as follows: SynL1 is semantically isomorphic to SynL2 iI ∀e1 ∈ Bp1 there is at least one e2 ∈ Bp2 such that EL01 (e1 ) ∩EL02 (e2 ) = ∅ and ∀r1 ∈ Rp1 there is at least one r2 ∈ Rp2 such that EL1 (Eval(r1 )) ∩EL2 (Eval(r2 )) = ∅. 5.2. Algebraic model of language translation Let L1 =
SemL ; SynL1 ; LL1 ⊆ SemL × SynL1 ; EL1 ⊆ SynL1 × SemL  and L2 =
SemL ; SynL2 ; LL2 ⊆ SemL × SynL2 ; EL2 ⊆ SynL2 × SemL  be two languages having the same universe of discourse, SemL . A translation of language L1 into the language L2 is a relation T ⊆ SynL1 × SynL2 such that (x; y) ∈ T iI EL1 (x) ∩EL2 (y) = ∅. Let SemL =
Bu ; Ru  be a speci#cation of SemL , SynL1 =
BL1 ; RL1  and SynL2 =
BL2 ; RL2  be speci#cations of SynL1 and SynL2 , respectively, de#ned by the unique speci#cation schema S. An algebraic translation of L1 into L2 is designed by the following approach: (1) Construct the two language evaluation relations EL1 : SynL1 → SemL , EL2 : SynL2 → SemL such that the consistency requirements provided in Section 5.1 are satis#ed. (2) De#ne a translation relation T ⊆ SynL1 × SynL2 by the following requirement: ∀(x; y) ∈ SynL1 × SynL2 , (x; y) ∈ T iI EL1 (x) ∩EL2 (y) = ∅. (3) Construct the translation T using the isomorphism h : SynL1 → SynL2 induced by EL1 and EL2 as follows: (a) ∀x ∈ BL1 de#ne T (x) = {y ∈ BL2 | EL1 (x) ∩ EL2 (y) = ∅}; (b) ∀r ∈ RL1 de#ne T (r) = {r  ∈ RL2 | EL1 (Eval(r)) ∩EL2 (Eval(r  )) = ∅}: Conventional design of an algebraic translation is based on the assumption that semantic elements of the language are abstract representations of valid word-expressions of the

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Fig. 7. Language processing components.

language syntax. Valid word-expressions are #nitely generated by the macro-processor MSyn from basic elements de#ning the language lexicon using macro-expansion (or rewriting) of SyntaxMacro-s. This process is described by syntax derivation trees, SynDT-s. On the other hand, meanings are #nitely generated by the macro-processor MSem from basic semantics entities using macro-expansion (or rewriting) of Semantics Macro-s. This process is described by semantic derivation trees, SemDT-s. In view of the three levels of language structuring, lexicon, syntax, semantics, an algebraic translation requires for each language to design three language processing components: a morphological component, Mrph = (MrfAnl; MrfGen), a syntax component, SynC = (SynAnl; SynGen), and a semantics component, SemC = (SemAnl; SemGen). These components of a language processing system are related by the equality MrfAnl ◦ SynAnl ◦ SemAnl = SemGen ◦ SynGen ◦ MrfGen, where ◦ is function composition, as shown in Fig. 7. Note, the analyzer of a component inputs syntax objects (such as a text) and produces their abstract representations (such as an abstract parsing tree), while the generator of a component inputs abstract representations and generates the syntactic entities they represent. Thus, the tuples Mrph, SynC, and SemC are de#ned as follows: (1) Mrph =
MrfAnl; MrfGen, is the morphological component of the system; the analyzer MrfAnl inputs words and generates lexical tokens (morphemes) and MrfGen inputs lexical tokens (morphemes) and generates words. (2) SynC =
SynAnl; SynGen, is the syntactic component of the system; SynAnl inputs tokens and outputs abstract parsing trees, APTsyn , representing syntactic structures, and SynGen inputs APTsyn representing syntactic structures and outputs tokens. (3) SemC =
SemAnl; SemGen, is the semantic component of the system; SemAnl inputs abstract parsing trees, APTsyn , representing syntactic structures and outputs abstract parsing tree, APTsem , representing the semantic structures of the input; SemGen inputs abstract parsing trees, APTsem , representing semantics structures and outputs the abstract parsing trees, APTsyn , representing the syntactic structure of the input. That is, the architecture of an algebraic translation system mapping a source language Ls into a target language Lt is shown in Fig. 8. This architecture simpli#es the complexity of a conventional translation system by using the same abstract representation for both the syntax and the semantics, provided by the abstract parsing trees, APT [41].

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Fig. 8. Algebraic implementation of a translator.

Fig. 9. Translator de#nition diagram.

5.3. Examples of algebraic translations Let Ls =
Sem; Syns ; Ls ⊆ Sem × Syns ; Es ⊆ Syns × Sem and Lt =
Sem; Synt ; Lt ⊆ Sem × Synt ; Et ⊆ Synt × Sem be two languages having the same semantic component, Sem. A translator T : Ls → Lt is a tuple (T1 ; T2 ), T1 : Sem → Sem, T2 : Syns → Synt such that the diagram in Fig. 9 is commutative. Rosetta [35] and TICS [40] are two examples of algebraic translator generators that #t this translator de#nition. The Rosetta system has been developed to handle natural language translation while the TICS system has been designed to handle compiler generation for programming languages. With Rosetta, the syntax of the source language, Syns =
Bs ; Rs , and the syntax of the target language, Synt =
Bt ; Rt , are speci#ed by Montague grammars [24], and language semantics, Sem =
B; R, is modeled by an algebra of sets. The semantic component, T1 , of the translator is the identity on Sem and the syntactic component, T2 , is the language isomorphism speci#ed by a grammar isomorphism (hb ; hr ), hb : Bs → Bt , hr : Rs → Rt , de#ned by the one-to-one correspondences hb1 : Bs → B, hb2 : B2 → B, hr1 : Rs → R, hr2 : Rt → R, using the following equalities: ∀bs ∈ Bs , hb (bs ) = bt ∈ Bt such that −1 bt = h−1 b2 (hb1 (bs )) and ∀rs ∈ Rs , hr (rs ) = rt ∈ Rt such that rt = hr2 (hr1 (rs )). Since Syns and Synt are in general diIerent, to construct the isomorphism T2 the generator sets Bs , Bt , and B of the algebras Syns , Synt , Sem, and the rules sets Rs , Rt , and R need to be equalized by a process called “grammar attuning”. While this process is diCcult to perform, there is no systematic way of doing it. With TICS, a language is a tuple L =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL  where LL and EL de#ne a Galois connection between SemL and SynL . The speci#cations Syns =
Bs ; Rs , Synt =
Bt ; Rt , Sem =
B; R are signatures of universal algebras. The language isomorphism T2 is de#ned by embedding the algebras Syns and Synt

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Fig. 10. TICS diagram of translator generation.

and Sem into a Semantic Expression Language, SEL [34]. Since SEL is a language, it is speci#ed by an appropriate schema as SEL =
SELsem ; SELsyn ; Lsel : SELsem → SELsyn ; Esel : SELsyn → SELsem . That is, TICS technology for translator implementation is shown in Fig. 10. The embeddings Embeds : Sems → SELsyn and Embedt : SELsem → Synt are implemented by semantic macro-operations as follows: (1) Let
Bs ; Rs  and
Bsel ; Rsel  be the speci#cations of Sems and SELsem , respectively, where ∀u : u ∈ Bs ∨ u ∈ Bsel , u is a constant speci#ed by a rule of the form lhs(u) = const, and ∀r : r ∈ Rs ∨ r ∈ Rsel , r is a construct speci#cation pattern of the form lhs(r) = t0 lhs1 t1 : : : tn−1 lhsn tn . (2) For each u ∈ Bs speci#ed by rsb : lhs(u) = const, construct the SEL-expression SEL(rsb ) of const using the notation provided by SELsyn ; for each u ∈ Bsel speci#ed b b by rsel : lhs(u) = const, construct the target expression T (rsel ) of const using the notation provided by Synt . (3) Associate each rs ∈ Rs , rs : lhs(r) = t0 lhs1 t1 : : : tn−1 lhsn tn , with a SEL macrooperation, SEL(rs ), that takes SEL-constructs of syntax categories lhs1 ; lhs2 ; : : : ; lhsn as parameters and expands them into the SEL-images of the constructs of syntax category lhs(r) speci#ed by rs ; associate each rsel ∈ Rsel , rsel : lhs(r) = t0 lhs1 t1 : : : tn−1 lhsn tn , with a target macro-operation, T (rsel ), that takes target constructs of syntax categories lhs1 ; lhs2 ; : : : ; lhsn as parameters and expands them into the target images of the constructs of syntax category lhs(r) speci#ed by rsel . s (4) Implement the macro-processor, Msel , that maps SEL macro-operations SEL(rs ) (@1 ; : : : ; @k ), where @1 ; : : : ; @k are SEL constructs of syntax categories lhs1 ; : : : ; lhsk , into SEL constructs of syntax category lhs(rs ) and thus embeds source language semantics into SELsyn . (5) Implement the macro-processor Mtsel that maps target macro-operations T (rsel )(@1 ; : : : ; @k ), where @1 ; : : : ; @k are target constructs of syntax categories lhs1 ; : : : ; lhsk , into target constructs of syntax category lhs(rs ) and thus embeds SELsem into Synt . Note, this methodology is Pexible and allows the translator speci#cation for languages that diIer at both components, syntax and semantics. The process of tuning is replaced by a systematic process of isomorphic embedding of one algebra within the other algebra using macro-operations. These macro-operations are called polynomials in algebra.

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Fig. 11. A correct implementation in SEL.

Fig. 12. Compiler correctness diagram.

5.4. Correctness of an algebraic translator A translator implementing the language L =
SemL ; SynL ; LL ⊆ SemL × SynL ; EL ⊆ SynL × SemL  in SEL is a tuple T = (T1 ; T2 ), T1 : SemL → SELsem , T2 : SynL → SELSyn such that the diagram in Fig. 9, where Lt is replaced by SEL, is commutative. The translator T =
T1 ; T2  is a correct implementation of the language L in SEL if T preserves the Galois connection of L in SEL. Fact 8. The tuple T =
T1 ; T2  where T1 = Embeds ◦ Esel , T2 = EL ◦ Embeds , and Embeds as de#ned in Section 5.3, is a correct implementation of L in SEL. Proof. Consider the translator diagram in Fig. 11 where T1 = Embeds ◦ Esel and T2 = EL ◦ Embeds . Let (o; wo ) ∈ SemL × SynL such that wo ∈LL (o) and o = EL (wo ). Mapping the o  where osel = Esel (Embeds (o)) and tuple (o; wo ) by the translator T we obtain (osel ; wsel o o wsel = Embeds (EL (wo )). Since EL (wo ) = o by assumption we have wsel = Embeds (o). o That is osel = Esel (wsel ). On the other hand, since osel = Esel (Embeds (o)) we have Lsel (osel ) = Lsel (Esel (Embeds (o)). Since Lsel and Esel form a Galois connections we o , which completes the have Lsel (Esel (Embeds (o)) = Embeds (o), that is Lsel (osel ) = wsel proof. A similar result holds for a correct implementation of SEL into a target language Lt . Hence, composing the correct implementation of Ls in SEL and the correct implementation of SEL in Lt we obtain the correct implementation of Ls in Lt shown in Fig. 10. Earlier research that approaches translator correctness using algebraic methods [20] do not rely on an algebraic concept of a language, where syntax and semantics are similar algebras related by a Galois connection. In this research, a translator is seen as a syntax to syntax mapping, T : Syns → Synt , and the translator correctness is expressed by the commutativity of the diagram in Fig. 12, where Sems , Semt are the

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Fig. 13. The diagram of a correct compiler for Example 2 in [20].

semantics algebras of source and target languages, respectively, and Ints , Intt , Dec, and Enc are homomorphisms. Since the corners of the diagram in Fig. 12 are not necessary similar algebras the mappings connecting them are not necessarily homomorphisms. Using embedding [41], as done in this paper, the diCculties signaled by Janssen [20] are removed. We show this in Fig. 13, which is the diagram of a correct compiler for Example 2 in [20], obtained from Fig. 10 by removing SEL details. Here we have: Ls (0) = {−0; +0}, Es (−0) = Es (+0) = 0, Lt (−0) = Et (−0) = −0, Lt (+0) = Et (+0) = +0, Embeds (0) = {−0; +0}. One can see that the class of source language constructs {−0; +0}, representing the source meaning 0, is mapped into the target constructs {−0; +0} and the source language meaning 0 is mapped into the target language {−0; +0} as expected. Acknowledgements Thanks are due to the anonymous reviewer, whose comments and suggestions helped me make this paper better, and to Pierre-Louis Curien, the editor of this volume. References [1] G. BirkhoI, Lattice Theory, American Mathematical Society, Providence, RI, 1948. [2] J. Bresnan, R.M. Kaplan, Grammars as mental representations of language, in: J. Bresnan (Ed.), The Mental Representation of Grammatical Relations, MIT Press, Cambridge, MA, 1982, p. 3. [3] R.M. Burstall, P.J. Landin, Programs and their proofs: an algebraic approach, Mach. Intell. 4 (1969) 17–43. [4] E.M. Clarke, E.A. Emerson, A.P. Sistla, Automatic veri#cation of #nite-state concurrent systems using temporal logic speci#cations, ACM TOPLAS 8 (2) (1986) 244–263. [5] N. Chomsky, Aspects of the Theory of Syntax, MIT Press, Cambridge, MA, 1965. [6] B. Courcelle, Graph rewriting: an algebraic and logic approach, in: J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, MIT Press=Elsevier, Cambridge, MA=Amsterdam, 1990, pp. 193–242. [7] P.-L. Curien, Categorical combinators, sequential algorithms, and functional programming, in: Research Notes in Theoretical Computer Science, Pitman, London, 1986. [8] N. Dershowitz, J.-P. Jouannaud, Rewrite systems, in: J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, MIT Press=Elsevier, Cambridge, MA=Amsterdam, 1990, pp. 243–320. [9] P. Deransart, M. Jourdan, B. Lorho, Attribute Grammars—De#nitions, Systems, and Bibliography, Lecture Notes in Computer Science, Vol. 322, Springer, Berlin, 1988. [10] G. Graetzer, Universal Algebra, The University Series in Higher Mathematics, Van Nostrand, Princeton, 1968, 2nd Edition, Springer, New York, 1979. [11] J.A. Goguen, J.W. Thatcher, E.G. Wagner, J.B. Wright, Initial algebra semantics and continuous algebras, J. Assoc. Comput. Mach. 24 (1977) 68–95. [12] R.R. Hausser, Foundations of Computational Linguistics, Springer, Berlin, 1999.

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www.elsevier.com/locate/tcs

Uni-transitional Watson–Crick D0L systems  Arto Salomaa Turku Centre for Computer Science, TUCS Lemminkaisenkatu 14, 20520 Turku, Finland

Abstract The phenomenon known as Watson–Crick complementarity is basic both in the experiments and theory of DNA computing. While the massive parallelism of DNA strands makes exhaustive searches possible, complementarity constitutes a powerful computational tool. It is also very fruitful to view complementarity as a language-theoretic operation: “bad” words obtained through a generative process are replaced by their complementary ones. This idea seems particularly suitable for Lindenmayer systems. D0L systems augmented with a speci3c complementarity transition, Watson–Crick D0L systems, have turned out to be a most interesting model and have already been extensively studied. A language is generated by a Watson–Crick D0L system as a sequence of words. Consequently, the systems can be applied also to compute functions in a natural way. In the present paper, attention is focused on uni-transitional systems, where at most one complementarity transition takes place in the generated sequence. In spite of their seeming simplicity, uni-transitional systems represent a vast extension of ordinary D0L systems. This becomes apparent in their capacity of de3ning functions. Quite remarkably, all basic decision problems for uni-transitional systems are algorithmically equivalent among themselves, as well as equivalent to a celebrated open problem. We investigate also a simpler case of systems with c 2002 Elsevier Science B.V. All rights regular triggers, as well as pose some open problems.
reserved. Keywords: Iterated morphisms; Watson–Crick complementarity; D0L sequences; DNA computing; Growth functions; Decidability



Dedicated to Maurice Nivat, a wonderful colleague and friend. Being the main originator of EATCS, ICALP and the Journal TCS, Maurice can be appropriately labeled as the Founding Father of the theoretical computer science community in Europe. I have had the privilege and pleasure of collaborating with him since the very early days. E-mail address: [email protected] (A. Salomaa). c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter
PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 6 - 9

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1. Introduction The interest in “DNA computing” has been growing rapidly since Adleman’s famous contribution [1] as apparent from the list of references in [12]. The impact of the resulting new notions and ideas to the theory of formal languages can be seen also from the recent Handbook [14]. A keynote in theoretical studies about DNA computing is a phenomenon known as Watson–Crick complementarity. DNA (deoxyribonucleic acid) consists of polymer chains, referred to as DNA strands. A chain is composed of nucleotides or bases. The four DNA bases are customarily denoted by A (adenine), C (cytosine), G (guanine) and T (thymine). A DNA strand can be viewed as a word over the DNA alphabet DNA = {A; C; G; T }. The familiar DNA double helix arises by the bondage of two strands. The Watson–Crick complementarity comes into the picture in the formation of such double strands. The bases A and T are complementary, and so are the bases C and G. Bonding occurs only if the bases in the corresponding positions in the two strands are complementary. ∗ Consider the letter-to-letter endomorphism h W of DNA de3ned by hW (A) = T;

hW (T ) = A;

hW (G) = C;

hW (C) = G:

The morphism h W will be referred to as the Watson–Crick morphism. Thus, a DNA strand x bonds with h W (x) to form a double strand. The complementarity of two strands leads (under appropriate conditions) to bondage. By encoding information on the original strands in a clever way, far-reaching conclusions can be made from the mere fact that bondage has occurred. This means that the phenomenon of complementarity provides computing power. The idea of using the fundamental knowledge, concerning how the double strands have possibly come into being, is central in Adleman’s experiment [1]. The idea is also behind the computational universality of many models of DNA computing [17,12]. Complementarity can be viewed also as a language-theoretic operation. As such h W is only a morphism of a special kind. However, the operational complementarity can be considered also as a tool in a developmental model: undesirable conditions in a string trigger a transition to the complementary string. Thus, the class of “bad” strings is somehow speci3ed. Whenever a bad string x is about to be produced by a generative process, the string h W (x) is taken instead of x. If the generative process produces a unique sequence of strings (words), the sequence continues from h W (x). Suppose that the class of bad strings satis3es the following soundness condition: whenever x is bad, the complementary string h W (x) is not bad. Then no bad strings are produced. While the operational complementarity can be investigated in connection with any generative process for words, it seems particularly suitable for Lindenmayer systems, the systems themselves being developmental models. The simplest L system, namely the D0L system, has been thoroughly investigated, [13,14]. A D0L system generates a sequence of words. When it is augmented with a trigger for complementarity transitions, as described above, the resulting sequences contain no bad words. The study of such “Watson–Crick D0L systems” was begun in [10,11,18], and continued in [2,7,20]. The present paper constitutes a further contribution to this topic.

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The formal de3nitions will be given below but an important remark should be made already at this stage. So far we have spoken only of the four-letter DNA alphabet but in our theoretical considerations below the size of the alphabet will be arbitrary. Indeed, we will consider DNA-like alphabets n = {a1 ; : : : ; an ; aJ1 ; : : : ; aJn }

(n ¿ 1)

and refer to the letters a i and aJi ; i = 1; : : : ; n, as complementary. The endomorphism h W of n∗ de3ned by hW (ai ) = aJi ;

hW (aJi ) = ai ; i = 1; : : : ; n;

is also now referred to as the Watson–Crick morphism. When we view the original DNA alphabet in this way, the association of letters is as follows: a1 = A;

a2 = G;

aJ1 = T;

aJ2 = C:

(Observe that this conforms with the two de3nitions of h W .) The nucleotides A and G are purines, whereas T and C are pyrimidines. This terminology is extended to concern DNA-like alphabets: the non-barred letters a1 ; : : : ; an are called purines, and the barred letters aJ1 ; : : : ; aJn are called pyrimidines. The class of bad words, considered most frequently in the sequel and denoted by PYR, consists of words where the pyrimidines form a majority. Thus PYR consists of words over the alphabet n , where the number of occurrences of barred letter exceeds that of non-barred letters. Clearly, PYR is a context-free non-regular language. In spite of their formal simplicity, Watson–Crick D0L systems have quite remarkable properties. This observation made already in [10,11,18] will be further substantiated in this paper. For the decision problems we will be concerned with, the following turns out to be very signi3cant. Problem Z pos . Decide whether or not a negative number appears in a given Z-rational sequence of integers. The decidability status of Z pos is open, although the problem is generally believed to be decidable. The input is of course assumed to be given by some eMective means such a linear recurrence with integer coeNcients, or a square matrix M with integer entries such that the sequence is read from the upper right corners of the powers M i ; i = 1; 2; 3; : : : : Further discussion about this problem and its diMerent representations can be found in [9,13,19]. We will use standard language-theoretic terminology and notation. In particular,  is the empty word, |w| is the length of the word w, and |w|a (resp. |w| ) is the number of occurrences of a (resp. letters of ) in w. Language-theoretic issues can be consulted from [16], and questions concerning Lindenmayer systems from [13,19]. The basic notion underlying our subsequent investigations is given in the following de3nition. Denition 1. A D0L system is a triple G = (; g; w0 ), where  is an alphabet, w0 ∈ ∗ (the axiom) and g is an endomorphism of ∗ . (In the sequel g is often de3ned in terms

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of productions, indicating the image of each letter.) A D0L system de3nes the sequence S(G) of words wi , i¿0, where wi+1 = g(wi ), for all i¿0. It de3nes also the language L(G), consisting of all words in S(G), the length sequence |wi |, i¿0, as well as the growth function f(i) = |wi |. Decision problems will be important below in our considerations. As regards D0L systems, the following general observation is signi3cant. Consider decision problems (for instance, the emptiness problem) involving languages of the form L ∩ K, where L is a D0L language and K is in one of the families of the Chomsky hierarchy. Such problems are usually decidable (resp. undecidable) if K ranges over regular (resp. context-sensitive) languages. If K ranges over context-free languages, the decidability is often hard to settle, although intuitively the problem might seem to be decidable. Such problems P are often algorithmically equivalent to the problem Z pos . By this we mean that any algorithm for solving the problem Z pos yields an algorithm for solving the problem P, and vice versa. In particular, various equivalence problems constitute a central chapter in the history of L systems, see [13,14]. We use here standard terminology. Thus, the sequence (resp. language, growth) equivalence problem for D0L systems consists of deciding of two given D0L systems whether or not they generate the same sequence (resp. language, growth function). For D0L systems, the decidability of the growth equivalence problem was settled 3rst. It was also shown quite early, that the decidability of the sequence equivalence implies the decidability of the language equivalence and vice versa, whereas the decidability itself remained as a celebrated open problem, until it was 3nally settled in the late 1970s, see [13,14]. Equivalence problems for Watson– Crick D0L systems will be one of the topics investigated below. It will be seen that in the most intriguing case the situation is similar to the one in the early history of D0L systems: very diMerent problems are algorithmically equivalent but the decidability itself remains open.

2. Watson–Crick D0L systems Consider a DNA-like alphabet n and the Watson–Crick morphism h W . A trigger TR is any recursive subset of n∗ . In many cases, the following condition is assumed to be satis3ed: whenever x is in TR, then h W (x) is in the complement of TR, that is, in n∗ − TR. According to our terminology in the Introduction, TR consists of “bad” strings. The restriction imposed on TR is our soundness condition: no “bad” strings result if emerging “bad” words are always replaced by their complementary ones. We now come to our central de3nitions. Denition 2. A Watson–Crick D0L system is a construct GW = (G; TR);

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where G = (n ; g; w0 ) is a D0L system, TR is a trigger and w0 ∈ n∗ − TR. The sequence S(G W ), consisting of words wi ; i = 0; 1; : : : ; is de3ned by the condition
hW (g(wi )) if g(wi ) ∈ TR; wi+1 = g(wi ) otherwise for all i¿0. The language, length sequence and growth function of G W are de3ned as in De3nition 1. Denition 3. Given a Watson–Crick D0L system G W , an in3nite word t1 t2 : : : over the alphabet {0; 1}, referred to as the Watson–Crick road or briePy road of G W , is de3ned as follows. Let wi ; i = 0; 1; : : : ; be the sequence of G W . Then, for all j¿1; tj = 0 (resp. tj = 1) if wj = g(wj−1 ) (resp. wj = h W (g(wj−1 ))). Thus, the road of G W indicates the positions in the sequence, where complementarity transitions take place. If the road equals 0 ! , then no complementarity transitions take place. The road 01010 ! tells us that in the sequence exactly w2 and w4 have been obtained by a complementarity transition. De3nition 3 has been formulated for the needs of this paper. The reader is referred to [18] for a more general framework in terms of Watson–Crick graphs and morphic equivalences [4]. Denition 4. A Watson–Crick D0L system is called stable if its road equals 0 ! . It is obvious that the properties of a Watson–Crick D0L system are largely determined by the trigger. Clearly, the trigger should not be too complicated. In a regular Watson–Crick D0L system the trigger is a regular language. We will be mostly interested in standard systems de3ned as follows. Denition 5. trigger.

A Watson–Crick D0L system is standard if the language PYR is its

As an illustration, we consider the following standard system. Its alphabet equals  = {a1 ; a2 ; a3 ; a1 ; a2 ; a3 }, it has the axiom a1 a2 a3 and its morphism is de3ned by the rules a1 → a1 ; a2 → a2 ; a3 → a3 ; a1 → a1 a2 ; a2 → a2 ; a3 → a3 3 : The beginning of the sequence looks as follows, where bold characters indicate that a complementarity transition has taken place: a1 a2 a3 ; a1 a2 a33 ; a1 a2 2 a33 ; a1 a23 a33 ; a1 a23 a39 ; a1 a2 4 a93 ; a1 a2 5 a93 ; a1 a2 6 a93 ; a1 a2 7 a93 ; a1 a2 8 a93 ; a1 a29 a39 ; a1 a29 a327 ; a1 a2 10 a27 3 ; after which there are 16 words before the next complementarity transition.

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Observe that only the barred letters induce growth. The length sequence is strictly growing: 3; 5; 6; 7; 13; 14; 15; 16; 17; 18; 19; 37; 38; : : : : The growth Puctuates between linear growth (from n to n + 1) and exponential jumps (from 1 + 2 × 3n to 1 + 3n + 3n+1 ). Indeed, when f(i) is of the form 1 + 2 × 3n ; n¿0; then f(i + 1) = 1 + 3n + 3n+1 . When f(i) (¿3) is not of the form 1 + 2 × 3n , then f(i + 1) = f(i) + 1. Thus, the strictly increasing growth function f assumes all values ¿3, except the values strictly between the numbers 1 + 2 × 3n and 1 + 3n + 3n+1 ; for all n¿0. It is not a D0L growth function and, in fact, it is not even a Z-rational function. Further details are given in [11]. The road of our example system begins with the word 101105 11017 11. In general, there is an exponentially growing sequence of 0s between words 11. Explicitly, after the 3rst position the bit 1 occurs exactly in positions 3i+1 + i and 3i+1 + i + 1, for all i¿0. As regards equivalence problems, the same ones can be formulated for Watson– Crick D0L systems as for ordinary D0L systems and, in addition, the problem of road equivalence: decide of two given systems whether or not they have the same road. The following theorem summarizes some results from [18,11]. If ’ is a recursive function mapping the set of positive integers into {0; 1}, we denote by t’ the in3nite binary word whose ith letter equals 1 exactly in case ’(i) = 1, for all i¿1. Theorem 1. For every recursive function ’, a Watson–Crick DOL system whose Watson–Crick road equals t’ can be e7ectively constructed. Every ultimately periodic in8nite binary word can be expressed as the road of a system with a 8nite trigger. The stability problem is decidable for Watson–Crick DOL systems with a regular trigger but undecidable for systems with a context-sensitive trigger. The stability problem for standard Watson–Crick DOL systems is algorithmically equivalent to the problem Z pos . The road, growth, sequence and language equivalence problems are all undecidable for Watson–Crick DOL systems with context-sensitive triggers. Any algorithm for solving the road, growth, sequence or language equivalence problem for standard Watson–Crick DOL systems can be converted into an algorithm for solving the problem Z pos .

3. Uni-transitional mode In this paper we will investigate of variant of Watson–Crick D0L systems, where at most one complementarity transition may take place in the sequence. In other words, only one correction is allowed. After a complementarity transition no further complementarity transitions may take place and, consequently, also bad words can be produced. Thus, intuitively, only such cases are considered, where one correction in the sequence suNces. We will see that, in spite of their seeming simplicity, such uni-transitional systems have amazing computational capabilities.

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Denition 6. A Watson–Crick D0L system G W = (G; TR), with the underlying D0L system G = (n ; g; w0 ), operates in the uni-transitional mode if its sequence wi ; i¿0 is de3ned by
hW (g(wi )) if i is the smallest index such that g(wi ) ∈ TR; wi+1 = g(wi ) otherwise; for all i¿0. Watson–Crick D0L systems operating in the uni-transitional mode are called uni-transitional systems or, briePy, UT-systems. Based on the sequence of an UT-system, its language, length sequence, growth function and road are introduced as in De3nitions 1–3. Clearly, the road equals either 0 ! , in which case the system is stable, or else 0 k 10 ! , where k¿0, in which case the number k + 1 is referred to as the transition point of the system. Thus, the transition point T (U) of a system U indicates the position of the bit 1 in the road of U. We will see that the problem of determining the transition point is algorithmically equivalent to many of the central problems about UT-systems, such as the sequence, language, growth and road equivalence problems. The latter problems are de3ned for UT-systems as for ordinary Watson–Crick D0L systems. Also De3nitions 4 and 5 about stable and standard systems are extended to concern UT-systems. Apart from the next section, we will consider only standard UT-systems in this paper. As an example, consider the following two standard UT-systems U1 and U2 over the DNA-alphabet {A; C; G; T }. (We prefer to use the DNA-notation for DNA-like alphabets n with n = 2. Recall that the pyrimidines C and T correspond to barred letters.) The axiom of U1 (resp. U2 ) is A (resp. TG 2 ). The productions of U1 are A → AC; C → C; G → ; T → A: The productions of U2 are the same except that T → A is replaced by T → T . Then the sequence of U1 is A; AC; TG 2 ; A; AC; AC 2 ; AC 3 ; : : : ; whereas the sequence of U2 is TG 2 ; A; AC; AC 2 ; AC 3 ; : : : : The roads (resp. transition points) of the two systems are 010 ! and 10 ! (resp. 2 and 1). Clearly, the two systems are language equivalent but not sequence, growth or road equivalent. Observe also that the sequence of U1 has repetitions, although its language is in3nite. This phenomenon is possible neither for D0L systems nor for Watson–Crick D0L systems. The sequence of a UT-system can be viewed as a D0L sequence, supplemented with some “initial mess”. Since the class of D0L growth functions is invariant under such a supplement, the following theorem is an immediate consequence of the de3nitions, and the fact that the language {AG; AC; TG; G} is not a D0L language but is generated by the standard UT-system with the axiom AC and productions A → T; T → ; G → G; C → C:

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Theorem 2. The class of sequences (resp. languages) of UT-systems includes properly the class of D0L sequences (resp. languages). The language of a UT-system is of the form F ∪ L, where F is 8nite and L is a D0L language. UT-systems and D0L systems de8ne the same class of growth functions. If we are dealing with standard UT-systems, it is not known how we can 3nd a D0L system de3ning the same growth function as a given UT-system. This and many other decision problems will be discussed in Section 5. In the case of regular triggers, such problems are decidable, as will be seen in the next section. Using the terminology customary for Lindenmayer systems, we will speak also of Watson–Crick D0L schemes and UT-schemes. A scheme is simply a system without the axiom. When a scheme is supplemented with an axiom, it becomes a system. Schemes give us the possibility of varying one item, the axiom, in the basic structure consisting of the alphabet, the morphism and the trigger. Such is possibility is needed when we want to compute functions in a more variegated fashion than what is provided by growth functions. In the following de3nitions, the argument for a function is varied by changing the axiom in a scheme. Denition 7. Consider a standard Watson–Crick D0L scheme G W , with the underlying D0L scheme G = (n ; g). A partial recursive function ’ mapping a subset of the set of non-negative integers into non-negative integers is computed by G W if the alphabet n contains the letters B; b; E; e with the productions E → E and e → e and satisfying the following condition. For all i¿0; the equation ’(i) = j holds exactly in case there is a derivation according to G W Bbi ⇒∗ Eej and, moreover, the letters E and e appear in this derivation at the last step only. A function ’ is Watson–Crick computable if it is computed by some standard Watson– Crick D0L scheme G W . Here we have used the customary yield relation ⇒ and its transitive closure ⇒∗ . Thus, the scheme G W is provided with the axiom Bb i , where i is the argument value. The function value j appears in the last word Ee j and, by our assumption about the productions, this word does not change any more. It is also clear from the notation that all of the special letters B; b; E; e are “non-barred” ones. De3nition 7 is due to SosRSk, [20]. He considers also functions with several variables. For our purposes in Section 6 below, we now modify the de3nition for UT-schemes. Because only one complementarity transition is possible, partial recursiveness might be needed only if the trigger is very complicated. Therefore, we restrict the attention to recursive functions. Denition 8. Consider a standard UT-scheme U, with the underlying D0L scheme G = (n ; g). A recursive function ’ mapping the set of non-negative integers into itself is computed by U if the alphabet n contains the letters B; b; E; e with the

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productions E → E and e → e and satisfying the following condition. For all i¿0; the equation ’(i) = j holds exactly in case there is a derivation according to U Bbi ⇒∗ Eej and, moreover, the letters E and e appear in this derivation at the last step only. A function ’ is UT-computable if it is computed by some standard UT-scheme U. We conclude this section with two examples of standard systems that will come into use also in Section 6. Consider 3rst the standard UT-scheme Ulog . We list the productions, from which it is also seen that the alphabet consists of the letters a; b; c; d; e; t; a1 ; c1 ; B; E; B1 , as well as of their barred versions: J B →; B1 dtta1 a; b → b; B1 → B1 ; tJ → tt; dJ → d; a → ac; c → c; a1 → a1 c1 ; c1 → c1 ; B1 → E; c1 → e; E → E; e → e; x →  otherwise: Here the last line takes care of all the remaining letters and their barred versions. Consider the derivation Bb10 ⇒ B1 dtta1 ab10 ⇒ B1 dtttta1 c1 acb10 ⇒ ⇒ B1 dtttttttta1 c1 c1 accb10 ⇒T B1 dt 16 a1 c13 ac3 b10 ⇒ Ee3 ; where we have indicated the complementarity transition by the lower index T in the yield relation. Observe that [log2 10] = 3. Indeed, it is not diNcult to see that our UT-scheme Ulog computes the function [log2 x], for x¿1. The (a; c)-part counts the number of steps before the power of t has exceeded the power of b given as the input. Then the complementarity transition will take place, because the (a; c)-part preserves the balance between purines and pyrimidines, and so does the (B; d)-part. Finally, the number of steps appears as the output. The principle behind our second standard UT-scheme Uroot is similar. The productions are now J B → B1 dta1 a; b → b; B1 → B1 ; dJ → d; tJ → tuvv; u → uvv; vJ → v; J a → ac; c → c; a1 → a1 c1 ; c1 → c1 ; B1 → E; c1 → e; E → E; e → e; x →  otherwise: Instead √ of powers of 2, the t-part computes now squares, yielding altogether the function [ x], for x¿0. Thus, we obtain the following result. √ Lemma 1. The functions [log2 x] and [ x] are UT-computable.

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4. Regular triggers The most challenging problems lie in the area of standard UT-systems and UT-schemes, that is, those having the language PYR as the trigger. Although the setup is very simple, many fundamental questions about standard UT-systems remain open. The situation is entirely diMerent if, instead of PYR, regular triggers are considered. The present section deals with regular triggers, whereas Sections 5 and 6 investigate standard systems. Thus, consider a UT-scheme U = (G; TR), with the underlying D0L scheme G = (n ; g) and regular trigger TR. We will establish a simple method of computing the transition point, after which all basic issues can be settled. The method works for UT-systems, that is, U is supplemented with an axiom, as well as for the setup where the axiom is not speci3ed. We will deduce for the transition point an upper bound depending only on n and the cardinality m of the state set of the minimal 3nite automaton accepting TR. Thus, the upper bound is independent of g and the chosen axiom, and depends on TR only in the way indicated. Our purpose is to establish the following result. Theorem 3. Consider two positive integers m and n. There is an e7ectively computable constant ) = )(m; n) such that, for any UT-system U = (G; TR), with the underlying D0L system G = (n ; g; w0 ) and with m being the cardinality of the state set for the minimal automaton accepting the regular trigger TR, the transition point of U (if it exists) satis8es T (U)6). The setup in Theorem 3 is similar to the one in [7]. As pointed out in [7], also some on the constructions in [5] become essentially simpler by these methods. Watson–Crick D0L systems and UT-systems can also be viewed as DT0L systems, [13], with two morphisms (tables) g and h W g. The following result can be inferred from Lemma 2.2 in [3]. We do not need the result because we will develop tools explicitly for our purposes. Lemma 2. Consider a DT0L system (; H; w0 ) and a regular language R ⊆ ∗ . Then the language {+ ∈ H ∗ |+(w0 ) ∈ R} is regular. We will consider 3nite deterministic automata A = (; Q; -; q0 ; F), where the items have their customary meaning: alphabet, state set, transition function, initial state, and 3nal state set. The domain of the transition function is extended to concern words in the usual way. Thus -(q; x) is the state where the word x takes the automaton from the state q. Denition 9. Given a 3nite deterministic automaton A = (; Q; -; q0 ; F) and a morphism g : ∗ → ∗ , the 3nite deterministic automaton g−1 (A) is de3ned by g−1 (A) = (; Q; -g−1 ; q0 ; F);

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where -g−1 (q; a) = -(q; g(a)), for all q ∈ Q and a ∈ . The sequence of automata S(A; g) is de3ned inductively: A0 = A; Ai+1 = g−1 (Ai ); i¿0. The length of the sequence, in symbols |S(A; g)|, is de3ned to be the smallest index i such that the automaton Ai appears already earlier in the sequence. The following lemma contains some immediate yet important observations. Lemma 3. An arbitrary word x is accepted by the automaton g−1 (A) exactly in case the word g(x) is accepted by the automaton A. The length of the sequence S(A; g) is always 8nite and satis8es the inequality |S(A; g)|6mmn ; where m and n are the cardinalities of Q and , respectively. Proof. The 3rst sentence is an immediate consequence of De3nition 9. The second sentence follows because only the transition function varies in the automata in the sequence S(A; g), and the (very rude) upper bound is given for all possible transition functions. Lemma 4. Consider a UT-system U = (G; TR), with the underlying D0L system G = (n ; g; w0 ) and regular trigger TR, accepted by the 8nite deterministic automaton A. The system U is stable if and only if the word w0 is not accepted by any of the automata in the sequence S(A; g). Otherwise, the transition point of U equals the smallest index i such that w0 is accepted by the automaton Ai in the sequence. Proof. The Lemma follows by de3nitions and Lemma 3. Observe that, by De3nition 2, the word w0 is not accepted by A. The following result is now an immediate consequence of Lemma 4. Lemma 5. Use the notation in the preceding Lemma. Then the transition point of the UT-system U satis8es the inequality T (U) ¡ |S(A; g)|: Clearly, Theorem 3 now follows by Lemmas 3 and 5. Moreover, we have given a method to compute the transition point. In general, the latter is much smaller than the upper bound given in Lemma 3. We turn next to the various equivalence problems for UT-systems with regular triggers. By Theorem 3, the situation is completely under control, and we can reduce the problems to known results concerning D0L systems. We still need the following auxiliary result. Lemma 6. Let Li ; i = 1; 2 (resp. Fi ; i = 1; 2) be e7ectively given D0L (resp. 8nite) languages. Then it is decidable whether or not L1 ∪ F1 = L2 ∪ F2 .

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Proof. The result follows because, by [13], membership is decidable for D0L languages and, by [15], inclusion is decidable for D0L languages. Theorem 4. Sequence, language, growth and road equivalence are all decidable for UT-systems with regular triggers. Proof. The Theorem follows by known results [13], for D0L systems, and by Theorem 3 and Lemma 6. The reader might also want to compare the situation to that concerning Watson–Crick D0L systems [7]. In the latter case, the language equivalence is considerably harder. Observe that, by Theorem 3, the construction claimed in the last sentence of Theorem 2 is eMective: For any given UT-system with a regular trigger, a D0L system de3ning the same growth function can be eMectively constructed. Also the following 3nal result in this section is a consequence of Theorem 3 (or, more explicitly of Lemma 4). We say that a UT-scheme U is universally stable if, for all possible axioms w0 , the UT-system resulting from U by supplementing it with w0 is stable. Theorem 5. It is decidable whether or not a given UT-scheme with a regular trigger is universally stable. 5. Decision problems for standard systems Nothing analogous to Theorem 3 can be obtained for standard UT-systems. On the contrary, we have the following result. Theorem 6. There exists an e7ectively constructable standard UT-scheme U with the following property. For any integer k, one can 8nd a word w0 such that the UTsystem obtained from U by supplementing it with the axiom w0 has the transition point ¿k. Proof. By [8], there exists (eMectively) a D0L system G = (; h; u0 ) whose length sequence xi = |ui |;

i = 0; 1; : : : ;

satis3es the following two conditions. (i) The set {i|xi ¿xi+1 } is in3nite. (ii) For each integer k, there exists an ik such that xik ¡ xik +1 ¡ · · · ¡ xik +k ¡ xik +k+1 : Based on G, we now construct a standard UT-scheme U as follows. We 3rst take primed versions a of the letters a ∈ , yielding the alphabet  . The alphabet n of U consists of the letters in  ∪  , as well as of their barred versions. (As usual, the barred letters are pyrimidines whose majority is decisive in the de3nition of the

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trigger PYR. A primed word, such as h(a) , indicates that primes are distributed to the individual letters.) The morphism g of U is de3ned as follows: g(a) = h(a); g(a ) = h(a) g(b) = b

for a ∈ ;

for b ∈ J ∪  :

Given k, we now choose the axiom w0 = uik +1 uik : We claim that the resulting standard UT-system satis3es Theorem 6. Indeed, the 3rst words in the sequence of the UT-system are w0 = uik +1 uik ;

w1 = uik +2 uik +1 ; : : : ; wk = uik +k+1 uik +k :

Thus, both the non-barred and barred parts in the words are obtained from the sequence of G, but the non-barred part runs one step ahead of the barred part. By (ii), none of the words w0 ; : : : ; wk belongs to the trigger PYR. On the other hand, by (i), some word wl with l¿k belongs to the trigger. This completes the proof of Theorem 6. Observe that the second line in the de3nition of g is arbitrary and does not aMect the proof. By changing this part of the de3nition, the resulting UT-system can be forced to have desired properties. As given, the de3nition makes the language of the UT-system 3nite. In the sequel we consider the following seven decision problems for standard UTsystems. While the decidability of the problems remains open, we show that they are algorithmically equivalent: an algorithm for solving any of them can be converted to an algorithm for solving any other. Thus, assume that we are given an arbitrary standard UT-system or two arbitrary standard UT-systems. We consider the following problems. 1. Determine the transition point. 2. Decide stability. 3. Decide sequence equivalence. 4. Decide language equivalence. 5. Decide growth equivalence. 6. Decide road equivalence. 7. Decide the 3niteness of the generated language. We begin with the following observation, showing why the situation here is essentially more complicated than in connection with D0L systems. For D0L systems, the sequence equivalence is decidable. However, no reasonable bound B is known such that, to decide the equivalence, it would suNce to compare the 3rst B words in the two given sequences. According to a well-known conjecture it suNces to choose B to be twice the size of the alphabet of the given systems. (All known examples satisfy this conjecture, see [6] for a recent contribution.) The following result shows that nothing analogous to this 2n-Conjecture can hold for UT-systems.

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Theorem 7. There are two (e7ectively constructable) standard UT-schemes, over the alphabet n and with the morphisms g1 and g2 , having the following property. For any integer k, one can 8nd a word w0 such that the sequences of the two standard UT-systems, resulting from the two schemes by supplementing them with the axiom w0 , coincide with respect to the 8rst k words but di7er later on. Proof. The proof is similar to that of Theorem 6. As already pointed out, the second line in the de3nition of g is arbitrary. We can de3ne it diMerently for the two UTschemes, to guarantee that the sequences will be diMerent after the transition point. (For instance, we can use the given de3nition for one scheme, and let all letters go to  in the other scheme.) Then the claim will be satis3ed. We will now discuss the seven problems listed above. Clearly, Problems 1 and 2 are algorithmically equivalent. On the other hand, it follows from the de3nitions that Problem 2 is algorithmically equivalent to the stability problem for Watson–Crick D0L systems. Hence, by Theorem 1, we obtain the following result. Lemma 7. Problems 1 and 2 are algorithmically equivalent to the problem Z pos . Lemma 8. Problems 1 and 6 are algorithmically equivalent. Proof. If we can determine the transition point, we can decide road equivalence by determining the two transition points. Suppose we can decide road equivalence. To determine the transition point of a given UT-system, we decide its road equivalence with some system whose road is known to be 0 ! . If the systems are not road equivalent, we know that the transition point exists. We then compute the road until the transition point is found. Lemma 9. Problems 2, 3 and 5 are algorithmically equivalent. Proof. Assuming 2, we can compute both sequences (resp. both length sequences) beyond the transition point, and then use the decidability result for D0L systems. Assume that we have an algorithm for Problem 5 and want to decide the stability of a given standard UT-system U. Two new standard UT-systems U1 and U2 are de3ned by adding the letters b1 ; b 2 ; b1 ; b 2 to the alphabet of U. The axiom of both systems is obtained by adding the pre3x b1 b 2 to the axiom of U. The productions b1 → b1 ; b2 → b2 are in both systems, whereas the productions b1 → b1 ; b2 → b2

(resp: b1 → ; b2 → )

are in U1 (resp. U2 ). Clearly, U is stable if and only if U1 and U2 are growth equivalent. The same argument shows also that the decidability of Problem 3 implies the decidability of Problem 2.

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Lemma 10. Problems 2 and 4 are algorithmically equivalent. Proof. Assume that we can decide Problem 4. The construction in the preceding lemma can be applied. The given system U is stable if and only if U1 and U2 are language equivalent. Conversely, if Problem 2 is decidable, we can use Lemma 6 to decide language equivalence. Lemma 11. Problems 2 and 7 are algorithmically equivalent. Proof. Again an algorithm for Problem 2 reduces Problem 7 to deciding the 3niteness of a given D0L language. Conversely, assume that an algorithm is known for Problem 7. Let U be a given standard UT-system, with the underlying D0L system (n ; g; w0 ). We want to decide the stability of U. Without loss of generality, we assume that the language of the underlying D0L system is in3nite. We construct a UT-system U1 with the underlying D0L system (1 ; g1 ; w0 ) as follows. Assume that n = {a1 ; : : : ; an ; a1 ; : : : ; an }: We consider new alphabets n (b) = {b1 ; : : : ; bn ; b1 ; : : : ; bn }; n (c) = {c1 ; : : : ; cn ; c1 ; : : : ; cn }: De3ne 1 = n ∪ n (b) ∪ n (c). Let f : n∗ → n (b)∗ ∪ n (c)∗ be the morphism de3ned by f(ai ) = bi ; f(ai ) = ci ;

1 6 i 6 n:

Now the morphism g1 is de3ned by g1 (a) = f(g(a)); g1 (bi ) = ai ;

a ∈ n ;

g1 (ci ) = ai ;

g1 (bi ) = g1 (ci ) = ;

1 6 i 6 n;

1 6 i 6 n:

It is easy to verify that the language of U1 is in3nite if and only if U is stable. Indeed, the beginning of the sequence of U1 is w0 ; f(g(w0 )); w1 ; f(g(w1 )); w2 ; f(g(w2 )); : : : ; where wi ; i = 0; 1; : : : ; is the sequence of U. An eventual complementarity transition takes place in a word f(g(wi )), after which the productions on the last line of the de3nition of g1 are applied, and the sequence ends with . Summarizing Lemmas 7–11 we now obtain the 3nal result. Theorem 8. Problems 1–7 are algorithmically equivalent among themselves and also algorithmically equivalent to the problem Z pos .

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6. Computing functions It has been shown in [20] that every partial recursive function is Watson–Crick computable. We now present some results about UT-computable functions, as introduced in De3nition 8. A more detailed discussion about UT-computability lies beyond the scope of this paper, and we hope to return to the matter in another context. The restriction of having at most one complementarity transition is an obvious limitation for computability. For instance, Watson–Crick computable functions are clearly closed under composition: the output can be viewed as a new input. This method does not work for UT-computability, since the new input gives rise to an ordinary D0L sequence. The following results can be obtained by arguments analogous to the ones used in Lemma 1. Although growth functions are de3ned very diMerently from UT-computable functions, we have the following theorem. Theorem 9. Every D0L growth function is UT-computable. Proof. Let f be the growth function of the D0L system (; h; w0 ). Thus, f(0) = |w0 | and f(1) = |h(w0 )|. We now construct a standard UT-scheme U with the underlying D0L scheme (n ; g) as follows. The alphabet n consists of the letters in , in  (the corresponding primed letters), in J and J , as well as of the other letters and their barred versions appearing in the productions below. (Our use of primes and bars should be clear. As always, the barred letters are pyrimidines.) The productions de3ning g are now listed. a → h(a); a → h(a)

for a ∈ ; J B → B1 tdw0 w0 ; b → b; dJ → d; B1 → B1 tJ; tJ → tJ; B1 → E; E → E; a → e for a ∈ ; e → e; x →  otherwise: It is easy to see that the scheme U computes the function f. The w-part computes the D0L sequence, keeping the balance between purines and pyrimidines. The t-part counts the number of steps, comparing it with the input i in b i . The length of the appropriate word in the D0L sequence appears as the output. Theorem 10. The class of UT-computable functions contains properly the class of N-rational functions. Proof. N-rational functions coincide with HD0L growth functions, [13]. Thus, we have to consider the sequence of an HD0L system, obtained from a D0L sequence by applying another morphism h1 . The construction in the preceding Theorem remains unchanged, except that the productions a → e have to be replaced by the productions a → e|h1 (a)| : The containment is proper because the functions in Lemma 1 are not even Z-rational.

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7. Conclusion The operation of complementarity transition, basic in DNA computing, alters drastically the simple structure of D0L systems. Even one complementarity transition, giving rise to UT-systems investigated in this paper, seems to open unexpected developmental possibilities. We have presented some fundamental results and constructions along these lines. As we have pointed out, many challenging problems remain still open. References [1] L. Adleman, Molecular computation of solutions to combinatorial problems, Science 266 (1994) 1021–1024. [2] E. Csuhaj-VarjRu, A. Salomaa, Networks of Watson–Crick D0L systems, in: M. Ito (Ed.), Proc. 3rd Internat. Colloq. on Words, Languages and Combinatorics, Kyoto, March 14 –18, 2000, to appear. [3] S. Ginsburg, G. Rozenberg, T0L schemes and control sets, Inform. and Control 27 (1975) 109–125. [4] J. Harrison, Morphic congruences and DOL languages, Theoret. Comput. Sci. 134 (1994) 537–544. [5] J. Harrison, Dynamical properties of PWD0L systems, Theoret. Comput. Sci. 143 (1995) 269–284. [6] J. Honkala, J. Easy cases of the D0L sequence equivalence problem, Discrete Appl. Math., to appear. [7] J. Honkala, A. Salomaa, Watson–Crick D0L systems with regular triggers, Theoret. Comput. Sci. 259 (2001) 689–698. [8] J. KarhumVaki, Two theorems concerning recognizable N -subsets of 2∗ , Theoret. Comput. Sci. 1 (1976) 317–323. [9] W. Kuich, A. Salomaa, Semirings, Automata, Languages, Springer, Berlin, Heidelberg, New York, 1986. [10] V. Mihalache, A. Salomaa, Lindenmayer and DNA: Watson–Crick DOL systems EATCS Bull. 62 (1997) 160–175. [11] V. Mihalache, A. Salomaa, Language-theoretic aspects of DNA complementarity, Theoret. Comput. Sci. 250 (2001) 163–178. [12] G. Paun, G. Rozenberg, A. Salomaa, DNA Computing—New Computing Paradigms, Springer, Berlin, Heidelberg, New York, 1998. [13] G. Rozenberg, A. Salomaa, The Mathematical Theory of L Systems, Academic Press, New York, 1980. [14] G. Rozenberg, A. Salomaa (Eds.), Handbook of Formal Languages, Vol. 1–3, Springer, Berlin, Heidelberg, New York, 1997. [15] K. Ruohonen, The inclusion problem for D0L languages, Elektron. Informat. Kybernet 15 (1979) 535– 548. [16] A. Salomaa, Formal Languages, Academic Press, New York, 1973. [17] A. Salomaa, Turing, Watson–Crick and Lindenmayer, Aspects of DNA complementarity, in: C. Calude, J. Casti, M. Dinneen (Eds.), Unconventional Models of Computation, Springer, Singapore, 1998, pp. 94–107. [18] A. Salomaa, Watson–Crick walks and roads on D0L graphs, Acta Cybernet. 14 (1999) 179–192. [19] A. Salomaa, M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer, Berlin, Heidelberg, New York, 1978. [20] P. SosRSk, Universal computation with Watson–Crick D0L systems, submitted for publication.

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www.elsevier.com/locate/tcs

L(A) = L(B)? A simpli$ed decidability proof G*eraud S*enizergues LaBRI and UFR Math-info, Universite Bordeaux1, 351 Cours de la liberation -33405- Talence Cedex, France

Abstract We give a proof of decidability of the equivalence problem for deterministic pushdown auc 2002 tomata, which simpli$es that of S*enizergues (Theoret. Comput. Sci. 251 (2000) 1).
Elsevier Science B.V. All rights reserved. Keywords: Deterministic pushdown automata; Rational languages; Finite dimensional vector spaces; Matrix semi-groups; Complete formal systems

1. Introduction 1.1. Result The so-called “equivalence problem for deterministic pushdown automata”, is the following decision problem: INSTANCE: two dpda A; B QUESTION: L(A) = L(B)? i.e. do the given automata recognize the same language? This work consists of a simpli$ed proof of the following Theorem 93. The equivalence problem for deterministic pushdown automata is decidable. This result was proved in [8; 11, Sections 1–9] and has been generalised in [11, Section 11; 9; 10] to other decision problems. Some simpli$cations of the method presented in Fax: 05-56-84-66-69. E-mail address: [email protected] (G. S*enizergues). URL: http://dept-info.labri.u-bordeaux.fr/∼ges/ c 2002 Elsevier Science B.V. All rights reserved. 0304-3975/02/$ - see front matter  PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 7 - 0

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[8] have been found by Stirling in [12,13]. Nevertheless, Stirling was not using the same framework nor style. We hope this new exposition of the ideas from [11] and [13] will combine the qualities of both works. Logics. Our solution consists in constructing a complete formal system, in the general sense taken by this word in mathematical logics i.e.: it consists of a set of wellformed assertions, a subset of basic assertions, the axioms, and a set of deduction rules allowing to derive new assertions from assertions which are already generated. The well-formed assertions we are considering are pairs (S; T ) of rational boolean series over the non-terminal alphabet V of some strict-deterministic grammar G =
X; V; P. Such an assertion is true when the two series S; T generate the same language over the terminal alphabet X , via the rules of G. 1.2. Main tools We still use the notions developed in [8,11] (1– 4) (1) the deduction systems (which were in turn inspired by [1]); (2) the deterministic boolean series (which were in turn inspired by [3]); (3) the notion of linear independence for such series (which is based on the ideas from [4,5]); (4) the analysis of the proof-trees generated by a suitable strategy (which was somehow similar with the analysis of the parallel computations, interspersed with replacement-moves, done in [14,7,6]). We bring into this framework some simpli$cations found by Stirling [12,13] (5–8): (5) the technical notion of “N-stacking sequence” is replaced by the slightly simpler notion of “B-stacking sequence”; (6) the analysis of Section 8 uses a choice of “generating set” which is simpler than the choice given in [11,9]; (7) a main simpli$cation linked with this more clever choice, is that one can restrict the proof to the case of a proper, reduced strict-deterministic grammar; (8) some “second level” undeterminates corresponding bijectively to the deterministic rational series of [8,11] are introduced; this trick allows to treat polynomials instead of general rational series in the crucial part of the proof (Section 8). 1.3. Contents We recall in Section 2 some basic de$nitions concerning automata, grammars, monoids and semi-rings. We introduce in Section 3 a notion of rational deterministic series, vectors and matrices; we then develop their basic algebraic properties. The set of these series can be seen as a set of automata-con7gurations endowed with nice algebraic operations. We introduce in Section 4 a kind of logical formal system and de$ne a particular system D0 allowing to deduce equivalences between series. In Section 5, we de$ne a triangulation process for linear equations over series. In Section 6 we give an overview of the constants used throughout the paper.

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Section 7 is devoted to the de$nition of strategies allowing to construct a formal equivalence-proof from a given pair of equivalent series. In Section 8 we analyze the trees produced by the strategies of Section 7. In Section 9 we prove the formal system D0 is in fact complete; this implies that the equivalence problem for dpda is decidable. In Section 10 we show that the in$nite set of “second-level” undeterminates can be safely eliminated from the formal system (though they were useful to ease the $rst completeness proof). 2. Preliminaries 2.1. Pushdown automata A pushdown automaton on the alphabet X is a 7-tuple M =
X; Z; Q; ; q0 ; z0 ; F where Z is the $nite stack-alphabet, Q is the $nite set of states, q0 ∈ Q is the initial state, z0 is the initial stack-symbol, F is a $nite subset of QZ ∗ , the set of 7nal con$gurations, and , the transition function, is a mapping : QZ × (X ∪ {}) → Pf (QZ ∗ ). Let q; q ∈ Q; !; ! ∈ Z ∗ ; z ∈ Z; f ∈ X ∗ and a ∈ X ∪ {}; we note (qz!; af) →M   ∗ (q ! !; f) if q ! ∈ (qz; a). →M is the reLexive and transitive closure of →M . f

∗ (q ! ; ). For every q!; q ! ∈ QZ ∗ and f ∈ X ∗ , we note q! →M q ! iM (q!; f) →M M is said deterministic iM, for every z ∈ Z; q ∈ Q; x ∈ X :

Card( (qz; )) ∈ {0; 1};

(1)

Card( (qz; )) = 1 ⇒ Card( (qz; x)) = 0;

(2)

Card( (qz; )) = 0 ⇒ Card( (qz; x)) 6 1:

(3)

M is said real-time iM, for every q ∈ Q; z ∈ Z; Card( (qz; )) = 0. A con$guration q! of M is said -bound iM there exists a con$guration q ! such that (q!; ) →M (q ! ; ); q! is said -free iM it is not -bound. A pda M is said normalized iM, it fu$lls conditions (1), (2) (see above) and (4) – (6): q 0 z0

is -free; F ⊆ Q

(4)

and for every q ∈ Q; z ∈ Z; x ∈ X : q ! ∈ (qz; x) ⇒ |! | 6 2;

(5)

q ! ∈ (qz; ) ⇒ |! | = 0:

(6)

The language recognized by M is w

L(M) = {w ∈ X ∗ | ∃ c ∈ F; q0 z0 →M c}:

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It is a “folklore” result that, given a deterministic pda M, one can eMectively compute another dpda M which is normalized and ful$lls: L(M) = L(M ) − {”}: 2.2. Deterministic context-free grammars Let M be some deterministic pushdown automaton (we suppose here that M is normalized). The variable alphabet VM associated to M is de$ned as VM = {[p; z; q] | p; q ∈ Q; z ∈ Z}: The context-free grammar GM associated to M is then GM =
X; VM ; PM ; where PM is the set of all the pairs of one of the following forms: ([p; z; q]; x[p ; z1 ; p ][p ; z2 ; q]); 



(7)



where p; q; p ; p ∈ Q; x ∈ X; p z1 z2 ∈ (pz; x) ([p; z; q]; x[p ; z  ; q]);

(8)

where p; q; p ∈ Q; x ∈ X; p z  ∈ (pz; x) ([p; z; q]; a);

(9)

where p; q; ∈ Q; a ∈ X ∪ {}; q ∈ (pz; a). GM is a strict-deterministic grammar (see De$nition 313). A general theory of this class of grammars is exposed in [2] and used in [3]. 2.3. Free monoids acting on semi-rings Semi-ring B

W . Let (B; +; ·; 0; 1) where B = {0; 1} denote the semi-ring of “booleans”. Let W be some alphabet. By (B

W ; +; ·; ∅; ) we denote the semi-ring of boolean series over W : ∗ the set B

W  is de$ned as BW ; the sum and product are de$ned as usual; each ∗ word w ∈ W ∗ can be identi$ed with the element of BW mapping the word w on 1 and every other word w = w on 0; every boolean series S ∈ B

W  can then be written in a unique way as
Sw · w; S= w∈W ∗

where, for every w ∈ W ∗ ; Sw ∈ B. The support of S is the language supp(S) = {w ∈ W ∗ | Sw = 0}: In the particular case where the semi-ring of coeOcients is B (which is the only case considered in this article) we sometimes identify the series S with its support. A series

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559

S ∈ B

W  is called a boolean polynomial over W if and only if its support is 7nite. The set of all boolean polynomials over W is denoted by B
W . We recall that for every S ∈ B

W ; S ∗ is the series de$ned by
n S∗ = S : (10) 06n

Given two alphabets W; W  , a map : B

W  → B

W   is said -additive iM it ful$lls: for every denumerable family (Si )i∈N of elements of B

W ,  

Si = (Si ): (11) i∈N

i∈N

A map : B

W  → B

W   which is both a semi-ring homomorphism and a -additive map is usually called a substitution. Actions of monoids. Given a semi-ring (S; +; ·; 0; 1) and a monoid (M; ·; 1M ), a map ◦ : S × M → S is called a right-action of the monoid M over the semi-ring S iM, for every S; T ∈ S; m; m ∈ M: 0 ◦ m = 0;

S ◦ 1M = S; 

(S + T ) ◦ m = (S ◦ m) + (T ◦ m)

and



S ◦ (m · m ) = (S ◦ m) ◦ m :

(12)

In the particular case where S = B

W ; ◦ is said to be a -right-action if it ful$lls the additional property that, for every denumerable family (Si )i∈N of elements of S and m ∈ M:  

Si ◦ m = (Si ◦ m): (13) i∈N

i∈N

The action of W ∗ on B

W . We recall the following classical -right-action • of the monoid W ∗ over the semi-ring B

W : for all S; S  ∈ B

W ; u ∈ W ∗ S • u = S  ⇔ ∀w ∈ W ∗ ; (Sw = Su·w ) (i.e. S • u is the left-quotient of S by u, or the residual of S by u). For every S ∈ B

W  we denote by Q(S) the set of residuals of S: Q(S) = {S • u | u ∈ W ∗ }: We recall that S is said rational iM the set Q(S) is 7nite. The reduced grammar G1 . The classical reduced and -free grammar associated with GM is G0 =
X; V0 ; P0  where ∗

V0 = {v ∈ VM | ∃ w ∈ X + ; v →PM w}; ’0 : B

V  → B

V0 

(14)

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is the unique substitution such that, for every v ∈ V : ’0 (v) = v (if v ∈ V0 );

∗

’0 (v) =  (if v →PM );

’0 (v) = ∅ (otherwise);

P0 = {(v; w ) ∈ V0 × (X ∪ V0 )+ | v ∈ V0 ; ∃ w ∈ (X ∪ VM )∗ ; (v; w) ∈ PM ; w = ’0 (w)}:

(15)

G0 is the reduced and -free form of GM . It is well-known that, for all v ∈ V0 : ∗

∃ w ∈ X + ; v →P0 w and ∗

∗

{w ∈ X ∗ ; v →PM w} = {w ∈ X ∗ ; v →P0 w}: For technical reasons (which will be made clear in Section 7), we introduce an alphabet of “marked variables” VP0 together with a $xed bijection: v → vP from V0 to VP0 . Let V1 = V0 ∪ VP0 . We denote by $e (letter e stands here for “erasing the marks”) the litteral morphism V ∗ → V0∗ de$ned by: for every v ∈ V0 , $e (v) = v;

$e (v) P = v:

Similarly, $Pe is the litteral morphism V ∗ → VP0∗ de$ned by: for every v ∈ V0 , $Pe (v) = v; P

$Pe (v) P = v: P

We denote also by $e ; $Pe the unique substitutions extending these monoid homomorphisms. At last, the grammar G1 is de$ned by, G1 =
X; V1 ; P1  where P1 = P0 ∪ {($Pe (v); $Pe (w) | (v; w) ∈ P0 }: In other words, the rules of G1 consist of the rules of the usual proper and reduced grammar associated with M to-gether with their marked copies. Let us consider the unique substitution ’1 : B

V1  → B

X  ful$lling: for every v ∈ V1 , ∗

’1 (v) = {u ∈ X ∗ | v →P1 u}; (in other words, ’1 maps every subset L ⊆ V1∗ on the language generated by the grammar G1 from the set of axioms L). Clearly, the equivalence problem for two diMerent dpda reduces to the equivalence problem for two diMerent con$gurations of a single normalized dpda M. This last problem reduces to the problem ’1 (S) = ’1 (T )? for some polynomials S; T ∈ B
V1 .

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561

3. Series and matrices 3.1. Deterministic series, vectors and matrices We introduce here a notion of deterministic series which, in the case of the alphabet VM associated to a dpda M, generalizes the classical notion of con7guration of M. The main advantage of this notion is that, unlike for con$gurations, we shall be able to de$ne nice algebraic operations on these series (see, in particular, Section 3.3). Let us consider a pair (W; ˆ) where W is an alphabet (i.e. a set1) and ˆ is an equivalence relation over W . We call (W; ˆ) a structured alphabet. The most classical examples are: • the case where W = VM , the variable alphabet associated to M and [p; z; q] ˆ [p ; z  ; q ] iM p = p and z = z  (see [2]) • the case where W = X , the terminal alphabet of M and x ˆ y holds for every x; y ∈ X (see [2]). 3.1.1. De7nitions Denition 31. Let S ∈ B

W . S is said left-deterministic iM either (1) S = ∅ or (2) S =  or (3) ∃ i0 ∈ [1; m]; Si0 = ∅ and ∀w; w ∈ W ∗ , Sw = Sw = 1 ⇒ [∃A; A ∈ W; w1 ; w1 ∈ W ∗ ; A ˆ A ; w = A · w1 and w = A · w1 ]: A left-deterministic series S is said to have the type ∅ (resp. , [A]ˆ ) if case (1) (resp. (2), (3)) occurs. Denition 32. Let S ∈ B

W . S is said deterministic iM, for every u ∈ W ∗ , S • u is left-deterministic. This notion is the straighforward extension to the in$nite case of the notion of ($nite) set of associates de$ned in [3, De$nition 32 p. 188]. We denote by DB

W  the subset of deterministic boolean series over W . Let us denote by Bn; m

W  the set of (n; m)-matrices with entries in the semi-ring B

W . Denition 33. Let m ∈ N; S ∈ B1; m

W : S = (S1 ; : : : ; Sm ). S is said left-deterministic iM either (1) ∀i ∈ [1; m]; Si = ∅ or (2) ∃ i0 ∈ [1; m]; Si0 =  and ∀i = i0 ; Si = ∅ or (3) ∀w; w ∈ W ∗ ; ∀i; j ∈ [1; m]; (Si )w = (Sj )w = 1 ⇒ [ ∃ A; A ∈ W; w1 ; w1 ∈ V ∗ ; A ˆ A ; w = A · w1 and w = A · w1 ]: 1

Notice that we do not suppose that W is $nite.

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A left-deterministic row-vector S is said to have the type ∅ (resp. (; i0 ); [A]ˆ ) if case (1) (resp. (2), (3)) occurs. The right-action • on B

W  is extended componentwise to Bn; m

W : for every S = (si; j ), u ∈ W ∗ , the matrix T = S • u is de$ned by ti;j = si;j • u: Denition 34. Let S ∈ B1; m

W . S is said deterministic iM, for every u ∈ W ∗ , S • u is left-deterministic. We denote by DB1; m

W  the subset of deterministic row-vectors of dimension m over B

W . Denition 35. Let S ∈ Bn; m

W . S is said deterministic iM, for every i ∈ [1; n], Si;: is a deterministic row-vector. Let us notice $rst an easy fact about deterministic series. Fact 36. For every S ∈ DB

W ; u ∈ W ∗ ; S • u ∈ DB

W . Norm. Let us generalize the classical de$nition of rationality of series in B

W  to matrices. Given M ∈ Bn; m

W  we denote by Q(M ) the set of residuals of M : Q(M ) = {M • u | u ∈ W ∗ }: Similarly, we denote by Qr (M ) the set of row-residuals of M :  Q(Mi;∗ ): Qr (M ) = 16i6n

M is said rational iM the set Q(M ) is $nite. One can check that it is equivalent to the property that every coeOcient Mi; j is rational, or to the property that Qr (M ) is $nite. We denote by RBn; m

W  (resp. DRBn; m

W ) the set of rational (resp. deterministic, rational) matrices over B

W . For every M ∈ RBn; m

W , we de$ne the norm of M as M  = Card(Qr (M )): 3.1.2. Residuals Lemma 37. Let S ∈ DB

W ; T ∈ B

W ; u ∈ W ∗ . If S • u = ∅ then (S · T ) • u = (S • u) · T . Proof. Let S ∈ DB

W ; T ∈ B

W ; u ∈ W ∗ , such that S•u = ∅. Let u ; u ∈ W ∗ such that u = u · u ; u =  and let w ∈ supp(S). If w • u =  then S • u =  (because S • u is left-deterministic), hence S • u =  • u = ∅, which would contradict the hypothesis. It follows that ∀u ≺ u; ∀w ∈ supp(S);

w • u = :

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

563

Hence ∀w1 ∈ supp(S); ∀w2 ∈ supp(T );

(w1 · w2 ) • u = (w1 • u) · w2 :

This proves that (S · T ) • u = (S • u) · T . Lemma 38. Let S ∈ DB

W ; T ∈ B

W ; u ∈ W ∗ and U = S ·T . Exactly one of the following cases is true: (1) S • u = ∅; in this case U • u = (S • u) · T . (2) S • u = ∅; ∃ u ; u ; u = u · u ; S • u = ; in this case U • u = T • u . (3) S • u = ∅; ∀u 4 u; S • u = ; in this case U • u = ∅ = (S • u) · T . Proof. Clearly, one of the hypotheses (1–3) must occur. Let us examine each one of these cases. In case (1), by Lemma 37, U • u = (S • u) · T . In case (2), U • u = (U • u ) • u and by case (1), U • u = (S • u ) · T . It follows that U • u = T • u . In case (3), if S = ∅, the conclusion of the lemma is clearly true. Let us suppose now that S = ∅ and let u ≺ u be the maximum pre$x of u such that S • u = ∅. A · u and there exist some Then, there exist some A ∈ W; u ∈ W ∗ such that u = u ·
 B1 ; : : : ; Bq ∈ W; S1 ; : : : ; Sq ∈ B

W  − {∅} such that S • u = 16i6q Bq · Sq and B1 ˆ · · · ˆ Bi ˆ · · · ˆ Bq (because S • u is left-deterministic). By maximality of u , A does not belong to {B1 ; : : : ; Bq }, hence   
Bi · Si · T • A • u = ∅ • u = ∅: U •u= 16i6q

Lemma 39. Let S ∈ DB1; m

W ; T ∈ Bm; s

W ; u ∈ W ∗ and U = S · T . Exactly one of the following cases is true: (1) ∃ j; Sj • u ∈ {∅; } in this case U • u = (S • u) · T . (2) ∃ j0 ; ∃ u ; u ; u = u · u ; Sj0 • u = ; in this case U • u = Tj0 • u . (3) ∀j; ∀u 4 u; Sj • u = ∅; Sj • u = ; in this case U • u = ∅ = (S • u) · T . This lemma is an easy extension of Lemma 38 to the matricial case. We leave to the reader the corresponding routine proof. Lemma 310. For every S ∈ DBn; m

W ; T ∈ DBm; s

W , 1. S · T ∈ DBn; s

W . 2. S · T 6S + T .

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Proof. As the notion of deterministic matrix is de$ned row by row, it is suOcient to prove this lemma in the particular case where n = 1. Let u ∈ W ∗ . Let us consider every one of the 3 cases considered in Lemma 39. In case (1) or (3), (S · T ) • u = (S • u) · T and in case (2), (S · T ) • u = T • u : In both cases, (S · T ) • u is left-deterministic. Moreover, the total number of residuals of the $rst (resp. second) form is 6S (resp. 6T ). 3.1.3. Operations on row-vectors Let us introduce two new operations on row-vectors and prove some technical lemmas about them. Given A; B ∈ B1; m

W  and 16j0 6m we de$ne the vector C = A∇j0 B as follows: if A = (a1 ; : : : ; aj ; : : : ; am ); B = (b1 ; : : : ; bj ; : : : ; bm ) then C = (c1 ; : : : ; cj ; : : : ; cm ), where cj = aj + aj0 · bj if j = j0 ;

cj = ∅ if j = j0 :

Lemma 311. Let A; B ∈ B1; m

W  and 16j0 6m. (1) if A; B are deterministic, then A∇j0 B is deterministic. (2) if A; B are deterministic, then A∇j0 B6A + B: Proof. Suppose that A; B are deterministic and let C = A∇j0 B, u ∈ W ∗ . Using Lemma 38 we obtain that one of the two following cases occurs: C • u = (A • u)∇j0 B;

(16)

C • u =
(B • u )|∅m j0 ;

(17)

m where ∅m ∅ and  have been exchanged and
∗; ∗ is the j0 is the row-vector j0 in which
m “scalar product” de$ned by
S; T  = j=1 Sj · Tj . These formulas show that C • u is left-deterministic. The number of residuals of the form (16) is bounded above by A and the number of residuals of the form (17) is bounded above by B. Hence C6A + B:

Given A ∈ DB1; m

W  and 16j0 6m we de$ne the vector A = ∇j∗0 (A) as follows: if A = (a1 ; : : : ; aj ; : : : ; am ) then A = (a1 ; : : : ; aj ; : : : ; am ) where aj = a∗j0 · aj if j = j0 ;

aj = ∅ if j = j0 :

Lemma 312. Let A ∈ DB1; m

W  and 16j0 6m. Then ∇j∗0 (A) ∈ DB1; m

W  and ∇j∗0 (A)6A:

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

565

Proof. Let us examine a residual A •u , for some u ∈ W ∗ . Let u = max{v 4 u | v ∈ a∗j0 }. Let u ∈ W ∗ such that u = u · u . One can check that A • u = (A • u )∇j0 A :

(18)

By formula (18), A • u is left-deterministic. Moreover, Card(Q(A ))6Card(Q(A)), i.e. A 6A. 3.2. The in7nite grammar G 3.2.1. Strict-deterministic grammars Denition 313. Let G =
X; V; P be a context-free grammar in Greibach normal form. G is said strict-deterministic iM there exists an equivalence relation ˆ over V ful$lling the following condition: for every x
∈ X , if (Ek )16k6m are distinct variables in V such that ∀k ∈ [1; m]; E1 ˆ Ek and Hk = (Ek ; h)∈P h • x, then (H1 ; H2 ; : : : ; Hm )

is a deterministic vector:

Any equivalence ˆ satisfying the above condition is said to be a strict equivalence for the grammar G. This de$nition is a reformulation of [2, De$nition 11.4.1 p. 347] adapted to the case of a Greibach normal-form. Theorem 314. Let G =
X; V; P be a strict-deterministic grammar. Then its reduced form G0 =
X; V0 ; P0 , as de7ned in formulas (14), (15), is strict-deterministic too. Moreover, if ˆ is a strict equivalence for G, its restriction over V0 is a strict equivalence for G0 . The proof would consist in slightly extending the proof of [2, Theorem 11.4.1 p. 350]. It is known that, given a dpda M, its associated grammar GM is strict-deterministic. By Theorem 314 G0 is strict-deterministic too. Let us consider the minimal strict equivalence ˆ for G0 and extend it to V1 by, ∀v; v ∈ V0 : vP ˆ vP ⇔ v ˆ v ;

vP ˆ v :

Then ˆ is a strict equivalence for G1 (the grammar G1 is de$ned in Section 2.3). This ensures that G1 is strict-deterministic. Marks. A word w ∈ V ∗ is said marked iM w ∈ V ∗ · VP0 · V ∗ ; it is said fully marked iM w ∈ VP0∗ . A series S ∈ B

V  is said marked iM ∃ w ∈ supp(S), w is marked; it is said fully marked iM ∀w ∈ supp(S), w is fully marked. It is said unmarked iM it is not marked. A matrix S ∈ Bm; n

V
 is said marked (resp. fully marked, unmarked) iM, for every n i ∈ [1; m], the series j=1 Si; j is marked (resp. fully marked, unmarked).

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Denition 315. Let d ∈ N. A vector S ∈ DB1; 5

V  is said d-marked iM there exists q ∈ N; 6 ∈ DRB1; q
V ; 7 ∈ DRBq; 5

V  such that S=

q
k=1

6k · 7 k

and

6 6 d;

and 7 is unmarked. Lemma 316. For every S ∈ DB1; 5

V  $e (S); $Pe (S) ∈ DB1; 5

V  Proof (sketch). Let us notice that the homomorphism $e : V ∗ → V ∗ preserves the equivalence ˆ: for every v; v ∈ V , if v ˆ v then $e (v) ˆ $e (v ). It follows that the corresponding substitution $e preserves determinism. The same argument applies on $Pe . Innite alphabet V. Let us introduce an in$nite alphabet V2 = {81 ; 82 ; : : : ; 8n ; : : :} together with a bijection: ’2 : V2 ∪ {} → DRB
V1  − {∅} such that ’2 () = : We suppose that the map n → ’2 (8n ) is computable. We then de$ne the (in$nite) structured alphabet (V; ˆ) as the following extension of (V1 ; ˆ): V = V1 ∪ V2 ; ∀v; v ∈ V; v ˆ v ⇔ (v; v ∈ V1 and v ˆ v ) or (v = v ∈ V2 ): In other words, the 8n are second-level variables which represent, bijectively, the deterministic rational boolean series over the set of 7rst-level variables V1 . We call linear polynomials the row-vectors of the form: (61 91 ; : : : ; 6i 9i ; : : : ; 6m 9m ) ∈ DB1;m
V 

(19)

such that ∀i ∈ [1; m]; 6i ∈ B1; m
V1 ; 9i ∈ V2 ∪ {}. We extend ’2 as a substitution B

V  → B
V1  by setting ∀v ∈ V1 ;

’2 (v) = v:

Let us de$ne a
set P2 of productions by q if ’2 (8n ) = k=1 Ek ·7k , where ∀k ∈ [1; q]; Ek ∈ V1 ; E1 ˆ Ek then, for every k ∈ [1; q] (8n ; x · (Ek  x) · ’−1 2 (7k ))

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is a rule of P2 . We extend now the set P1 by de$ning P = P1 ∪ P2 . One can check that the (in$nite) grammar G =
X; V; P is still strict-deterministic. Let us consider the unique substitution ’ : B

V  → B

X  ful$lling: for every v∈V, ∗

’(v) = {u ∈ X ∗ | v →P u}:

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567

The action of X ∗ on B

V . Let (V; ˆ) be the in$nite structured alphabet associated with M in the above paragraph. We de$ne the right-action  as the unique -right-action of the monoid X ∗ over the semi-ring B

V  such that for every v ∈ V; 9 ∈ V ∗; x ∈ X  
h • x · 9; (21) (v · 9)  x = (v;h)∈P

  x = ∅:

(22)

Lemma 317. For every S ∈ B

V ; u ∈ X ∗ , (1) ’(S  u) = ’(S) • u (i.e. ’ is a morphism of right-actions). (2) ’ = ’1 ◦ ’2 Proof. (1) Let v ∈ V; 9 ∈ V ∗ ; x ∈ X . One can then check on formulas (21), (22) that ’(  x) = ’() • x

and

’((v · 9)  x) = ’(v · 9) • x:

By -additivity of ’, point 1 of the lemma is true in the case where |u|61. By induction on |u| the general case follows. (2) It suOces to prove that, for every 8 ∈ V , ’(8) = ’1 (’2 (8)):

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(2.1) If 8 ∈ V1 , (23) is true because ’2 (8) = 8 and the only rules appearing in a ∗ derivation 8 →P u belong to P1 .
q (2.2) Suppose 8 ∈ V2 and ’2 (8) = k=1 Ek · 7k . Let us show, by induction on |u|, that for every u ∈ X ∗ , ’2 (8  u) = ’2 (8)  u:

(24)

If u = , this is clear. If u = x for some x ∈ X : q
8x = (Ek  x) · ’−1 2 (7k ) k=1

while

 ’2 (8)  x =

q
k=1

 Ek · 7k

x =

q
k=1

(Ek  x) · 7k :

Hence, equality (24) is true. If u = u x for some u ∈ X ∗ ; x ∈ X : ’2 (8  u x) = ’2 ((8  u )  x) = (’2 (8  u ))  x (by the case above) = (’2 (8)  u )  x (by induction hypothesis) = ’2 (8)  u x: Equality (24) is established. Using point 1 of the lemma, we get that:

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

u ∈ ’(8) ⇔  ∈ ’(8)•u ⇔  ∈ ’(8u) ⇔ 8u =  (because G is proper) ⇔ ’2 (8 u) =  ⇔  ∈ ’2 (8) • u ⇔ u ∈ ’(’2 (8)) ⇔ u ∈ ’1 (’2 (8)) (because ’; ’1 coincide on V1 ): Equivalence of vectors. We denote by ≡ the kernel of ’ i.e.: for every S; T ∈ B

V , S ≡ T ⇔ ’(S) = ’(T ): One can notice that, by Lemma 317 point (2), for every S ∈ B

V , S ≡ ’2 (S):

(25)

For every integer n, we denote by ≡n the following approximation of ≡: S ≡n T ⇔ ’(S) ∩ X 6n = ’(T ) ∩ X 6n : We extend the substitutions ’1 ; ’2 ; ’, the action  and the equivalence ≡, componentwise, to Bn; m

V  (for every integers n; m¿0). Right-action and product. Let us de$ne here handful notations for some particular vec j. tors or matrices. We use the Kronecker symbol i; j meaning  if i = j and ∅ if i = For every 16n; 16i6n, we de$ne the row-vector in as: n in = (i;j )16j6n

n where ∀j; i;j = i;j :

We call unit row-vector any vector of the form in . For every 16n, we denote by ∅n ∈ DB1; n

V  the row-vector: ∅n = (∅; : : : ; ∅): Let us de$ne K0 = max{(E1 ; E2 ; : : : ; En )  x | (Ei )16i6n is a bijective numbering of some class in V= ˆ; x ∈ X }:

(26)

Lemma 318. For every S ∈ DB1; 5

V ; u ∈ X ∗ , (1) S  u ∈ DB1; 5

V  (2) S  u6S + K0 · |u|: Proof. We treat $rst the case where u is just a letter. Let S ∈ DB1; 5

V  and x ∈ X . If S = ∅5 or S = ”5i (for some i ∈ [1; 5]), then Sx = ∅5 and points (1), (2) are both true. Otherwise S=

q
k=1

E k · 7k

for some q ∈ N; 7k ∈ DB1; 5

V ; (Ek )16k6q bijective numbering of some class of V= ˆ.

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By Eq. (21) , which de$nes the right-action , S x =

q
k=1

(Ek  x) · 7k

hence S  x has the form H · 7 where H ∈ DB1; q

V  (see De$nition 313), H 6K0 (see Eq. (26)) and 7 ∈ DBq; 5

V . By Lemma 310, H · 7 is deterministic and by Lemma 2 H · 767 + K0 . As every 7k ∈ Qr (S) we obtain H · 7 6 7 + K0 6 S + K0 : Both points (1), (2) are proved. The general case where u is any word of X ∗ can be deduced by induction on |u| from this particular case. Lemma 319. Let 5 ∈ N − {0}; S ∈ DRB1; 5

V ; u ∈ X ∗ . One of the three following cases must occur: (1) S  u = ∅5 ; (2) S  u = j5 for some j ∈ [1; 5], (3) ∃ u1 ; u2 ∈ X ∗ ; v1 ∈ V ∗ ; q ∈ N; E1 ; : : : ; Ek ; : : : ; Eq ∈ V; 7 ∈ DRBq; 5

V  such that u = u1 · u2 ; S  u1 = S • v1 =

q
k=1

Ek · 7 k ; S  u =

q


(Ek  u2 ) · 7k ; and

k=1

∀k ∈ [1; q]; Ek ˆ E1 ; Ek  u2 ∈= {; ∅}: Proof. Let u ∈ X ∗ . Let us prove the lemma by induction on |u|. u = : if S ∈ ∅5 ∪ {j5 | 16j65} then clearly the conclusion of case (1) or (2) is realized.
q Otherwise, as S is left-deterministic, S has a decomposition as S = k=1 Ek ·7k such that the conclusion of case (3) is realized with u1 = u2 = ; v1 = , the given integer q and the letters E1 ˆ · · · ˆ Eq ∈ V . u = u0 · a; a ∈ X : Let us consider the u1 ; u2 ; v1 ; q; (Ek )16k6q ; (7k )16k6q given by the induction hypothesis on u0 .   q
(Ek  u2 ) · 7k  a and (S  u)  a = k=1

∀k ∈ [1; q]; Ek  u2  ¿ 3: Case 1: ∀k ∈ [1;
q]; Ek  u2 a¿3. q Then S  ua = k=1 (Ek  u2 a) · 7k . Hence conclusion (3) of the lemma is ful$lled by u1 = u1 ; u2 = u2 a; v1 = v1 ; q = q; Ek = Ek ; 7k = 7k . Case 2: ∃ r ∈ [1; q]; Er  u2 a = 2. In other words: there exists some r ∈ [1; q] such that Er  u2 a = , hence S  ua = 7r :

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Subcase 1: 7r ∈ {∅5 } ∪ {j5 | 16j65}. Conclusion (1) or (2) of the lemma is then realized.
r  Subcase 2: 7r = ‘=1 F‘ · 8‘ for some r  ∈ N; F1 ˆ · · · Fr  ∈ V; 8 ∈ DRBr  ; 5

V . Then 

S  ua =

r
‘=1



F‘ · 8‘ ;

S • (v1 Er ) = 7r =

r
‘=1

F ‘ · 8‘ :

Conclusion (3) of the lemma is then realized by u1 = ua; u2 = ; v1 = v1 Er ; q = r  ; Ek = Fk ; 7 = 8. Case 3: ∀k ∈ [1; q]; Ek  u2 a = 1. This means that E  u2 a = ∅q , hence that case (1) is realized. Corollary 320. Let 5 ∈ N − {0}; S; S  ∈ DB1; 5

V . Then S ≡ S  if and only if, ∀u ∈ X ∗ ; ∀j ∈ [1; 5]; S  u = j5 ⇔ S   u = j5 : Proof. (1) Let us show that for every T ∈ DB1; 5

V ; j ∈ [1; 5] and u ∈ X ∗ : T  u = j5 ⇔ u ∈ ’(Tj ):

(27)

Suppose T  u = j5 . Then Tj  u = . and by Lemma 317, point (1), ’(Tj ) • u = , hence u ∈ ’(Tj ). Suppose now that u ∈ ’(Tj ). Then Tj  u =  (same arguments as above) and, by Lemma 318 point (1), T  u is a deterministic vector, hence T  u = j5 . (2) From equivalence (27), the lemma follows. We give now an adaptation of Lemma 39 to the action  in place of •. Lemma 321. Let S ∈ DB1; m

V ; T ∈ Bm; s

V ; u ∈ X ∗ and U = S · T . Exactly one of the following cases is true: (1) S  u ∈ {∅m } ∪ {jm | 16j6m} in this case U  u = (S  u) · T . (2) ∃ j0 ; ∃ u ; u ; u = u · u ; S  u = js0 ; in this case U  u = Tj0  u . (3) ∀j; ∀u 4 u; S  u = ∅m and S  u = jm ; in this case U  u = ∅s = (S  u) · T . Proof. The arguments used in the proofs of Lemmas 37–39 can be adapted to  in place of •. The only non-trivial adaptation is that of lines 6 –7 of the proof of Lemma 37: let us suppose that u ∈ X ∗ is such that ∀u ≺ u;

S  u = 

(28)

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

571

and let us prove that (S · T )  u = (S  u) · T:

(29)

We prove by induction on |u| that (28) implies (29). |u| = 0: by de$nition of a right-action, ∀S  ∈ DB

V ; S    = S  . Hence conclusion (29) is true. u = u0 · a, where u0 ∈ X ∗ ; a ∈ X : Hypothesis (28) is ful$lled by u0 too, hence, by induction hypothesis (S · T )  u0 = (S  u0 ) · T:

(30)

If S  u0 = ∅, then, by the above equality (S · T )  u0 = ∅ too, hence (S · T )  u0 a = ∅ = (S  u0 a) · T; hence (29) is true. Otherwise, by hypothesis (28) Su0 ∈ {∅; }, hence there exists q ∈ N; E1 ˆ · · · ˆ Eq ∈ V; 7 ∈ DBm; s

V  such that S  u0 =

q
k=1

E k · 7k :

(31)

By De$nition (21) and the fact that  is a -action (Ek · 7k )  a = (Ek  a) · 7k hence, by -additivity  q  q

Ek · 7 k  a = (Ek  a) · 7k k=1

k=1

and by product by T : (S  u0 a) · T =

q


(Ek  a) · 7k · T:

k=1

(32)

Let us examine now (ST )  u0 a. By (30) (S · T )  u0 =

q
k=1

Ek · 7k · T:

By De$nition (21) and the fact that  is a -action (Ek · 7k · T )  a = (Ek  a) · 7k · T hence, by -additivity  q  q

Ek · 7 k · T  a = (Ek  a) · 7k · T k=1

k=1

(33)

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Using (33) this last equality can be read (ST )  u0 a =

q
k=1

(Ek  a) · 7k · T:

(34)

As equalities (34), (32) have the same right-hand side, we conclude that (29) is true. 3.3. Linear independence We adapt here the key-idea of [4,5] to deterministic series in DRB

V .
Let us call linear combination of the series S1 ; : : : ; Sj ; : : : ; Sm any series of the form 16j6m 6j · Sj where
∈ DRB1; m

V . Following an analogy with classical linear algebra, we develop a notion corresponding to a kind of linear independence of the classes (mod ≡) of the given series. Lemma 322. Let S1 ; : : : ; Sj ; : : : ; Sm ∈ DRB

V
. The following
are equivalent (1) ∃
;  ∈ DRB1; m

V ;
≡ , such that 16j6m 6j · Sj ≡ 16j6m 9j · Sj ;
(2) ∃ j0 ∈ [1; m]; ∃  ∈ DRB1; m

V ;  ≡ jm0 , such that Sj0 ≡ 16j6m =j · Sj ;
(3) ∃ j0 ∈ [1; m]; ∃  ∈ DRB1; m

V ; =j0 ≡ ∅, such that Sj0 ≡ 16j6m =j · Sj : The equivalence between (1), (2) and (3) was $rst proved in [4, Lemma 11 p. 589], 2 in the case where the Sj ’s are con$gurations qj !, with the same !. Proof. Let use the notation S = (Sj )16j6m ∈ DRBm;1

V . (1) ⇒ (2): Let us consider u = min{’(
)S’()}: By Corollary 320, ∃ j0 ∈ [1; m], such that
 u = jm0 ⇔   u = jm0 : Let us suppose, for example, that
 u = jm0 while   u = jm0 and let  =   u. As ≡ is preserved by the action  (see Lemma 317) (
· S)  u ≡ ( · S)  u:

(35)

Using Lemma 321 we obtain (
· S)  u = Sj0 :

(36)

Let us examine now the right-hand side of equality (35). Let u ≺ u. By minimality of u,   u is a unit iM
 u is a unit. But if
 u is a unit, then
 u = ∅m , which 2

Numbering of the english version.

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

573

is false. Hence   u is not a unit. By Lemma 321 ( · S)  u = (  u) · S:

(37)

Let us plug equalities (36) and (37) in equivalence (35) and let us de$ne  =   u. We obtain Sj0 ≡  · S;

where  ≡ jm0 :

(2) ⇒ (3):



Sj0 ≡ =j0 · Sj0 +




j=j0

 ; =j · Sj

;

=j0 ≡ :

By the well-known Arden’s lemma (see Corollary 49 point (C1)), we can deduce that
∗ Sj0 ≡ =j0 =j · Sj = ∇∗j0 (=) · S: j=j0

Taking = = ∇j∗0 (=) we obtain Sj0 ≡ = · S

where =j0 = ∅:

(3) ⇒ (1): (3) asserts that
j;j0 Sj ≡ 16j6m


16j6m

=j · Sj :

The fact that j0 ; j0 =  ≡ ∅ shows that ( 1; j0 ; : : : ; m; j0 ) ≡ (=1 ; : : : ; =m ). Hence (1) is true. 3.4. Derivations For every u ∈ X ∗ we de$ne the binary relation ↑ (u) over DB1; 5

V  by: for every S; S  ∈ DB1; 5

V ; S ↑ (u)S  ⇔ ∃ q ∈ N; ∃ E1 ; : : : ; Ek ; : : : ; Eq ∈ V; 7 ∈ DBq; 5

V  such that S=

q
k=1

E k · 7k ;

S =

q
k=1

(Ek  u) · 7k

and ∀k ∈ [1; q]; E1 ˆ Ek ; Ek  u ∈= {∅; }. It is clear that if S ↑ (u)S  then S u = S  and that the converse is not true in general. A sequence of deterministic row-vectors S0 ; S1 ; : : : ; Sn is a derivation iM there exist x1 ; : : : ; xn ∈ X such that S0  x1 = S1 ; : : : ; Sn−1  xn = Sn . The length of this derivation is n. If u = x1 · x2 · : : : · xn we call S0 ; S1 ; : : : ; Sn the derivation associated with (S; u). We u denote this derivation by S0 → Sn . A derivation S0 ; S1 ; : : : ; Sn is said to be stacking iM it is the derivation associated to a pair (S; u) such that S = S0 and S0 ↑ (u)Sn . A derivation S0 ; S1 ; : : : ; Sn is said to be

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

a sub-derivation of a derivation S0 ; S1 ; : : : ; Sm iM there exists some i ∈ [0; m] such that,  ∀j ∈ [1; n]; Sj = Si+j . Lemma 323. Let S ∈ DRB1; 5
V ; w ∈ X ∗ , such that ∀u 4 w; S  u¿S. w Then the derivation S → S  w is stacking. Proof. S is left-deterministic. If it has type ∅ or (; j), the lemma is trivially true. Otherwise S=

q
k=1

Ek · 7 k

for some class of letter [E1 ]ˆ = {E1 ; : : : ; Eq } and some matrix 7 ∈ DRBq; 5

V . Suppose that for some pre$x u 4 w and k ∈ [1; q], Ek  u = :

(38)

Then, S  u = 7k so that S  u67¡S which contradicts the hypothesis. Let us apply now Lemma 321 to the expression (E · 7)  w: case (2) is impossible, hence (E · 7)  w = (E  w) · 7; which is equivalent to S ↑ (w)S  w: Lemma 324. Let S ∈ DRB1; 5
V ; w ∈ X ∗ ; k ∈ N, such that S  w¿S + k · K0 + 1. w Then the derivation S → S  w contains some stacking sub-derivation of length k. Proof (sketch). Let S = S0 ; : : : ; Si ; : : : ; Sn be the derivation associated to (S; w). Let i0 = max{i ∈ [0; n] | Si  = min{Sj  | 06j6n}}. Let w = w0 w where |w0 | = i0 . As Sn  − Si0 ¿k · K0 + 1; by Lemma 318 we must have |w |¿k. Let w = w2 w3 with |w2 | = k. By de$nition of i0 , ∀i ∈ [i0 + 1; i0 + k]; Si ¿Si0  + 1. By Lemma 323, the sub-derivation Si0 ; : : : ; Si0 +k (associated to (Si0 ; w2 )) is stacking. Lemma 325. Let S; S  ∈ DRB

V ; w ∈ X ∗ ; k; d; d ∈ N, such that S is d-marked and: w (1) the derivation S → S  contains no stacking sub-derivation of length k. (2) |w|¿d · k. Then S  is unmarked. Proof. By hypothesis S=

q
k=1

6k · 7 k

for some
∈ DRB1; q
V ; 7 ∈ DRBq;1

V , 
6d; 7 unmarked.

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

575

w

Let S → S  = (S0 ; : : : ; Sn ). By induction on ‘, using hypothesis (1) and Lemma 323 one can show that: for every ‘ ∈ [0; d], there exists some pre$x w‘ of w, with length |w‘ |6k · ‘ such that either S  w‘ =

q
k=1

(6k  w‘ ) · 7k ;

with 
 w‘  ¡ 6 − ‘

(39)

or there exists an integer k ∈ [1; q] such that S  w‘ = 7k :

(40)

Let us apply this property to ‘ = d: inequality (39) is not possible for this value of ‘ because, by hypothesis 
 − ‘60. Hence (40) is true and, as 7 is unmarked, 7k is unmarked, so that S  w is unmarked. 4. Deduction systems 4.1. General formal systems We follow here the general philosophy of [3,1]. For any set E, we denote by P(E) the set of its subsets and by Pf (E) the set of it 7nite subsets. Let us call formal system any triple D =
A; H; |−−  where A is a denumerable set called the set of assertions, H , the cost function is a mapping A → N ∪ {∞} and |−− , the deduction relation is a subset of Pf (A) × A. A is given with a $xed bijection with N (an “encoding” or “GTodel numbering”) so that the notions of recursive subset, recursively enumerable subset, recursive function; : : : over A; Pf (A); : : : are de$ned, up to this $xed bijection; we assume that D satis$es the following axiom: (A 1) ∀(P; A) ∈ |−− , (min{H (p); p ∈ P}¡H (A)) or (H (A) = ∞). (We let min(∅) = ∞). We call D a deduction system iM D is a formal system satisfying the additional axiom: (A 2) |−− is recursively enumerable. In the sequel we use the notation P |−− A for (P; A) ∈ |−− . We call proof in the system D, relative to the set of hypotheses H ⊆ A, any subset P ⊆ A ful$lling: ∀p ∈ P;

(∃Q ⊆ P; Q |−− p) or

(p ∈ H):

We call P a proof iM ∀p ∈ P;

(∃Q ⊆ P; Q |−− p)

(i.e. iM P is a proof relative to ∅). Let us de$ne the total map ? : A → {0; 1} and the partial map ?P : A → {0; 1} by ?(A) = 1 if H (A) = ∞;

?(A) = 0 if H (A) ¡ ∞;

?(A) P = 1 if H (A) = ∞;

?P is unde$ned if H (A) ¡ ∞:

(? is the “truth-value function”, ?P is the “1-value function”).

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Lemma 41. Let P be a proof relative to H ⊆ H −1 (∞) and A ∈ P. Then ?(A) = 1. In other words: if an assertion is provable from true hypotheses, then it is true. Proof. Let P be a proof. We prove by induction on n that P(n): ∀p ∈ P; H (p) ¿ n: It is clear that, ∀p ∈ P; H (p)¿0. Suppose that P(n) is true. Let p ∈ P − H: ∃ Q ⊆ P; Q |−− p. By induction hypothesis, ∀q ∈ Q; H (q)¿n and by (A1), H (p)¿n + 1. It follows that: ∀p ∈ P − H; H (p) = ∞. But by hypothesis, ∀p ∈ H; H (p) = ∞. A formal system D will be said complete iM, conversely, ∀A ∈ A; ?(A) = 1 ⇒ there exists some 7nite proof P such that A ∈ P. (In other words, D is complete iM every true assertion is “$nitely” provable). Lemma 42. If D is a complete deduction system, ?P is a recursive partial map. Proof. The property “H (A) = ∞” is semi-decidable just because the property “there exists a $nite P such that P is a D-proof and A ∈ P” is semi-decidable too. In order to de$ne deduction relations from more elementary ones, we set the following de$nitions. Let |−− ⊆ Pf (A) × A. For every P; Q ∈ Pf (A) we set: [0]

— P |−− Q iM P ⊇ Q [1]

— P |−− Q iM ∀q ∈ Q; ∃ R ⊆ P; R |−− q 0

[0]

— P |−− Q iM P |−− Q 1

— P |−− Q iM ∀q ∈ Q; (∃ R ⊆ P; R |−− q) or (q ∈ P) n+1

1

n

— P |−− Q iM ∃ R ∈ Pf (A); P |−− R and R |−− Q (for every n¿0). ∗ n  — |−− = n¿0 |−− . Given |−−1 ; |−−2 ⊆ Pf (A) × Pf (A), for every P; Q ∈ Pf (A) we set: P( |−− 1 ◦ |−− 2 )Q

iM

∃R ⊆ A; (P |−− 1 R) ∧ (R |−− 2 Q):

The particular deduction systems Di =
Ai ; Hi ; |−−Di  (i ∈ [0; 1]), that we shall introduce in Sections 4.3 and 10, will always be de$ned from simpler binary relations ||−−j by means of the above constructions. The key-statement of this work is that a particular deduction system, D0 (de$ned in Section 4.3), is complete (Theorem 92). We prove this completeness result by exhibiting a “strategy” S which, for every true assertion constructs a $nite D0 -proof of this assertion.

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

577

4.2. Strategies Let D =
A; H; |−−  be a deduction system. We call a strategy for D any partial map S : A+ → A∗ such that: (S1) if S(A1 A2 · · · An ) = B1 · · · Bm then ∃ Q ⊆ {Ai | 16i6n − 1} such that {Bj | 1 6 j 6 m} ∪ Q |−− An ; (S2) if S(A1 A2 · · · An ) = B1 · · · Bm then min{H (Ai ) | 1 6 i 6 n} = ∞ ⇒ min{H (Bj ) | 1 6 j 6 m} = ∞: Given a strategy S, we de$ne T(S; A), the proof-tree associated to the strategy S and the assertion A as the unique tree t such that: ” ∈ dom(t);

t(”) = A

and, for every path x0 x1 ; · · · xn−1 in t, with labels t(xi ) = Ai+1 (for 06i6n − 1) if xn−1 has m sons xn−1 · 1; · · · :xn−1 · m ∈ dom(t) with labels t(xn−1 · j) = Bj (for 16j6m) then (∀i ∈ [1; n − 1]; Ai = An and S(A1 · · · An ) = B1 · · · Bm ) or ( ∃ i ∈ [1; n − 1]; Ai = An and m = 0) or (A1 · · · An ∈= dom(S) and m = 0):

(41)

Notice that xn−1 is a leaf (i.e. m = 0) iM: (S(A1 · · · An ) = ”) or ( ∃ i ∈ [1; n − 1]; Ai = An ) or (A1 · · · An ∈= dom(S)): (42) Let us say that S terminates iM, ∀A ∈ ?−1 (1); T(S; A) is $nite; S is said closed iM, ∀W ∈ (?−1 (1))+ ; W ∈ dom(S) (i.e. S is de$ned on every non-empty sequence of true assertions). For every tree t let us de$ne L(t) = {t(x) | ∀y ∈ dom(t); x4y ⇒ x = y}; I(t) = {t(x) | ∃ y ∈ dom(t); x ≺ y}: (Here L stands for “leaves” and I stands for “internal labels”). Lemma 43. If S is a strategy for the deduction-system D then, for every true assertion A: (1) the set of labels of T(S; A) is a D-proof, relative to the set L(T(S; A)) − I(T(S; A)). (2) every label of a leaf is true. Proof. Let us suppose that H (A) = ∞. Let t = T(S; A); P = im(t) (the set of labels of t), H = L(T(S; A)) − I(T(S; A)). Using (S2), one can prove by induction on the depth of x ∈ dom(t) that, H (t(x)) = ∞. Point (2) is then proved. Let x be an internal node of t, with sons x · 1; x · 2; · · · ; x · m (m¿0), and with ancestors y1 ; y2 ; : : : ; yn−1 ; yn = x (n¿1), such that t(y1 ) · · · t(yn ) = A1 · · · An ;

t(x · 1)t(x · 2) · · · t(x · m) = B1 · B2 · · · Bm :

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

By de$nition of T(S; A), S(A1 · · · An ) = B1 · · · Bm and by condition (S1): ∃Q ⊆ {Ai | 1 6 i 6 n − 1};

such that {Bj | 1 6 j 6 m} ∪ Q |−− An :

It follows that for every p ∈= H; ∃ R ⊆ P; R |−− p, hence ∀p ∈ P;

(∃R ⊆ P; R |−− p) or

p ∈ H:

Point (1) is proved. Lemma 44. If S is a closed strategy for D, then, for every true assertion A, the set of labels of T(S; A) is a D-proof. Proof. Let us suppose that H (A) = ∞. Let t = T(S; A) and let P; H be de$ned as above. By Lemma 43, P is a D-proof relative to H. By Lemma 43 point (2) and Lemma 41, every label of a node of t is true. By the de$nition of a closed strategy, if p ∈ H and x is a leaf of t such that p = t(x) then, the only possible true assertion in clause (42) is “S(A1 · · · An ) = ””, which implies that ∃Q ⊆ P;

Q |−− t(x):

Lemma 43 point (1) and this fact show that P is a proof. Lemma 45. If D admits some terminating, closed strategy then D is complete. Proof. Clear from Lemma 44. 4.3. System D0 Let us de$ne here a particular deduction system D0 “Taylored for the equivalence problem for dpda’s”. Given a $xed dpda M over the terminal alphabet X , we consider the variable alphabet V associated to M (see Section 3.1) and the set DRB

V  (the set of Deterministic Rational Boolean series over V ∗ ). The set of assertions is de$ned by A = N × DRB

V  × DRB

V  i.e. an assertion is here a weighted equation over DRB

V . The “cost-function” H : A → N ∪ {∞} is de$ned by H (n; S; S  ) = n + 2 · Div(S; S  ); where Div(S; S  ), the divergence between S and S  , is de$ned by Div(S; S  ) = inf {|u| | u ∈ ’(S)!’(S  )}:

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

579

Let us notice that here ?(n; S; S  ) = 1 ⇔ S ≡ S  : We de$ne a binary relation ||−− ⊂ Pf (A) × A, the elementary deduction relation, as the set of all the pairs having one of the following forms: (R0) {(p; S; T )} ||−− (p + 1; S; T ) (R1)

for p ∈ N; S; T ∈ DRB

V , {(p; S; T )} ||−− (p; T; S)

(R2)

for p ∈ N; S; T ∈ DRB

V , {(p; S; S  ); (p; S  ; S  )} ||−− (p; S; S  )

(R3)

for p ∈ N; S; S  ; S  ∈ DRB

V , ∅ ||−− (0; S; S)

(R  3)

for S ∈ DRB

V , ∅ ||−− (0; S; T )

(R4)

for S ∈ DRB

V ; T ∈ {∅; }; S ≡ T , {(p + 1; S  x; T  x) | x ∈ X } ||−− (p; S; T )

(R5)

for p ∈ N; S; T ∈ DRB

V ; (S ≡  ∧ T ≡ ), {(p; S; S  )} ||−− (p + 2; S  x; S   x)

(R6)

for p ∈ N; S; T ∈ DRB

V ; x ∈ X; {(p; S · T  + S  ; T  )} ||−− (p; S ∗ · S  ; T  )

(R7)

for p ∈ N; (S; S  ) ∈ DRB1; 2

V ; T  ∈ DRB

V ; S ≡ ; {(p; S; S  ); (p; T; T  )} ||−− (p; S + T; S  + T  ) for p ∈ N; (S; T ); (S  ; T  ) ∈ DRB1; 2

V ,

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

(R8) {(p; S; S  )} ||−− (p; S · T; S  · T ) for p ∈ N; S; S  ; T ∈ DRB

V ;

(R9)

{(p; T; T  )} ||−− (p; S · T; S · T  ) (R10)

for p ∈ N; S; T; T  ∈ DRB

V , ∅ ||−− (0; S; $e (S))

(R11)

for S ∈ DRB

V ; ∅ ||−− (0; S; ’2 (S))

for S ∈ DRB

V : Remark 46. One can check that, by the results of Section 3, the above rules really belong to Pf (A) × A. Lemma 47. Let P ∈ Pf (A); A ∈ A such that P ||−− A. Then min{H (p) | p ∈ P}6 H (A). Proof. The only non-trivial checks of this property are for rules of type (R4) or (R6). (R4) Let S; T ∈ DRB

V ; S ≡ ”; T ≡ ”. If Div(S; T ) = ∞ the required inequality is true. If Div(S; T ) = n ∈ N, let us consider some u ∈ ’(S)!’(T ); |u| = n. We can suppose, for example, that S  u = ”; T  u = ”. As S ≡ ”; ∃ x0 ∈ X; ∃ v ∈ X ∗ ; u = x0 · v. Hence (S  x0 )  v = ”; (T  x0 )  v = ”; Div(S  x0 ; T  x0 )6|v|. Hence we have min{H (p + 1; S  x; T  x) | x ∈ X )} 6 H (p + 1; S  x0 ; T  x0 ) 6 (p + 1) + 2|v| 6 (p + 1) + 2 · Div(S; T ) − 2 ¡ H (p; S; T ): (R6) We are reduced to prove that, for every n¿0; (S; S  ) ∈ DRB1; 2

V ; T  ∈ DRB

V ; S ≡ ”, S · T  + S  ≡n T  ⇒ S ∗ · S  ≡n T  :

(43)

Let us suppose that S · T  + S  ≡n T  . By de$nition of the star operation: S n+1 · S ∗ · S  +

n
k=0

S k · S  = S ∗ · S :

(44)

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

581

And by the properties established in the treatment of (R7), (R9): S n+1 · T  +

n
k=0

S k · S  ≡n T  :

(45)

Let u ∈ X 6n . As S ≡ ; ∀u 4 u; S n+1  u = . By Lemma 321, for every U ∈ B

V , (S n+1 · U )  u = (S n+1  u) · U = : Using now equations (44), (45) we obtain that S∗ · S  u =  ⇔

n
k=0

S k · S   u =  ⇔ T   u = :

hence S ∗ · S  ≡n T  : This ends the proof of implication (43). Let us de$ne |−− by: for every P ∈ Pf (A); A ∈ A, ∗

[1]

P |−− A ⇔ P ||−− ◦ ||−−

0;3;4;10;11 ◦

∗

||−− {A};

where ||−−0; 3; 4; 10; 11 is the relation de$ned by (R0); (R3); (R  3); (R4); (R10); (R11) only. We let D0 =
A; H; |−− : Lemma 48. D0 is a deduction system. Proof. From the well-known decidability properties of $nite automata and the hypothesis that ’2 is a computable bijection, we conclude that |−− is recursively enumerable. Using Lemma 47, one can show by induction on n that n

P ||−− Q ⇒ ∀q ∈ Q;

min{H (A) | A ∈ P} 6 H (q):

The proof of Lemma 47 also reveals that P ||−− {0;3;4;10;11} q ⇒ (min{H (p) | p ∈ P} ¡ H (q)) or H (q) = ∞: It follows that, for every m; n¿0: n

[1]

P ||−− Q ||−−

0;3;4;10;11 R

m

||−− q ⇒ (min{H (p) | p ∈ P} ¡ H (q)) or

H (q) = ∞: Both axioms (A1); (A2) are ful$lled. Let us remark the following algebraic corollaries of Lemma 47.

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Corollary 49. (C1) ∀(S; S  ) ∈ DRB1; 2

V ; T  ∈ DRB

V ; S ≡ ; S · T  + S ≡ T  ⇒ S∗ · S ≡ T  (C2) ∀S; S  ∈ DRB

V ; T ∈ DRB

V , [S · T ≡ S  · T and T ≡ ∅] ⇒ S ≡ S  : Proof. Statement (C1) is a direct corollary of the fact that the value of H at the lefthand side of rule (R6) is smaller or equal to the value of H at the right-hand side of rule (R6). Let us prove (C2): let us consider S; S  ∈ DRB

V ; T ∈ DRB

V , such that S · T ≡ S · T

S ≡ S  :

and

(46)

Let u = min{v ∈ X ∗ | (S  v = ) ⇔ (S   v = )}: From the hypothesis that S · T ≡ S  · T , we get that, for every v ∈ X ∗ , (S · T )  v ≡ (S  · T )  v and by the choice of u we obtain that T ≡ (S   u) · T

or

(S  u) · T ≡ T;

which, by (C1), implies T ≡ (S   u)∗ · ∅ or

(S  u)∗ · ∅ ≡ T;

i.e. T ≡ ∅:

(47)

We have proved that (46) implies (47), hence (C2). 5. Triangulations Let S1 ; S2 ; : : : ; Sd be a family of deterministic rational boolean series over the structured alphabet V (i.e. Si ∈ DRB

V ). We recall V is the in$nite alphabet associated with some dpda M as de$ned in Section 2.2. Let us consider a sequence S of n “weighted” linear equations (Ei ): pi ;

d
j=1

6i;j Sj ;

d
j=1

9i;j Sj ;

(48)

where pi ∈ N − {0}, and A = (6i; j ); B = (9i; j ) are deterministic rational matrices of dimension (n; d), with indices m6i6m + n − 1; 16j6d. The coeOcients 6i; j ; 9i; j , are

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

583

supposed to belong to DRB

V1  (i.e. they do not contain any occurrence of the “second-level” variables). For any weighted equation, E = (p; S; S  ), we recall the “cost” of this equation is: H (E) = p + 2 · Div(S; S  ). We associate to every system (48) another system of weighted equations, INV(S), which “translates the equations of S into equations over the coeOcients (6i; j ; 9i; j ) only”. 3 The general idea of the construction of INV consists in iterating the transformation used in the proof of (1) ⇒ (2) ⇒ (3) in Lemma 322, i.e. the classical idea of triangulating a system of linear equations. We assume here that ∀j ∈ [1; d];

Sj ≡ ∅:

(49)

Let us de$ne INV(S), W(S) ∈ N ∪ {⊥}; D(S) ∈ N, by induction on n. W(S) is the weight of S while D(S) is the weak codimension of S. Case 1: 6m; ∗ ≡ 9m; ∗ INV(S) = ((W(S); 6m;j ; 9m;j ))16j6d ;

W(S) = pm − 1;

D(S) = 0:

Case 2: 6m; ∗ ≡ 9m; ∗ ; n¿2; pm+1 − pm ¿2 · Div(6m; ∗ ; 9m; ∗ ) + 1 Let us consider u = min{v ∈ X ∗ | ∃j ∈ [1; d]; (6m;∗  v = jd ) ⇔ (9m;∗  v = jd )}:

(50)

(Corollary 320 ensures the existence of such a word u). Let j0 ∈ [1; n] such that (6m; ∗  u = jd0 ) ⇔ (9m; ∗  u = jd0 ). Subcase 1: 6m; j0  u = ”; 9m; j0  u = : Let us consider the equation  ): pm + 2 · |u|; Sj0 ; (Em

d


j=1

(9m;j0  u)∗ (9m;j  u)Sj

j=j0

and de$ne a new system of weighted equations S = (Ei )m+16i6m+n−1 by
[6i;j + 6i;j0 (9m;j0  u)∗ (9m;j  u)] · Sj ; (Ei ): pi ;
j=j0

j=j0

[9i;j + 9i;j0 (9m;j0  u)∗ (9m;j  u)] · Sj :

(The above equation is seen as an equation between two linear combinations of the Si ’s, 16i6d, where the j0 th coeOcient is ∅ on both sides). We then de$ne INV(S) = INV(S );

W(S) = W(S );

D(S) = D(S ) + 1:

3 This function INV is an “elaborated version” of the inverse systems de$ned in [5, Eq. (2.8), p. 586, english version] or [5, Eq. (2.8), p. 677, english version] in the case of a single equation.

584

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Subcase 2: 6m; j0  u = ”; 9m; j0  u = ”: (analogous to subcase 1). Case 3: 6m; ∗ ≡ 9m; ∗ ; n = 1. We then de$ne INV(S) = ⊥;

W(S) = ⊥;

D(S) = 0;

where ⊥ is a special symbol which can be understood as meaning “unde$ned”. Case 4: 6m; ∗ ≡ 9m; ∗ ; n¿2; pm+1 − pm 62 · Div(6m; ∗ ; 9m; ∗ ): We then de$ne INV(S) = ⊥;

W(S) = ⊥;

D(S) = 0:

Lemma 51. Let S be a system of weighted linear equations with deterministic rational coe
INV(S) ∪ {Ei | m 6 i 6 m + D(S) − 1} ||−− A−1 (Em+D(S) );

(51)

where, for every integer k ∈ Z, Ak : {(p; S; S  ) ∈ A | p¿ − k} → A is the translation map on the weights: Ak (p; S; S  ) = (p + k; S; S  ). if D(S) = 0: as INV(S) = ⊥; S must ful$ll the hypothesis of case 1.   d d

Em = pm ; 6m;j Sj ; 9m;j Sj = Em+D(S) j=1

j=1

INV(S) = ((pm − 1; 6m;j ; 9m;j ))16j6d : Using rules (R7), (R8) we obtain

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

585

Fig. 1. Proof of Lemma 5:2.

∗

INV(S) ||−−

 pm − 1;

d
j=1

6m;j Sj ;

d
j=1

 9m;j Sj

= A−1 (Em ):

if D(S) = n + 1; n¿0: S must ful$ll case 2. • Suppose case 2, subcase 1 occurs. Using |u| times (R5) and then (R6) (this is possible because 9m; j0  u ≡ ), we obtain a deduction |u|+1

 : Em ||−− Em

Using (R7) – (R9) we get that, for every i ∈ [m + 1; m + D(S)], ∗

 } ||−− (max{pi ; pm + 2|u|}; {Ei ; Em

(52)

586

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608


j=j0


j=j0

(6i;j + 6i;j0 (9m;j0  u)∗ (9m;j  u)) · Sj ; (9i;j + 9i;j0 (9m;j0  u)∗ (9m;j  u)) · Sj ):

The hypothesis of case 2 implies that max{pm+1 ; pm + 2|u|} = pm+1 and the fact that INV(S ) is de$ned implies that ∀i ∈ [m+1; m+D(S)]; pi ¿pm+1 , hence, max{pi ; pm + 2|u|} = pi and the right-hand side of the above deduction is exactly Ei . Hence ∀i ∈ [m + 1; m + D(S)];

∗

 {Ei ; Em } ||−− Ei :

(53)

Using deductions (52) and (53), we obtain that ∗

{Ei | m 6 i 6 m + D(S) − 1} ||−− {Ei | m 6 i 6 m + D(S) − 1}:

(54)

By induction hypothesis ∗

 INV(S ) ∪ {Ei | m + 1 6 i 6 m + 1 + D(S ) − 1} ||−− A−1 (Em+1+D(S ))

which is equivalent to ∗

 ): INV(S) ∪ {Ei | m + 1 6 i 6 m + D(S) − 1} ||−− A−1 (Em+D(S)

(55)

As pm + 2 · |u|6pm+1 − 16pm+D(S) − 1, we have also the following inverse deduction (which is similar to deduction (53)) ∗

  {Em ; A−1 (Em+D(S) )} ||−− A−1 (Em+D(S) ):

(56)

Combining together deductions (54) – (56), we have proved (51). Using rule (R0), this last deduction leads to point (1) of the lemma. • Suppose that case 2, subcase 2 occurs: this case can be treated in the same way as subcase 1 just by exchanging the roles of
; . Let us prove statement (2) of the lemma. We prove by induction on D(S) the statement: min{H (Ei ) | m 6 i 6 m + D(S)} = ∞ ⇒ H (INV(S)) = ∞:

(57)

if D(S) = 0: as INV(S) = ⊥, case 1 must occur. 6m; ∗ ≡ 9m; ∗ implies that H (INV(S)) = ∞, hence the statement is true. if D(S) = p + 1; p¿0: as D(S)¿1 and INV(S) = ⊥, case 2 must occur. Using deductions (52) and (53) established above we obtain that ∗

{Ei | m 6 i 6 m + D(S)} ||−− {Ei | m + 1 6 i 6 m + 1 + D(S )};

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

587

which proves that min{H (Ei ) | m 6 i 6 m + D(S)} 6 min{H (Ei ) | m + 1 6 i 6 m + 1 + D(S )}:

(58)

As D(S ) = D(S) − 1, we can use the induction hypothesis: min{H (Ei ) | m + 1 6 i 6 m + 1 + D(S )} = ∞ ⇒ H (INV(S )) = ∞: (59) As INV(S) = INV(S ), (58), (59) imply statement (57). Lemma 53. Let S be a system of linear equations satisfying the hypothesis of case 2. Then; ∀i ∈ [m + 1; m + n − 1], 6i; ∗ 66i; ∗  + 9m; ∗  + K0 |u|; 9i; ∗ 69i; ∗  + 9m; ∗  + K0 |u|: Proof. The formula de$ning S from S show that  = 6i;∗ ∇j0 (∇∗j0 (9m;∗  u)); 6i;∗

 9i;∗ = 9i;∗ ∇j0 (∇∗j0 (9m;∗  u)):

From these equalities and Lemmas 311, 312, 318, the inequalities on the norm follow. Let us consider the function F de$ned by F(d; n) = max{Div(A; B) | A; B ∈ DRB1; d

V1 ; A 6 n; B 6 n; A ≡ B}: (60) For every integer parameters K0 ; K1 ; K2 ; K3 ; K4 ∈ N − {0}, we de$ne integer sequences ( i ; ‘i ; Li ; si ; Si ; Ci )m6i6m+n − 1 by m = 0;

‘m = 0;

Lm = K2 ;

sm = K3 · K2 + K4 ;

Sm = 0;

Cm = 0;

(61)

i+1 = 2 · F(d; si + Ci ) + 1; ‘i+1 = 2 · i+1 + 3; Li+1 = K1 · (Li + ‘i+1 ) + K2 ; si+1 = K3 · Li+1 + K4 ; Si+1 = si + Ci + K0 F(d; si + Ci ); Ci+1 = Ci + Si+1

(62)

for m6i6m + n − 2. These sequences are intended to have the following meanings when K0 ; K1 ; K2 ; K3 ; K4 are chosen to be the constants de$ned in Section 6 and the equations (Ei ) are labelling nodes of a B-stacking sequence (see Section 8.2): i+1 6increase of weight between Ei ; Ei+1 , ‘i+1 ¿increase of depth between Ei ; Ei+1 , Li+1 ¿increase of depth between Em ; Ei+1 , si+1 ¿size of the coeOcients of Ei+1 ,

588

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Fig. 2. Proof of Lemma 5:4. (i+1−m) Si+1 ¿size of the coeOcients of Ei+1 (these systems are introduced below in the proof of Lemma 54), Ci+1 ¿increase of the coeOcients between Ek(i−m) ; Ek(i+1−m) ( for k¿i + 1).
d
d For every linear equation E = (p; j=1 6j Sj ; j=1 9j Sj ), we de$ne

|||E||| = max{(61 ; : : : ; 6d ); (91 ; : : : ; 9d )}: Lemma 54. Let S = (Ei )m6i6m+d−1 be a system of d linear equations such that H (Ei ) = ∞ ( for every i) and (1) ∀i ∈ [m; m + d − 1]; |||Ei |||6si , (2) ∀i ∈ [m; m + d − 2]; W(Ei+1 ) − W(Ei )¿ i+1 : Then INV(S) = ⊥; D(S)6d − 1; ∀E ∈ INV(S); |||E|||6Cm+D(S) + sm+D(S) . Proof. (Fig. 2 might help the reader to follow the de$nitions below). Let us de$ne a sequence of systems S(i−m) = (Ek(i−m) )m6i6k6m+d−1 , where i ∈ [m; m + D(S)], by induction: — Ek(0) = Ek for m6k6m + d − 1; — if case 1 or case 3 or case 4 is realized, D(S) = 0, hence S(i−m) is well-de$ned for m6i6m + D(S); — if case 2 is realized then we set: ∀i¿m+1; Ek(i−m) = (Ek )(i−m−1) , for m+16k6m+ d − 1:

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

589

Let us prove by induction on i ∈ [m; m + D(S)] that, ∀k ∈ [i; m + d − 1]: |||Ek(i−m) |||6sk + Ci :

(63)

i = m: in this case |||Ek(i−m) ||| = |||Ek |||6sk = sk + Cm : i + 16m + D(S): in this case, by Lemma 53 |||Ek(i+1−m) ||| 6 |||Ek(i−m) ||| + |||Ei(i−m) ||| + K0 |ui |; where ui = min{v ∈ X ∗ | ∃j ∈ [1; d]; (6i;(i−m)  v = jd ) ⇔ (9i;(i−m)  v = jd )}: ∗ ∗

(64)

By de$nition of F and the induction hypothesis |ui | 6 F(d; |||Ei(i−m) |||) 6 F(d; si + Ci ): Hence |||Ek(i+1−m) ||| 6 (sk + Ci ) + (si + Ci ) + K0 F(d; si + Ci ) = (sk + Ci ) + Si+1 = sk + Ci+1 : Let us notice that D(S) is always an integer and that this proof is valid for m6i6m+ D(S); i6k6m + d − 1. Let us prove now that INV(S) = ⊥. Let us consider the system (Ek(D(S)) )m+D(S)6k6m+d−1 . (D(S)) ) consists of a single equation, it must ful$ll If D(S) = d − 1, as the system (ED(S) either case 1 or case 3 of the de$nition of INV. Using the successive deductions (52), (53) established in the proof of Lemma 52 we get that ∗

(d−1) }: {Ei | m 6 i 6 m + d − 1} ||−− {Em+d−1

Using now the hypothesis that H (Ei ) = ∞ ( for m6i6m + d − 1), we obtain (d−1) H (Em+d−1 ) = ∞:

(65)

For any system of equations S, let us de$ne the column-support of the system as   m+n−1
csupp(S) = j ∈ [1; d] | 6i;j + 9i;j = ∅ : i=m

Let us consider = Card(csupp(S(d−1) ). One can prove by induction on i that Card(csupp(S(i−m) )) 6 d − i + m

590

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

hence = Card(csupp(S(d−1) )) 6 d − (d − 1) = 1: — If = 1; csupp(S(d−1) ) = {j0 }, for some j0 ∈ [1; d]. By Corollary 49 point (C2), and hypothesis (49), the implication (d−1) (d−1) (d−1) (d−1) [6m+d−1;j Sj0 ≡ 9m+d−1;j Sj0 ] ⇒ 6m+d−1;j ≡ 9m+d−1;j 0 0 0 0 (d−1) (d−1) (d−1) holds. Hence, by (65), 6m+d−1; ful$lls case 1, so that j0 ≡ 9m+d−1; j0 , i.e. S

INV(S) = INV(S(d−1) ) = ⊥: — If = 0; csupp(S) = ∅. (d−1) (d−1) d (d−1) ful$lls case 1. Then 6m+d−1; ∗ = 9m+d−1; ∗ = ∅ . Here also S If D(S)¡d − 1, by hypothesis W(Em+D(S)+1 ) − W(Em+D(S) ) ¿ m+D(S)+1 = 2F(d; sm+D(S) + Cm+D(S) ) + 1: D(S) D(S) (D(S)) If 6m+D(S); ∗ ≡ 9m+D(S); ∗ , then Em+D(S) ful$lls case 1 of the de$nition of INV, hence INV(S) = ⊥. Otherwise, let us consider: (D(S)) (D(S)) u = min{v ∈ X ∗ | ∃j ∈ [1; d]; (6m+D(S);∗  v = jd ) ⇔ (9m+D(S);∗  v = jd )}:

(66) By de$nition of F and inequality (63) (D(S)) |u| 6 F(d; |||Em+D(S) |||) 6 F(d; sm+D(S) + Cm+D(S) ):

Hence pm+D(S)+1 − pm+D(S) ¿2|u| + 1, i.e. the hypothesis of case 2 is realized. This proves that D(S(D(S)) )¿1 while in fact, D(S(D(S)) ) = 0. This contradiction shows that (D(S)) this last case (D(S)¡d − 1 and Em+D(S) not ful$lling case 1 of de$nition of INV) is impossible. We have proved point (2) of the lemma. 6. Constants Let us 7x a dpda M. This short section is devoted to the de$nition of some integer constants: these integers depend on the dpda M only. The motivation of each of these de$nitions will appear later on, in diMerent places for the diMerent constants. The equations below provide an overview of the dependencies between these constants and allow to check that the de$nitions are sound (i.e. there is no hidden loop in these dependencies). k0 = max{D(v) | v ∈ V };

k1 = max{2k0 + 1; 3};

(67)

K0 = max{(E1 ; E2 ; : : : ; En )  x | (Ei )16i6n is a bijective numbering of some class in V= ˆ; x ∈ X }:

(68)

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

591

K0 serves as an upper-bound on the possible increase of norm under the right-action of a single letter x ∈ X , see Lemma 318. D1 = k0 · K0 + |V1 | + 2;

k2 = D1 · k1 · K0 + 2 · k1 · K0 + K0 :

(69)

k1 is used in the de$nition of strategy TB (Section 7), D1 appears as an upper-bound on the marked part of series and k2 is used in Lemma 85. k3 = k2 + k1 · K0 ;

k4 = (k3 + 1) · k0 + k1 :

(70)

k3 appears in Lemma 86, k4 is used in de$nition (98) of the polynomial space V0 . K1 = k1 · K0 + 1; K2 = k12 · D1 · K0 + k12 · K0 + 2 · k1 · K0 + D1 · k1 + 2 · k1 + 4:

(71)

These constants K1 ; K2 appear in Lemma 88. K3 = k0 |V1 |;

K4 = D1 :

(72)

These constants K3 ; K4 appear in Lemma 89. d0 = Card(X 6k4 );

(73)

d0 appears as an upper-bound on the dimension of the polynomial space V0 de$ned by Eq. (98) and used in Lemma 88. We consider now the integer sequences ( i ; ‘i ; Li ; si ; Si ; Ci )m6i6m+n−1 de$ned by the relations (62) of Section 5 where the parameters K0 ; K1 ; : : : ; K4 are chosen to be the above constants and m = 1; n = d = d0 . Equivalently, they are de$ned by 1 = 0;

‘1 = 0;

L1 = K2 ;

s1 = K3 · K2 + K4 ;

S1 = 0;

C1 = 0;

(74)

i+1 = 2 · F(d0 ; si + Ci ) + 1; ‘i+1 = 2 · i+1 + 3; Li+1 = K1 · (Li + ‘i+1 ) + K2 ; si+1 = K3 · Li+1 + K4 ; Si+1 = si + Ci + K0 · F(d0 ; si + Ci ); Ci+1 = Ci + Si+1

(75)

for 16i6d0 − 1. The function F is de$ned in Section 5. The constants si ; i

(1 6 i 6 d0 )

(76)

appear in the hypothesis of Lemma 54 when we take d = d0 . The constant Cd0 + sd0 serves as an upper-bound on the norm of the equations produced by the strategy TC (proof of Lemma 91). N0 = k3 + 4:

(77)

N0 appears as a lower bound for the norm in the de$nition of a B-stacking sequence (Section 8.2, condition (88)).

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

7. Strategies for D0 Let us de$ne strategies for the particular system D0 . We de$ne $rst auxiliary strategies Tcut ; T∅ ; T” ; TA ; TB ; TC and then derive some closed strategies from them. Let us $x here some total ordering on X and also some total ordering 6 of type ! on A (inherited from the usual well-ordering of N by the $xed encoding). From these orderings one can construct in the usual way an ordering of type ! on the sets X ∗ ; A∗ and N∗ × (DRB

V )∗ . Tcut : Tcut (A1 · · · An ) = B1 · · · Bm iM ∃i ∈ [1; n − 1]; ∃S; T; Ai = (pi ; S; T );

An = (pn ; S; T );

p i ¡ pn

and

m = 0:4

T∅ : T∅ (A1 A2 · · · An ) = B1 · · · Bm iM ∃S; T; An = (p; S; T ); p¿0; S = T = ∅ and m = 0 T” : T” (A1 · · · An ) = B1 · · · Bm iM An = (p; S; T ); p¿0; S = T = ” and m = 0 TA : TA (A1 · · · An ) = B1 · · · Bm iM An = (p; S; T );

m = |X |;

B1 = (p + 1; S  x1 ; T  x1 ); : : : ;

Bm = (p + 1; S  xm ; T  xm ); where S ≡ ”; T ≡ ” TB+ : TB+ (A1 · · · An ) = B1 · · · Bm iM n¿k1 + 1; An−k1 = (F; UP ; U  ), (where UP is unmarked) U =

q
k=1

E k · 7k

for some q ∈ N; Ek ∈ V1 ;

(Ek )16k6 bijective numbering of a class in V1 = ˆ, 7k ∈ DRB

V  Ai = (F + k1 + i − n; Ui ; Ui ) for n − k1 6i6n, (Ui )n−k1 6i6n is a derivation, (Ui )n−k1 6i6n is a “stacking derivation” (see de$nitions in Section 3.4), Un =

q
k=1

(Ek  u) · 7k ;

for some u ∈ X ∗ ;

m = 1; B1 = (F + k1 − 1; V; V  ); V = Un , V =

q
k=1

$Pe (Ek  u) · (UP  uk );

where ∀k ∈ [1; q]; uk = min(’(Ek )). TB− : TB− is de$ned in the same way as TB+ by exchanging the left series (S − ) and right series (S + ) in every assertion (p; S − ; S + ). TC : TC (A1 · · · An ) = B1 · · · Bm iM there exists d ∈ [1; d0 ]; D ∈ [0; d − 1]; S1 ; S2 ; : : : ; Sd ∈ DRB

V  − {∅}; 16G1 ¡G2 ¡ · · · ¡GD+1 = n; such that, (C1) every equation Ei = AGi = (pGi Sp−Gi ; Sp+Gi ) is a weighted equation over S1 ; S2 ; : : : ; Sd , with coeOcients in DB1; d
V1  and pGi ¿1, 4

I.e. B1 · · · Bm = .

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

593

(C2) S = (Ei )16i6D+1 is such that, INV(S) = ⊥; D(S) = D, |||Ei |||6si ( for 16i6D + 1) and pGi+1 − pGi ¿ i+1 ( for 16i6D), (C3) (G1 ; G2 ; : : : ; GD+1 ; S1 ; : : : ; Sd ) ∈ N∗ × (DRB

V )∗ is the minimal vector satisfying conditions (C1),(C2) for the given sequence (A1 · · · An ) and −1 (C4) B1 · · · Bm = ’−1 is the obvious extension of ’−1 to 2 (INV(S)) (where ’2 2 pairs of series and then to sequences of weighted equations; in other words the result of TC is INV(S) where the series are replaced by the corresponding second-level variables). Lemma 71. Tcut ; T∅ ; T” ; TA are D0 -strategies. Proof. Tcut : (S1) is true by rule R0. (S2) is trivially true. T∅ ; T” : (S1) is true by rule R3. (S2) is trivially true. TA : by rule (R4), {Bj | 16j6m} ||−−4 An , which proves (S1). Suppose H (An ) = ∞, i.e. S ≡ T . Then, ∀j ∈ [1; m]; S  xj ≡ T  xj , so that min{H (Bj ) | 16j6m} = ∞. (S2) is proved. Lemma 72. TB+ ; TB− are D0 -strategies. Proof. Let us show that TB+ is a D0 -strategy. Let us use the notation of the de$nition of TB+ . Let H = {(F; UP ; U  ); (F + k1 − 1; V;  V )}. Let us show that ∗

H ||−− D0 (F + k1 − 1; Un ; Un ):

(78)

Using rule (R5) we obtain: ∀k ∈ [1; q], 

 ∗ q
{(F; UP ; U  )} = F; UP ; Ej · 7 j ||−− R5 (F + 2 · |uk |; UP  uk ; U   uk ) j=1

∗

||−− R0 (F + 2 · k0 ; UP  uk ; U   uk ) = (F + 2 · k0 ; UP  uk ; 7k ): (79) Using rule (R1), (R10), for every k ∈ [1; q]: ∗

∅ ||−− D0 (0; Ek  u; $Pe (Ek  u)):

(80)

Using (80), (79) and rules (R0), (R2), (R7), (R8), we obtain ∗

{(F; UP ; U  )} ||−− D0 =

 F + 2k0 ;

q
k=1

(Ek  u) · 7k ; ∗

q
k=1



$Pe (Ek  u) · (UP  uk )

{(F; UP ; U  )} ||−− D0 (F + 2k0 ; Un ; V  ):

(81)

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Let us recall that Un = V . Hence, by (R0), (R1), (R2) ∗

{(F + k1 − 1; V; V  ); (F + 2k0 ; Un ; V  )} ||−− D0 (F + k1 − 1; Un ; Un ):

(82)

By (81), (82), (78) is proved. Using now (78) and rule (R0), we obtain ∗

H ||−− D0 (F + k1 − 1; Un ; Un ) |−− R0 (F + k1 ; Un ; Un );

(83)

i.e. TB+ ful$lls (S1). Let us suppose now that ∀i ∈ [n − k1 ; n]; Ui ≡ Ui . Then, by (81), Un ≡ V  and by hypothesis V = Un ≡ Un . Hence V ≡ V  . This shows that TB+ ful$lls (S2). An analogous proof can obviously be written for TB− . Lemma 73. Let (p; S; S  ) be a weighted equation; i.e. p ∈ N; S; S  ∈ DRB

V . Then ∗

∗

{(p; S; S  )} ||−−D0 {(p; ’2 (S); ’2 (S  ))} and {(p; ’2 (S); ’2 (S  ))} ||−−D0 {(p; S; S  )}. Proof. Follows easily from (R1), (R2), (R11). Lemma 74. TC is a D0 -strategy. Proof. By Lemma 52 point (1), combined with Lemma 73, (S1) is proved. By Lemma 52 point (2), combined with Lemma 73, (S2) is proved. Let us de$ne the strategy SAB by: for every W = A1 A2 · · · An , (0) if W ∈ dom(Tcut ), then SAB (W ) = Tcut (W ) (1) elsif W ∈ dom(T∅ ), then SAB (W ) = T∅ (W ); (2) elsif W ∈ dom(T” ), then SAB (W ) = T” (W ); (4) elsif W ∈ dom(TB+ ), then SAB (W ) = TB+ (W ); (5) elsif W ∈ dom(TB− ), then SAB (W ) = TB− (W ), (6) elsif W ∈ dom(TA ), then SAB (W ) = TA (W ); (7) else SAB (W ) is unde$ned. The strategy SABC is obtained by inserting “(3) elsif W ∈ dom(TC ), then SABC (W ) = TC (W )” in the above list of cases. Lemma 75. SABC ; SAB are closed. Proof. Given any true assertion An = (F; S; T ) and any word W = A1 · · · An , at least one of T” ; TA is de$ned on W . 8. Tree analysis This section is devoted to the analysis of the proof-trees A produced by the strategy SAB de$ned in Section 7. The main results are Lemmas 810 and 811 whose combination asserts that if some branch of A is in$nite, then there exists some $nite pre$x on which

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595

TC is de$ned. This key technical result will ensure termination of strategy SABC (see Section 9). We $x throughout this section a tree A = T(SAB ; (F0 ; U0− ; U0+ )) (i.e. the proof tree associated to the assertion (F0 ; U0− ; U0+ ) by the strategy SAB ). We suppose that U0− ; U0+ are both linear polynomials:

(84)

(i.e. elements of DB
V1  · (V2 ∪ {})) U0− ; U0+ are both unmarked

(85)

U0− ≡ U0+ :

(86)

We recall that, formally, A is a map dom(A) → N × DRB

V  × DRB

V  such that dom(A) ⊆ {1; : : : ; |X |}∗ is closed under pre$x and under “left-brother” (i.e. w · (i + 1) ∈ dom(A) ⇒ w · i ∈ dom(A)). 8.1. Depth and weight In this paragraph we check that the weight and the depth of a given node are closely related. Let us say that the strategy T “occurs at” node x iM, A(x) = T (A(x[0]) · A(x[1]) · · · A(x[|x| − 1])); i.e. the label of x belongs to the image of the path from  (included) to x (excluded) by the strategy T . Lemma 81. Let 6 ∈ {−; +}; A1 ; : : : ; An ∈ A such that TB6 (A1 · · · An ) is de7ned. Then, ∀i ∈ [n − k1 + 1; n]; Ai = TB (A1 · · · Ai−1 ). In other words: if TB occurs at node x of A, it cannot occur at any of its k1 above immediate ancestors. Proof. Suppose that ∃i ∈ [n − k1 + 1; n]; Ai = TB (A1 · · · Ai−1 ). Hence Fi = Fi−1 − 1¡ Fn−k1 + i, contradicting one of the hypothesis under which TB (A1 · · · An ) is de$ned. Lemma 82. Let A be a proof-tree associated to the strategy SAB . Let x; x ∈ dom(A); x 4 x . Then |W (x ) − W (x)|6|x | − |x|62 · (W (x ) − W (x)) + 3. (We recall the depth of a node x is just its length |x|). We denote by W (x) the weight of x which we de$ne as the $rst component of A(x), i.e. the weight of the equation labelling x). Proof. Let x; x be such that |x | = |x| + 1. Then W (x ) − W (x) ∈ {− 1; +1}, hence the inequality |W (x ) − W (x)|6|x | − |x| is ful$lled by such nodes. The general case follows by induction on (|x | − |x|).

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Lemma 81 ensures that, in every branch (xi )i∈I and for every interval [n+1; n+4] ⊆ I , at most one integer j is such that TB occurs at j. The second inequality follows from this remark. Lemma 83. There exists some 7nite subset VP2 ⊆ V2 ; such that; for every label (F; U − ; U + ) of the tree A; U − ; U + ∈ DB
V1  · (VP2 ∪ {”}): Proof. Let 8− ; 8+ be the variables in V2 appearing in the two polynomials U0− ; U0+ . −1 − ∗ + ∗ Let us de$ne VP2 = {’−1 2 (’2 (8 ) • v) | v ∈ V1 } ∪ {’2 (’2 (8 ) • v) | v ∈ V1 }. This set − + P V2 is $nite because ’2 (8 ); ’2 (8 ) are rational. The property is true at the root of A by hypothesis (84), and this property is preserved by the maps TA ; TB− ; TB+ . 8.2. B-stacking sequences We establish here that every in$nite branch must contain an in$nite suOx (a “Bstacking sequence”) where at least d0 labels (U; U  ) are belonging to the same polynomial space V0 of dimension 6d0 with coordinates not greater than sd0 (over some $xed generating family of cardinality 6d0 ). Let  = (xi )i∈I be a path in A, where I = [i0 ; ∞[ and let (xi )i¿0 be the unique branch of A containing . Let us note A(xi ) = (Fi ; Ui− ; Ui+ ). We call  a B-stacking sequence iM: there exists some 60 ∈ {−; +} such that TB60 occurs at xi0 +k1 +1

(87)

and, for every i ∈ I; 6 ∈ {−; +}, if TB6 occurs at xi+k1 +1 then 0 Ui−6  ¿ Ui−6  ¿ N0 : 0

(88)

From now on and until Lemma 811, we $x a B-stacking sequence  = (xi )i∈I and we denote by S0 the series Ui0−60 . Lemma 84. There exists some word u0 ∈ X ∗ and some sign 60 ∈ {−; +} such that 6

S0 = U0 0  u0 . Proof. One can prove by induction on i ∈ N that, for every 6 ∈ {−; +}, Ui6 has one of the two following forms:  (1) Ui6 = U06  u for some 6 ∈ {−; +}; |u|6i,
 q (2) Ui6 = k=1 9k · (U06  uuk ) for some deterministic polynomial vector ∈ DB1;q
V1 , 6 ∈{−; +}; |u·uk |6i; |uk |6k0 : Lemma 85. Suppose that i0 6j¡i; no TB occurs in [ j + 1; i]; Uj−6 is D1 -marked and Uj6 is unmarked. Then; for every j  ∈ [ j; i]; Uj6 ¿Ui6  − k2 . Proof. Let i; j ful$ll the hypothesis of the lemma. (1) Let us treat $rst the case where j  = j.

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

597

If (i − j)6(D1 + 1)k1 then, by Lemma 318 Ui6  6 Uj6  + (D1 + 1) · k1 · K0 6 k2 hence the lemma is true. Let suppose now that (i − j)¿(D1 + 1)k1 + 1. We can then de$ne the integers j¡i1 ¡i2 ¡i by i1 = j + D1 · k1 ;

i2 = i − k1 − 1:

By Lemma 318 we know that Ui61  6 Uj6  + D1 · k1 · K0

and

Ui6  6 Ui62  + (k1 + 1) · K0 :

(89)

If there was some stacking subderivation of length k1 in Uj−6 → Ui1−6 , as all the Uk6 ( for k ∈ [j; i]) are unmarked, TB would occur at some integer in [ j + k1 + 1; i1 + 1], which is untrue. Hence there is no such stacking subderivation, and by Lemma 325 Ui1−6 is unmarked. If there was some stacking subderivation of length k1 in Ui16 → Ui26 , as all the Uk−6 ( for k ∈ [i1 ; i]) are unmarked, TB would occur at some integer in [i1 + k1 + 1; i], which is untrue. Hence there is no such stacking subderivation, and by Lemma 324 Ui62  6 Ui61  + k1 · K0 :

(90)

Adding inequalities (89), (90) we obtain Ui6  6 Uj6  + (D1 · k1 + 2 · k1 + 1) · K0 = Uj6  + k2 ; which was to be proved. (2) Let us suppose now that j6j  6i. If (i − j)6(D1 + 1)k1 , the same inequality is true for i − j  and the conclusion is true for j  . Otherwise, if j  6i1 , (89), (90) are still true for j  instead of j, hence the conclusion too. Otherwise, by the arguments of part 1, Uj−6 ; Uj6 are both unmarked. Hence the   hypothesis of part 1 are met by ( j ; i) instead of ( j; i), hence the conclusion is met too. (We illustrate our argument in Fig. 3). Lemma 86. Let i ∈ I; 6 ∈ {−; +} such that TB6 occurs at i + k1 + 1. Then; there exists u ∈ X ∗ ; |u|6(i − i0 ); Ui−6 = S0  u and; for every pre7x w 4 u, S0  w ¿ S0  − k3 : Proof. We prove the lemma by induction on i ∈ [i0 ; ∞[. Basis. i = i0 . Choosing u = , the lemma is true.  Induction step. i0 6i ¡i, TB6 occurs at i + k1 + 1, TB6 occurs at i + k1 + 1 and TB does not occur in [i + k1 + 2; i + k1 ].

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Fig. 3. Uj6  too small is impossible.

By induction hypothesis, there exists some u ∈ X ∗ ; |u |6(i − i0 ) ful$lling 

= S0  u ; Ui−6 

(91)

∀w 4u ; S0  w  ¿ S0  − k3 :

(92)

Let us de$ne j = i + k1 + 1. Let uP ∈ X ∗ be the word such that uP

Uj−6 → Ui−6

(93)

is the derivation described by the −6 component of the path from xj to xi . Case 1. 6 = 6. 

 u1 Uj−6 = Ui−6  for some u1 ∈ X ∗ ; |u1 | = k1 and Uj6 is D1 -marked. Let us choose u = u · u1 · u. P Hence Ui−6 = S0  u: Let us consider some pre$x w of u. Subcase 1. w 4 u . By (92) we know that S0  w¿S0  − k3 :

(94)

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

599

Subcase 2. w = u · u1 · u , for some u 4 u. P By Lemma 85 we know that S0  w¿Ui6  − k2 , and by de$nition of a B-stacking sequence we also know that Ui6 ¿S0 . Hence S0  w ¿ S0  − k2 : Subcase 3. w = u · u1 , where u1 is a pre$x of u1 . Then, by Lemma 318 and the above inequality we get S0  w ¿ S0  u u1  − k1 · K0 ¿ S0  − k3 : Case 2: −6 = 6. Uj−6 =

q
k=1

9k · (Ui6  uk );

where 9 is a deterministic polynomial marked and every |uk |6k0 .
q which is fully P · (Ui6  uk ) or there exists a decompoBy Lemma 321 either Ui−6 = k=1 (9k  u) sition uP = uP1 · uP2

(95)

and an integer k ∈ [1; q] such that Ui−6 = Ui6  uk uP2 :

(96)

But, as Ui−6 is unmarked (by de$nition of TB6 ), the $rst formula is impossible unless 9  uP is unitary or nul. Hence (95), (96) is the only possibility. Let us choose u = u · uk · uP 2 . It is clear from (96) that Ui−6 = S0  u: Let us consider some pre$x w of u. Subcase 1: w 4 u . Same arguments as in case1, subcase1. Subcase 2: w = u · uk · u , for some u 4 uP 2 . By Lemma 85 applied on the interval [ j + |uP 1 | + 1; i], we can conclude that S0  w ¿ S0  − k3 : Subcase 3: w = u · uk , where uk is a pre$x of uk . Same arguments as in case1, subcase3. Given any subset G ⊆ DRB

V , we de$ne the polynomial space generated by G as the set

m
V(G) = 6j · Sj | m ¿ 0; ∀j ∈ [1; m] Sj ∈ G; 6 ∈ DB1;m
V1  : j=1

It follows from  Lemma 310 that V(G) is closed under linear combinations with coef$cients in 5¿0 DB1; 5
V1 .

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The following generating set of series and polynomial space of series will be used in the sequel. G0 = {S0  u | u ∈ X ∗ ; |u| 6 k4 };

(97)

V0 = V(G0 ):

(98)

Lemma 87. Let i¿i0 such that TB occurs at i. Then; Ui− ; Ui+ ∈ V0 . −6 Proof. Let us suppose that TB6 occurs at i. By Lemma 86, Ui−k = S0  u and, for 1 −1 every pre$x w 4 u,

S0  w ¿ S0  − k3 ¿ 4:

(99)

By Lemma 319, ∃u1 ; u2 ∈ X ∗ ; v1 ∈ V ∗ ; E1 ; : : : ; Ek ∈ V; E1 ˆ E2 : : : ˆ Ek ; 7 ∈ DRBq; 1

V , such that u = u1 · u2 , q


S0  u1 = S0 • v1 = S0  u =

q
k=1

k=1

Ek · 7k

(100)

(Ek  u2 ) · 7k :

(101)

As S0 is a polynomial (by Lemma 83) |v1 | 6 S0  − S0 • v1  6 k3 : Formula (101) can be rewritten −6 Ui−k = 1 −1

q
k=1

(Ek  u2 ) · (S0 • v1 Ek ) =

q
k=1

(Ek  u2 ) · (S0  uPk );

where uP k ∈ X ∗ is the minimal word such that v1 Ek  uP k = . The fact that inequality (99) holds for S0  u1 implies that no Ek belongs to V2 , hence that all v1 Ek belong to V1∗ . It follows that |uPk | 6 (k3 + 1) · k0 : −6 Using Lemmas 321 and 310 we can deduce from the above form of Ui−k that 1 −1

Ui6 ∈ V({S0  w | w ∈ X ∗ ; |w| 6 (k3 + 1) · k0 + k0 }); Ui−6 ∈ V({S0  w | w ∈ X ∗ ; |w| 6 (k3 + 1) · k0 + k1 }); hence that both Ui−6 ; Ui6 belong to V0 . We recall that K1 = k1 · K0 + 1;

K2 = k12 · D1 · K0 + k12 · K0 + 2 · k1 · K0 + D1 · k1 + 2 · k1 + 4:

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

601

Lemma 88. For every L¿0 there exists i ∈ [i0 + L; i0 + K1 · L + K2 ] such that; Ui− ; Ui+ ∈ V0 . Proof. Let us establish that ∃i ∈ [i0 + L; i0 + K1 · L + K2 − k1 − 1]; ∃6 ∈ {−; +}; TB6 occurs at i + k1 + 1:

(102)

Let L¿0 and let i ¿i0 be the greatest integer in [i0 ; i0 + L] such that TB occurs at i + k1 + 1. Let j = i + k1 + 1. We then have 

Uj6 =

q
k=1



9k · (Ui−6  uk );  

where 96D1 and Uj−6 is unmarked. Case 1: there exists i ∈ [ j; j + k1 · D1 ], such that TB occurs at i + k1 + 1. In this case the small constants K1 = 0; K2 = k1 · D1 + k1 + 1 would be suOcient to satisfy (102). A fortiori the given constants satisfy (102). Case 2: there exists no i ∈ [ j; j + k1 · D1 ], such that TB occurs at i + k1 + 1.  6 Then, there is no stacking subderivation of length k1 in Uj6 → Uj+k . By 1 · D1 6 Lemma 325 it follows that both Uj+D are unmarked. 1 ·k1 (1) Let j1 = j + D1 · k1 and let us show that there exists some i¿j1 such that TB occurs at i + k1 + 1. If such an i does not exist then, for every 6 ∈ {−; +}, the in$nite derivation Uj61 → Uj61 +1 → · · · does not contain any stacking sequence of length k1 . By Lemma 324 we would have ∀k ¿ j1 ;

Uk6  6 Uj61  + k1 · K0 :

As the set {Uk6 ; k¿j1 ; 6 ∈ {−; +}} is $nite, by Lemma 83, the set {Uk6 ; k¿j1 ; 6 ∈ {−; +}} is $nite too, hence there would be a repetition (Uk− ; Uk+ ) = (Uk− ; Uk+ )

with j1 6 k¡k  and Fk ¡Fk  ;

so that Tcut would have been de$ned on some $nite pre$x of the branch, contradicting the hypothesis that the branch is in$nite. (2) Let i¿i be the smallest integer (in [ j1 ; ∞[) ful$lling point 1 above and suppose that TB6 occurs at i + k1 + 1. By an argument analogous to that used in Lemma 84 we see that Uj1−6 = S0  u for some |u|6( j1 − i0 ), and by Lemma 318 we get  6 (j1 − i0 ) · K0 + S0 : Uj−6 1

(103)

We also know that −6  + K0 : S0  6 Ui−6  6 Ui−1

(104)

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

−6 As the derivation Uj1−6 → Ui−1 contains no stacking sub-derivation of length k1 , by Lemma 323 we obtain −6 Ui−1  6 Uj−6  − (i − j1 − 2)=k1 : 1

(105)

Combining the three inequalities (103), (104), (105) we get successively S0  6 S0  + (j1 − i0 + 1) · K0 − (i − j1 − 2)=k1 ; (i − j1 − 2) 6 (j1 − i0 + 1) · k1 K0 : (i − i ) = (i − j1 − 2) + (j1 − i + 2) 6 (j1 − i0 + 1) · k1 · K0 + D1 · k1 + k1 + 3 = (i − i0 ) · k1 · K0 + k12 · D1 · K0 + k12 · K0 + 2 · k1 · K0 + D1 · k1 + k1 + 3 = (K1 − 1)(i − i0 ) + K2 − k1 − 1:

(106)

(3) By the choice of i ; i, we know that i 6i0 + L6i. Using (106) we obtain i 6 i + (K1 − 1)(i − i0 ) + K2 − k1 − 1; i 6 i0 + K1 · L + K2 − k1 − 1: Assertion (102) is now established for case 2 as well as for case 1. From (102) and Lemma 87 the lemma follows. (We illustrate our argument in Fig. 4). Let us give now a stronger version of Lemma 88 where we analyze the size of the coe
K4 = D1 :

Let us $x a total ordering on G0 : G0 = {J1 ; J2 ; : : : ; Jd };

where d = Card(G0 ):

Let us remark that d6Card(X 6k4 ) = d0 : Lemma 89. Let L¿0. There exists i ∈ [i0 +L; i0 +K1 · L+K2 ] and; for every 6 ∈ {−; +}; there exists a deterministic polynomial family (9i;6j )16j6d ful7lling
d (1) Ui6 = j=1 9i;6j · Jj , (2) 9i;6∗ 6K3 · (i − i0 ) + K4 : Proof. By Lemma 88 there exists i ∈ [i0 + L; i0 + K1 · L + K2 ] and 6 ∈ {−; +} such that TB6 occurs at i. Let us use the notation of the proof of Lemma 87 and compute

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

603

Fig. 4. Two successive TB .

upper-bounds on the coeOcients of Ui−6 ; Ui6 expressed as linear combinations of the vectors of G0 . −6 Coe3cients of Ui−6 : Ui−6 = Ui−k  u , for some u ∈ X ∗ ; |u | = k1 . By Lemma 321, 1 −1 −6 Ui can be expressed in one of the two following forms: Ui−6 = S0  (uPk u ) Ui−6 =

q
k=1

where u is a suOx of u ;

(Ek  u2 u ) · (S0  uPk ):

(107) (108)

d In case (107) we can choose as vector of coordinates: 9i;−6 ? = j0 . We then have 9i; ?  = 26K4 .  In case (108), we can choose: 9i;−6 ? = E  u2 u (completed with ∅ in all the columns j not corresponding to some vector S0  uP k of G0 ). We then have

9i;?  = E  u2 u  6 K0 · (i − i0 ) 6 K3 · (i − i0 ): Coe3cients of Ui6 : By de$nition of TB6 Ui6 =

r
‘=1

−6 A‘ · (Ui−k  wP ‘ ); 1 −1

where A6D1 ; |wP ‘ |6k0 .

(109)

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G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

Replacing u by wP ‘ in the above analysis, we get −6 Ui−k  wP ‘ = 1 −1

∀‘ ∈ [1; r];

d
j=1

=‘;j · Jj

(110)

with =‘;? 6K0 · (i − i0 ). Equalities (109), (110) show that Ui6 = A · = · J; where A; =; J are deterministic rational matrices of dimensions, respectively (1; r); (r; d); (d; 1). Let us choose 9i; ? = (A · =). 9i;?  6 A + = 6 D1 + r · K0 · (i − i0 ) 6 D1 + |V1 | · K0 · (i − i0 ) = K3 · (i − i0 ) + K4 : Lemma 810. There exists i0 6G1 ¡G2 ¡ · · · ¡Gd and deterministic rational vectors (9i;6 j )16j6d ( for every i ∈ [1; d]) such that (0) (1) (2) (3)

W (G1 )¿1;
d ∀i; ∀6; UG6i = j=1 9i;6j Jj ∈ V0 ; ∀i; ∀6; 9i;6∗ 6si ; ∀i; W (Gi+1 ) − W (Gi )¿ i+1 ;

where the sequences ( i ; ‘i ; Li ; si ; Si ; i ) are those de$ned by relations (74), (75) in Section 6. Proof. Let us consider the additional property (4) Gi − i0 6Li . We prove by induction on i the conjunction (1) ∧ (2) ∧ (3) ∧ (4). i = 1: By Lemma 89, there exists G1 ∈ [i0 ; i0 + K2 ] such that ∀6 ∈ {−; +}; ∃ a deterministic vector (91;6 j )16j6d , such that UG61 =

d
j=1

6 91;j Jj

and in addition 91;6 ∗ 6K3 K2 + K4 = s1 . i → i + 1: Suppose that G1 ¡G2 ¡ · · · ¡Gi are ful$lling (1) ∧ (2) ∧ (3) ∧ (4). By Lemma 89, there exists Gi+1 ∈ [i0 + Li + ‘i+1 ; i0 + K1 (Li + ‘i+1 ) + K2 ] such that ∀6 ∈ {−; +}; ∃ a 6 deterministic polynomial vector (9i+1; j )16j6d , such that UG6i+1 =

d
j=1

6 9i+1; j Jj

(111)

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

605

and in addition 6 9i+1;∗  6 K3 (K1 (Li + ‘i+1 ) + K2 ) + K4 = K3 Li+1 + K4

= si+1

(112)

By Lemma 82 2(W (Gi+1 ) − W (Gi )) + 3 ¿ Gi+1 − Gi ¿ ‘i+1 = 2 i+1 + 3 hence W (Gi+1 ) − W (Gi ) ¿ i+1 :

(113)

At last Gi+1 − i0 6 K1 (Li + li+1 ) + K2 = Li+1 :

(114)

The above properties (111) – (114) prove the required conjunction. It remains to prove point (0): the integer G1 introduced by Lemma 89 is such that TB occurs at G1 , hence W (G1 ) = W (G1 − k1 − 1) + k1 − 1 ¿ W (G1 − k1 − 1) + 2 ¿ 1: Lemma 811. Let b = (xi )i∈N be an in7nite branch of A. Then there exists some i0 ∈ N such that (xi )i¿i0 is a B-stacking sequence. Proof. Let us distinguish, a priori, several cases, and see that only the case where b admits a B-stacking sequence is possible. Case 1: TB occurs $nitely often on A. Let j be the largest integer such that TB occurs at j. By the arguments used in the − + proof of Lemma 88, assertion (102), case 2, we know that Uj+k ; Uj+k are both 1 ·D1 1 ·D1 unmarked, and that ∀k ¿ j + k1 · D1 ; ∀6 ∈ {−; +};

6 Uk6  6 Uj+k  + k 1 · K0 : 1 ·D1

By Lemma 83 the set of values of (Uk− ; Uk+ ) would be $nite. Hence the branch b would have a $nite pre$x on which Tcut is de$ned: this is impossible on an in$nite branch. Case 2: For some sign 6, there are in$nitely many integers i such that [TB6 occurs at i + k1 + 1 and Ui−6 ¡N0 ]. In this case there would exist an in$nite sequence of integers i1 ¡i2 ¡ · · · ¡i‘ ¡ such that ∀‘ ¿ 1;

Ui−6 = Ui−6 : 1 ‘

606

G. Senizergues / Theoretical Computer Science 281 (2002) 555 – 608

− For a given Ui−6 , only a $nite number of values are possible for the pair (Ui+k ; 1 +1 +  ). Hence there exist integers ‘¡‘ such that Ui+k 1 +1

‘ ¡ ‘ ;

F‘ ¡ F‘

− + (U‘+k ; U‘+k ) = (U‘− +k1 +1 ; U‘+ +k1 +1 ): 1 +1 1 +1

and

Here again Tcut would have a non-empty value on some pre$x of A, which is impossible. Case 3: TB occurs in$nitely often on A and, for every sign 6, there are only $nitely many integers i such that [TB6 occurs at i + k1 + 1 and Ui−6 ¡N0 ]. Let us consider the set I0 of the integers i such that, there exists a sign 6i such that [TB6i occurs at i + k1 + 1 and Ui−6i  ¿ N0 ]: By the hypothesis of case 3, I0 = ∅. Let i0 such that −6i0

Ui0

 = min{Ui−6i  | i ∈ I0 }:

Then (xi )i¿i0 is a B-stacking sequence. 9. Completeness of D0 We show that, up to some slight details, SABC is terminating. An assertion A = (M; S − ; S + ) is said simple iM it ful$lls the 3 conditions below S − ; S + are both linear polynomials;

(115)

S − ; S + are both unmarked;

(116)

S − ≡ S +:

(117)

Lemma 91. Let A0 be some simple assertion. Then the tree T(SABC ; A0 ) is 7nite. Proof. Suppose A0 = (M0 ; S0− ; S0+ ) is simple and t = T(SABC ; A0 ) is in$nite. By Koenig’s lemma, t contains an in$nite branch whose (in$nite) labelling word is A0 A1 · · · An · · ·. Condition (C2) in the de$nition of TC together with Lemma 54, applied to m = 1 and d6d0 , shows that the equations Bj = (Fj ; Tj ; Uj ) produced by TC must ful$ll: ’2 (Tj ) 6 Cd0 + sd0 ;

’2 (Uj ) 6 Cd0 + sd0

hence that the number of possible pairs (Tj ; Uj ) produced by TC is 7nite. Hence TC occurs only a $nite number of times on this branch (otherwise Tcut would occur on this branch, which is impossible on an in$nite branch). Let n0 be the last point where TC occurs or n0 = 0 if TC does not occur on this branch. (An0 +i )i¿0 is a branch of the tree t  = T(SAB ; An0 ). If n0 ¿0; An0 is a result of TC , hence t  ful$lls hypotheses (84), (85) of Section 8.2. As A0 is true and the strategies TA ; TB ; TC preserve truth, An0 is also true, hence t  ful$lls hypothesis (86) of Section 8.2. If n0 = 0, as A0 is simple, t  ful$lls the hypotheses (84), (85), (86) assumed in Section 8.2. We can apply now the results of Section 8.2.

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607

By Lemma 811, the branch (An0 +i )i¿0 must contain a B-stacking sequence . Let us remark that, as T∅ does not occur (otherwise the branch would be $nite) every equation (F; U − ; U + ) labelling this branch is such that U − ≡ ∅; U + ≡ ∅. By Lemma 810 such a B-stacking sequence contains a subsequence (AG1 ; AG2 ; : : : ; AGi ; : : : ; AGd ) with d6d0 , ful$lling hypotheses (1; 2) of Lemma 54. Hence some pre$x (AG1 ; AG2 ; : : : ; AGD+1 ful$lls conditions (C1,C2) of the de$nition of TC (it suOces to choose D = D(AG1 ; AG2 ; : : : ; AGi ; : : : ; AGd )). Hence the sequence of assertions (Ai )06i6GD+1 belongs to dom(TC ). The priority ordering given in the de$nition of SABC then implies that either Tcut ; T∅ ; T” or TC occurs at some n0 + i. But the three $rst cases cannot occur on an in$nite branch and the fourth one contradicts the maximality of n0 . Theorem 92. The system D0 is complete. Proof. By Lemma 75 SABC is a strategy for D0 which is closed and by Lemma 91 SABC is terminating on every simple assertion. By a slight variant of Lemma 45, every simple assertion has a D0 -proof. −1 − Let A = (M; S − ; S + ) be some true assertion. Let A = (M; ’−1 2 (’2 (S )); ’2 (’2 +   (S ))). As A is simple, there exists a $nite D0 -proof P of A . By rule (R11), P ∪ {A} is a $nite D0 -proof of A. Theorem 93. The equivalence problem for deterministic pushdown automata is decidable. Proof. Let M be some dpda. The equivalence relation ≡ on DRB

V  (where V is the structured alphabet associated to the given M) has a recursively enumerable complement (this is well-known). By Theorem 92 and Lemma 42 ≡ is recursively enumerable too. Hence ≡ is recursive. In addition, the system D0 associated with M is computable from M, hence the theorem follows. 10. Elimination Let D1 =
A1 ; H1 ; |−−D1  where A1 = N × DRB

V1  × DRB

V1 , H1 is the restriction of H to A1 , and the elementary deduction relation ||−−D1 is the set of all instances of the metarules (R0); (R1); (R2); (R3); (R4); (R6); (R7); (R8); (R9), where the series are taken in DRB

V1  (i.e. we eliminate all the second-order variables and also the rule dealing with the marks). The deduction relation |−−D1 is now de$ned by ∗

[1]

|−− D1 = ||−− D1 ◦ ||−−

R0;R3;R4 ◦

∗

||−− D1 :

Lemma 101. D1 is a deduction system.

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Proof (sketch). As |−−D1 ⊆ |−−D0 , property (A1) is ful$lled by |−−D1 . By the wellknown decidability properties for $nite-automata, rules (R0) – (R9), are recursively enumerable. Hence property (A2) is ful$lled by D1 . Theorem 102. D1 is a complete deduction system. Proof. Let A = (M; S − ; S + ) be some true assertion with S − ; S + ∈ DRB

V1 . A admits some $nite D0 -proof P. Let P  = $e (’2 (P)). One can check that every elementary rule of D0 is mapped, by $e ◦ ’2 , into an elementary rule of D1 . References [1] B. Courcelle, An axiomatic approach to the Korenjac–Hopcroft algorithms, Math. Systems Theory (1983) 191–231. [2] M.A. Harrison, Introduction to Formal Language Theory, Addison-Wesley, Reading, MA, 1978. [3] M.A. Harrison, I.M. Havel, A. Yehudai, On equivalence of grammars through transformation trees, Theoret. Comput. Sci. 9 (1979) 173–205. [4] Y.V. Meitus, The equivalence problem for real-time strict deterministic pushdown automata, Kibernetika 5 (1989) 14–25 (in Russian, English translation in Cybern. Systems Anal.). [5] Y.V. Meitus, Decidability of the equivalence problem for deterministic pushdown automata, Kibernetika 5 (1992) 20–45 (in Russian, English translation in Cybern.Systems Anal.). [6] M. Oyamaguchi, The equivalence problem for real-time d.p.d.a’s, J. Assoc. Comput. Mach. 34 (1987) 731–760. [7] V.Yu. Romanovskii, Equivalence problem for real-time deterministic pushdown automata, Kibernetika 2 (1985) 13–23. [8] G. S*enizergues, The equivalence problem for deterministic pushdown automata is decidable, in: Proc. ICALP 97, Lecture Notes in Computer Science, Vol. 1256, Springer, Berlin, 1997, pp. 671– 681. [9] G. S*enizergues, Decidability of bisimulation equivalence for equational graphs of $nite out-degree, in: R. Motwani (Ed.), Proc. FOCS’98, IEEE Computer Society Press, Silver Spring, MD, 1998, pp. 120–129. [10] G. S*enizergues, T(A) = T(B)? in: Proc. ICALP 99, Lecture Notes in Computer Science, Vol. 1644, Springer, Berlin, 1999, pp. 665 – 675. Full proofs in Technical Report 1209-99 of LaBRI, T (A) = T (B)? pp. 1– 61. [11] G. S*enizergues, L(A) = L(B)? decidability results from complete formal systems, Theoret. Comput. Sci. 251 (2001) 1–166. [12] C. Stirling, Decidability of dpda’s equivalence, Technical Report, Edinburgh ECS-LFCS-99-411, 1999, pp. 1–25. [13] C. Stirling, Decidability of dpda’s equivalence, Theoret. Comput. Sci. 255 (2001) 1–31. [14] L.G. Valiant, The equivalence problem for deterministic $nite-turn pushdown automata, Inform. Control 25 (1974) 123–133.

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www.elsevier.com/locate/tcs

 ! Yantra Geometry Sr Gerard Huet INRIA-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France

0. Introduction  ! Yantra (pronounced Shri Yantra) is a sacred diagram of Tantric Hinduism. Its Sr symbolism is explained in Zimmer [18,20]. It consists of three concentric parts: • An inner 2gure of interpenetrating triangles. This 2gure, symmetric in its vertical central axis, contains both upward-pointing triangles (vahni), symbolizing the male element (“Purusha”, the Cosmic Person), and downward-pointing triangles (#sakti), symbolizing the female aspect (“Prakr. iti”, i.e. energy) of divinity. There are 4 male triangles, and 5 female ones. Their inter-penetration symbolizes the complementarity of the opposite principles in creating the illusion of duality through ignorance, whereas the general symmetry and balance of the whole 2gure symbolizes the more profound reality of Unity of God through its various unfoldings. • Two concentric rings, holding a regular lotus (padma) design. The inner pattern bears 8 lotus petals, the outer one 16. These patterns symbolize the sanctity of the inner diagram, used as instrument (yantra) in yoga meditation; such diagrams are abstract analogues of the more profusely decorated tibetan thankas [20]. • A triple Greek-like key-pattern, the bh(upura, or s#i#sirita (shivered) frame. It symbolizes the ground-plan of a temple with 4 square openings, in the 4 directions of space. This sanctuary is the seat (p().t ha) of one’s chosen deity (is. .t a-devat(a), representing one’s Higher Self.  ! Yantra diagram is a sacred symbol of Tantric Hinduism, and it is traditionally The Sr used in yoga meditation exercises. The original edition of Zimmer [18] had erroneous  ! Yantra, according to the later English translation [20]. Unfortunately, descriptions of Sr that one is also marred with errors in diagrams, and unclear references to inside-out construction instructions for the drawing of the triangular central pattern. The present  ! Yantra. paper presents a more systematic approach to the formal de2nition of Sr

E-mail address: [email protected] (G. Huet). c 2002 Published by Elsevier Science B.V. 0304-3975/02/$ - see front matter
PII: S 0 3 0 4 - 3 9 7 5 ( 0 2 ) 0 0 0 2 8 - 2

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  Yantra 1. In search of Sr 1.1. Drawing experiments  ! Yantra Our 2rst approach was completely experimental: the author tried to draw Sr in free hand, and failed. A more systematic attempt with a computer drawing system also failed. The only reliable model available then was the frontispiece 2gure in [20], all the other diagrams in the book being obviously erroneous, or at least signi2cantly diGerent from the frontispiece 2gure. Instructions for drawing the diagram led to various  ! Yantra shown in Fig. 1. inexact drawings, such as the False Sr Using the Macintosh package FreeHand, and by precisely measuring Zimmer’s fron ! Yantra was then produced, shown below as Fig. 2, and tispiece diagram, a correct Sr measured to serve as an initial approximate con2guration. 1.2. A more rigorous geometric analysis  ! The diJculty of the above experiments had left us undecided as to whether Sr Yantra was indeed uniquely de2ned in the real plane, under-speci2ed, or even impossible. Using the notation XA for the horizontal coordinate of point A in Fig. 1, and

 ! Yantra. Fig. 1. False Sr

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611

T V

S G

U

d I

F E

D b W

R

a

i

Q

M

e

N h

C P B

H

c

J g

A

K

Z f L

O

 ! Yantra. Fig. 2. Approximate Sr

YA for its vertical coordinate, and writing A&B → c for “points A and B determine the line c joining them” and a×b→C for “lines a and b determine their intersection C”, we compute a system of real coordinate constraints as follows.  is the inner circle of the diagram. We choose YQ, XF, YP, XA and YJ as parameters, and we compute: YQ× → Q; Q&O → e; YF = YQ; YJ × → J ; J &T → h; XA×e → A; YJ ×e → H ; YQ×h → I ; XU = 0; YU = YQ; U &A → f; YP×f → C; F&H → g; 0T ×g → V ; XW = 0; YW = YP; F&W → a; a×f → D; a×h → G; YJ ×f → B; XZ = 0; YZ = YA; C&Z → c; c×g → M ; I &B → d; d×0 → YL; f×YL → L; d×YP → P; a×YM → E; P&E → i; a×YV → S; g× YA → K; i×YD → N ; d×YG → R. The parameters are consistent if point (0; YG) is on line i, which gives one extra constraint. We thus obtain a problem with 4 degrees of freedom, which admits several solutions. However, we have to take into account the further constraints that the various points of the diagrams should lie inside the circumscribing circle , we thus have a very shallow range for the 4 real parameters, leading to solutions which are Osthetically close to Fig. 2. A standard Newton approximation solution of the constraint on G from various initial solutions leads to the following particular solution, where the diameter of circle  is taken as unit length:

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 ! Yantras. Fig. 3. Two exact Sr

# ) Yantra is de2ned by: YF = 0:668, XF = 0:126, YP = 0:463, Denition. Classical Sr( XA = 0:187, YJ = 0:398, YL = 0:165, YA = 0:265, YG = 0:769, YV = 0:887, YM = 0:603, YD = 0:551. This investigation solved our query: # ) Yantra is an under-determined Euclidean plane geometry problem with Theorem. Sr( # ) Yantra. 4 real parameters, admitting an in
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613

 ! Yantra. Fig. 4. Classical Sr

diagram. The main worry was the seemingly mandatory inSexion in the slope variation of the upward diagonal lines e, d, c, b, a, and symmetry (but to a lesser extent) of the downward diagonal lines f, i, g, and h. This essentially experimental study converged  ! Yantra, and the corresponding rendition to the de2nition given above of Classical Sr is given below in Fig. 4. 1.5. Bibliographic search The initial hope of the above mathematical analysis of the yantra was to formally describe a parametric situation admitting multiple solutions which could be optimized according to an aesthetic criterion. However, even though the 2rst part of the conclusion was reached, see the Theorem above, the shallow range of solutions made it absolutely impossible to optimize the diagram to the extent, for instance, that the various triangle slopes vary in a monotonous fashion. Doubts thus began to enter the mind of the author as to the precise de2nition  ! Yantra. Even a serious study such as [20] contained inconsistencies. It deof Sr 2nes descriptions of it, culminating in Fig. 10, which is clearly diGerent from its 2nal colour rendition presented in the frontispiece. The frontispiece 2gure conforms to the mathematical analysis given above, and thus we may ascertain that it is a  ! Yantra. But the awkward sloping of the precise graphical rendition of Classical Sr innermost shakti triangle of the latter makes it less harmonious in some sense than

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 ! Yantra shown the smoother design in Fig. 10 of this work, similar to the False Sr above. The inside-out instructions, attributed to Bhaskararaya’s Nity(as#oda#sik(arnava, are clearly misleading, since there is no hope, except by extraordinary luck, to get points J and Q on the circle  determined by its diameter 0T . Actually, this text can only be  ! Yantra, and by no means as precise considered as an approximate description of Sr instructions for its geometrical construction. It was not clear at this point as to which of the two designs was the traditional one. It was not a priori obvious whether the more exact, or the more harmonious drawing, was to be preferred. The frontispiece in [20] is credited to [12], but was added by the translators, and is absent from the original edition [18]. The 2rst lead was to follow Zimmer himself, who later consigned his observations on Myths and Symbols in Indian Art and Civilization in a volume edited after his death by Joseph Campbell [19]. There, Fig. 36 (also reproduced on the front cover) is a version of  ! Yantra. The credits attribute this 2gure to Sir John WoodroGe, judge at the False Sr the Supreme Court of Calcutta, who published tantric works under the pseudonym Arthur Avalon. Indeed, this 2gure may be found on the cover of [17], published in 1914. The next available reference was the work of Danielou [3]. The relevant diagram is represented in the French edition on p. 537, which we shall call the Reverse Classical  ! Yantra, since it is indeed a geometrically precise yantra, but it is shown upside Sr down. The diagram is called Shrˆ! Cakra in this work, i.e. Fortunate Circle, while alluding to its other name of Shrˆ! Yantra, translated as Diagram of Beauty or Harmony. It is said that the Shrˆ! Cakra represents the Universal Goddess. At this point it seemed  ! Yantra was indeed the correct rendition, but the vertical orientation that the Classical Sr was in doubt, especially with respect to the remarks on incorrect orientation in footnote q p. 161 of [20]. One of the most confusing sources is Rao, who discusses the diagram under the  !chakra, in the chapter Devi of the 2rst volume of his extensive study of name Sr  ! Hindu iconography [4]. In this work are presented a hand-drawn Reverse Classical Sr  ! Yantra metallic Yantra as Plate XCVII, and the photograph of a genuine Classical Sr  !˙ngeri Mat.ha as Plate XCVIII. plate engraving from the Sr Evidence for the correct orientation was found in Renou [14], article 1166, which says that the most eJcient yantra, c?rˆ)yantra, is composed of 5 downward and 4 up ! Yantra, ward triangles. The accompanying Fig. 28, p. 581 is, however, a False Sr apparently identical to Avalon’s original one. The text refers to Zimmer as its main source. Further evidence for the correct orientation was found in a book on Tantra Art [7],  ! Yantra in the form of a Rajasthan painting from the late 18th where a Classical Sr century is reproduced as Plate 7. It is mentioned that, according to the Tantrasara, this yantra should not be drawn during nighttime. Plate 51 of the same source shows a copper plate rendition from South India, dated 17th–18th century. It is stated that the ultimate object of meditation on this yantra is to realize the unity of the meditator’s individual soul with the Goddess. The text of this work gives mysterious instructions for drawing the diagram: “Taking the positive pentagon (and its negative ally,

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the pentacle) and drawing diagonals from (a) centre to angles, and (b) from centre to sides, two series of ten lines, having two lengths only, appear. This design is called Sri Yantra.” It goes on in describing still more mysterious relationships between the proportions of the diagram and the musical scales (ragas and raginis) of Indian music. Finally, most doubts were dispelled by reading Pott’s scholarly work on Yoga and Tantra [11]. It is remarked there that Zimmer’s 2gures were incorrect, but it is asserted rather strangely that the drawing instructions are correct! A correct diagram is presented decorated with letters of the Devanagar! alphabet, as Fig. 7. We note that this diagram is pretty close to the left diagram in our Fig. 3, with point S almost on circle . We also remark that the bindu is drawn there as a small circle, following Rao. We note  ! Yantra in Kundangar [5]. that Pott refers to a good illustration of Sr Another book on the art of Tantra, by Rawson [13], presents 2rst a copper plate, as Fig. 49, and then a Rajasthan drawing, c. 1800, decorated with mantras (magic  ! Yantra, this drawing is rather remote from formulas), as Fig. 50. Although called Sr the classical one, since it possesses only 3 vahni triangles, and 4 s#akti ones. But  ! Yantra drawn as a gouache painting on cloth, Fig. 65 shows a splendid Classical Sr from Nepal, c. 1700. Described as “a diagram of the continuous process of Creative Generation”, this beautiful painting is illustrated with images of gods in various geometric positions. Its colour rendition is signi2cantly diGerent from the frontispiece from [20]. In particular, the triangular sub-patterns are all red, and not alternating red and blue. The text presents a metaphysical interpretation of the diagram, based on the K(amakal(avil(asa. This text, whose complete translation is given in the appendix, is obscure, to say the least. Further information is provided in The Tantric Way [8]. Besides previously referenced representations, we note a 3-dimensional bronze rendition. It corresponds to what  ! Yantra [4]. Similar 3-dimensional crystal or metal Rao calls the Meru form of the Sr representations may be found in North India and Nepal.  ! Yantra, but they We 2nally mention that numerous books on symbolism mention Sr  ! Yantra usually show incorrect representations of it, either reproducing the False Sr from [19] (e.g. Campbell), or its upside-down inverse (e.g. Jung). Another geometric study of the diagram has come recently to our attention [2]. But this study mentions only approximate constructions, and dubious angular relationships with the Great Pyramid of Cheops.   Yantra 2. Semantics of Sr We shall now brieSy comment on the ritual signi2cance of the diagram. 2.1. A sacred symbol of Dev() We 2rst follow Rao [4]: “The worship of yantras is common throughout India; perhaps the most important of # )chakra. It generally consists of forty-three triangles interestingly these yantras is the Sr(

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Fig. 5. Vahni triangle.

Fig. 6. Arrow.

arranged in a plane and may also be produced in three other diGerent forms called Meru, # )chakra in plan; but the various Kailasa and Bhu. The Meru is the same as the plane Sr( triangles, surrounding the innermost one, are piled one over another in diGerent planes so that the whole becomes shaped into the form of a pyramid. The topmost layer of the Meru contains a circle called the bindu. If associated with the eight m(atr. ik(a deities the Meru becomes the Kailasa; and with the V(asin() deities it becomes the Bhu. The # ( Sruti or Vedic revelation itself supports the worship of yantras (cf. Taitt. Aran.). # )chakra is given herein ... This The drawing of the famous yantra known as the Sr( and other yantras are generally engraved on some metallic plate, preferably one of # )chakra engraved on gold; silver and copper also are often enough employed. The Sr( metallic plates is an object of worship. In South Indian temples of the medieval and # later periods, there are shrines called by the name of Sakti-p( ).t h(alayas, in which there is a p().t h(a or smaller altar very much resembling the common bali-p().t ha whereon the oblations of formal worship are usually oGered in temples. It is said that these bali# p().t has associated with the Sakti-p( ).t h(alayas contain inside them the plate on which the # )chakra is engraved. Regular p(uj(a is oGered to the Sakti-p( # Sr( ).t ha at least twice a day. In as much as this is thus an object of worship, it has been treated in this work as an icon. The other yantras are engraved upon thin silver, gold, silver or copper plates, which are rolled into a cylinder and then put into a golden or other metallic case so that they may be worn on the body of the person with a view to avoid diseases, possession by devils, and other such evils, which, it is supposed, they have the power to ward oG. Occasional worship is also oGered to this case containing the magical yantra, and the wearer’s faith in its eJcacy may well eGect cures in many cases.” 2.2. Analysing the three basic symbols  ! Yantra consists of concentric areas, formed from The inner triangular area of Sr three basic symbols: the triangle, the arrow, and the 2r tree, depicted, respectively, in (Figs. 5–7). These three basic 2gures, together with their upside-down symmetric duals, form by successive intersections the concentric areas into which the triangular pattern de-

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Fig. 7. Fir tree.

Fig. 8. Trailokya-mohana chakra.

composes itself. Such an area is called a chakra or wheel (Figs. 8–10). The 2rst  ! Yantra consists in identifying these areas, instead of the step of initiation in Sr confusing jumble of triangles. For instance, Fig. 14 is formed from a triangle and a reversed arrow. Figs. 12 and 13 are formed with two opposite arrows, and Fig. 11 is formed from two opposite 2r trees. It is remarkable that our Postscript program (see appendix) actually uses this analysis for the colour rendering of these rings.

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Fig. 9. Sarv(as#a( paripuraka chakra.

Fig. 10. Sarva-#sa˙nkshobhana chakra.

Let us now examine these concentric rings in an outside-in approach, corresponding  ! Yantra in meditation practice. to the standard use of Sr 2.3. An outside-in walk An elaborate description of the constituents of the diagram, basically consistent with the terminology from Zimmer [20], is given in The Tantric Way [8]. It goes as follows.

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Fig. 11. Sarva-saubh(agyad(ayaka chakra.

Fig. 12. Sarvartha-s(adhaka chakra.

Fig. 13. Sarvaraksh(ak(ara chakra.

Fig. 14. Sarva-rogahara chakra.

 ! Yantra is called ‘Nava Chakra’ since it is composed of nine circuits, count“The Sr  ! Yantra, ing from the outer plane to the bindu. Through contemplation on the Sr the adept can rediscover his primordial sources. The nine circuits symbolically indicate the successive phases in the process of becoming. They rank from the earthly plane and rise slowly step by step to the 2nal point, the state of supreme joy. By entering into the e lan vital of the yantra, the adept reintegrates with it. The nine

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Fig. 15. Sarva-siddh()prada chakra.

Fig. 16. Bindu : Sarva-(anandamaya chakra.

 ! Yantra move from the gross and tangible to the sublime and subtle circuits within Sr realms. The outermost periphery consists of a square, with four gates, coloured white, red  ! Yantra. and yellow. This is the Bhupura, the ground-plan, of the Sr Inside the square are three concentric circles, girdles [mekhal(a]. The space between the square and three girdles is the Trailokya-mohana, or the Enchantress of the Triple World, chakra; at this stage the adept is infatuated by aspirations and desires. Next are two concentric rings of sixteen and eight lotus petals, respectively. They are called Sarvasaparipuraka chakra and Sarva-sa˙nkshobhan . a chakra, indicating ful2llment of desire. The fourth chakra, Sarva-saubhagyadayaka, or Giver of Auspiciousness, projects the realm of possibilities in spiritual ascent. It consists of the fourteen triangles forming the outer rim of the complex interlocking of triangles. The next two chakras are each constructed of ten triangles. Called Sarvartha-sadhaka and Sarvarakshakara, Accomplisher of All purpose and Giver of Protection, they indicate a stage when inner realization begins to dawn. The seventh chakra, consisting of eight triangles, is called Sarva-rogahara, Remover of All Desires and Ills, and represents the stage when the adept is free from earthly bonds and is at the threshold of the inner circle of realization. An inverted triangle is the eighth chakra, Giver of All Accomplishments, of Sarvasiddh!prada; it denotes a stage before the consummation of realization. All the triangular chakras are coloured red, to represent radiant energy or the dynamic and 2ery element of the cosmos. The last chakra, the Bindu, is known as Sarva-anandamaya, Full of Bliss. It is the sanctum sanctorum, abounding in joy, in which the adept participates in union. The point is light itself, beyond all colours, and is therefore represented as colourless” (Figs. 15 and 16). At the end of this spiritual search, one feels like the pilgrim who has completed his ascent of the Borobudur 3-dimensional mandala, and 2nally contemplates the 2nal crowning stupa, symbolizing Emptiness. We note that the colour descriptions diGer from the frontispiece 2gure in [20], where the chakras alternate blue and red, the outer one being yellow. Also, we note that the bindu in both works is described as a mere dot, whereas Rao [14] describes it as a small circle. 2.4. Yantra as an instrument of worship Now let us turn to Zimmer [19]: “In Hindu devotional tradition, “yantra” is the general term for instruments of worship, namely, idols, pictures, or geometrical diagrams. A yantra may serve as (1) a

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representation of some personi2cation or aspect of the divine, (2) a model for the worship of a divinity immediately within the heart, after the paraphernalia of outward devotion (idol, perfumes, oGerings, audibly uttered formulO) have been discarded by the advanced initiate, (3) a kind of chart or schedule for the gradual evolution of a vision, while identifying the Self with its slowly varying contents, that is to say, with the divinity in all its phases of transformation. In this case the yantra contains dynamic elements. We may say, then, that a yantra is an instrument designed to curb the psychic forces by concentrating them on a pattern, and in such a way that this pattern becomes reproduced by the worshiper’s visualizing power. It is a machine to stimulate inner visualizations, meditations, and experiences. The given pattern may suggest a static vision of the divinity to be worshiped, the superhuman presence to be realized, or it may develop a series of visualizations growing and unfolding from each other as the links or steps of a process”. The text goes on elaborating on the processes of enfolding and unfolding visions arising from meditation on linear yantras. Much more information on linear yantras  ! Yantra in particular, are given in Zimmer’s learned treatise on in general, and Sr the relationship between Art and Yoga in the Sacred Images of India [20], where a full chapter is devoted to linear yantras. This work also explains that yantras must be consecrated by a breath-giving ceremony (pr(a.napratis. .t h(a), where the deity, seen in one’s own self by meditation, is installed as a vivifying element into the material image. Following the Gandharva Tantra [16], after Zimmer [20]: “Next, after performing Pran . ayama [the preparatory breathing exercises], the Sadhaka [devotee] should take handfuls of Sowers. The Dev! should never be invoked without handfuls of Sowers. The Sadhaka who controlled his Pran . a will meditate on the Paramesvar! [that is, the Highest Ruler] as above described, in his heart, and seeing by Her grace that image, the substance of which is consciousness in his heart, let him think of the identity between the image manifested within and the image without. Next, the energy [tejas] of consciousness within should be taken without by means of the Vayu-B!ja [the mantra “yam”] with the breath along the nostrils, and infused into the handful of Sowers. Thus, issuing with the breath, the Devata enters into the Sowers [he is holding to his nostrils]. The Sadhaka should then establish the Devata in the image or Yantra by touching it with those Sowers”. 2.5. Mantras  ! Yantra The photographic illustration given as Plate XCVIII in Rao [4] shows a Sr bearing devanagar! letters. This is analysed in Pott [11], which we cite here: “Within the outline of the bh(upara, the square with projections, is a three-fold circle inside which is a sixteen-petalled lotus which in its turn envelopes an eightpetalled lotus. Within this latter lotus there is a 2gure consisting of nine juxtaposed triangles—the 2gure is indeed called by the name navayonicakra—which together make up the total of 43 small triangles of which Rao speaks. In these small

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triangles, just as on the petals of the 16-petalled lotus and with the exception of the nine innermost triangles, are the characters of the nagar! alphabet, while on the petals of the eight-petalled lotus there are groups of characters, each letter provided with a bindu”.  ! Yantra with nagar! Pott gives at this point, as Fig. 7, a graphic rendition of the Sr  ! Yantra, resembling the one represented letters. This hand-drawn 2gure is an exact Sr above as the left picture in Fig. 3. Actually, point S almost touches circle . It is not a faithful rendition of Plate XCVIII from Rao [4], which is much closer to the Classical  ! Yantra, contains further multi-syllabic mantras in the eight-petalled lotus, and 2nally Sr shows all the inscriptions oriented circle-wise in the 2gure, as opposed to horizontally as in Pott [11]. This last author continues in describing the precise placement of letters on the diagram. # ) Yantra 2.6. Mysteries of Sr( It is clear that this investigation leaves many questions unanswered. The precise  ! Yantra and similar diagrams depend mantras, mudras, and divinities associated with Sr on sectarian traditions. This esoteric tradition is still largely hermetic. A symposium organized by CNRS in 1984 on mantras and tantras led to a published volume [9], in which an article by Tara Michael [6] describes one tradition of ritual use, taken from the Saundarya-Lahar!, or Wave of Beauty hymn. The most complete erudite account available so far is the Yogin()hr. daya (Heart of the Yogin!), published recently in French by Andre Padoux [10]. Although it is hinted in several sources that this symbol is very old, the author does not know of any published representation anterior to the 17th century, leaving open its date of creation. But the really puzzling enigma is: How could such a diJcult geometric construction be invented at all?  ! Yantra, in its formal We cannot resist quoting again The Tantric Way [8]. “Sr content, is a visual masterpiece of abstraction, and must have been created through revelation rather than by human ingenuity and craft”.

3. Conclusion  ! Yantra than was previously availThis study oGers a more precise analysis of Sr able in the published literature. We oGer this study as evidence of the non-triviality of bridging the cultural gap between Humanities and Science on the one hand, and Eastern–Western knowledge, or sensibility, on the other.

Appendix  ! Yantra (Fig 17). We give here the full Postscript source of the Classical Sr

G. Huet / Theoretical Computer Science 281 (2002) 609 – 628

Fig. 17.

% Shri Yantra in Color Postscript Copyright G. Huet 1990 % Lines settings 0.52 setlinewidth % 0.35 for printer 0 setlinecap 0 setlinejoin 1.5 setmiterlimit 0 0 0 setrgbcolor % Scaling 3.937 0.72 mul dup scale /UN 100 def

% % % % %

% Triangles 105 90 translate

10 cm in inches 72 units in 1 inch 1 mm in units new unit=1mm unit=10cm % Origin at O

/YO 0 def /XO 0 def /YT UN def /XT 0 def

% O=(0,0) % T=(0,1)

% Inputs - Obtained by Newton approximation computed by Caml program /YQ 66.77 def /XF 12.6 def /YP 46.27 def /XA 18.7 def /YJ 39.8 def

623

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G. Huet / Theoretical Computer Science 281 (2002) 609 – 628

% Coordinates computations % circle alpha 4*X^2+(2*Y-1)^2=1 /sq {dup mul} def % squaring /alpha {UN sq Y 2 mul UN sub sq sub sqrt 2 div} def

% X=alpha(Y)>0

/XQ /Y YQ def alpha def /YF YQ /XJ /Y /YA YQ /XH YJ /XI XJ /XC XA /YV YF /T1 /T2 /YD /T3 /T4 /YG /T5 /XB

XA XF T1 XJ XF UN YQ XA

/YM YP XC /XM YM /YL /XL /XP /XE

YJ XA XB XF

/XS /XK /XN /XR /XD /XG

XF XF XP XB YD YG

% Q on alpha; Q determined % Q & O determine e: Y*XQ=X*YQ def % F determined YJ def alpha def % J on alpha; J determined % J & T determine h: (1-Y)*XJ=X*(1-YJ) XA mul XQ div def % A on e determined XQ mul YQ div def % YH=YJ, H on e determined UN YQ sub UN YJ sub div mul def % YI=YQ, I on h determined % XU=0, YU=YQ, U & A determine f: XA*(Y-YQ)=X*(YA-YQ) YQ YP sub YQ YA sub div mul def % YC=YP, C on f determined % F & H determine g: (Y-YF)*(XH-XF)=(YJ-YF)*(X-XF) YF YJ sub XF mul XH XF sub div add def % XV=0, V on g determined % XW=0, YW=YP determine W % F & W determine a: (Y-YP)*XF=(YF-YP)*X YF YP sub mul def YQ YA sub mul def YQ mul T2 YP mul add T1 T2 add div def % D determined as a*f YQ YP sub mul def UN YJ sub mul def T3 mul T4 YP mul add T3 T4 add div def % G determined as a*h YJ sub def T5 mul YQ YA sub div def % YB=YJ, B on f determined % XZ=0, YZ=YA determine Z % C & Z determine c: (Y-YA)*XC=X*(YP-YA) YA sub XF T5 mul YQ XH XF sub mul add mul T5 YA mul XC mul add T5 mul YP YA sub XH XF sub mul add div def % M determined as c*g YA sub XC mul YP YA sub div def % I & B determine d: (Y-YJ)*(XI-XB)=(YQ-YJ)*(X-XB) T5 XB mul XI XB sub div sub def % (0,YL) sur d YL YQ sub mul YA YQ sub div def % L on f determined YP YJ sub XI XB sub mul T5 div add def % P on d determined YM YP sub mul YF YP sub div def % YE=YM, E on a determined % P & E determine i: (Y-YP)*(XE-XP)=(YM-YP)*(X-XP) YV YP sub mul YF YP sub div def % YS=YV, S on a determined YA YF sub XH XF sub mul YJ YF sub div add def % YK=YA, K on g determined YD YP sub XE XP sub mul YM YP sub div add def % YN=YD, N on i determined YG YJ sub XI XB sub mul T5 div add def % YR=YG, R on d determined YP sub XF mul YF YP sub div def % D on a determined YP sub XF mul YF YP sub div def % G on a determined

% Input is correct if (0,YG) on i: YG=YP-XP*(YM-YP)/(XE-XP) /red {gsave 1 0 0.1 setrgbcolor fill grestore} def % coloring in red /blue {gsave 0 0.8 1 setrgbcolor fill grestore} def % coloring in blue /yellow {gsave 1 1 0.1 setrgbcolor fill grestore} def % coloring in yellow /eored {gsave 1 0 0.1 setrgbcolor eofill grestore} def % eo-coloring red /eoblue {gsave 0 0.8 1 setrgbcolor eofill grestore} def % eo-coloring blue /eoyellow {gsave 1 1 0.1 setrgbcolor eofill grestore} def % eo-coloring yellow

G. Huet / Theoretical Computer Science 281 (2002) 609 – 628 % Bindu - The bindu is not at the center of alpha, but at the center of the % innermost triangle, assumed to be equilateral /BIN YP YD 2 mul add 3 div def newpath 0.6 BIN moveto 0 BIN 0.6 0 360 arc closepath red % Drawing triangle (trikona) newpath 0 YP moveto XD YD lineto XD neg YD lineto closepath stroke % Drawing 8 star (vasukona) newpath XN YD moveto XD YD lineto XE YM lineto XE neg YM lineto XD neg YD lineto XN neg YD lineto 0 YJ lineto closepath 0 YQ moveto XC YP lineto XC neg YP lineto closepath red stroke % Drawing first 10 star (dasharayugma) newpath 0 YG moveto XP YP lineto XC YP lineto XB YJ lineto XB neg YJ lineto XC neg YP lineto XP neg YP lineto closepath 0 YA moveto XM YM lineto XE YM lineto XF YQ lineto XF neg YQ lineto XE neg YM lineto XM neg YM lineto closepath blue stroke % Drawing second 10 star newpath 0 YV moveto XH YJ lineto XB YJ lineto XA YA lineto XA neg YA lineto XB neg YJ lineto XH neg YJ lineto closepath 0 YL moveto XI YQ lineto XF YQ lineto XG YG lineto XG neg YG lineto XF neg YQ lineto XI neg YQ lineto closepath red stroke % Drawing 14 star (manvashra) newpath 0 0 moveto XQ YQ lineto XI YQ lineto XR YG lineto XG YG lineto XS YV lineto XS neg YV lineto XG neg YG lineto XR neg YG lineto XI neg YQ lineto XQ neg YQ lineto closepath 0 YT moveto XJ YJ lineto XH YJ lineto XK YA lineto XA YA lineto XL YL lineto XL neg YL lineto XA neg YA lineto XK neg YA lineto XH neg YJ lineto XJ neg YJ lineto closepath blue stroke % Padmas /PI 180 def /D1 7 /R2 R1 D1 add def /R4 R2 D3 add def /R6 R5 D4 add def

def /R3 /R5 /R7

/D2 12 def /D3 2 def /D4 2 def /R1 50 def R2 D2 add def R3 D4 add def R6 D4 add def

0 R1 translate % Origin at center of circles /petal { % We assume we start at X3 Y3 neg X2 Y2 neg X1 Y1 neg R 0 curveto X1 Y1 X2 Y2 X3 Y3 curveto

625

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G. Huet / Theoretical Computer Science 281 (2002) 609 – 628 } def % One padma petal

/THETA0 PI 4 div def % Inner 8-petal padma /R R2 def /THETA1 PI 8 div def /U1 5 def /V1 8 def /DELTA1 20 def /GAMMA1 5 def /X1 R2 U1 DELTA1 cos mul sub def /Y1 U1 DELTA1 sin mul def /X3 R1 THETA1 cos mul def /Y3 R1 THETA1 sin mul def /X2 X3 V1 THETA1 DELTA1 sub cos mul add def /Y2 Y3 V1 THETA1 DELTA1 sub sin mul add def gsave newpath X3 Y3 neg moveto 8 {petal THETA0 rotate} repeat closepath % Drawing circle alpha R1 0 moveto 0 0 R1 0 360 arc closepath eored stroke grestore % Outer 16-petal padma /R R3 def /THETA2 PI 16 div def /U2 4 def /V2 6 def /DELTA2 15 def /GAMMA2 5 def /X1 R3 U2 DELTA2 cos mul sub def /Y1 U2 DELTA2 sin mul def /X3 R4 THETA2 cos mul def /Y3 R4 THETA2 sin mul def /X2 X3 V2 THETA2 DELTA2 sub cos mul add def /Y2 Y3 V2 THETA2 DELTA2 sub sin mul add def gsave newpath X3 Y3 neg moveto 16 {petal THETA1 rotate} repeat closepath % Drawing circle beta R2 0 moveto 0 0 R2 0 360 arc closepath eoblue stroke grestore % Drawing circle gamma newpath R3 0 moveto 0 0 R3 0 360 arc closepath % Drawing 3 outer circles R5 0 moveto 0 0 R5 0 360 arc closepath

G. Huet / Theoretical Computer Science 281 (2002) 609 – 628

627

R6 0 moveto 0 0 R6 0 360 arc closepath stroke newpath R7 0 moveto 0 0 R7 0 360 arc closepath % Squares /greek {L1 L1 neg moveto 4 {L1 L3 neg lineto L1 L2 add L3 neg lineto L1 L2 add L3 L4 add neg lineto L1 L2 add L5 add L3 L4 add neg lineto L1 L2 add L5 add L3 L4 add lineto L1 L2 add L3 L4 add lineto L1 L2 add L3 lineto L1 L3 lineto L1 L1 lineto 90 rotate} repeat closepath} def % Drawing a greek frieze /W1 2 def % Spacing between greeks /W2 3 def % Inner greek /L1 R7 def /L2 W1 4 mul W2 add def /L3 L1 3 div def /L4 L1 L3 sub 2 div def /L5 W2 def greek eoyellow stroke /transform {/L1 /L2 /L3 /L5

L1 L2 L3 L5

W1 W1 W1 W1

add def 2 mul sub def add def 2 mul add def} def

% Middle greek transform greek stroke % Outer greek transform greek stroke showpage

References [1] Adobe Systems Incorporated, Postscript Language Reference Manual, Addison-Wesley, Reading, MA, 1985.  !-Yantra, Religion 7 (1) (1977) 66–85. [2] N.J. Bolton, D.N.G. Macleod, The geometry of the Sr [3] A. Danielou, Le Polytheisme Hindou, Buchet=Chastel, 1960. [4] T.A. Gopinatha Rao, Elements of Hindu Iconography, Madras, 1914, Repr. Motilal Banarsidass, Delhi, 1985. [5] K.G. Kundangar, Notes on Shri Maha Lakshmi Temple, Kolhapur, Bombay, 1929. [6] T. Michael, Le sr!cakra dans la Saundarya-Lahar!, in: A. Padoux (Ed.), Mantras et diagrammes rituels dans l’Hindouisme, Editions du CNRS, Paris, 1986, pp. 127–139.

628 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

G. Huet / Theoretical Computer Science 281 (2002) 609 – 628 A. Mookerjee, Tantra Art. Publ. Ravi Kumar, Random House, New York, 1971. A. Mookerjee, M. Khanna, The Tantric Way, Thames and Hudson, London, 1977. A. Padoux (Ed.), Mantras et diagrammes rituels dans l’Hindouisme, Editions du CNRS, Paris, 1986. A. Padoux, Le C_ur de la Yogin! —Yogin!hrdaya, avec le commentaire D!pika d’Amrtananda, Publications de l’Institut de Civilisation Indienne, Coll`ege de France, 1994 (diGusion de Boccard). P.H. Pott, Yoga and Yantra, Their Interrelation and their Signi2cance for Indian Archaeology, Martinus NijhoG, The Hague, 1966. Punyananda-Natha, Kama-Kala-Vilasa, With the commentary of Natanananda-Natha, trans. with commentary by A. Avalon. Ganesh & co, Madras, 1961. P. Rawson, The Art of Tantra, Thames and Hudson, London, 1973. L. Renou, L’Inde Classique, Tome 1, Ist edition, Payot, Paris, 1947, Repr. Maisonneuve, Paris, 1985.  ! cakrasambharatantra in: A. Avalon, Tantrik Texts, vol. VII, Luzac, Calcutta, K.D. Samdup (Ed.), Sr London, 1919. Shr!yukta Shiva Chandra Vidyarnava Bhattacharya Mahodaya, Principles of Tantra (Tantra Tattva). With an introduction and commentary by A. Avalon, Luzac, London, 1916. T. Vidyaratna (Ed.), Prapa˜ncasaratantra, in: A. Avalon, Tantrik Texts, Vol. III, Luzac, Calcutta, London, 1914. H. Zimmer, Kunsform und Yoga im indischen Kultbild, Frankfurter Verlags-Anstalt, Berlin, 1926. H. Zimmer, (ed. by J. Campbel) Myths and Symbols in Indian Art and Civilization, Bollingen Series, Vol. VI, Princeton University Press, Princeton, NJ, 1946, Paperback edition, 1972. H. Zimmer, Artistic Form and Yoga in the Sacred Images of India, Princeton University Press, Princeton, NJ, 1984. (trans. by G. Chapple, J.B. Lawson)

Theoretical Computer Science 281 (2002) 629–630

Author index volume 281 (2002) (The issue number is given in front of the page numbers)

Arnold, A., Nivat’s processes and their synchronization Ausiello, G., M. Mislove and D. Sannella, Preface Banderier, C. and P. Flajolet, Basic analytic combinatorics of directed lattice paths Beauquier, D. and J.-C. Fournier, Groups and tilings Berstel, J. and L. Vuillon, Coding rotations on intervals Boudol, G. and I. Castellani, Noninterference for concurrent programs and thread systems Bruni, R. and U. Montanari, Dynamic connectors for concurrency Castellani, I., see G. Boudol Courcelle, B. and T. Knapik, The evaluation of first-order substitution is monadic second-order compatible Cousineau, G., Tilings as a programming exercise Curien, P.-L., Une br"eve biographie scientifique de Maurice Nivat Dauchet, M., S. Tison and M. Tommasi, R!eduction de la non-lin!earit!e des morphismes d’arbres Recognizable tree-languages and non-linear morphisms de Vries, S., see P. Gritzmann Del Lungo, A., Reconstructing permutation matrices from diagonal sums Delorme, M. and J. Mazoyer, Reconnaissance parall"ele des langages rationnels sur automates cellulaires plans D!ıaz, J., M. Serna and D.M. Thilikos, Counting H-colorings of partial k-trees Durand, B., De la logique aux pavages Ehrenfeucht, A., T. Harju and G. Rozenberg, Gene assembly through cyclic graph decomposition Ferrari, L., E. Pergola, R. Pinzani and S. Rinaldi, An algebraic characterization of the set of succession rules Flajolet, P., see C. Banderier Fournier, J.-C., see D. Beauquier Gastin, P. and M. Mislove, A truly concurrent semantics for a process algebra using resource pomsets Grigorieff, S., Modelization of deterministic rational relations Gritzmann, P. and S. de Vries, On the algorithmic inversion of the discrete Radon transform Harju, T., see A. Ehrenfeucht ! ı Yantra Geometry Huet, G., Sr% Knapik, T., see B. Courcelle Mazoyer, J., see M. Delorme Mislove, M., see G. Ausiello, Mislove, M., see P. Gastin Montanari, U., see R. Bruni Okada, M., A uniform semantic proof for cut-elimination and completeness of various first and higher order logics Pergola, E., see L. Ferrari

(1–2) (1–2) (1–2) (1–2) (1–2)

31– 36 1– 1 37– 80 81– 97 99–107

(1–2) 109–130 (1–2) 131–176 (1–2) 109–130 (1–2) 177–206 (1–2) 207–217 (1–2) 3– 23 (1–2) 219–233 (1–2) 455–469 (1–2) 235–249 (1–2) 251–289 (1–2) 291–309 (1–2) 311–324 (1–2) 325–349 (1–2) 351–367 (1–2) 37– 80 (1–2) 81– 97 (1–2) 369–421 (1–2) 423–453 (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2)

455–469 325–349 609–628 177–206 251–289 1– 1 369–421 131–176

(1–2) 471–498 (1–2) 351–367

630

Author index volume 281 / Theoretical Computer Science 281 (2002) 629–630

Pinzani, R., see L. Ferrari Rinaldi, S., see L. Ferrari Rozenberg, G. and A. Salomaa, ICALP, EATCS and Maurice Nivat Rozenberg, G., see A. Ehrenfeucht Rus, T., A unified language processing methodology Salomaa, A., see G. Rozenberg Salomaa, A., Uni-transitional Watson–Crick D0L systems Sannella, D., see G. Ausiello, S!enizergues, G., LðAÞ ¼ LðBÞ? A simplified decidability proof Serna, M., see J. D!ıaz Thilikos, D.M., see J. D!ıaz Tison, S., see M. Dauchet Tommasi, M., see M. Dauchet Vuillon, L., see J. Berstel

(1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2) (1–2)

351–367 351–367 25– 30 325–349 499–536 25– 30 537–553 1– 1 555–608 291–309 291–309 219–233 219–233 99–107