Computational Analysis of One-Dimensiona

COMPUTATIONAL ANALYSIS OF ONE-DIMENSIONAL CELLULAR AUTOMATA

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley

Series A. MONOGRAPHS AND TREATISES Published Titles Volume 1:

From Order to Chaos L. P. Kadanoff

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Smooth Invariant Manifolds and Normal Forms 1. U. Bronstein and A. Ya. Kopanskii

Volume 8: Dynamical Chaos: Models , Experiments , and Applications V. S. Anishchenko

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A. M. Samoilenko and N. A. Perestyuk Volume 16: Turbulence, Strange Attractors and Chaos D. Ruelle Volume 17: The Analysis of Complex Nonlinear Mechanical Systems: A Computer Algebra Assisted Approach M. Lesser

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Series A Vol. 15

Series Editor: Leon 0. Chua

COMPUTATIONAL ANALYSIS OF ONE-DIMENSIONAL CELLULAR AUTOMOTH

Burton H. Voorhees Professor of Mathematics Athabasca University

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V

Preface

The growth of interest in the study of complex systems which has occurred over the past twenty five years is a prime example of the explosive development of an idea whose time has come. The data processing capacities of modern computers have made possible work that could not even be imagined before their invention. For the theoretician vast new possibilities have opened in the continual search for understanding. And from the practical direction, the rapid integration of societies, economies, and cultures in the shrinking world; and the problems concomitant on explosive economic growth, have made the understanding of complexity an essential criterion for survival. The goal of a united world and a sustainable economy will be achieved, but if we do not understand the processes in which we are involved, and the requirements which they place on us, the sustainable economy may be that of Ethiopia rather than Utopia. The problem with complex systems is captured in the name. Complex systems are--complex. This means that many of the usual tools which have been used to study simple systems may not apply. New methods of analysis, new conceptual approaches, and new synthesizing perspectives are required, and their development is the focus of much of the current research on "complexity." Cellular automata provide one of the most interesting avenues into the study of complex systems in general, as well as having an intrinsic interest of their own. Because of their mathematical simplicity and representational robustness they have been used to model economic, political, biological, ecological, chemical, and physical systems. Almost any system which can be treated in terms of a discrete representation space in which the dynamics is based on local interaction rules can be modelled by a cellular automata.

vi

The intent of this book is to give an introduction to the analysis of cellular automata ( CA) in terms of an approach in which CA rules are viewed as elements of a non -linear operator algebra, which can be expressed in component form much as ordinary vectors are in vector algebra. Although a variety of different topics are covered , this viewpoint provides the underlying theme. The actual mathematics used is not hard, and the material should be available to anyone with a junior level university background , and a certain degree of mathematical maturity. Each chapter concludes with an exercise section . The problems in these sections have been chosen to provide basic exercise in the kind of analysis, and the kinds of thinking involved in working with cellular automata of the kind studied in this book . The suggestion, for serious students , is to do each exercise by hand. In some cases it would be a trivial task to work a problem on a computer , but this is not the point. The idea is to develop expertise , and intuition in working with cellular automata , not with computers. That expertise can be found elsewhere. There are many loose ends in this book as well. In writing it, it was a constant struggle to stick to the immediate point rather than chase off after another tantalizing lead that might produce some further results. It is my hope that at least a few readers will be stimulated , or provoked, to take up these unfinished lines of work. Perhaps the best illustration of the way I view this book is given in the following tale:

One fine morning Nasrudin announced to his friends at the tea house that he was setting out to climb the high mountains which lay several days travel from his village . Having proclaimed his intention, he strode away from the village whistling a merry tune. Five days later he returned , somewhat worse for the wear, and made his way to the tea house. After drinking some tea to quench his thirst, he proceeded to dazzle those who were present with descriptions of the wonders he had seen while in the mountains. "Mighty waterfalls , and vast forests ! Cliffs that climbed beyond the sky! And animals, such wonders as you have never seen!"

VII

His audience was enchanted, and Nasrudin was warming to his topic when a young rascal seated in the front row interrupted. "Excuse me, Nasrudin" He said, "but please, tell me how you had time to see all of those wonderful things? It is a two day walk to the mountains , and two days back , and you were only gone five days." "Your are right," Nasrudin replied , "In the time I was gone, I only was able to reach the foothills . But from that vantage point I could easily see all of the things which I am describing." My hope is that in describing the little that I have been able to see, it will encourage others to go further. Vancouver, B.C. November 29, 1994

This page is intentionally left blank

ix

Table of Contents Preface Introduction Chapter 1 The Operator Algebra of Cellular Automata 1. Basic Definitions and Notation 2. Boolean Forms of CA Rules 3. The Canonical Basis Operators 4. Symmetry Transformations on CA Rules 5. CA Rules as Maps of the Interval 6. Exercises for Chapter 1 Chapter 2 Cellular Automata Arithmetic 1. Products of CA Rules 2. The Division Algorithm 3. Residue Arithmetic of Cellular Automata 4. Exercises for Chapter 2 Chapter 3 Fixed Points and Cycles 1. Fixed Points and Shift Cycles via Rule Decomposition 2. Jen's Invariance Matrix Method 3. Exercises for Chapter 3 Chapter 4 Commutation of CA Rules 1. Computation of Commutator Sets 2. Idempotence 3. Ito Relations 4. Some Interesting Sub-Groups 5. Exercises for Chapter 4

67 71 72 77 85

Chapter 5 Additive Rules: I. Basic Analysis 1. Conditions for Additivity

87

2. Cyclotomic Analysis 3. Injectivity 4. Another View of Injectivity 5. Exercises for Chapter 5 Chapter 6 Additive Rules: II. Cycle Structures and Entropy 1. State Transition Diagrams 2. Cycle Periods 3. Reachability 4. Conditions for States on Cycles 5. Entropy Reduction 6. Exercises for Chapter 6 Chapter 7 Additive Rules: III. Computation of Predecessors 1. Predecessors of 3-Site Rules 2. k-Site Rules 3. Examples 4. The Operator B 5. Exercises for Chapter 7

v 1 26 29 31 33 33 35 42 44 47 52 54 60 65

94 98 100 101 119 121 122 124 131 132 134 138 140 144 149

X

Chapter 8 The Binary Difference Rule 1. Basic Properties of D 2. The Graph of D:[0 , 1]-+[0,1] 3. Some Numerical Relations 4. A Density Result 5. Exercises for Chapter 6 Chapter 9 Computation of Pre-Images 1. Pre-Images and Predecessors 2. The Rule Graph and Basic Matrix 3. Computation of Pre - Images From the Basic Matrix 4. Pre-Images and the Jen Recurrence Relations 5. Exercises for Chapter 9 Chapter 10 The Garden of Eden 1. GE(X) and GE*(X) 2. Computation of GE*(X) 3. GE*(X) for 3-Site Rules 4. Classification With GE* 5. Exercises for Chapter 10 Chapter 11 Time Series Simulation 1. Cellular Automata Generating Time Series 2. Statistics of Time Series 3. Exercises for Chapter 11 Chapter 12 Surjectivity of Cellular Automata Rules 1. A Kitchen Sink Theorem 2. The de Bruijn Diagram 3. The Subset Diagram 4. The Semi-Group G(X) 5. The Subset Matrix and Some Replacement Diagrams 6. Exercises for Chapter 12 Appendix 1 Boolean Expressions for 2 and 3 -Site Rules Appendix 2 Canonical Forms and Decompositions of 3-Site Rules Appendix 3 Strongly Legal 3-Site Non- Generative Rules Appendix 4 The Mod( 2) Pascal Triangle Appendix 5 GE*(X) for 3-Site T-Equivalence Class Exemulars Appendix 6 U(X) for Selected Rules References

150 154 158 159 160 164 165 169 171 176 193 195 198 203 207 208 213 218 220 224 230 235 236 243 245 249 254 256 260 266 271

1

Introduction A.M. Turing, in a classic paper, "Systems of logic based on ordinals," names three abilities which are required for good mathematical reasoning. These are intuition, ingenuity, and, what he considered most important of all, the ability to distinguish between interesting and uninteresting Questions [1]. At the end of his introduction to the proceedings of the 1989 Los Alamos Workshop on Cellular Automata, H.A. Gutowitz quotes Turing, and then asks, "What makes a cellular automata interesting?" [2]. This question can be viewed in two ways. What is is that makes cellular automata an interesting class of mathematical objects, and what is it that distinguishes particular cellular automata for special attention? Of course, many different answers to these questions can be given. What is of interest to a particular individual, or research community, may not be so to another individual or community. For a pure mathematician, for example, cellular automata may have an intrinsic interest relating to formal aspects of their behavior. For a biologist, cellular automata may provide an interesting way to model biological phenomena such as cell sorting. There are many different individuals and groups, each finding cellular automata interesting from their own particular viewpoint, and these individuals and groups are often not in close communication with each other. Another way of saying this is to say that the boundaries around the field which includes cellular automata, the field of complex systems, are fuzzy. In this sense, the study of complex systems, and of cellular automata is in what Thomas Kuhn calls a pre-paradigmatic state. The threads are there, but the carpet has yet to be woven. What unites the various threads of interest, integrating them into the larger tapestry of science, is the goal of understanding. Both for the individual and for the community, science begins with questions, with the desire to understand. An eloquent expression of this underlying passion for understanding is found in a passage from Issac Newton's Optiks: "What is there in places almost empty of matter, and whence is it that the sun and planets gravitate towards one another, without dense matter between them? Whence is it that nature doth nothing in vain; and whence arises all that order and beauty which we see in the world? To what end are comets, and whence is it that planets move all one and the same way in

2 orbs concentric , while comets move all manner of ways in orbs very eccentric , and what hinders the fixed stars from falling upon one another? How came the bodies of animals to be contrived with so much art, and for what ends were their several parts? Was the eye contrived without skill in optics, or the ear without knowledge of sounds? How do the motions of the body follow from the will, and whence is the instinct in animals?" In this light, cellular automata are interesting because of the belief that they may contribute , perhaps substantially, to the more general goal of understanding ; that is, to the development of a view of the world in which that which is seen to occur --is seen to occur naturally . The reason for this belief is the fact that cellular automata are capable of generating complex patterns of global behavior on the basis of very simple local interaction rules. Because of this, cellular automata have the twin virtues of mathematical tractability, and representational robustness. The basic cellular automata modelling paradigm involves either an initially discrete representation space , or the course grain partition of a representation space into individual cells, each of which may display one of a finite number of possible values . A system configuration is an assignment of a value to each cell, and system evolution is described in discrete time via local interaction functions between cells . These local functions define the cellular automata rule. All cells are updated synchronically, and the value assigned to a cell at a given time step depends only on its value, and the values of certain neighboring cells at the preceeding time step. For example, in the Greenberg-Hastings [3,4] model for pattern formation in chemical reaction-diffusion systems, each cell in a 2-dimensional square lattice can be in one of three possible states: quiescent, active, or refractory. The model can be considered as strictly formal, or it can be supposed to have a physical analog in which cells are assumed small enough that chemical concentration dynamics within cells can be course grained into these three catagories , and all concentration gradients are zero, except at cell boundaries where discrete jumps may occur. The neighborhood of a cell, that is, the set of all other cells able to directly influence its evolution, is the von Neumann neighborhood, consisting of the cell itself together with the four cells directly above, below , to the right, and to the left, as indicated in Figure 1 below . Local interaction rules are

3 defined in terms of the influence of cells in the neighborhood of a cell on the updated value to be placed in that cell. i j+1 i-1 j

i+1j ij-1

Figure 1 von Neumann Neighborhood of Cell ij

The local interaction rules in the Greenberg-Hastings model implement the following dynamics: 1. If a cell is quiescent it remains so unless one or more of its neighbors is active. In this case, it becomes active at the next time step. 2. Active cells become refractory in the next time step. 3. Refractory cells return to quiescence in the next time step. Even though these evolution rules are exceedingly simple, they produce spiral wave patterns which mimic the behavior of such chemical reaction-diffusion systems as are found in the Belousov-Zhabotinsky reaction. An example, with only active cells highlighted, is shown in Figure 2.

Figure 2 Sample Pattern Generated by Greenberg- Hastings Model

4

The robustness of cellular automata as a modelling class is exhibited in the wide range of applications in which they have been employed since their introduction by J. von Neumann and S. Ulam [5,6] . They have been used in a variety of biological applications, including modelling heart fibrillation [71, aspects of development [8,9], and testing evolution theories [ 10]. In physics, cellular automata have been used to model non-linear chemical reaction systems [3 ,4], the evolution of spiral galaxies [11], dendritic crystal growth [ 12], and spin exchange systems [13,14], to list but a few cases. In computation theory , cellular automata can be considered as parallel computers [ 15,16, 171, and have been used as parallel multipliers [ 18,19], sorters [20], and prime number sieves [ 21]. They have also been extremely useful in image processing and pattern recognition [22,23,24]; and it is known that some cellular automata are universal computers , hence equivalent to a universal Turing machine [25,26]. Cellular automata have been studied under a variety of names: "cellular spaces," "cellular structures," "homogeneous structures," "tessellation structures ," "tessellation automata," and "iterative arrays." Over the past dozen years the term cellular automata has come to predominate , coincident with the major increase of interest in these systems stimulated by the ground breaking work of S. Wolfram [27 ,28,29,30,31,321. In terms of the question of interest, Wolfram shifted the focus from the possible relevence of cellular automata for modelling to the discovery of particular cellular automata with interesting properties. This question was posed with the term "complexity engineering " [32]; that is , the design of complex systems exhibiting predetermined properties. Within this overall direction,Gutowitz [2] distinguishes two lines of research , relating to what he terms the forward problem, and the inverse problem. The forward problem is to determine the characteristic properties of a given cellular automaton . In its basic form, the forward problem involves analysis of the behavior of particular cellular automata rules. This may involve such things as determination of cycles and /or fixed points; statistical analysis of space- time patterns generated; and computation of conditions for configurations to he on cycles. In its more general form, the forward problem involves development of analytic techniques which can be applied to such analysis . The forward problem has been a focus of much of the recent research on cellular automata, and many useful techniques have been

5

developed; e.g., the polynomial representations introduced by O. Martin, A.M. Odlyzko, and S. Wolfram [33] for studying cycle periods; the recurrence relations derived by E. Jen [34,35] for counting pre-images; the statistical and entropic measures introduced by Wolfram [27], and D.A. Lind [361; the mean field analysis emphasized by Gutowitz [37]; graph theoretic approaches [e.g., 38,39]; computation of fractal dimensions [40,41]; and the 2, and Z parameters [42,43]. These, and other methods of analysis provides a rich toolkit. The inverse problem, on the other hand, has received relatively little attention outside of the practical modelling literature. As already indicated, there have been specific applications of cellular automata in modelling; e.g., J.H. Conway's well known "Game of Life" [44] is an example of the construction of a cellular automaton to mimic some properties of living systems, but little has appeared on the abstract problem of model construction, or of determining classes of cellular automata which satisfy preestablished constraints. Work to date on these problems has been aimed at determination of global behavior in the space of all cellular automata rules; e.g., phase transitions and criticality [45,46,47]. From another, more purely mathematical direction, L.P. Hurd [48] has constructed cellular automata exhibiting various degrees of computational complexity. Nevertheless, the inverse problem remains less well researched than the forward problem. Part of the aim of this book is to provide a framework for further work on inverse questions by writing cellular automata rules in terms of variable coefficients which can become variables in constraint equations whose solution sets define the sets of rules having pre-determined properties.

Particular

Universal

Forward Problem Determination of characteristics and evolutionary behavior of given individual cellular automata Development of techniques and

Inverse Problem Design of cellular automata for specific computational or modelling tasks

measures for

satisfying general

Determination of cellular automata

analysis of sets of precellular automata specified constraints Table 1 Particular and Universal Aspects of Forward and Inverse Problems

6 A general overview of the main issues in the forward and inverse problems is shown in Table 1. Attention in this book is on 1-dimensional cellular automata with cells taking binary values. Although this is the simplest possible case, it one which has been the focus of much investigation. The configuration space is now a 1-dimensional lattice, either finite or infinite. If infinite it is either doubly infinite, or half-infinite; while if finite, boundary conditions are specified at the end points, or the lattice is taken as a circle. Neighborhoods are blocks of k cell sites. A 3-site rule, for example, has neighborhoods consisting of 3 cells. The 1-dimensional k-site rules are labeled by assigning them a binary number, which is generated by listing the 2k possible neighborhoods in ascending numerical order and treating the listing of 0's and 1's obtained by specifying which neighborhoods map to 0 and which to 1 (called the rule table) as a binary number. For example, for the 3-site rule 90 the neighborhood table is 000 001 010 011 100 101 110 111 0

1

0

1

1

0

1

0

and the binary number 01011010 equals the denary (base 10) number 90. The generality of the 1-dimensional case is shown by an early result of G.A. Hedlund [49], to the effect that 1-dimensional cellular automata are the shift commuting endomorphisms of the shift dynamical system. From a purely mathematical point of view, one of the interesting aspects of cellular automata is that they may point the way toward a reconceptualization of the idea of a function. If the configuration space is taken as the set E+ of all right half-infinite binary sequences then it maps to the interval [0,1] in a natural way by associating to each sequence µ1µ2µ3••• the point I µ; 2-'. Thus, cellular automata acting on E+ also define maps of [0,1]. Although there is a technical issue related to sequences which consist of all 1's after some index value, cellular automata are "almost" functions on [0,1], but they are not defined in the ordinary way, as point functions in which the value of the function at each point is specified. Instead,they are defined by their local action on finite sequence segments. This leads to selfsimilarity in their graphs. The graphs defined by cellular automata are generalized linear graphs, with self-similarity taken into account. The question is whether non-linear generalizations can be defined, perhaps by making alterations in the neighborhood structures.

7

Historically speaking, much of the original interest in 1-dimensional cellular automata was stimulated by the complexity of the spatio-temporal evolution patterns which they are able to generate. Patterns generated by some of the rules discussed in this book are shown in the figures at the end of this introduction. Inspection of these patterns leads to several observations, and raises a number of questions. Some rules, such as rules 46, 72, 172, and 116 produce very simple patterns, either identically reproducing themselves after the first few transient iterations, or emulate left or right shifts. Rules 184 and 226 are slightly more complicated in their behavior, giving the appearence of "particle" dislocations travelling either right or left against a fixed checkerboard background. Use of the canonical decompositions introduced in Chapter 3 shows that these two rules generate mirror image behaviors. Rule 184 emulates a right shift on configurations on which rule 226 a left shift, and vice versa. More interesting are rules such as rules 18, 22, 30, and 126 which produce space-time patterns characterized by an apparently random distribution of inverted triangle structures of varying sizes . One of the early problems in studying these and similar rules was to characterize the distribution of the inverted triangles, and determine the fractal dimension of the patterns generated. These rules fall into Wolfram 's class III, "chaotic" rules. Rule 30 has also been suggested as a possible random sequence generator [501. Rule 60 also shows a distribution of triangles. Initially the appearence of these inverted triangles was likened to a distribution induced by statistical fluctuations. Rule 60 is additive, however: if g and g' are given as distinct configurations then rule 60 acting on their sum is equal to the sum of rule 60 acting on each configuration independently. Further, it is known that starting from an initial configuration containing only a single 1, rule 60 will generate the mod(2) Pascal triangle. Thus the action of this rule on any configuration can only give an XOR superposition of mod(2) Pascal triangles. In the figure shown, this fact shows up in the eventual synchronization which occurs at the bottom, due to the use of periodic boundary conditions. Finally, the figure for rule 54 shown an apparent Brownian motion of particles against a background. Analysis of this rule [51] has verified this interpretation by identifying the background, and two distinct types of "particles."

8 The most widely studied rule is rule 90, which is just the square of rule 60. It is the simplest non-trivial 3-site additive rule (rule 60 is really a 2-site rule in disguise). The best known of the non-additive rules is rule 18, which has been shown to reduce to rule 90 on the invariant subset of configurations space consisting of all sequences containing only isolated 1's separated by an odd number of 0's [51,52]. Two distinct points of view can be taken with regard to the order generating properties of cellular automata, and these different viewpoints lead to different questions, and different choices of analytic tools. Many studies of cellular automata have concerned themselves with analysis of statistical features of their output, and have at least implicitely used an analogy to statistical and thermodynamical systems. Some times this is very explicit, as in Wolfram's paper "Statistical mechanics of cellular automata," [27] which contains the statement: "The two dimensional picture formed by the succession of configurations in time is characteristically peppered with triangle structures. These triangles are formed when a long sequence of sites which suddenly all attain the same value, as if by a fluctuation, is progressively reduced in length by 'ambient noise."' As already indicated, however, the appearence of order in these rules is known to be completely deterministic. Although statistical mechanical and thermodynamical analogies have been often used in studies of cellular automata, and have proved extremelt useful, the better analogy is perhaps to deterministic chaotic systems'. There is a similarity, for example, between the long term unpredictability of chaotic systems and the undecidability results of K. Culik, L.P. Hurd, & S. Yu [53] which show that membership in behavioral classes defined on the basis of long term behavior is undecidable. To be more precise, Culik & Yu [54] define four classes of cellular automata. For 1-dimensional binary cellular automata which map the neighborhood consisting of all O's to 0, these are defined by the four conditions: 1. All finite configurations evolve to the configuration of all O's. 2. All finite configurations evolve to an ultimately periodic configuration. lIn the second view, the order generated by cellular automata pre-exists in the particular evolution rule, and is a manifestation of symmetry breaking induced by the choice of initial configuration.

9 3. For any pair of configurations c l and c2 is is decidable whether c l evolves to c24. All other cellular automata. For an arbritrary given cellular automaton, however, membership in any of these classes is undecidable. Thus, the general long term behavior of cellular automata, starting from a given finite initial configuration, has about it an undecidability which is comparable to that found in chaotic systems, for which the long term evolution is unpredictable. This book deals with work which has been carried out over the past eight years on certain aspects of both the forward and inverse problems. As indicated in the title, the emphasis in this book is on the development of various analytic and computational tools and methods for analysis of cellular automata. Chapter 1 introduces the basic definitions and notation , discusses two different forms of representation, defines symmetry equivalence classes of cellular automata, and indicates the way that cellular automata rules define maps of the unit interval. Chapter 2 introduces a division algorithms for cellular automata, and studies their residue arithmetic. Chapter 3 presents a method for finding fixed points and shift cycles, and gives a review of an invariance matrix method developed by Jen [551 to count fixed points and cycles. In chapter 4 a method for computing commutator sets of specified rules is derived. This also allows an understanding of a result due to H. Ito [56], which in the formalism introduced in the chapter, generalizes to shed light on the question of reconstructability raised by Crutchfield [571. Chapters 5, 6, 7, and 8 deal with additive rules . In chapter 5 the conditions for additivity of a rule are formulated in terms of an "obstruction matrix." This is followed by a representation of additive cellular automata rules in terms of complex polynomials, a formalism which allows simple conditions to be written for the injectivity of additive rules . In chapter 6 results of Martin, Odlyzko, & Wolfram [33], and of Jen [35] relating to cycle periods and reachability are reviewed, and a method for finding conditions for states to lie on cycles is derived. The chapter concludes with a review of some results on entropy reduction. Chapter 7 is devoted to a method for computing predecessors for arbritrary configurations under arbritrary additive rules. Chapter 8 presents a detailed analysis of the binary difference rule, which is equivalent to a discrete derivative with respect to sequence index.

10 Chapter 9 deals with the question of computing pre-images for finite sequences when the evolution rule is not additive. A matrix technique is introduced which allows such computations in arbritrary cases, and connections are shown to Jen's work on recurrence relations and enumeration [34,35]. Chapter 10 considers the "Garden-of-Eden" of a cellular automaton, that is, the set of configurations which have no predecessors under the given rule. Particular results are derived for the set of all 3-site rules, and it is shown that this provides a means of classifying rules. Chapter 11 considers the use of cellular automata to generate time series. Methods are presented for determining if a given cellular automata could have generated a specified time series, and if so what are the possible initial configurations. This method is statistically unbiased, however, only in the case of rules having an empty Garden-of-Eden. Finally, Chapter 12 considers the question of surjectivity, which is equivalent to having an empty Garden-of-Eden. A number of other properties equivalent to surjectivity are investigated, and methods are introduced for determination of whether or not given rules are surjective. Acknowledgements The origin of this work can be traced to a 1987 conversation with Ralph Abraham at the University ofCalifornia, Santa Cruz. At that point I was looking for new research problems, and mentioned that neural networks seemed interesting. Ralph replied that if I wanted to study neural networks, it would be a good idea to first study cellular automata. I took that advice, became fascinated, and this book is the result. Portions of the work reported here have been published in various journals over the past eight years. The canonical basis operators of chapter 1, together with their use to determine fixed points and shift cycles as described in chapter 3, and the results reported in chapters 9 and 10, were published in Physica D. The work reported in chapter 7 was published in a paper in Communications in Mathematical Physics for 3-site rules , and a subsequent paper in Physica D for the general case . The division algorithm and residue arighmetic given in chapter 2; the commutation computations of chapter 4; the use of complex polynomials to determine injectivity of additive rules in chapter 5; and the analysis of the binary difference rule given in chapter 6 were originally reported in Complex Systems. The results on entropy

11

reduction given in chapter 6 were reported in the International Journal of Theoretical Physics . References to these papers are given in the biblography. Needless to say, the work reported here has benefited from conversations and interactions with many others . As already indicated, it was originally stimulated by Ralph Abraham , and further encouraged during four months of a sabbatical which were spent in Santa Cruz . The method given for computing predecessors in chapter 7 grew out of a conversation at the Santa Fe Institute with Wentian Li and Mats Nordahl. Much of the material in chapter 12 was first presented as a seminar at the Santa Fe Institute. The work in chapter 5 dealing with Ito relations , and much of the work in chapter 11 was stimulated by a number of conversations with Jim Crutchfield and Jim Hanson . The work in chapters 7, 9, and 10 was reported at a meeting in June , 1992 in Oberwolfach , Germany. Conversations there with Hans Otto Peitgen and Fritz von Haeseler were stimulating and the meeting itself was wonderful . Correspondences with Harold McIntosh have shed light on a number of issues relating to uses of the de Bruijn and subset diagrams discussed in chapter 12, and some of the results reported in chapter 4 were suggested in conversations with Erica Jen. This work was also assisted by a number of summer research assistants . Scott Bradshaw ( 1991 ), Winslow Lacesso ( 1992), Hong Nugyun and Marcus Molenda ( 1993), and Robert Newton ( 1994), all of whom made siginficant contributions . Assistance in programming was given by Ron Haukenfrers and John Ramsden of Athabasca University Computer Services. Completion of this book has been assisted by an Athabasca University Presidents Award for Research and Scholarly Excellence which provided the time to finish it close to the editorial deadline . Financial support has been provided by a 1989 grant from the Athabasca University Academic Research Committee , and by operating grant OGP - 0024871 from the National Science and Engineering Research Council of Canada.

12 Biblography 1. Voorhees , Burton Cellular automata , Pascal' s triangle, and generation of order. Physica D 31, E1988) 2. Voorhees , Burton . Predecessor states for certain cellular automata evolutions . Communications in Mathematical Physics 117 , ( 1988). 3. Voorhees , Burton . Entropy of additive cellular automata. International Journal of Theoretical Physics 28 , No. 11 , ( 1989). 4. Voorhees , Burton . Period multiplying operators on integer sequences modulo a prime . Complex Systems 3, (1989). 5. Voorhees, Burton. Nearest neighbor cellular automata over Z2 with periodic boundary conditions . Physica D 45, (1990). 6. Voorhees, Burton . Division algorithm for cellular automata rules. Complex Systems 4, (1990). 7. Voorhees , Burton . Geometry and arithmetic of a simple cellular automata . Complex Systems 5, (1991). 8. Voorhees , Burton . Symmetric group modelling of visual information. Il Nuovo Cimento 106B No . 10, (1991). 9. Voorhees, Burton . Determining fixed points and shift cycles for nearest neighbor cellular automata. Journal of Statistical Physics 66 Nos. 5/6 (1992). 10. Voorhees, Burton . Predecessors of cellular automata states : I. Additive automata. Physica D 68, (1993). 11. Voorhees, Burton. Predecessors of cellular automata states : II. Preimages of finite sequences . Physica D 73, (1994). 12. Voorhees , Burton & Bradshaw , Scott. Predecessors of cellular automata states : III. Garden- of-Eden classification of cellular automata rules. Physica D 73, (1994). 13. Voorhees , Burton . Commutation of cellular automata rules. To appear in Complex Systems. 14. Voorhees , Burton. A note on the injectivity of additive cellular automata . To appear in Complex Systems.

13 Graphs of Space-Time Output for Sample CA Rules Rules generating figures shown are nearest neighbor with periodic boundary conditions on a 128x128 lattice. Rule 46: Emulates left shift (Complimentary rule 116 emulates right shift). Rules 72 and 172: Emulate identity Rules 184: White "particles" travelling right, black "particles" travelling left against checkerboard background. Anihilation bitwise at collisions with "thickest" particle winning. Rule 226: Mirror image of rule 184. Rules 18, 22, 30, 126: Apparently random patterns of inverted triangles of varying sizes. Rule 60: Superposition of copies of mod(2) Pascal's triangle. Rule 54: Apparent "Brownian motion" of particles against a background.

14

Time Sequence Rule 46

15

Time Sequence Rule 116

16

Time Sequence Rule 72

17

R

vr

rr

IF

Time Sequence Rule 172

OF

OF

IF

V

r

18

Time Sequence Rule 184

19

Time Sequence Rule 226

20

Time Sequence Rule 18

21

Time Sequence Rule 22

22

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Time Sequence Rule 30

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Time Sequence Rule 126

24

Time Sequence Rule 60

25

Time Sequence Rule 54

26

Chapter 1 The Operator Algebra of Cellular Automata Cellular automata can be specified in a variety of ways: as local neighborhood maps, as global maps of a configuration space, as Boolean equations implementing truth conditions, as arithmetic recurrences, as finite replacement schemes, as maps of the interval, etc. This chapter provides an introduction to some of the standard forms in which cellular automata are presented. It also introduces the idea of representing the global map defined by a cellular automata in terms of abstract components defined with respect to a canonical basis. 1. Basic Definitions and Notation Cellular automata are discrete symbolic dynamical systems defined in terms of a lattice of sites, L; an alphabet of symbols, K; and an evolution rule X, which maps configurations at any given time t to new configurations at time t+1. A configuration, or state, is an assignment of a symbol from K to every site of the lattice L. The set of all possible configurations is the state, or configuration space, which will be denoted E in the generic case. Given a configuration µ(t) the evolution rule generates a new configuration µ(t+1) by synchronically assigning to every site of L a symbol choosen from the alphabet K on the basis of the symbols in a neighborhood of that site in the given configuration. In a fundamental paper, published in 1968, Hedlund [49] has shown that cellular automata are the shift commuting endomorphisms of the shift dynamical system. In all that follows, the lattice is taken as either a finite set of n sites located on the circumference of a circle, or as a right half-infinite one dimensional array of sites. The first case gives what Jen [55] calls a cylindrical cellular automata, because its evolution can be visualized as taking place on a cylinder. The configuration space in this case will be denoted En. It consists of all periodic sequences of symbols with period a divisor of n. In the second case, the configuration space is the set of all right half-infinite symbol sequences, and is denoted E+. When the configuration space is taken generically, it will be denoted by E. All considerations in this book will be restricted to binary cellular automata, for which the alphabet is the set K = 10, 1).

27 The view taken is that cellular automata are defined in terms of particular kinds of operators on configuration space. Configurations will generally be represented by lower case Greek letters , and operators by upper case Latin letters. If µe E and X:E->E then µi (1
28 The local map fX will be denoted by the same symbol, X, as the global rule which it defines. Thus fX(iO...ik-1) is written as X(i0...ik-1). The set {i0...ik_11 ise {0,1}}, listed in ascending numerical order, will be called the standard k-site neighborhood list. If X is the operator representing a k-site rule then the components of X with respect to the standard k-site neighborhood list are defined as xi = X(i0...ik_1) (1.1) where i is the denary (base 10) form of iO...ik_1. The expression X = (xOxl...x2k_1) is called the component form of X. As a general notation j and jO...jn will indicate respectively the denary (base 10) and binary forms of the number "j". Wolfram [27] introduced the practice of numerically labeling rules in terms of their components'. If X = (xoxl...x2k_1), then the numerical label assigned to Xis given by 2k_1 xs2s

N(X) _ I

(1.2)

S=0

If X is a 3-site rule, for example, then N(X) = x0 + 2x1 + 4x2 + 8x3 + 16x4 + 32x5 + 64x6 + 128x7 The component form of X is just the standard representation of X in terms of its local action on neighborhoods. It is important, however, to allow the entries in the component form to be variables. This allows questions to be posed in terms of finding rules for which certain conditions, specified in terms of the components of those rules, are satisfied. The left shift operator on E is defined by [a(g)]i = µi+1 (1.3) The right shift is defined as gi_1 Periodic boundary conditions [a 1(µ)]i = 0 if i = 1 (1.4) Left justified rules µi-l

if i > 0

'Wolfram lists neighborhoods in descending numerical order, so the listing of rules by components used in this book is the reverse of that used by Wolfram. For example, Wolfram gives rule 30 as 00011110 while in the notation of this book rule 30 is 01111000. The present convention is used as a means of emphasizing the connection between CAs and maps of [0,1] since each neighborhood can be viewed as equivalent to a sub-interval, as briefly discussed in section 4 of this chapter.

29 A k-site CA rule is said to be additive if it can be represented as a sum of shifts: k-1 X = a-r I asas (1.5) s=0

If neighborhoods are left justified in (1.5) then r=0. For symmetric rules r = (k-1)/2. A rule is linear if it can be written in the form k-1 X=a1+cr'laws (1.6) s=0 where 1 is the rule that maps all neighborhoods to 1. If a = 0 the rule is homogeneous. Otherwise it is inhomogeneous . Comparison with equation (1.5) shows that the k-site additive rules are just the k-site linear homogeneous rules.

Table 1.1 lists the additive 3-site rules in nearest neighbor form. Their expression in the form (1.5), and the symbols which will be used to denote these operators are also given. The rule that maps all neighborhoods to 0 is denoted 0, and the rule that maps all neighborhoods to 1 is denoted 1. This latter is not an additive rule since 1(µ+µ') =.1 while 1(µ)+1(µ') = 1+1= Q. The left justified form of these rules is obtained by multiplying each rule by the left shift a. Rule

Symbol

Components

Shift Form

0 240

0 a-1

00000000 00001111

0 0-1

right shift

204

I

00110011

I

identity

170 102

a

01010101

a

left shift

D*

01100110

I+0

60

D

00111100

I+c-1

90

S

01011010

c+0-1

150

A

01101001

I+a+a-1

Table 1.1 Nearest neighbor form of the 3-site Additive Rules

2. Boolean Forms of CA Rules Cellular automata can also be defined in terms of multinomial Boolean functions. If X:E-E is a k-site rule then Xis represented by a Boolean

30 function in the variables is (0:5s:5k-1) which is of degree at most k. Given the component expression for a rule, there is a simple proceedure for construction of its Boolean form in terms of the AND and XOR (exclusive or) operations. These are represented respectively by multiplication (ab = a AND b) and binary addition (a+b mod(2) = a XOR b). Given a one dimensional CA rule X:E-+E, written in its component form X = (xOxl...x2k_1), the Boolean expression for this rule is constructed as follows:

Algorithm 1.1: 1. For each iO...ik_1 in the neighborhood list write X(iO...ik-1) = A(iO...ik-2)ik-1 + Y(iO...ik-2) mod(2) with Y(iO...ik-2) = x2i, where i = iO...ik_2, and A(i0 ••ik-2) = 0 x2i = x2i+1 (1.7) 1 x2i # x2i+1

2. If the components of A are all ones, return to step 1 for the rule Y. If the components of A are not all ones , apply the decomposition of step 1 to both A and Y. 3. Continue to iterate this process until the rules A and Y found depend only on a single variable. At this point the expressions for A and Y can be easily determined. 4. Recombine the expressions found in this process to obtain the Boolean expression for the rule X. As an example, the Boolean form for the 3-site rule 18 will be computed. The rule table for rule 18 is 000 001 010 011 100 101 110 111 0 1 0 0 1 0 0 0 Taking X(iOili2) = A(i0il)i2 + Y(i0il) mod(2), application of step 1 of the algorithm yields the rule tables for A and Y as 11 10 00 01 A

1

0

1

0 = (1010)

1 0 = (0010) Y 0 0 Repeating the proceedure for Y gives Y(i0i1) = A'(iO)i1 + Y(i0) mod(2) with both A' and Yequal to (01). Hence X(i0) = Y'(ip) = i0. Applying the proceedure to A(i0il) = A"(iO)il + Y'(i0) mod(2) yields A" = Y" = (11) = 1. Hence the Boolean expression for rule 18 is given by X(i0ili2) = [A"(i0)il + Y"(iO)li2 + [A'(i0)il + Y'(i0)] mod(2)

31 = (1+i1)i2 + ioi1 + i0 =(1+il)(iO+i2) Boolean forms computed in this way for all two and three site rules are given in Appendix 1. Note: In order to simplify notation, in the future all sums of (0.1)-sequences. CA configurations. or CA rules will automatically be taken mod( 2) unless otherwise specified. 3. The Canonical Basis Operators Let {i0 ...ik-1} be the k-site neighborhood list with i the denary form of i0...ik- 1, and define a set { vk)) of 2k k-site rules by (i)

1 i = j (1.9) J 10 That is , vk) is the global rule having local form that maps the neighborhood i = i0...ik- 1 to 1 and all other neighborhoods to 0. The numerical label for vk) is 2i and it is clear that the global operator X for every k-site rule can be written as a linear combination of these basis operators: 2k_1 X = xivk) i=0 where xi is the i-th component of the rule X.

(1.10)

As a matter of convienience, a different notation will be used for the vk) when k = 3, as indicated in Table 1.2. 000

001

010

011

100

101

110

111

(0) v3

(1) v3

(2) v3

(3) v3

(4) v3

(5) v3

(6) v3

(7) v3

K

n+

t

R-

a

P+

X

il+ Table 1.2

Notation Used for 3-Site Basis Operators

The set { vk) } provides a basis for the non-linear algebra of k-site CA rules . The rule form given in equation (1.10) is just the operator form of Wolframs numerical labeling, as given in equation (1.2). It will be called the canonical form of a rule. Canonical forms are listed in Appendix 2 for all 3site rules.

32 The canonical form of a rule X can be determined either directly from the component form of the rule, or from its numerical label N(X) by expressing this as a sum of powers of 2. Canonical representations can be added directly, with coefficients reduced mod(2), and the component forms of rules can be added component-wise, mod(2). A k-site rule X = (xOxl...x2k-1) can always be extended to a k+m site rule. To do this, the generic k-site neighborhood iO...ik_1 is mapped to the set of k+m site neighborhoods Ni = fr0...rs-1i0...ik-IJO.- jm-s-1 } for arbritrary s and m-s digit blocks ro...rs_l and jO...jm_s_1. Every neighborhood in the k+m site neighborhood list which belongs to Ni is assigned the value X(i0...ik-1). This defines the (s,m-s) extension of X. The (0,m) and (m,0) extensions are called respectively the m-site right extension of X, and the m-site left extension of X. If m = 1 the terms right and left extension will be used. The mapping site remains fixed during extension. As an example, the right and left extensions of the 2-site binary difference rule (0110) are, respectively, (00111100) (rule 60) and (01100110) (rule 102). Since each of the canonical basis operators is associated with a single neighborhood, it is possible to consider the way in which the set of k-site basis operators maps to k+1 site basis operators under extension. This mapping is indicated in Figure 1.1.

Figure 1.1 Hierarchy of Canonical Basis Operators for 1 , 2, and 3-Site Neighborhoods This figure shows the form of a binary tree. At each level, each of the k-site basis operators is at the top of its own binary tree. Thus, if X is a k-site CA rule with left justified neighborhoods, each coefficient xi in equation (1.10) selects out the binary tree headed by vk). The (0,m) right extension of X to

33

k+m sites is then obtained by associating the coefficient xi to every basis operator in the intersection of the set { vk+m 10< _ r<_ 2k+m-1} with the tree headed by vk). 4. Symmetry Transformations on CA Rules The set of all CA rules (CA rule space) can be partitioned into equivalence classes defined in terms of symmetry transformations on the canonical basis operators, or equivalently, on the neighborhood lists. The three operations which can be defined are: T1X(i0...ik-1) = X(ik-1...iO) Right/Left Reversal T2X(9) = X(1+9) State Complementation T3X(9) =1+X(µ) Rule Complementation The notation is that (1+B)i = 1+Bi. Li and Packard [58] have used the operators T1 and T3T2 to partition the 256 3-site rules into 88 equivalence classes. Use of all three rules yields 47 equivalence classes. In terms of components these transformations are given by [T1X]i = xj with js = ik-s-1 [T2X]i = x2k-i-1 [T3X]i = 1+xi mod(2) (1.11) The transformation T3 in equation ( 1.11) applies to CA rules, but it could equally well apply to CA states as binary complementation in the form: T3(9) = 1+µ. With this interpretation , the relationship between T2 and T3 can be expressed in the equations (T2X)n = (XT3)n (T3T2X)n = 1 + T2Xn (1.12) A. Wuensche & M. Lesser provide a good discussion of symmetry transformations in their atlas of basins of attraction [43]. 5. CA Rules as Maps of the Interval The left justified form of every CA rule defined on E+ defines a map of the interval [0,1]. Let f: E+-+[0,1] be given by f(µ) = Jµs2-s s=1

( 1.13)and for a rule X :E+-E+ define X:[0,1]-4[0,1] by X(f(4)) = f(X(µ)).

34 Note that while Xis a function on E+ it is not generally so on [0,1] because, while f(91...11n0i) = f('i..•gn1Q), this equality does not hold for most CA rules. Instead, X:[0,1]-*[0, 1] will usually be double valued on all points of [0,1] having the form K/2s. The condition that X:[0,1]-.[0,1] be a function is that all blocks of k digits satisfy X(i0...ir00 ... 0) = X(i0...ir01...1) for all iO ...ir with 05x- 0 with equality if and only if g =.Q. Thus it is only necessary to show that f satisfies the triangle inequality: f(µ+µ') <_ f(µ) + f(µ'). But f(µ+µ') is given by

f(µ +µ') =

1((4s

+ µ's) mod(2))2-s = 1 µs 2-s + I µ's 2 S - 2 1 µsµ'8 2-s s=1 s=1 s=1 s=1

= f(µ) + f(µ') - 21 µ8µ's 2-s s=1 and the final term is negative unless µs * µ's for all s, in which case it is 0, µ+µ' = 1 and fU) = 1.1 The figures 1.2 at the end of this chapter show graphs of several of the nearest neighbor rules which are considered in this book, for fixed boundary conditions at the left boundary of E+. Two forms of the graph are shown, a point form which emphasizes the fractal distribution of geometric forms, and a line form which has a more standard appearence. A metric on E+ can then be defined by taking g(µ,µ') = f(µ+µ'). If the set {vk)} of k-site basis operators is thought of in terms of its defining neighborhoods, it gives a uniform partition of the interval [0,1] into 2k sub-intervals. Each set {vk'+m } then defines a refinement of this partition. Thus, the set {vk^ I 1<- k < oo} provides a basis for the neighborhood topology on [0,1]. If the state space is En then the Hamming distance between states is defined as the number of Ot->1 interchanges required to transform one state into the other. More formally, if µE En let #(µ) be the number of 1's contained in µ. The Hamming distance between states µ and µ' is then defined as #(µ+µ') where the addition is mod(2) in each coordinate position. The

35 maximum possible distance occurs when µ'i = 1+µi mod(2) for all i. The Hamming distance is often used to measure the distance between CA rules. 6. Exercises for Chapter 1. Compute the Boolean form for the following 4-site rules: a) 0010110100001111 b) 0011100100110011 c) 0011001101100011 d) 0000111101001011 2. Write out the canonical forms of the 3 -site rules listed below: 18, 22 , 30, 54 , 60, 90 , 108, 110, 120, 150, 170, 204 , 232, and 240 3. Compute the sums of the following 3 -site rules: Rule 18 + Rule 72 Rule 18 + Rule 54 Rule 22 + Rule 90 Rule 43 + Rule 201 Rule 180 + Rule 47 4. Compute the group table for the set of sixteen 2-site rules , with the group operation of component -wise binary addition. 5. Find the (2,0), (1,1 ), and (0 ,2) extensions of the 2-site rules (0110), (0010), and (0111). 6. Partition the set of 2-site rules into equivalence classes based on the T1, T2, and T3 symmetry operations. 7. Prove the validity of equation (1.12) 8. Make use of the formula for a geometric series a(1)n = a

n=0 r 1-r in order to compute the values in [0,1] of g , X(µ), X2(µ), and X3 (µ) for 3site rules 18, 54 , 60, 90, 150 , and 170 , for the following periodic configurations is E+: a)µ=201 b)µ= 10011 c)µ= 100110 9. Compute the distances between µ, X(µ), X2 (µ), and X3 (µ) for the rules and configurations given in exercise 8.

36 10. Sketch a graph of D[0,1]e [0,1] where D is the 2-site binary difference rule (0110 ). (Hint: Partition [0,1] into blocks and assign each block the value of D acting on the binary string which characterizes the left end point of the block.)

37

Figures 1.2 Graphs of 3-site rules 18, 22, 30, 46, 54, 90, 110, 150. The graph of rule 60 is shown in Chapter 8. Graphs are for nearest neighbor rules acting on 12 digit sequences representing points of [0,1] with fixed end point. That is, each sequence has the form 0.µ1µ2...µ12 so the first digit in the image sequence is the X(041µ2)• Use of periodic boundary conditions will change form of graphs. Each graph is shown in two forms. In the first only points on the graph are plotted. In the second these points are connected.

38

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42

Chapter 2 Cellular Automata Arithmetic This chapter introduces a division algorithm , which for given rules Q and X, allows determination of rules A and R such that Q = AX+R, with R being as small as possible . Some results in the arithmetic of residues are also derived. 1. Products of CA Rules An important tool in the analysis of CA rules is the list obtained by applying a k-site rule to the k+m site neighborhood list. By definition, if a ksite rule X is applied to the k-site neighborhood list, the result is just the component form of that rule. When applied to the k+m site neighborhood list the result is a list of length 2k+m, of m+1 digit binary numbers, which will be denoted L(k,m;X). The i-th element of L(k,m;X) is given in terms of the components of X by Li(k,m;X) = xioxil ...xim

(2.1)

where the subscripts is are dete rmined by (2 . 2) 05 s _ m 2m mod(2k) =[ sI where i is the denary form of iOi1...ik-1, the i-th term in the k+m site neighborhood list, and Fri indicating the greatest integer less than or equal is

to r. Recalling the discussion of the extension of a k-site rule to a k+m site rule, it is easy to see that the following is true: Lemma 2.1 The rule obtained by reading the j-th digit of each entry in L(k,m;X) is the (j,m-j) extension of X. Note that reading the j-th digit of the k-site neighborhood list gives the components of aj-r, where r is the designated mapping site for neighborhoods. If x is a binary variable then the binary compliment of x is defined as the variable x' = 1+x mod(2). The complete Boolean product (also called the minimal product) of a set of binary variables is the set of products of these variables having the property that each element of the set contains all variables, either in primed or unprimed form, and the set is exhaustive with respect to this property. For example, the complete Boolean product of the set {xO,x1) is the set {x'Ox'1,X Ox1,xOX 1,xOx1).

43 The list L(k,m;X) contains 2k+m entries , Li(k,m;X), 05 i < 2k+m, each a product of m+1 of the components of the rule X. Each entry can be expanded as a complete Boolean product which contains 2m+1 elements. The elements of these products are labeled with an index j = jo...jm such that the xs term in the j-th element is primed or unprimed as is is 0 or 1 respectively. The 2k + mx2m+ 1 matrix with ij element given by the j-th element of the complete Boolean product of Li(k,m;X ) will be denoted X(k,m). X(k,0 ) consists of two 2k digit columns, the first containing the primed and the second the unprimed components of X. Since only one member of a Boolean product is 1 while all others are 0, each row of X(k,m) contains a single 1 . For example, if X = (xOxlx2x3) is a 2site rule then , noting that for all cases xx' = 0, X (2,0) and X(2,1) are given by 0

(XI I

0

x0x0

xoxi xO xl xoxi xoxi xix2 x ix 2 xlx2 x lx 2 X(2,0) =

X(2,1) =

x1xg xlx3 xlx'3 xlx3

x2xo x2xi

x2x0 x2x'0 x2x0

(2.3)

x2xi x2xi x2x1

xgx2 x3'x2 x3x

2

x3xg

x3x3

0

0

x3 x2

The matrix X(k,m) can be used to compute the components of the composition product of two rules. If X is a k-site rule and A is an r-site rule, then AX will be a k+r-1 site rule. In constructing the composition of the rules A and X, the list of k+r-1 site neighborhoods is first acted on by X , and the resulting list L(k,r-1;X) of r-site neighborhoods (which is not, in general, the full r-site neighborhood list) is acted on by A. By definition , L(k,r-1;X) consists of a list of 2k+r-1 r- digit neighborhoods on which A will operate in construction of the product AX. It is not difficult to see that this action is given by the second column of the matrix product (AX)(k+r- 1,0) = X(k,r- 1)A(r,0) (2.4) (It is easier, however, to use a = (a0al ... a2-kl )T and write equation (2.4) in the form AX = X(k, r-1)a, bearing in mind that this is now a column vector.) This is illustrated in Table 2 . 1 for the case in which both A and X are 2-site rules. The first column lists the eight 3-site neighborhoods. The second column gives the operation of a 2-site rule X on these neighborhoods,

44 and the third column shows the operation of a 2-site rule A on the 2-site neighborhoods defined in the second column in terms of the components of the rule X. The third column is constructed by considering how A acts on each 2-site neighborhood, and multiplying the as component by the combination of primed and unprimed terms which will yield the neighborhood which maps to as. iQili2 X(iQi1i2 ) A(X(iQiili2)) 000 xOxO aox' Ox'O + a3xO xO 001 xOxl aox'Ox'l + alx'Oxl + a2xOx' l + a3xOxl 001 xlx2 aox' lx'2 + alx' lx2 + a2xlx'2 + a3xlx2 001 xlx3 aOx' lx'3 + alx' lx3 + a2xlx'3 + a3xlx3 001 x2x0 aOx ' 2x'O + alx'2xO + a2x2x' O + a3x2xO 001 x2xl aOx'2x'l + alx'2xl + a2x2x' l + a3x2xl 001 x3x2 aOx'3x'2 + alx'3x2 + a2x3x '2 + a3x3x2 001 x3x3 a0x'3x 3 + a3x3x3 Table 2.1 Action of AX on 3-Site Neighborhoods

2. The Division Algorithm The Matrix X(k,m) can also be used to derive a division algorithm for CA rules. The algorithm is best illustrated by the simplest case, in which a 3site rule Q is divided by a 2-site rule X. That is, given Q and X, the task is to find rules A and R such that Q = AX+R, with R being as small as possible in some yet to be defined sense. Before carrying out this division, some remarks on neighborhood structure are necessary. Since Q and X are given, their neighborhoods are known, as is the designated mapping site. The rule R will have the same neighborhood structure as Q, but the neighborhood structure for the rule A needs to be choosen in such a way that the A and X neighborhoods, taken together, cover the Q neighborhood, preserving its mapping site. Thus, if Q is a nearest neighbor rule, and the neighborhoods for X are {ili2} then the mapping site for X must be il, and A must have neighborhoods {ioil} with it the designated mapping site. If all rules are left justified, no difficulties of this nature arise.

45

The proceedure will be to look for a 2-site rule A (with the appropriate neighborhood structure ) such that Q = AX+R, and then to minimize R. From Table 2. 1 the action of AX on 3-site neighborhoods is known. The components of A are now to be choosen so as to fit the entries in the final column of this Table as closely as possible to Q = (qoqlq2q3q4q5q6q7). Taking q as the column vector with components ( goglq2q3q4q5q6q7)T, consider the equation X(2, 1)A(2,0) = q. Let ci be the i -th column vector of X(2,1), and cTi its transpose . Then, since each row of X(2,1) contains only a single 1 , the scalar product is given by = njsij (2.5) where Sij is the Kronecker symbol and nj is the number of l's contained in cj. Thus , multiplication of X(2 , 1)A(2,0 ) = q by the transpose XT(2, 1) yields the set of equations

niai = (2.6) and the term on the right in this equation is just the number of 1's which ci and q have in common. The algorithm for choice of the ai , and the remainder R is now defined as follows: Algorithm 2 . 2 (Division) 1. If = 0, set ai = 0 2. If ni = * 0, set ai = 1 3. If ni ^ ;e 0 then: a. If < ni/2 set ai = 0 b. If >_ ni/2 set ai = 1 4. If a component ai is not contained in equations (2.6) set it to 0 5. Determine the rule R by R = Q+AX mod(2). This algorithm minimizes R in the sense that the components of R contain the fewest possible number of 1 's, although as will be seen later, there is an indeterminacy which arises resulting in the need to define larger equivalence classes than just residue classes. As an example of the division algorithm , the nearest neighbor rule 90 will be divided by the 2 -site rules (0110) and (0010 ). Rule 90 has been studied extensively [e.g., 33 ,59,60 ,61,62] as one of the simplest additive rules. For rule 90, q = (01011010)T, and for X = (0110)

46

(1 0 0 0) 0

1 0 0

0

0 0

1

0 0 1 0 X(2,1) = 0 0 1 0 0 0 0 1 0

1 0

(2.7)

0

1 0 0 0 so that equations (2.7) become 2a0=0 2a1=2 2a2

=

2

(2.8)

2a3 = 0 This indicates that A = (0110) as well. Note, however, that if rule 90 is taken in nearest neighbor form then the neighborhoods for X are {i 1i2} while for A they are {ioil}. The question of neighborhood choice shows up, for example, in the fact that the identity operator for {ili2} neighborhoods is (0011), while (0101) is the identity operator for {i0i1} neighborhoods , and the nearest neighbor identity, (00110011) is the product of (0101 ) and (0011). With this cavet, the 2-site rule D = (0110) is the square root of rule 90. (With left justified neighborhoods, the binary difference rule D is exactly the square root of rule 90 .) Note that D is the binary difference rule, with 3-site extension given by rule 60 for {ioil } neighborhoods, and by rule 102 for {ili2} neighborhoods (see Table 1 . 1 for expressions of these rules). If X = (0010) then 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 X(2,1) = 0 0 1 0

(2.9)

0 0 1 0 0 1 0 0 1 0 0 0 and equations (2.6) become 4a0 = 2 gal=1

(2.10)

47

2a2 = 1 Application of the division algorithm gives A = (1110). Computation now yields AX = (11111111), so there is a remainder, given by (10100101); i.e., rule 165. Thus rule 90 is the product of the rules (1110) and (0010) with a remainder given by rule 165. This second example serves to illustrate a significant point. In the division algorithm as given, ai is set to 1 if = ni/2. A remainder with an equal number of 1 components would be obtained, however, by taking ai equal to 0 in this case. When such a situation occurs, it will be called a case of equivocation. The choice made will be called positive equivocation, while setting ai to 0 for = ni/2 will be called negative equivocation. The possibility of equivocation has significant consequences for the arithmetic of residue classes. It should be clear that the division algorithm can be applied for any pair of rules X and Q so long as the neighborhood structure of Xis such that L(k,m;X) can be constructed. Even if Q is defined for neighborhoods of fewer sites than X, it can still be divided by X by first extending it to a rule defined for neighborhoods with more sites than X. The algorithm can also be generalized to rules defined over alphabets other than {0,1}, and to higher dimensional lattices. All that is required is the ability to construct the appropriate X(k,m) matrix, and choose the neighborhoods so that they overlap in the required way. 3. Residue Arithmetic of Cellular Automata If Q = AX+R, then by analogy with ordinary residue arithmetic, Q will be said to be congruent to R modulo X, written Q - R mod(X). Lemma 2.3 Congruence modulo X satisfies 1. Q Q mod(X) 2. Q = R mod(X) t= R Q mod(X) This asserts that congruence modulo X has the reflexive and symmetric properties. The possibility of equivocation, however, leads to cases in which transitivity does not hold. Consider, for example, rules Q, R, S, and X such that Q - R mod(X) and R = S mod(X). Then there are rules A and B such that Q = AX+R and R = BX+S. Thus Q = AX+BX+S = (A+B)X+S which formally looks as if Q - S mod(X).

48 Suppose, however, that Q = (00101110) while X = (0110). The matrix X(2,1) is given by equation (2.7) and application of the division algorithm yields A = (0111), R = (01010000). But it is easy to show that the rule R can be written as (01010000) = (0110)(0110)+(00001010). Thus, Q can be written as the combination Q = [(0111)+(0110)1(0110)+(00001010) = (0001)(0110)+(00001010) This equation is true , but it does not follow from the division algorithm unless negative equivocation is choosen. Further analysis indicates that each column of the matrix X(k,m) may or may not be equivocal for a given rule Q, and it is necessary to consider all possible combinations of equivocation. In order to deal with this kind of situation, the equivocation class of a rule Q is defined as the set of all rules congruent to Q under all possible choices of equivocation. Let c(X,m) be the number of columns of the matrix X(k,m) which do not contain all 0's. Let e(Q,X,m) be the number of columns of this matrix which are equivocal for Q = AX+R. Lemma 2.4 Let X be a given k-site rule. The number of k+m site rules which are congruent to 0 mod(X) is given by 2c(X,m) with 1<_ c(X,m) < 2m+1. Further, the set (Oil of all rules Q congruent to 0 mod(X) is a group with respect to component-wise binary addition. Proof If Q is congruent to 0 mod(X) then there is an A such that Q = AX. That is, the binary label of Q decomposes exactly into a sum of columns of the matrix X(k,m). The total number of columns in this matrix is 2m+1. Since c(X,m) is the number of non - zero columns in this matrix there are 2c(X,m) possible distinct combinations of columns. To see that {Oi} is a group note that Oi = AiX and there is no equivocation in this product. Thus Oi+Oj = (Ai+Aj)X which corresponds to another combination of the columns of X(k,m) and hence is also in [Oil. The zero element of this group is 0, which is always congruent to 0, and each element is its own inverse. I Lemma 2.5 Let rules Q and X be given, with Q = AX+R. The number of residue classes modulo X in the equivocation class of Q is given by 2e(Q,X,m), and for each residue class there will be 2c(X,m)-e(Q,X,m) distinct values of A.

49 Proof If the equivocation of Q when divided by Xis e(Q,X,m) there are e(Q,X,m) columns of X(k,m) which are equivocal. For each of these columns the corresponding component ai of A can be taken as either 0 or 1, without changing the total number of 1's in the remainder R. Each such choice determines a distinct rule A, and a distinct remainder. Thus the equivocation class of Q will contain 2e(Q,X,m) distinct residue classes . c(X,m)-e(Q,X,m) is the number of columns of X(k,m) which are not equivocal, so each residue class contains 2c(X,m)-e(Q,X,m) elements, each with a different value of A. I Theorem 2.6 Let X be a k-site rule, with {Oi I 05 i < 2c(X,m)} the set of k+m site rules which are congruent to 0 modulo X. Let Q be any given k+m site rule. Then the full equivocation class of Q modulo Xis the set Eq(Q) = {R I R = Q+Oi, 0<_ i < 2c(X,m)) Proof Q = A(Q)X+R and Oi = AiX. Hence Q+Oi = [A(Q)+Ai]X+R. Write A(Q) as U(Q)+E(Q) where U(Q) is the part of A(Q) for which Q is not equivocal and E(Q) is the part for which Q is equivocal. Note that the remainder R comes entirely from the E(Q)X contribution: a component of R will be 1 either to compensate for an extra 1 in E(Q)X, or to inclued a 1 which is contained in Q but not in E(Q)X. Also note that Oi is conpletely unequivocal. Thus, if addition of Oi to A(Q) changes only U(Q) there will be no change in the remainder R. If there is a change in E(Q), however, this will correspond to a change in equivocation. Since the set (Oil contains all possible combinations of non-zero columns of X(k,m), it will also exhaust the possible combinations of equivocation which can occur for Q. I Ai (0000) (0010)

Oi (00000000) (00001100)

(0100)

(00100010)

(0110) (1000) (1010) (1100) (1110)

(00101110) (11010001) (11011101) (11110011) (11111111) Table 2.2

3-Site Rules Equivalent to 0 Mod(0010)

50 As an example, consider the 2-site rule X = (0010). The X(2,1) matrix for this rule is given by equation (2.9). For this matrix c(X,1) = 3 so the set {Oi} contains eight members. These are listed in Table 2.2, together with the corresponding Ai rules. If all 3-site rules are divided by (0010) there are 31 additional equivocation classes . These are listed, in terms of their component residue classes and the associated A matricies, in Table 2.3. E(Q )

0000

0010 0100 0110 1000 1010 1100 1110

Zo

0

x

x

x

x

x

x

x

Zl Z2 Z3 Dl D2 D3 D4 D5 Bl B2 B3 B4 B5 El E2 E3 E4 E5 Cl C2 C3 Hl H2 H3 Fl F2 F3 Gi G2 G3

1 X X 16 X X 128 X X (4,8) X (5,9) X (20,24) X (68,72) X (132,136) X (2,32) (3,33) (18,48) (66,96) (130,160) (6,10,36,40) (7,11,37,41) (22,26,52,56) (70,74,100,104) (134,138,164,168) (17,192) (65,144) (80,129) (21,25,196,200) (69,73,148,152) (84,88,133,137) (19,49,194,224) (67,97,146,176) (82,112,131,161) (23,27,53,57,198,202,228,232) (71,75,101,105,150,154,180,184) (86,90,116,120,135,139,165,169)

X X X

X X X X X X X X X X X X X X X X X X

X X X

X X X X X X X X

X X X

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

x

X X X X X

X X X X X

X X X

X X X X X X

X X X

X X X

Table 2.3 Equivocation Classes for 3 -Site Rules Divided by X = (0010)

In this Table, an X in a given A matrix column indicates that that value of A is a coefficient in the equation Q = AX+R. The residue classes in each equivocation class are listed under Eq(Q), together with a lable for each equivocation class. For example, equivocation class D3 = (20,24) indicating that the residue classes of the 3-site rules 20 and 24 are contained in the

51

same equivocation class . By Theorem 2.6, there are eight rules in each equivocation class, hence there are four rules in each of the two residue classes in equivocation class D3. Thus there are four rules which are congruent to rule 20 mod(X) and four rules which are congruent to rule 24 mod(X). In addition, the values of A which appear in this equivocation class are (0010), (0110), (1010), and (1110). Rule 20 has component form (00101000) and rule 24 has component form (00011000). The rules in each residue class are listed in Table 2.4. Congruent to Rule 20

Congruent to Rule 24

(00100100) rule 36

(00010100) rule 40

(00000110) rule 96

(00110110) rule 108

(11011101) rule 187

(11000101) rule 163 (11100111) rule 231

(11010111) rule 237 Table 2.4

Rules Congruent to Rules 20 and 24 modulo X = (0010)

If rules Q and Q' are in the same equivocation class this will be indicated with Q = Q' emod(X). The emod relation is reflexive, symmetric, and transitive. Theorem 2.7 Let Q = R emod(X), R = S emod(X), and Q'-= S emod(X). Then 1. Q+Q' = R+S emod(X) 2. Q = Q emod(X) 3. Q= R emod(X) r* R u Q emod(X) 4. Q = S emod(X) The set of equivocation classes is a group under component-wise binary addition. There are also several sub-groups. For the equivocation classes of Table 2.3, the sets C = {C1,C2,C3}, (Z,C) = {Z0,Z1,Z2,Z3,C1,C2,C3}, and (Z,C,E,G) = (ZO,Z1,Z2,Z3,C1,C2,C3,E1,E2,E3,E4,E5,G1,G2,G3) are subgroups. If (Z,C) is taken as the identity, then {(Z,C),(E,G),(B,F),(D,H)) forms a group with group table isomorphic to that of {00,01,10,11}, while the set {(Z,C,E,G),(B,F,D,H)} forms a group with group table isomorphic to that of {0,1}. If a rule is not congruent to 0 modulo any X other than itself and the identity rule, they it will be said to be prime. In contrast to the case with integers, however, almost all CA rules are prime.

52 A rough estimate of the percentage of composite k-site rules can be gained by computing an upper bound on the number of possible products of rules which yield k-site rules . A k-site rule can be divided by any r-site rule for which 2:5 r < k. There are k-2 possible values for r, and each value defines 22r rules. Each r- site rule will multiply a k-r+1 site rule to give a ksite rule. Ignoring cases in which rules commute, or in which different r-site rules yield the same product , there are then kc,122r+2k-r+l

L^

(2.11)

r=2 possible products . On the other hand , there are a total of 22k k-site rules. This includes those rules which are the extensions of lower site value rules, but these are only a small portion of all the k - site rules . Dividing the sum in equation (2.11) by the number of k - site rules then gives a rough approximation of the percentage of composite k-site rules as -122k ( 2r-k+21-r-1)

(2.12)

rll==2 2 The actual number of composite rules will be less than this. For example , for r = 2 , k = 3 the value of equation ( 2.12) is 1, but in fact only 61 of the 3 - site rules are composite . Further, as k increases , the percentage of composite rules decreases rapidally . For example, for a 4 - site rule (2.12) is .125, while for a 5-site rule it is 2-11+2-16.

4. Exercises for Chapter 2 1. Write out the complete Boolean product of {xi0,xi1,xi2}. 2. Compute the component expression for the product AX where A and X are given by A

X

(0111)

(01001000)

(0110)

(01101000)

(0100)

(01111110)

(1001)

(00110110)

3. Prove Lemma 2.3. 4. Divide the given rules Q by the given rules X: Q X (01001000 ) (0111)

53 (01001000)

(0110)

(01111110)

(0110)

(01111000)

(1101)

(01111000)

(0010)

5. Compute the set of 3-site rules which are congruent to 0 mod(X) for X given by (0110) and (0111). 6. Compute the equivocation classes mod(X) for the 3-site rules below, with X given by the 2-site rules of exercise 5. a) 01001000 b) 01101000 c) 01101100 d) 01111110 e) 01011010 7. Compute the rules in the residue classes for each of the equivocation classes D4, B3, El, C3, and F2 as given in Table 2.3 8. Find all composite 3-site rules.

54

Chapter 3 Fixed Points and Cycles Two approaches to the study of fixed points and cycles (i.e., attractors) are given. The first involves separation of CA rules into linear and non-linear parts, allowing computation of conditions for a state to be a fixed point, or to lie on a shift cycle. The second approach extends work initiated by Jen on the ennumeration of fixed points and cycles, by providing a means of counting the number of fixed points and cycles a rule will have in terms of its components. 1. Fixed Points and Shift Cycles Via Rule Decomposition Considerations will be restricted to 3-site nearest neighbor rules, defined on En. The techniques introduced, however, apply generally to arbritrary k-site rules. The notation of Table 1.1 is used for the basic 3-site additive rules, and the notation of Table 1.2 for the 3-site basis operators. Lemma 3.1 For all states gE En: 1. 13+(µ) =11+(_1+µ) 2. 9(µ) = t(l+µ) 3. X(g) = x(l+µ) 4. (B±)2(µ) = 0 5. t2(µ) = t(µ) Table 3.1 given the canonical representations for the nearest neighbor additive 3-site rules. This form is what allows partition of general 3-site rules into additive and non-additive parts. Rule

Symbol

Canonical Representation

240

0 a-1

13++r1-+x+A

204

I

13++13-+x+t

170

a

f3-+rl++x+9

102

D

fi++11++O+t

90 60

S D*

13++13-+rl++rl 13'+r1-+9+t

A

rl++rl-+x+t

0

150

Table 3.1 Canonical Representations of 3-Site Additive Operators

55

For left justified forms of the rules in Table 3.1 each rule is multiplied by the left shift a. Let A be one of the seven non-zero additive operators in Table 3.1, and let X be the global operator for an arbritrary 3-site rule. Then F = X+A defines a decomposition of X in terms of the additive operator A: X = A+F. If F(g) = Q then X reduces to A on the state g. If X reduces to A on some nonempty subset of En-{Q,1} the decomposition is called legal. If the can onical representation of X contains the basis operator x then X is said to be generative . Legal decompositions of all non -generative 3-site rules are given in Appendix 2. If rules X and Y are related by the transformation T1 defined in Chapter 1, they will be called conjugate. If rules X and Y are conjugate this will be indicated by writing (X,Y). For a given rule X, a configuration gE En is on a cycle of period p(n) of X if Xp(n)(g) = g, and p(n) is the smallest number such that this is true. In the case p(n) = 1, g is a fixed point of X. A question of fundamental interest in studies of cellular automata is determination of their fixed points, and cycling behavior. In general, it is difficult to determine such parameters as the maximal cycle period, even for additive rules. It is known [33] that for additive rules defined on En, with n = 2mnO (no odd), the maximum cycle period b(n) divides the number b*(n) defined by b * (n) = min

2k - 2m 1 (21-2m)n0

(3.1)

where k and 1 are respectively the smallest integers such that 2k = 2m mod(n) and 21= ±2m mod(n). It is also known that the periods of all cycles divides the maximum cycle period b(n), and that in many cases b(n) = b*(n). Jen [551 has shown that the exceptions to b(n) = b*(n) result when certain values of n allow "anomalous shifts," which act to reduce the maximum cycle period. In the same paper, Jen has also demonstrated the fundamental role of shifts for cycles of additive CA rules. Theorem 3.2 (Jen, [55]) A configuration g lies on a cycle of an additive cellular automata X:En-*En if and only if there exist integers r and s such that Xr(g) = a-s(g). (Note that r = b(n) and s = n are possibilities.) The fundamental role of shifts focuses attention on shift cycles, that is, cycles on which a rule acts as a left or right shift. Legal decompositions of CA

56 rules can be used to determine both shift cycles, and fixed points. Suppose that X = A+F is a decomposition of X with additive part A, and let En(X,F) denote the kernel of F in En. Then, by definition, X reduces to the additive rule A on En(X,F). Thus, if A is the identity, configurations in En(X,F) will be fixed points of X, while if A is a or a-1 then configurations in En(X,F) will lie on either left or right shift cycles. The criteria for the 3-site basis operators to map a configuration other than .Q to .1 to .Q are given in Table 3.2.

Basis Operator States Mapped to 0 B± States without 11 sequences 11± States without 00 sequences x States without 111 sequences 0 States without isolated 0's t States without isolated 1's x States without 000 sequences Table 3.2 Criteria for a Configuration to Map to .Q Under a 3-Site Basis Operator All non-generative rules have .Q as a fixed point, and all generative rules either have 1 as a fixed point (if their canonical representation contains x), or have QHl as a period two cycle (if their canonical representation does not contain x). More generally, inspection of table 3.2 allows distinction of four different cases of legal decompositions:

1. X reduces to A on a highly restricted subset of En-(Q,1}. In these cases, F will contain conbinations of the basis operators which include B++rl±, tl±+x, r]±+x+t, B±+x, and B±+O+x. These will be called weakly legal decompositions. All other cases will be called strongly legal. Strongly legal decompositions of all non-generative 3-site rules are given in Appendix 2. 2. F involves B±, B±+x, rl±, or rl±+x. 3. F involves B±+O, B±+x+O, rl±+t, or rl±+t+x. 4. F involves x, 0, t, x, x +O, x +t,O+t, x +O+t, t+x, O+x, O+t+x. For each of these cases it is easy, based on Table 3.2, to work out the conditions for a state to belong to En(X,F). This, in turn, gives all of the possible cases in which 3-site rules reduce to additive rules. Table 3.3 gives the conditions for membership in En(X,F) for each possible case. In this table, the form of the non-additive rule F is taken to include all possible

57

combinations of the basis operators listed. For example, when 13++t1+ is listed, this is understood to include 13++r1+, 13++T1-, 13'+il+, and 13'+Tl-. Case: F

Condition for Membership in E„ (X,F)

1. 13++r1+

gE {Qi,1Q}

,l++x+t

gE 10 11 , 1 1 0 ,19-1)

J3±+9+K

4E (001,0 1 0,120j

71++x

g has no 00 or 111 sequences

13++K 2. 3+, or 13++x

g has no 11 or 000 sequences

Tl or TI++K 3. 13++0, or 13++9+x Tl++t, or Tl++t+K 4. X 0

g has no 11 sequences g has no 00 sequences g has no 11 sequences, or isolated 0's g has no 00 sequences, or isolated 1's g has no 111 sequences g has no isolated 0's

t

g has no isolated 1's

K

g has no 000 sequences

x+9 x+t

g has no 111 sequences, no isolated 0's g has only 11 pairs

0+t x+9+t

g has no isolated 0's or 1's

O+K

g has only 00 pairs

L+K

g has no 000 sequences, no isolated 1's g has only 00 pairs , no isolated 1's

9+t+K

g has only 11 pairs , no isolated 0's

Table 3.3 Conditions for Membership in En(%, F) for Given F

Lemma 3.1, together with Table 3.3 and some elementary inclusion arguments yields the following theorem: Theorem 3.3 Let binary coefficients (a, b, c) and (d, e, f) be not (0,0,0). Then 1. En(X,aTl++bq-+cc) = 1+En(d13++e13-+fx) 2. En(X,O) = 1+En(X,t) 3. En(X,x+t)nEn(X,x+9) = En(X,x+t)nEn(X,9+t) =En(X,x+9)nEn(X,O+t) = En(X,x+O+t) 4. En(X,9)nEn(X,x) = En(X,x+O)

5. En(X,t)nEn(X,9) = En(X,O+t)

58 6. En(X,x)cEn(X,aB++bB-+ cX+d0) (a,b,c,d not all 0) The fixed points for a rule are found by decomposing it so that its additive part is the identity operator , while the right and left shift cycles are found by decomposing with the additive part a-1, or a respectively. Appendix 3 lists all non- generative 3-site nearest neighbor rules having fixed points and/or shift cycles in En-{Q,1}. There are several illustrative cases worth further comment: 1. The conjugate rules ( 184,226) have shift decompositions given respectively by a-1+B++B- = a+il++11- (rule 184 ) and a+B++B- = a-l+rl++rl(rule 226). This indicates that under rule 184 , all configurations having only isolated l's lie on right shift cycles, while all configurations having only isolated 0's he on left shift cycles, with conjugate results for rule 226. 2. The conjugate rules (172,228 ) have identity decompositions given by I+13++0 (rule 172 ) and I+8-+0 (rule 228), while they have shift decompositions a+rl++t (rule 172 ) and a-l+rl -+t (rule 228 ). Hence they have as fixed points all configurations with isolated l's separated by two or more 0's; while configurations having only isolated 0 ' s separated by two or more l's lie on left shift cycles of rule 172 and on right shift cycles of rule 228. 3. Conjugate rules (202,216) have identity decompositions I+il++t (rule 202) and I+il -+t (rule 216 ); and shift decompositions given by a +B++O (rule 202) and a- 1+13++0 (rule 216). Thus , fixed points of (202 , 216) lie on cycles of ( 172,228 ), and vice versa. Theorem 3.4 With the exceptions of rules (44,100), ( 60,102), (106,120 ), ( 74,88), (142,212), and (14,84) all 3-site nearest neighbor rules having strongly legal shift decompositions have no cycles other than shift cycles. Proof The exceptional rules can be determined by computation of cycles. For the remaining rules, the proof that only shift cycles exist follows the proof for rule 248. The canonical representation of this rule is B++13-+rl-+x+0. This has right shift decomposition a-1+13-, indicating that this rule acts as a right shift on all configurations having only isolated l's. Hence, in En these configurations must lie on right shift cycles with period p I n. On the other hand , suppose that a state g contains a string of the form ...011...110..., containing at least two adjacent 1's. After a single iteration of

59 the rule, this will become either ... 111...111 ... or ...011...111... That is, the length of the string has increased irreversibly by at least one unit . Thus any state containing two or more adjacent l's will eventually iterate to the fixed point 1,.1 In cases (2) and ( 3) of rules with strongly legal decompositions, the rule reduces to its additive part on configurations containing only isolated l's (0's) separated by two or more 0's (1's). There is a simple recursion relation which allows the number of such configurations to be counted . Let Sn(r;s) be the subset of En consisting of states composed of blocks of exactly r l's separated by s or more 0's, and let Nn (r;s) be the number of elements in this set. Lemma 3.5 Nn(r;s) = Nn(r-m;s+m), 1<_m
60 a. If this final 1 is not the only 1, then there are at least s 0's separating it from the next 1 on its left. Elimination of this 1 (i.e., the first encountered to the left of the terminal 1), together with s 0's to its right, results in an element of Sn-s(l;s) which has a final digit of 1. This proceedure generates all elements of Sn-s(l;s) having a terminal digit of 1. b. The unique element (0...01)e Sn+1( 1;s) is all that remains since elimination of the final 1 in this string results in a string of all 0's, which is not contained in either Sn(l;s) or in Sn-s(l;s). I Remark Clearly the results of these two lemmas remain unaltered if the role of 1's and 0's are interchanged. Corollary 3.7 For n>0, the number of states in En having only isolated 1's, or only isolated 0's, is given by Ln-1 where Ln is the n-th Lucas number. Proof By Lemma 3.6, Nn+1(1;1) = Nn(1;1)+Nn-1(1;1)+1. Addition of 1 to each side of this equality yields Nn+1(1;1)+1 = (Nn(1;1)+1) + (Nn- 0; 1)+ 1), and this is the Fibonacci recursion relation for the quantity Fn = Nn(1;1)+1. Further, N1(1;1) = 0 and N2(1;1) = 2, so the sequence generated for Fn is given by the Lucas sequence. I The technique of identifying configurations which are fixed points, or which lie on shift cycles can be applied to k-site rules as well as 3-site rules. All that is required is construction of tables similar to Tables 3.2 and 3.3, although this construction and its analysis becomes progressively more difficult as the neighborhood size increases. 2. Jen' s Invariance Matrix Method A way of counting fixed points and cycles for rules with periodic boundary conditions has been developed by Jeri [55]. For a symmetric k=2r+ 1 site rule X, a 2l -1x2k-1 invariance matrix, W(X) is defined such that the ij entry of this matrix is 1 if the binary expressions of i and j can be overlapped to form a k-site neighborhood which leaves the value at the mapping site invariant under X, and is 0 otherwise. To construct W(X), i and j are written in binary form i = i-ri-r+1 ...ir-1 and j = j-r+IJ-r+2... jr. W(X) is given by s - Js (-r + 1 <_ s < r -1) A [X(i_rJ-r+1... Jr = iol [W(X)]ii = f 1 i 0 otherwise

61 This definition generalizes to arbritrary non-symmetric neighborhoods. With X any k-site rule, write i = i0il...ik-2 and j = jOj l.. jk-2. These two digit strings can be overlapped to form a k-site neighborhoods if and only if they satisfy i1...ik-2 =iO...ik-3. Let is be the digit located at the designated mapping site for the rule X. Then W(X) is defined by

[W(X)]u

=

11

il...ik-2 = JO... Jk-3 A LX ( i o ...ik-2Jk-2) = is] otherwi se

(3.2)

Theorem 3.8 (Jeri [55]) Let X be a one dimensional k-site CA rule with invariance matrix W(X). Then the number of fixed points of X on En is given by Tr(Wn(X)). Proof The matrix W(X) is the adjacency matrix of a graph having 2k-1 verticies, labeled by i = i0...ik-2. The ij component of Wn(X) is equal to the number of length n paths in this graph which begin at vertex i and end at vertex j = j0.. jk-2• Each pair of verticies connected by an edge in this path can be combined to form a neighborhood which maps to the value is under the rule X. Thus, each such path corresponds to a fixed point. Further, since periodic boundary conditions are assumed, each length n path which corresponds to a state in En must begin and end at the same vertex. Thus the fixed points of X on En are counted by the diagonal elements of Wn(X), and the total number of fixed points is the sum of these diagonal elements; i.e., the trace of this matrix. I Since configurations on cycles of X with period p are fixed points of XP Jen's method allows counting of cycles, as well. That is, the number of cycles of period d I p is given by Tr(Wn(XP)). Tables 3.4 and 3.5 shows the invariance matricies for the 3-site basis operators of Table 1.2, and the 3-site additive operators of Table 1.1. Jen's results can be generalized. Given a k-site rule X, define V(X) by xi-1 j = 2i -1 xi j = 2i [V(X)]ij -

1<-i<2k-2

and

(3.3) xi + 2k -1 j = 2i -1

xi+2k-1 j = 2i

0 otherwise For example, if X is a 3-site rule then

2k-2 < i < 2k-1

62 xo xl 0 0 0 0 x2 x3

(3.4)

V(X) = X4 X5 0 0 0 0 X6 X7

The matrix V(X) is easier to work with than the W(X) matrix since it directly involves only the canonical components of a rule , and it is clear from the definition of V(X) that for each basis operator vk ), V(vk )) contains only a single non-zero entry.

W(B+) =

1 0 0

1 0 0

0 0 0

0 0 0

1 0 0

W(X) =

0 1 0 1 0 0

W(1-) =

0 0 1 100

W(0) =

0 0 0 1 0 0

W(t)=

0 0 0 1 0 0

000

0 1 0 1 0 0 0 0 0 1 0 0

0 0 0 1 0 0

W(i1 )=

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0

W(TI+) =

100 0 0 1 1 0 0

WOO =

0 0 0

0 0 0 1 0 0 0 0 0

Table 3.4 Invariance Matricies for 3 -Site Basis Rules

Lemma 3.9 Let X be a k-site CA rule expressed in terms of its canonical components , as in equation (1.10). Then 2k-1 V(X) _ xsV(vk)) S=O

(3.5)

The next theorem gives the relation between W(X) and V(X). Theorem 3.10 Let X be a k-site CA rule. Then W(X) = W(O)+V(X) mod(2)

(3.6)

63 Proof W(O) contains a 1 at every position where the neighborhood formed from the binary strings for i and j has a 0 at the mapping site is , and has 0's at all other positions . V(X) has a 1 at every position ij when xr = 1, where r is the denary form of the binary neighborhood formed from the binary strings for i and j . Therefore, the mod(2) sum of these matricies has 1 's only at ij positions for which either xr = 1 (with r as before ) and the ij neighborhood has is = 1 (with is the mapping site ), or xr = 0 and is = 0. But this is just the definition of W(X). I Combination of equations (3.5) and (3.6) gives the theorem: Theorem 3.11 Let X be given in terms of the canonical basis operators. Then 2k-1 2k-1 W(X) = 1+ Yxs W(0)+ Ix5W(vk)) s=0 s=0

W(0) =

1 1 0 0

1 0 0 0

0 0 0 0

0 0 1 0

1 1 0 0

W(D) =

0 0 1 1 1 1 0 0

W(S) =

0 0 0 1

1000 0001

W(e) =

0 0 1 0

0 1 0 0 0 0 1 1

0 0 0 0 0 0 0 0

0 1 0 0

0 0 0 1 1 1 0 0

(1 1 0 0

W(6 1) =

0 0 0 1 0 0 1 0 (1 0 0 0

0 0 1 1 1 0 0 0 WW =

1 0 0 0 0 0 1 0 1 0 0 0

0 0 0 0 1 1 0 0

W(I) =

(3.7)

W(D*) =

0 0 1 1

0 0 0 0 0 0 0 0

Table 3.5 Invariance Matricies for 3 -Site Additive Rules

Theorems 3.10 and 3.11 give expressions for W(X) in terms of the components of X. The exact form of this expression, however, will depend on the location of the mapping site. For example, if Xis a 3-site rule then

64 x0 x1 0 0

xp x1 0 0

W(X) _

0 0 X2 X3 X4 x5 0 0

W(X) _

0 0 x2 xg

X4 X5 0 0 l 0 0 x6 x7 ) l 0 0 x6 X7) nearest neighbor left justified

where, as always, x' = 1+x mod(2). Equation (2.1) in Chapter 2 gives the i-th entry in L(k,m ;X), the listing which results when a k-site rule Xis applied to the k+m site neighborhood list. If M is a matrix , define List(M) as the list obtained by reading across the rows of M , starting at the top. Define List *(M) as the list obtained from List( M) by obmiting all entries which are automatically 0. Thus , for the matrix V(X) of equation (3.4), List(V(X)) = (xOx10000x2x3x4x50000x6x7)T (transpose by convention ) while List*(V(X)) = (xOxlx2x3x4x5x6x7)T. The next theorem is a direct consequence of the definition of V(X). Theorem 3.12 1. For 1!5s k-1. 1. Divide the 2k+s-1 entries of L(k, s;X) into 2k- 1 blocks of 2s elements each, starting from the top. Label each block with an index i = 1,...,2k-1. 2. Divide each block into 2s-k+l sub-blocks, each with 2k-1 elements. Label the sub-blocks with an index r = 1,...,2s-k+1, and label elements in each sub-block with an index jr = 1,...,2k-1, 3. Denote the j-th element of the r-th sub-block of the i-th block of L(k,s;X) by [L(k, s;X)]ijr. 4. Then Vs(X) is given by 2s-k+1 [VS(X)]ij = ,[L(k,s ;X)]ijr (3.8) r=1

Proof Ifs = k-1 then 2s-k+1 = 1 and equation (3.8) follows directly from Theorem 3 .12. On the other hand, suppose that equation (3.8) is true for s. But Vs+l(X) = V(X)VS(X) which by equation (3.3) gives the components of the product as [Vs+1(X)] ij = xi- 1[VLS(X)]ij + xi[Vs(X)]i+1j 1 :5i<2k-2 [Vs+l(X)] ij = xi+2k-1-1[Vs(X)]ij + xi+2k -1[Vs(X)] i+lj 2k - 25i<2k-1

65 But on an induction hypothesis this becomes [Vs+l(X)]ij = xi-1Y,[L(k,s;X)lijr + x ,[L(k,s;X)]i+ljr [V s+1 (X)]ij = xi+2k -1-1E[L( k,s;X)lijr + xi+2k-17-[L(k,s;X) li+ljr

1<_i<2k-2 2 k-2< i<2 k-1

and this is just the transformation which takes L(k,s;X) to L(k,s+1;X). I Equation (3.8) also yields an immediate formula for Tr(Vs(X)): Theorem 3.14 2k-1 2s-k+1 Tr(VS(X))= 1:[L(k,s;X)liir (3.9) i=1 r=1

The only difference between V(X) and W(X) is that some entries in W(X) will be primed. As already indicated, just which entries these are will depend on the choice of mapping site. What this means, however, is that Theorems 3.12 and 3.14; and Algorithm 3.13 are valid for W(X) as well, with the appropriate xs written as x's instead. By Theorem 3.8, the number of fixed points of a rule X acting on En is given by Tr(Wn(X)), and hence the number of cycles of period d I p will be given by Tr(Wn(XP)). That is, Tr(Wn(Xp)) _ Tr(Wn(Xd)) (3.10) dip

Computation of the components of XP is simplified by making use of the set of complete Boolean functions constructed from L(k,s;X). When written as the matrix X(k,s), this gives a means to compute the components of the product of two rules, as indicated in equation (2.4). Taking A = X in this equation, noting that the components of XP are given by L(p(k-1)+1,0;XP), and iterating the product gives the formula of the next theorem. Theorem 3.15 P-1 L(p(k -1)+ 1,0; XP) = fl X(k,s) (3.11) S=O

This formula allows easy computation of the components of XP. 3. Exercises for Chapter 3 1. Compute the action of the 3-site rules 30 and 90 on E6, listing all configurations which are fixed points, or which lie on cycles. 2. Show that theorem 3.4 is valid for rules 48 and 208. 3. Use lemmas 3.5 and 3.6 to count the number of configurations in E18 which consist of blocks of two 1's separated by two or more 0's.

66 4. Determine the conditions for a configuration to be a fixed point of the following rules: a) Rule 196 b) Rule 236 c) Rule 72 d) Rule 180 e) Rule 202 5. If each rule in exercise 4 acts on E12 how many fixed points will it have? How many if it acts on E 15? 6. Compute the matrix W(X) where X is taken as the following rules: 18, 22, 30, 54, 81, 105. 7. Give a proof of Theorem 3.12 8. Compute V12 (X) for the 3 -site rules of exercise 4. 9. Use the results of exercise 8 to determine the number of fixed points for each of these rules on E12. 10. How many cycles with period a divisor of 24 does the 2-site binary difference rule (0110 ) have on E24? 11. Prove that Tr(VT(X)V(X)) = Tr(V(X)VT( X)) gives the number of 1's in the rule table for X. 12. What conditions must a 3 -site rule X satisfy for: a) VT(X)V(X) to be diagonal?

b) V(X)VT(X) to be diagonal?

67

Chapter 4 Commutation of CA Rules One of the advantages of the use of abstract components is that it is possible to state general questions which can be answered by computing the solution sets of Diophantine equations in which the variables are rule components. This chapter begins with derivation of a set of such equations which, for a given k-site rule X, has solution set corresponding to all ksite rules which commute with X. The idea of commuting diagrams is introduced and used to analyze conditions under which a CA rule might be thought of as reproducing the output of a measuring instrument with output also represented by a CA rule. Finally, the division algorithm of Chapter 2 is combined with the commutation results to distinguish several interesting subgroups of the CA rule space. 1. Computation of Commutator Sets Let [X,A] = XA+AX denote the commutator of rules X and A. (Note that the + sign can be used since addition is mod (2).) In this chapter a set of Diophantine equations is derived which allows computation of the set (A I [X,A]=01 of all k-site rules which commutate with a given k-site rule X. If both X and A are k-site rules then [X ,A] will be a 2k- 1 site rule. Thus, for a given X, equation (2.4) yields [X,A] = X(k,k-1)A(k,0) + A(k,k- 1)X(k,0) (4.1) and setting this to 0 yields a set of 22k - 1 equations for the components of the rule A. The i-th equation in this system has the form (x^0 ...X;k-2xlk 1)a0 +(x10 ...Xik 1 )al+...+(x ; 0 ...x;k-2Xlk - 2Xik - 1)a2k-1

=(a^0 ...aik-talk-1 )x0 +(a10 ...aik-2alk-1)xl+...+(a;0...a;k-2alk-1)x2k-1 which can be written as k-1 L..1

E

r=0...0

1..1

s=0

k-1

x rrl ( 1 +rs +a;$)

arll( l+rs+x ;s)= r=0...0

(4.2)

s=0

where r = rO ...rk-1, rs is the s-th entry in r, and is is given by equation (2.2) with m=k-1. Lemma 4.1 L..1 k-1 L..1 1-n 1n 2k-n -1 2k -n-1 1 ar11 ( 1+rs+x;3 )= I xi0 °...Xik-1 -l I( Jar (4.3) r=0...0 s=0 n=0...0 r=0 lr J

68

where ( ) indicates the (q+1) entry in the p-th row of the mod (2) Pascal q

triangle. Proof The result claimed will follow if the coefficients of ar are equal on both sides of equation (4.3). For a given value of r define two subsets of the numbers 0,1,...,k-1: R0={s I rs=0} and R1={s I rs=1}. The left hand side of equation (4.3) then has the form ar [I x;3 [J( 1+x;e ) SERI sERO

This indicates that the coefficient of ar consists of a sum over all products of the xis which contain the product over Rl as a factor. On the right hand side of the equation, for the same fixed r, ar appears for each value of n such that r<- 2k-n-1. This condition selects out those n values for which n_<2k-r-1 . If n = 0 ... 0 there is only the single term xi0 ...xik-1, while taking n = 2k-r-1 gives the term xis . This follows , since the binary sER1

expression for 2k-1 consists entirely of 1's so that the binary form of 2k-r-1 has 0 ' s whenever the binary form of r has a 1, and has 1's in all other positions. Now, suppose that n = 2k- 1-(r+b) for0
b . These satisfy the condition that +b )

they are 0 if the binary forms of r and b have a 1 in the same position, and are 1 otherwise . This is a consequence of a theorem of Kummer [631 which states that the exponent of 2 in the prime factorization of

Cp

I* (the star

q

denotes that these binary coefficients are expressed in base 10 ) is equal to the number of borrows required in the binary subtraction p-q. Application of this result to (r+b l* indicates that this coefficient is the number of borrows in r J th e binary subtraction ( r+b)-r, and this is 0, yielding an odd coefficient, hence equal to 1 mod( 2), if and only if the binary form of r+b has a 1 in every position in which the binary form of r has a 1. Or, equivalently, the binary forms of r and b have no l 's in the same position . This means that the binary

69 form of n = 2k-1-(r+b) has 0's in every position that r has a 1. Thus, as b ranges over its set of possible values, there will be 0's occuring in all possible combinations of the remaining digits of n. This argument shows that the coefficient of ar on the right of equation (4.3) consists of all possible products of the xis which contain [1 xis as a sER1

factor, which is the same as the coefficient of ar on the left side of the equation. Since this is true for all r, the result follows. I Equation (4.3) allows the commutation conditions to be written as 1no 1-nk-1 2k_n-1^2k-n-11 x• ..x a n=0...0 ^^ ,k 1 r=0 p r k -nk -n- 11x (4.4) _ I a1-n0.. a1-nk-12-n-1(2

n=0.. .0 10

'k-1 r=O r r

The left hand side of equation (4.4) was derived from the expression for the product AX. This can be expressed in terms of the action of A on the list L(k,k-1;X) which was defined in equation (2.1). If x(i) is defined as the denary form of the binary number xi0xil...xik-1 where is is defined by equation (2.2) with m = k-1, then the i-th component of AX is just equal to ax(i), the x(i)-th component of A. This gives a system of Diophantine equations for the components of all k-site rules which commute with a given k-site rule X. Theorem 4.2 Let X be a given k-site CA rule. The set of k-site rules A which commute with Xis given by the solution set of the equations ax(i)

=

LA k -1 I - n - l xr L. a 0 n0 .. a k nl -1 n=0...0 r=0 r J

J

(4.5)

for 0-
(4.6)

which has formal uses, but is not computationally helpful since the a(i) values are not given a priori. Theorem 4.2 is not restricted to rules defined on neighborhoods of the same size . If X is a k-site rule and A is a k'-site rule then equation (4.5) can

70 be used by extending X to a k'-site rule (if k'>k), or by extending A to a k-site rule (k>k'). Theorem 4.3 Let X be a k-site rule, A a k'-site rule. Without lose of generality, assume that k'
71

a4 = a5(1+a3) 2. ala2 = ala4 = a2a4 = a2a5 = 0 al(1+a3) = 0 a3(1+a7) = 0 All equations in the second set with the exception of the last are satisfied identically as a result of the equations in the first set. The full solution set contains the left and right extensions of all 2-site rules commuting with X, as well as three additional rules. As must be the case, it contains both the right and left shifts, the identity, and both right and left extensions of (0010). This is indicated in Figure 4.1, with arrows indicating right (upper) and left (lower) extensions. (0000) (00000000) (0010) r _>(00100010)

(00001100) (0101 )r >(01010101) (00110011) (0011 )(00001111) (00100011) (00001110) (01010001) Figure 4.1 Two and Three Site Commutators of (0010)

2. Idem n otence The method used to compute commutator sets can also be used to determine rules which are idempotent. The question of which k-site rules are idempotent requires computation of the solution set to the equation X2 = X*, where X* is the extension of X to 2k-1 sites. The only technicality involved is to insure that the mapping site is preserved on both sides of the equation. If X is a k-site rule defined such that [X(g)]i = X(µi-r...µi+k-r-1) then X2 will be a 2k-1 site rule with [X2(µ)]i = X2(9i-2r•••9i+2k-2r-2)• Thus X must be extended

72 so that [X*( g)]i = X*(µi-2r..•µi+2k-2r-2). This requires the X neighborhoods to be extended as j1 •••jriO ...ik-1sl ... sk-r-1• Theorem 4.4 Let X be a k-site CA rule , and let s denote the mapping site of X2. Then X is idempotent if and only if 1 k 11 2k - n - llxr xis = x ^no...xlknl 2 J n=0...0 r=0 l r

J

(4.7)

As an example , consider the problem of determining all of the left justified 2-site idempotent rules. Expansion of equation (4.7) yields the set of independent equations 0 = x0(1+x3) 0 = xl+x0 ( 1+x1+x2 )+x0x1(x2+x3) 0 = x0(1+x1 )( 1+x2)+xl ( 1+x2x3) 0 = x0(1+xi )( 1+x3)+xl(1+x2+x3)+x1x2x3 (4.8) 0 = x0(xl+x2+xlx2+x2x3) 0 = x0(1+x1 )( 1+x2)+x1(1+x2x3) 0 = x0(1+x2 )( 1+x3)+x2(xl+x3)+x3(1+xlx2) After some computation , solution of equations ( 4.8) gives the complete set of left justified 2-site idempotent rules as 1(0000), (0010), (0011), (1011), (1111 )}. Similarly, the set of right justified 2-site idempotent rules is determined to be {(0000), (0100), (0101), ( 1101), (1111)}. 3. Ito Relations A comparative study of the 3-site rules 18 and 126 was carried out by Ito [56] , motivated by the remarkable similarity of the state transition diagrams for these two rules. He found that this similarity was due to the fact that rules 18 and 126 are related via the 3-site rule 252 . This relation is expressed by commutativity of the diagram of Figure 4.2. Relations of this type will be called Ito relations . In this diagram X stands for rule 18, Y for rule 126, and T for rule 252 . If the diagram commutes then YT = TX, and rules X and Y will be said to be Ito related via T. Theorem 4.5 shows that the Ito relation is determined in terms of the division algorithm of Chapter 2, and the commutation relations studied in this chapter.

73

E T {>E X

Y

0

V

-----> E E T Figure 4.2

Ito Relation of Rules 18 and 126

Theorem 4.5 For given CA rules X and T, there exists a rule Y such that X and Y are Ito related via T if and only if T I [X,T]. Proof If such a Y exists, then TX = YT. But TX = XT + [X,T], so this may be written as XT + YT = (X+Y)T = [X,T], and T does divide the commutator. On the other hand, if T I [X,T] there is an A such that [X,T] = XT+TX = AT, which may be written as TX = (X+A)T so Y is taken as X+A. I The same arguments leading to equation (4.6) prove the next theorem. Theorem 4.6 Let rules X and T be given. The set of rules A such that X and Y = A+X are Ito related via T is determined by the solution set of at(i) = xt(i)+tx(i) (4.9) For example, suppose that X = (01001000) (rule 18) and T = (0111) (note that rule 252 is just the 3-site right extension of T). For these rules, equations (4.9) become a0 = x0+t0 = 0 al=xl+tl=0 a3 = x3+t2 = 1 a4 = x4+t2 = 0 a5 = x5+t3 = 1 a6 = x6+t1 = 1 a7 = x7+tO = 0 The most general solution of these equations is A = (00a210110). In the case a2 = 1 this gives X+A = (01111110), which is rule 126, the case studied by Ito. If a2 = 0 then X+A = (01011110), indicating that rule 122 is also Ito related to rule 18.

74 The question of when , given two rules X and T, there is a third rule Y such that the diagram of Figure 4.2 commutes is important in connection with an analysis carried out by Crutchfield [57] on the possible use of cellular automata to infer underlying spatio -temporal dynamics from data series. In his approach the rule T is a "measurement function," which describes the action of the measuring instrument . The rule X is the actual underlying dynamics , and the rule Y is the observed dynamics. The conclusion reached was that in most cases cellular automata would misrepresent underlying dynamics . The reason, in terms of the present results, being that in most cases , for a given X and T, there is no rule Y which is Ito related to X via T. Thus it is quite important to be able to determine under just what sets of conditions it is possible to find rules for which the diagram of Figure 4.2 does commute. If X and T are given, then the possible rules Y, if any exist, can be found by solution of equation (4.9) and use of Y = A+X. On the other hand, if T and Y are given, which would be the more usual experimental situation, equation (4.9) is not much help. Instead, it is necessary to return to equation (4.5), in the form k

(2k-n-1 x1-no -1 1tr Yt(i) = 1 is. xl-nk-12 ik-1 r JI n=0...0 r=0

(4.10)

to obtain a set of equations which may be solved for the components xi of X. Another question which is of interest is to determine all possible rules X which satisfy T I [X,T] for a given rule T . In this case the equation YT = TX, written in the form yt (i) = tx(i), can yield a set of logical constraints on the components of X. If T is a k-site rule , while X and Y are r-site rules , then YT and TX will both be k+r- 1 site rules. The yt(i) are easily computed from the k+r-1 site neighborhood list, while the tx(i) will be given in terms of the list Li(k,r-1;X) defined in equations (2.1) and (2.2). The condition that yt(i) = Yt(j) whenever t(i) = t(j) then yields conditions on the possible values of the xi. For example, if T is the 2-site binary difference rule D = (0110), and X is assumed to be a 3-site rule , the initial step in solving for Xis indicated in Table 4.1. Inspection of this table indicates that there are eight constraint equations to be satisfied : txOx0 = tx7x7; ttOx1 = tx7x6; tx1x2 = tx6x5; tx1x3 = tx6x4; tx2x4 = tx5x3 , tx2x5 = tx5x2, tx3x6 = tx4x1 ; and tx3x7 = tx4x0.

75 In this notation, txrxs indicates the component of T labeled with the denary form of the binary number xrxs. i

vtf,

0000 0001 0010 0011 0100 0101 0110 0000

yO

i txOx0

yl Y3

tx0x1 txlx2

Y2 Y6

tx2x4

Y7

tx2x5

Y5

tx3x6

YO

i

1000 1001 1010 1011 1100 1101 1110 1000

txlx3

tx3x7

Yt(i)

Lx(i)

y4 Y5 y7 Y6 Y2

tx4x0 tx4x1 tx5x2 tx5x3 tx6x4

Y3

tx6x5

yl y4

tx7x7

tx7x6

Table 4.1 Yt(i) and tx (i) When T is the Binary Difference Rule

Working out the conditions which are imposed on the xi is simplified by use of Figure 4.3, which indicates the cases in which trs = tr's'.

Figure 4.3 Cases In Which trs = tij For T = D (Cases of equality shaded)

This figure, in conjunction with the constraint equations, indicates that txOx0 = tx7x7 and tx2x5 = tx5x2 are always satisfied, while the remainder of the constraints can be formulated in the form of the binary truth conditions [(xi=xj)n(xr=xs)]V[(xi=x j)n(xr=x's)] . In these expressions the indicies (i,j,r,s) take on the following sets of values el = (0,1,7,6), e2 = (1,2,6,5), e3 = (1,3,6,4), e4 = (2,4,5,3), e5 = (3,6,4,1), and e6 = (3,7,4,0). The rules X for which there exists a Y with YD = DX can now be computed via the flow chart of in Figure 4.4.

76

Figure 4.4 Computation of X Such That D I [X,D]

Table 4.2 lists the solutions for X having x0 = 0, together with the corresponding values of Y, which can be computed, once X is known. (Rules commuting with the binary difference rule D are indicated with a * in this table.)

0 24 36 60 66 90 102 126 142 150 170 178 204 212 232 240

X 00000000 00011000 00100100 00111100 01000010 01011010 01100110 01111110 01110001 01101001 01010101 01001101 00110011 00101011 00010111 00001111

0 116 72

Y 00000000* 00101110 00010010

60

00111100*

46 90 102 18 226 150 170 222 204 184

01110100 01011010* 01100110* 01001000 01000111 01101001* 01010101* 01111011 00110011* 00011101

132

00100001

240

00001111*

Table 4.2 Non-Generative 3-Site Rules Satisfying YD = DX (* Indicates Rules Commuting With D)

77

4. Some Interesting S ibgr,ps In Chapter 2 it was noted that the set of k-site CA rules formed an abelian group under component-wise binary addition. The division and commutation results which have been presented allow some interesting subgroups to be distinguished. This will be illustrated for the case in which both Q and T are 2-site rules. Table 4.3 lists the 3-site products of all 2-site rules. In this table, the row and column lables are the numerical labels for 2-site rules, indicated with an astrick to distinguish them from the 3-site numerical labels. The entry in the cell with row label ab* and column label cd* is the 3-site numerical label for the composite rule (ab*)(cd*). 9*_

1*

2-**

2

4*

E**

fL

Z

li*

9*

9_ *

0

0

0

0

0

0

0

0

0

0

1*

0

128

0

192 0

136 36

236 1

129 17

209 3

48

48

12

68

16

66

34

12

68

2*

4*

0

11*

12* =

14* IV

0

0

0 0

0

139 55 255

0

64

0

192 48

240

12

204 60

252 3

195 51

243

15

207 63 255

12

34

34

2

24

12

48

48

64 0

187

119 255

24

2

34

8 0

0

8

0

136 68

9*

0

72

116 60

7*

0

200

116 252 46

li*

255 55

139 3

255 183

139 195 209 153 165 237 237

165 153 209

195 139

183 255

255 119

187 51

102 170 34

204 68

136 0

68

204 34 48

66

16

68

170 102 238 17

153 85

102 90

90

18

18

238 126 254 19

209 17

221 85

129

153

1

17

102 46

60

219 119 255 63

236 36

238

221 51

136 0

116 72 0 255

192 0

11* 255 247

187 243 221 221 189 253 239 231 187 243 207 207

12* 255 63

207 240 243 51

195 3

252 60

204 12

127 255 128 0

191 255

240 48

192 0 1± 255 191 207 207 243 187 231 239 253 189 221 221 243 187 247 255 14* 255 127 255 63 255 119 219 19 254 126 238 46 252 116 200 0 ib± 255 255 255 255 255 255 255 255 255 255 255 255 255 255 255 255

Table 4.3 3-Site Produces of 2-Site Rules (ab* indicates numerical label of 2-site rule)

Referring to this table, consider the diagram of Figure 4.5, in which X and Y are 3-site rules; Q and T are 2-site rules; and Q is taken as a linear rule, excluding 0 and 1. Thus the rules considered for Q are the rules given by: rule 3*, (0011), which is the left justified identity I; rule 5*, (0101), which

78 is the left shift a; rule 6*, (0110), which is the binary difference rule D; rule 9*, (1001), 1+D; rule 10*, (1010), 1+a; and rule 12, (1100), 1+I. Also, in the diagram, X = QT and Y = TQ, so the commutator is [Q,T] = X+Y.

T ENE Y

V

T

DE

Figure 4.5 Commutator Diagram

If Q is taken as a, or 1+a then reading across the corresponding rows, and down the corresponding columns of Table 4.3 selects out the rules listed in Table 4.4a. Rule Co=onents

Cannonical Form

Additive Form

a+13-+x

34

(01000100)

TI++A

68 136 238 221 187

(00100010) (00010001) (01110111) (10111011) (11011101)

13++t

I+f3-+x

13-+x 13++13-+Tl++X+8+t 6++13-+r1-+X+t+x

0+13-+x D+13-+X 1+a+13-+X

13-+T1++Tl-+x+O+x

1+I+f3-+x

119 17 0 102 170 204 255 153 85 51

(11101110) (10001000) (00000000) (01100110) (01010101) (00110011) (11111111) (10011001) (10101010) (11001100)

13++11++T1-+8+t+x ^-+x

1+D+13-+

13++11++A+t 13-+T1++X+9

0 D a

13++13-+X+9

I

!3-+Tl-+X+x 13++T1-+t+x Tl++71-+9+x

1+f3-+x

1 1+D

1+a 1+I

Table 4.4a Composite 3-Site Rules With Factor a or 1+a

The shift a commutes with all rules, but 1+a does not. However, its commutators are already included among the rules in Table 4.4a. (If this were not the case , it would be necessary to include them, as will be the case when Q is taken as the binary difference rule D.)

79 Using the same notation as Table 4.3, the non-zero commutators of 1+0 are given by: [1+0,0*] = 255; [1+0,1*] = 102; [1+0,2*] = 153; [1+0,4*] = 153; [1+a,6*] = 255; [1+a,7*] = 102; [1+0,8*] = 102; [1+a,9*] = 255; [1+0,11*] = 153; [1+0,13*] = 153; [1+a,14*] = 102; [1+0,1] = 255. The remaining four commutators are 0. Under the operation of binary addition, the rules listed in Table 4.4a form a sub-group of the abelian group of all CA rules. This sub-group is defined by the group table given in Figure 4.6a. This table is given in compressed form in that the rules in the table could also be listed as indices for the table. This is unnecessary, however, since if rules X, Y, and Z satisfy X+Y = Z, then Y+Z = X and X+Z = Y.

34 68

34

68

136

238

221

187

119

17

0 102

102

204

255

153

0

170 204

170

153

255

85 51

51 85

136

170

204

0

102

85

51

255

153

238 221

204

170

102

0

51

153

255

153

85

51

0

85 102

170

255 204

187

153

255

51

85

102

0

204

170

119

85 51

51 85

255 153

153 255

170 204

204

17

0 102

0

170

102

Figure 4.6a Group Table for 3-Site Rules With Factor a or 1+a

If Q is taken as I, or 1+1, the rules to be considered are those listed in Table 4.4b. Similarity to the rules from Table 4.4a should be immediately obvious. In fact, the rules in these two tables are related to each other by the symmetry transformation Ti defined in equation (1.11) of Chapter 1, and thus correspond on En to two different orientations on the circle. The identity I commutes with all rules, but 1+I does not. As before, however, the commutators are among the rules in Table 4.4b. The non-zero commutators of 1+1 are given by: [1+1,0*] = 255; [1+I,1*] = 60; [1+I,2*] = 195; [1+1,3*] = 255; [1+1,4*] = 195; [1+1,6*] = 255; [1+1,7*] = 60; [1+1,8*] = 60; [1+1,9*] = 255; [1+1,11*] = 195; [1+1,13*] = 195; [1+1,14*] = 60; [1+1,1] = 255. Rule Components Cannonical Form Additive Form 12 (00110000) (3-+1 1+13++x

80

252 (00111111)

rl- +e a-1 + B+ + x B ++x o +B ++x 8+ +B-+il- +x +e +t D*+13+ + x

243 (11001111)

B++rl++rl- +x +e+x 1+I+B+ + x

207 (11110011)

B-+rl++rl- +x +t+x I+a-1 +13 + +x

63 (11111100)

B- +rl ++rl- +e +t+x 1+B+ + x

3 (11000000)

r1++K 1+D*+B++X

0 (00000000) 60 (00111100)

0 B-+q-+e+t D*

204 (00110011)

B++B-+x+e I

240 (00001111)

B++rl-+x+e a-1

255 (11111111) 195 (11000011)

1 B++rl++x+K 1+D*

51 (11001100) 15 (11110000)

Tl++rl-+e+K 1+I B-+rl++t+x 1+a-1

48 (00001100) 192 (00000011)

Table 4.4b Composite 3-Site Rules With Factor I or 1+I

As in the previous case, the rules listed in Table 4.4b form a sub-group of the abelian group of all CA rules. This sub-group is defined by the group table of Figure 4.6b. 12

48

192

252

243

207

63

3

255

195

187

15

12

0

60

204

240

68

60

0

240

204

195

255

15

187

204

240

0

60

187

15

255

195

252

240

204

60

0

15

187

195

243

255

195

187

15

0

60

204

255 240

207

195

255

15

187

60

0

240

204

240

0

60

204

60

0

192

63

187

15

255

195

204

3

15

187

195

255

240

Figure 4.6b Group Table for 3-Site Rules With Factor I or 1+I

Finally, for Q equal to D or 1+D, the rules which must be considered are listed in Table 4.4c. Since D, in contrast to a and I, does not commute with most other 2-site rules, this table is twice the size of Tables 4.4a and 4.4b.

81

Rule Components

Cannonical Form

Additive Form

18

(01001000)

A+x+t = 8+B++B-

46

(01110100)

71++71B++11++e+t

72

(00010010)

B++B-

a+x+t I +x +t

116

(00101110)

B++r1-+O+t

a-1+x+t

126

(01111110)

B++B-+r1++rI -+9+t e+p++p-+x+o =

8+0 +t

66

(01000010 )

13++r1+

0+(3++(3-+x+e

36

(00100100 )

A+t

I+(3++(3- +x +A

24

(00011000 )

237

(10110111 )

B--M8 + +B- +x +9+t+x

1+0 + x+t

209

(10001011) (11101101 )

B ++r1- +x +x

1+a + x+t

183

71++11- +x +8 +t +x

1+I +x +t

139

(11010001 )

B-+11+ +x +x

1+a-l+x+t

129

(10000001 )

x +x

1+0+ (3++(3-+x+e

189

(10111101 )

J3-+i1- +x +8+t+x

219

(11011011 )

13++i3-+T1++T1-+x+x 1+I+0++p-+x+O

231

(11100111 )

B++r ++X+8+t+x

0 60 90

(00000000) (00111100 ) (01011010 )

B-+r1-+9+t 13++13-+r1++i1-

102

(01100110)

B++11++O+t

D

255 195

(11111111) (11000011 )

B++r1+ +x+x

1 1+D*

165

(10100101 )

x +8+t+x

1+8

153

(10011001 )

B-+ri- +x +x

1+D

}++ + +

1+a+p++R-+x+8 1+a-1±+±a-±X±Q 0 D* 8

Table 4.4c Composite 3-Site Rules With Factor D or 1+D

The group table for these rules is given in Figure 4.6c. Unlike the group tables given in Figures 4.6a,b, this figure contains additional rules not listed in Table 4.4c. These are due to the non-commutativity of D. These additional rules are listed in Table 4.4d. It is interesting to note that the only rules which are not either commutators or products in this group table are rules 10, 80, 245, and 175.

82

116 72

46 46

0

116 90

18

90

102 60

0

60

72

102 60

18

60

0

102 90

24

36

126 209

139 183 237

189 231 219 129

108 54

165 153 195

147 201 245 175

66

10

80

255

102 54

108 80

10

165 255 195 153 201 147

175 245

90

10

80

108 54

195 255 165 345 175

147 201

0

80

10

54

66

108 54

10

80

0

90

102 60

24

54

108 80

10

90

0

60

36

10

80

108 54

126 80

10

54

102 60

108 60

0

102 90

147 201 245

153

108 195

153 165 255

147 201 245 175 255 165 147 175 245

165 255

90

245

175 147 201

153 195 255 165

0

175 245 201 147

153 195

139 165 255

195 153 201 147 175 245 90

175 0

90

102 60

0

60

183 153 195 255 165 245 175 147 201 102 60 165 255

175 245 201

147 60

0

102 90

195 153

195 153

165 255

108 54

10

80

108 80

10

102 54 90

10

80

108 54

0

80

10

54

108

10

80

0

90

102 60

153 54

108 80

10

90

0

60

102

165

10

80

108 54

0

90

195 153 165 255 80

10

54

189 147 201 245 175 255 165 153 231 201 147

175 245

165 255 195

219 245 175

147 201

153

129 175 245 201 147

153 195

102 201

209 255 165

237 195 153

175 245 201 147

195 255

195 108 54

102 60

108 60

102 90

0

Figure 4.6c Group Table Generated By 3-Site Rules With Factors D or 1+D

Rule Components Cannonical Form 54

(01101100) it ++1T-+0+t (00110110) f3++13-+O+t

= A+x+e

Comment 54 = [v,1+D], v = 1*,7*,11*,13*

10

(01010000) 13-+ii+

= I +x +A = a+x+9

80

(00001010) f3++rl-

= a-1+x+8

201 147

(10010011) 13++13- +x +1( (11001001) rl++il- +x +x

201 = [v,1+D], v = 2*,4*,8*,14* 147 = [v,1+D], v =8*,11*,13*,14*

245

(10101111) f3++Ty-+x+A+t+x

neither commutator nor product

175

(11110101) f3-+il+ +x +8+t+x

neither commutator nor product

108

108 = [v,1+D], v = 1*,2*,4*,7* neither commutator nor product neither commutator nor product

Table 4.4d Rules From Figure 4.6c Not Included in Table 4.4c

For each sub-group, inspection of the additive and cannonical forms shows that the elements separate into classes of four such that any three elements in a class sum to the fourth. For example, in Table 4.4c rules 18, 46, 72, and 116 form such a class, the additive forms of rules in this class being, respectively: 4+x+t, a+x+t, I+x+t, and a-l+x+t. The other such rule

83 classes in Table 4.4c are (24,36,66,126), (139,183,209,237), (129,189,219,231), (0,60,90,102), and (153,165,195,255). In addition, the rules in each sub-group are related by the symmetry transformations T1, T2, and T3 defined in Chapter 1. These relations are illustrated in Figures 4.7a,b.

Generating Rules Fzo n Identity Sub.Group

Generating Rules From Shift SubGroup

Figure 4.7a Symmetry Relations of Identity and Shift Sub-Group Rules (Full Sub-Groups Include Binary Compliments)

In Figure 4.7a only half of the sub-group elements are shown, the remaining half being the binary compliments of these, and sharing exactly the same symmetry relations. Figure 4.7b shows all of the sub-group elements. Rules 54, 108, 147, and 201 are commutators, but are not composite rules, while rules 10, 80, 175, and 245 are neither composite nor commutators, as indicated in Table 4.4d. The remaining rules in the subgroup are all composite. Of particular interest is the much discussed rule 18.

84

Figure 4.7 b Symmetry Relations Between Elements of D; 1+D Sub-Group

85 Rule 18 is the composition of the binary difference rule D on the 2-site rule (0111). Thus, the non-additive properties of rule 18 are determined by the rule (0111), which maps all 2-site neighborhoods except 00 to 1. Since [D(t)]i is 1 only when µi # µi+l this indicates that rule 18 yields 1's only at the boundaries of strings containing only isolated 0's. Examination of the additive decompositions of the rules in these different sub-groups suggests definition of domains of the state space on which the rules act as additive rules. For the rules of Tables 4.4a,b the nonlinear part of the rule is given by either B±+X, or 1+B+-+x, indicating that these rules reduce to their additive ((B±+x)(µ) = 0), or linear ((1+B±+x)(4) = 0) parts on all states which contain only isolated 1's. For the rules of Table 4.4c there are four domains which are defined: D1= (ge E I g contains only pairs of 1's) D2 = {µE E I g contains only pairs of 1's, and no isolated O's} D3 = (µE E I t contains isolated 1's or pairs of 1's, and no isolated 0's) D4 = (µE E I t contains only isolated 1's) D5 = {µE E 14 contains no isolated 0's or 1's) By inspection, D2cD1, D2cD5, and D4CD3. Also, the various nonlinear parts of the rules from Table 4.4c satisfy the conditions:

X+t I D1= X+t I DZ = B++13-+x+9 I DZ = x+6 I D3 = x+t I D1= B++B I D4 = A+t I D5 = 0. Domains D4 and D5 are those on which rules 18 and 126 respectively reduce to rule 90. It should be noted, however, that this reduction to an additive rule does not, in general, survive under more than a single iteration of the rule. For rule 18, for example, Hanson and Crutchfield [64] have shown that the domain on which this rule reduces to rule 90 regardless of the number of iterations is the set of states which consist of only isolated 1's separated by an odd number of 0's, and this domain is a proper subset of D4. The question of on what domain a rule reduces to an additive rule for all iterations is difficult, although an elegant empirical technique to search for such domains has been developed [64]. 5. Exercises for Chapter 4 1. Compute the commutator set for the 2-site rule (0111). 2. Compute the commutator set for the left and right extensions of (0111); e.g., for the rules (00111111) and (01110111).

86 3. Use equation (4.7) to determine whether or not the following 3-site nearest neighbor rules are idempotent: a) (00100000) b) (00010011) c) (01110100) d) (00111010) 4. Find the rules which are Ito relted to X via T = (0110) when X is: a) (00011000) b) (01001101) c) (01110001) d) (01000010) 5. Find all rules X such that T I [X,T] for T = (01111011) and (01011010). 6. Suppose that rules X, Y, and T in Figure 4.2 satisfy TX = YT+R. Derive a formula for TXn. 7. Show that rules 122 and 126 are Ito related to rule 18 via rule 238, as well as via rule 252. 8. Show that if T is rule 30 then the only possible rules X, defined for fewer than 4 sites, for which there is a rule Y such that the diagram of Figure 4.2 commutes are 0, a, a2, and rule 30. 9. Let X and Y be respectively k and r-site rules having the property that a) X(0il...ik-1) = 1+X(lil...ik-1) Y(Oil...ir_1) = 1+Y(lil...ir-1) b) X(Oil...ik-1) = 1+X(lil...ik-1) Y(iO...ir-20) = l+Y(iO...ir-2l) Show that in each case their commutator also has this property. 10. Rule 18 is equal to rule 90 plus rule 72, and is also equal to rule 150 plus rule 132. a) Show that rule 18 is identical to rule 90 for all iterations (i.e., for all n, Xn18 = 8n) on configurations containing only isolated l's separated by an odd number of 0's. b) Show that there is no configuration other then 0 on which rule 18 is identical to rule 150 for all iterations.

87

Chapter 5 Additive Rules: I. Basic Analysis In this chapter the conditions which a CA rule must satisfy in order to be additive are formulated in terms of a map from the Cartesian product of the configuration space with itself to the configuration space. This map is represented for k-site rules by a 2kx2k matrix. CA rules can be partitioned into equivalence classes on the basis of their additivity properties, as characterized in this matrix. An approach to the analysis of additive rules in terms of complex polynomials is also introduced, and used to derive a condition for the injective of additive rules. 1. Conditions for Additivity In Chapter 1, a cellular automata rule was defined as additive if it was linear and homogeneous; or equivalently, if it could be expressed as a sum of shifts, as in equation (1.5). The more common definition of additivity is that a rule Xis additive if X(µ+µ') = X(µ) + X(µ') V µ,µ'e E (5.1) In Theorem 5.2 it is shown that these two definitions of additivity are equivalent. As a preliminary, some results on the component form of additive rules are necessary. Equation (5.1) can be stated in terms of the components of X by using the fact that the neighborhood set for X is an additive group under site-wise addition mod(2). For example, Figure 5.1 gives the group table for 3-site neiehborhoods. 000

010

011

100

101

110

111

000

001

000

001

010

011

100

101

110

111

010

101

100

111

110

001

110 111

111 110

100 101

100

001

001

000

011

010 011

010 011

011

000 001

010

000

101

100

101

110

111

000

001

010

011

100

111

110

001

000

011

010

110

101 110

111

100

101

010

011

000

001

111

111

110

101

100

011

010

001

000

100 101

Figure 5.1 Group Table for 3-Site Neighborhoods

88 By definition, however, for a k-site rule X the neighborhood iO...ik-1 maps to xi under X. If X is to be additive, its action on the sum of two neighborhoods must give the same result as the sum of its action on each neighborhood separately. Hence the group table for k-site neighborhoods defines an addition table for the components of X. If X is additive then X(iO...ik-1) + X(jo...jk-1) = X(iO...ik-1 +iO...ik-1) for all neighborhoods. This condition will be written as Xi + Xj

+ Xi+j = 0

(5.2)

where the sum i+j is understood as the denary value of the site-wise addition of the binary forms of i and j. The table which this defines for 3-site rules is shown in Figure 5.2. X0 xp X1 x2 x3 x4 X5 xg x7

xl

x2

X3

x4

X5

x6

X7

0

0

0

0

0

0

0

0

o

0

xl +x2+x3 x1+x2+x3 x1 +x4+x5 X1+X4+X5 x1 +x6+x7 xl+x6+x7 0

xl +x2 +x3 x2+x4+x6 X2+X5+X7 x2+x4+ x6 X2+X5+X7

0

X1 +X2+X3

0

X1 +X2+X3 X1+X2+X3

0

+x4+x6 x3 +x4+x7 x1 +x4+x5 x2

0

x1 +x4+x5 X2+X5 +X7 x3+x5+x6 x1+x4+x5

0

+x5+x6 +x4+x6 x3 +x5+x6 x 2+x4+x6 x3 x1 +x6+x7 x2

p

xl +x6 +x7 x2+x5+x7 x3+x4+x7 x3+x4+x7 x2+x5+x7 xl+x6+x7

0

X3+X4 +X7 x3 +x5 +x6 x3 +x5+x6 x3+x4+x7 0

+x 4+x +x4+x6 x3 x1 +x4+x5 x2 7 0

x3+x5+x6 X2+X5+X7 0

x1 +x 6 +x 7 0

Figure 5.2 Component Sums Defined by Neighborhood Table of Figure 5.1 Continuing the 3-site example, equation (5.2) yields the set of conditons

x0 = xl+x2+x3 = xl+x4+x5 = xl+x6+x7 =x2+x4+x6 = x2+x5+x7 = x3+x4+x7 = x3+x5+x6 =0

(5.3)

and the independent equations in this set may be taken as x0=0 xl+x2+x3 = 0 xl+x4+x5 = 0

(5.4)

xl+x6+x7 = 0 x2+x4+x6 = 0 giving the most general 3-site rule satisfying equation (5.1) as X = (0,x1,x2,x1+x2,x4,xl+x4,x2+x4,xl+x2+x4) The next theorem generalizes this form to general k-site rules. Theorem 5.1

(5.5)

89 A k-site rule X satisfies equation (5.2) if and only if k-1 Xi = Xik -s-1x23 `d i = i0...ik-1 (5.6) s=0

Proof By equation (5.5) the claim is true for k = 3, and it is easily verified for k = 2. Suppose that the claim is true for k-site rules. Let G(k) denote the ksite neighborhood group table; and G(0)(k), G(1)(k) denote this table with all elements extended on the left by the addition of a 0 or a 1 respectively. Then G(k+ 1) = G(o)(k) G(1)(k)

G(1)(k) G(o)(k)) Equations (5.2), as applied to G(k+1), can be divided into two sets: a) Those arising from terms of the form OiO...ik_1 + G(0)(k). These are the equations which determine the components of the k-site additive rules. Thus, they will determine the first 2k components of the k+1 site additive rules , which will have the same form as for the k-site additive rules. These will be denoted xi, with 0<_ i:5 2k-1. b) Consider equations of the form 10...0 + [G(1)(k)]Oj = [G(0)(k)loj which come from the 2k+1 row of G(k+1). These yield the component equations xi+x2k+xi+2k = 0 0<_i<_2k-1 (5.7) which specify xi+2k in terms of xi and x2k. But by the induction hypothesis, the xi satisfy equation (5.6), hence by equation (5.7) the second 2k components of X are just xi+2k = xi + x2k and these are of the form required in equation (5.6) since i+2k = liO...ik_ 1. Theorem 5.1 gives a quick way to obtain the components of the general k+1 site additive rule from those of the general k-site additive rule. List the latter twice, and add x2k to every term in the second list. This also provides an immediate proof of the equivalence of the two definitions of additivity. Theorem 5.2 A k-site rule X satisfies equation (5.1) if and only if it satisfies equation (1.5). Proof Assume that X satisfies equation (5.1). Then, by Theorem 5.1, the components of X are given by equation (5.6). Thus, the expression for xi contains a term x28 if and only if ik-s-1 = 1, where i = i0...ik-1. Suppose that

90 x 2 s =1 and all other independent components of X are 0. Then the component form of X will consist of alternating blocks of 0's and 1's, each of length 2k-s-1. But this is just the component form of the shift 6 s-r+1 where it is the designated mapping site . Thus, rules with components satisfying equation (5.1) define rules which also satisfy equation (1.5) with the coefficients as defined by a 1 x2k_s_1 = 1 0< s <- k s 0 otherwise

2

(5.8)

Likewise , if X satisfies equation ( 1.5) then (5.8) determines the nonzero x2k _s_1, and a rule is obtained which satisfies equation (5.1). Definition 5.3 For a given k -site rule X, define a map U(X):ExE-E by [U(X;µ,µ')]i = xi(µ)+xi(µ')+xi(µ+g') mod(2) where i(p) is taken as the denary form of.tiµi+1 ... µi+k-1 • U(X) can be specified in a state independent form via a 2kx2k matrix with elements [U(X)]ij = xi + xj + xi+j mod(2) where i+j is understood as the denary form of the mod(2) site-wise addition of the binary forms of i and j. By definition, U(X) satisfies X(µ+µ') = X(µ) + X(µ') + U(X;p t'). (5.9) The matrix U(X) will be called the obstruction to the additivity of X. Theorem 5.4 2k-1 Let X = I xsvk) be a k- site CA rule defined in terms of the set of s=0

canonical basis operators { vk)). Then 1. U(X) = 0 a Xis additive. k_ k_1 2. U(X) = U 2^,xsvk) =IxsUlvk)) S=O s=0

Proof The first part of the theorem follows immediatly on comparison of equations (5.1) and (5.9). To demonstrate the second claim, write k 21 xsU^^'k)

S=O

2

k_

I l xs( (vk))i(µ)+(vk))i(µ')+( vk))i(µ+µ')) (5.10) s=0

and note that (v(s) )i = is so that equations (5.9) and (5.10) are identical.

91 Theorem 5.4 allows derivation of a variety of identities relating combinations of the k-site basis operators . For example, the 3 -site nearest neighbor identity rule is I = B++13-+x+t. Hence, U(I) = 0 yields the identity U(B++B-) = U(x+t). More generally, if A and A' are additive, and X = A+F, while X' = A'+F where F is not additive, then U(X) = U(X') = U(F). A graphical representation of U(X) is easily constructed. If Xis a k-site rule, construct a 2nx2n matrix with n_k, and with ij entry given by the value of X(iO...in-1)+X(O...jn-1)+X(i0...in-1+jO...jn-1)• If this matrix is viewed as a partition of [0,1]x[0,1], and the entries are taken as coefficients for expansions in powers of 1/2 , then in the limit n-+oo U(X) will be represented as a graph over the unit square with maximum height equal to 1. Figure 5.3 illustrates the self-similar nature of this process for the 2site rule X = (0100), as applied to a 16x16 grid. Notice that all non-additive 2-site rules with x0 = 0 (non-generative ) have the same obstruction, given by 0 0 0 0 U(X) =

0

1

0 10 1 0 1 0 1 1 0

The first square is viewed as overlaying the 16x16 grid, so that each of its sub-squares covers sixteen of the original grid squares. The first digit of entries in each square covered by one of these larger squares equals the digit shown in that square. In the next step, the 16x16 grid is quartered into four 8x8 grids, and each of these is again covered by an overlay, now with each overlay square covering four squares of the original grid . The indicated digit in the overlay square gives the second digit for the original grid squares covered. Finally, each of the 8x8 sub-divisions of the original grid is itself subdivided into four 4x4 grids, whose squares now coincide with the squares of the original 16x16 grid. The third digit for the value assigned by U(X) to each of the original grid squares is given by the number indicated in each of these final squares. So, for example, the original grid square which is labeled (0110,1010) maps under U(X) to the value 110, obtained from the (01,10) square of U(X) which is 1; the (11,10) square of U(X) which is also 1; and the (10,10) square, which is 0.

92

0 0 0 0 ---- - - ----- L ----1 1 0 0 0 1 _----r-----

0 1

0 1 ----------

1 0

0:0:0:0 0;0;1;1 0:1 i0 :1 0;1:1:0

0:0 :0 :0 0:0:1:1 0i1i01 0:1:1:0

0:0 :0:0

0:0 ;0 :0

0:0:1:1

0:0'1,1

0:1:0:1 0,1,1:0

0;1:0:1 0:1:1,0

Second Digits of U(X)

First Digit of U(X)

Third Digits of U(X) (black = 1, White = 0) Figure 5.3 Representation of U(0100) Acting on 4-Digits

The identities which can be derived from Theorem 5.3 allow rules to be grouped into additivity classes, each rule in a class being non-additive in essentially the same way. The fact that U(X) is the same for all non-additive 2-site rules for which x0 = 0, for example, indicates that all of these rules belong to a single additivity class. All 2-site additive rules have x0 = 0, and contain two 1's. The non-additive 2-site rules with x0 = 0 all have either a single 1, or three 1's, and all can be represented as an additive rule plus a rule containing only a single 1. Table 5.1 gives the obstructions to additivity for all of the 3-site basis operators.

93

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

U((3+)

0

0 0

0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0

0 0 0 0 0 0 1

0

1

0

1

1

1 0 0 0 0 0 0 0 0 0

0

0

0

1 0

1

0

1 0 1 0 1 0 1 0 1

1

0

0

0 0

0 0

0

0

1

1 0

1 0

0

1

0

0

0 0

1 0

1 0

0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 0

0

1 0

0

0

0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

1

0

0 0

0

0

0 0

0

1 0

1

0

0 0 1 0 0 0 1 0 0 0 1 0 0 0 0

1 1

_ 0 U(9) 0

1

0

0 0

0

1

1 0 0

0

0

0

1 0

0

1

0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 1 0 1 1

0

1 0 0

0

0 0

1

0

0

0

1 0

1 0 0

0

1

1

1

1

1

0

0

0

1

0

0

1

0

0

0

0

0 0

0 0

1

1

1

1

1

1

1

1

1 0 0 0 0

1

1 0

0

0 0

0 0 0

U(r) =

0

0 U01-) = 0 0

1 0

_ 0 0 U(x) 0 0

0 0

0 0

0 1 0 0 1 0 0 0

0

1

0 0 0 0

0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0

0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0

1

1 0 1 0 0 1 0 0 0

U(0_)

1 0 0 0

1

1 0 0

1

0 U01+) _ 0 0

1

1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0

1

1 0

0 0

1

1

0 0

0 0

1

1 0

1

0 0 1

1

1

1

1

0

1

1 0

0 0 0 0

0

0

1 0

0 0

1 0

0

0

1 0

0

0

0

0

1 0 1 0

0

0 0

1 0 U(x)

0

0

1 0

0

0

0

0

1 0

0

0

0

0

0

1 0

0

_ 1 0 0 1 0 0

0

1 0

1

1 0 0

0

0

1 0 0 0 1 0 0

1 0 0

0

0

0

1 0 0

0

0

0

0

1 0 1

0

Table 5.1 Obstructions to Additivity for 3-Site Basis Operators

Counting the eight 3-site additive rules as a 0-class, there are a total of sixteen additivity classes for the non-generative 3-site rules. These classes

94 are listed in Table 5.2. Approximations to the graphs of U(X) for each of the fifteen non-trivial additivity classes are shown in the figures at the end of this chapter. Class Form Rules 0 Q 0, 60, 90, 102, 150, 170, 204, 240 1 Q+B+ 26, 38, 64, 124, 140, 176, 214, 234 2 Q+B- 8, 52, 82, 110, 158, 162, 196, 248 3 Q+11+ 2, 62, 88, 100, 148, 168, 206, 242 4 Q+11- 16, 44, 74, 118, 134, 186, 220, 224 5 Q+x 22, 42, 76, 112, 128, 188, 218, 230 6 Q+O 28, 32, 70, 122, 138, 182, 208, 236 7 Q+t 4, 56, 94, 98, 146, 174, 200, 244 8 Q+B++x 12, 48, 86, 106, 154, 166, 192, 252 9 Q+13-+x 30, 34, 68, 120, 136, 180, 210, 238 10 Q+B++O 6, 58, 92, 96, 144, 172, 202, 246 11 Q+13-+8 20, 40, 78, 114, 130, 190, 216, 228 12 Q+x+0 10, 54, 80, 108, 156, 160, 198, 250 13 Q+x+t 18, 46, 72, 116, 132, 184, 222, 226 14 Q+O+t 24, 36, 66, 126, 142, 178, 212, 232 15 Q+x+O+t 14, 50, 84, 104, 152, 164, 194, 254

Table 5.2 Additivity Classes of Non-Generative 3-Site Rules

2. Cvclotomic Analysis Their simplicity of representation has made additive rules fertile ground for research, and the properties of this type of rule have been extensively studied. Generally, this work has been restricted to rules defined on the finite state space En. Basic results on cycle periods and maximum number of iterations necessary to reach a cycle were obtained in the now classic paper by O. Martin, A.M. Odlyzko, and S. Wolfram [33]. Analysis in this paper was carried out in the language of dipolynomials defined over finite fields. Major contributions have also been made by E. Jen [34,351, working in terms of recursion relations over finite fields, an approach also used by J. Urias [65]. Another approach to additive CAs defined on En, employed by P. Guan and Y. He [66], makes use of a representation in terms of circulant matricies.

95 All of these different approaches are particular representations of the abstract operator formalism used in this book. After a brief review of the dipolynomial and circulant formalisms , some results will be given which make use of representation in terms of complex polynomials. The immediate result is a criterion for the injectivity of additive rules. In the dipolynomial approach , configurations are represented as polynomials. If µ = 41...4ne En then 4 defines a polynomial µ(y) by n µ(Y) _ Dui-1 (5.11) i=1

Multiplication of this polynomial by y±s, with the resulting dipolynomial reduced modulo yn-1 according to n

n

JNiy'-1mod(yn -1) = i=1

I [j

4 +jnjy'1 (5.12)

i=1

clearly corresponds to the action of a±s on g. Hence, every additive operator on En can be represented as multiplication by a dipolynomial. The diploynomial representation is found by direct substitution oft fora in equation ( 1.5). This form of representation allows application of well known division and congruence properties of dipolynomials . If neighborhoods are left justified, then the representation is in terms of polynomials rather than dipolynomials. Taking a different approach, P. Guan and Y. He [66] represent elements of En as n-dimensional column vectors . Then consider left justified rules, and write the evolution equations as µ(t+1) = X(µ(t)) (5.13) where X is a representation of the rule X in terms of an nxn circulant matrix, obtained from the shift representation of equation (1.5) by substitution of the circulant representation of the left shift: (0 1 0 0 ... 01

1 0 0 1 0 ... 0 a=circ(010...0)=

(5.14) 10 0 0 0 ... 1 1 0 0 0 ... 0

As in the dipolynomial case , the value of the circulant representation lies in the fact that many properties of circulant matricies are known.

96 A direct connection between dipolynomial and circulant formalisms can be made in terms of certain complex polynomials p(wn) where wn = e2m/n is an n-th root of unity. Generally the subscript n will be supressed in what follows, understanding w to be defined in terms of whatever value of n is being used. Replacing the variable yin equation (5.11) by w gives the representation of a configuration g as n µ(w) = J giwi-1 (5.15) i=1

and the action of an additive rule X takes the form of multiplication by the complex conjugate of the polynomial n-1 p(w) _ I aiw' (5.16) i=O

where ai is the i-th entry in circ(a0al ... an- 1), the circulant matrix which represents X. In this form, the reduction modulo wn-1 is automatic as a result of the identity wn = 1. The following results are taken from a survey of circulant matricies by P.J. Davis [67]. For further properties of circulant matricies the reader is refered to this very excellent survey. Lemma 5.5 (Davis, [671) An nxn matrix A is circulant if and only if [A,a] = 0. Lemma 5.6 (Davis, [67]) An nxn matrix A is circulant if and only if A = pA(a) for some polynomial PA. If A = circ(a0al ... an- 1) then n-1 PA(a) _ Iaia' i=0

Definition 5.7 The Fourier matrix of order n is the matrix 1 1 1 ... 1 1 wn 1 wn-2 ... w F = = 1 wnn

2 wn-4 w2

(5.17)

w w2 ... wn-l

The Hermitian conjugate (i.e., transpose complex conjugate) of this matrix will be denoted F*.

97

Lemma 5.8 (Davis, [67]) 1. F is unitary: FF* = F*F = I. 2. The eigenvalues of F are ± 1 and ±i with multiplicities which depend on n. 3. The characteristic polynomials of F* are: (x-1)2(x-i)(x+1)(x4-1)(n/4)-1 n = 0 mod(4) (x-1)(x4.1 )( n-1)/4 n = 1 mod(4) (x2-1)(x4.1 )(n-2)/4 n = 2 mod(4) (x-i)(x2-1)(x4.1)(n-3)/4 n 3 mod(4) Lemma 5 .9 (Davis, [67]) Let A = circ(a0a 1 ... an-1) have associated polynomial pA and let A(A) denote the diagonal matrix diag(pA(1),pA(w),...,pA(wn-1)). Then A

=

F*A(A)F

(5.18)

Corollary 5.10 The eigenvalues of A are given by Xi = pA(wi) Since [a(g)]i = µi+1 the shift operator is equivalent to multiplication by vn-1, the complex conjugate of w. Hence the action of the rule defined by the circulant matrix A is represented by multiplication by pA(wn-l), the n-th eigenvalue of A. Corollary 5.11 If A is non-singular then A-1 = F*A-1(A)F (5.19) It is also possible to carry out a spectral decomposition of additive rules. If ni = diag(0,0,...,0,1,0,...,0) with the 1 being in the i-th position, and matricies Ili are defined by Ili = F*niF, then the matrix A defined by equation (5.18) can be expressed in the form n-1 A= IpA(w')IIi (5.20) i=0

The matricies Ili are Hermitian , and satisfy the conditions r Ilinj 0 i#j = jl Rj i=J n-1 :IIi=1 i=0

That is, they are idempotent, orthogonal, and form a resolution of unity. Thus, they are projection matricies.

98 3. Infectivity A cellular automata rule X:E-E is surjective if every configuration has a predecessor. That is, for every Be E there is a gE E such that X(µ) =13. If in addition this predecessor state is unique, the rule is in' i . It is known that for cellular automata injectivity is equivalent to reversibility . Hence, if a rule Xis injective then there is another rule X-1 such that if Be E, X(µ) = B, then X- 1(B) = g. It is also known that the question of whether or not a particular CA rule is injective is decidable only in dimension one [68,69]. When considering injectivity, attention can be restricted to those rules defined on En since Hedlund [49] has shown that a rule defined on E+ will be injective if and only if it is injective on En for all n. Since reversibility and injectivity are equivalent , an additive rule represented by a circulant matrix X will be injective if and only if X-1 exists. Since the Fourier matrix is non- singular, and A(X) is diagonal, equation (5.19) implies that X-1 exists if and only if none of the diagonal entries in A(X) are zero . Recalling that these entries are to be reduced mod(2) yields the condition for injectivity of additive CAs. In what follows , the polynomial px is allowed to take on arbritrary complex values z. Theorem 5.12 Let X: En-4En be an additive CA rule represented by a circulant matrix X = circ(a0a1 ... an-1). Then X will be injective on En if and only if no n-th root of unity is a root of the complex polynomial px(z) mod(2). Since wn = 1 is an n - th root of unity, this condition requires that an odd number of the coefficients as be non -zero. Also, the roots of complex polynomials come in complex conjugate pairs , so if wr is a root then w -r is a root as well. Theorem 5 . 12 shows that the injectivity of X:En -+En is equivalent to the irreducibility of px(z) with respect to the n -th roots of unity . If X is to be injective in general , then it must be so for all n. That is, pX(z) must be irreducible with respect to all roots of unity. Theorem 5.13 An additive CA rule X will be injective on E+ if and only if PX(z) z) dz = 0 (5.21) lim 2ni f e-0 c(c) PX (

99 where p'X(z) is the derivative of pX(z), and c(c) is the contour consisting of a circle of radius 1+c and a circle of radius 1-c. Proof For any closed contour C the integral of equation (5.21) counts the number of zeros minus the number of poles of the function pX(z) which are contained within the contour. Since pX(z) is a polynomial it has no poles, and only isolated zeros. Hence the quantity in (5.21) counts the number of zeros of pX(z) which lie on the unit circle. Any such zero would be a root of unity for some n, hence X can be injective for all n if and only if equation (5.21) is satisfied. I The 3-site rule 150 provides a good example. For this rule, in left justified form, X = circ(1110...0). The action of this rule on a configuration µ(w) is given by the product (1+wn-2+vn-1)µ(w), and pX(z) = 1+z+z2. The roots of pX(z) are given by Z=-1-i43-

2 2 which is equal to e2ni/3 (+) or e4xi/3 (-). If n is a multiple of 3 then each of these will be equal to some power of w, hence rule 150 is injective on En if and only if n is not a multiple of 3. An extension of this well known result is given by the next theorem. Theorem 5.14 Let X:En-*En be a k-site additive CA for which every coefficient as in equation (1.5) equals 1. If k is even then Xis never injective. If k is odd then X will be injective for all values of n which are not divisible by k. Proof If all of the coefficients in equation (1.5) are 1 then pX(z) = 1+z+z2+...+zk-1 If k is even there are an even number of coefficients and pX(1) = 0 mod(2). Hence X cannot be injective. If k is odd then pX(1) = 1 mod(2), but w = e2m/r is a root for 1<-r<_k. Hence for n = mk, e2xxmi/n will be a root. Since pX(z) has degree k-1 there are only k-1 roots, so no other values of n can yield roots. Hence X will be injective so long as n # mk. I Table 5.2 lists the additive rules which are injective for at least some n, for k<-5. With the exception of the final rule in this table, which

100 illustrates Theorem 5.14, all injective rules are combinations of rule 150 and shifts. k 5 4 3 5 2 5 5 4

InectivitvConditions always always always n#3m always always always n#3m

00001 00010 00100 00111 01000 01011 01101 01110

Shift Form a4 a3 a2 a2A a a(I+(;2+a3) a(I+a+(y3) GA

10000

I

1

always

10011 10101

I+a3+a4 A2

5 5

always n*3m

10110 11001 11010 11100 11111

I+a2+a3 I+a+a4 I+a+a3 A I+a+a2+a3+a4

4 5 4 3 5

always

ag

always always n#3m n*5m

Table 5.2 Injective Additive Rules fork < 5

4. Another View of Infectivity The connection between the complex variable approach taken here, and the polynomial approach is extremely close. The configuration space En is isomorphic to the finite field F2n defined over Z2. Let ge En and consider the extension Z2(w) where w is the primitive n-th root of unity e21riln. Since wn-1 = 0, wr is algebraic over Z2(w). Theorem 5.15 (Lidl & Pilz, [701) For no > 0 an odd integer: 1. There exists a finite extension Z2(w) of Z2 which contains a primitive n0-th root of unity. 2. Z2(w) is the splitting field of xn0-1 over Z2. 3. xn0-1 has exactly no distinct roots in Z2(w). These roots form a cyclic group generated by the primitive n0-th roots of unity. There are ^(n0) primitive n0-th roots of unity given by wr with gcd(n0,r) >_ 1, where 0 is the Euler function.

101

Z2(w) is the associated cyclotomic field to Z2. Definition 5.16 The polynomial qn (z) = 11(z - wd ), with coefficients in Z2, d < n, d\n

where d\n indicates that d is not a divisor of n, is the n-th cyclotomic polynomial over Z2(w). Theorem 5.17 (Lidl & Pilz, [70]) zn-1=[Jgd(z) din The qn(z) are, in turn, determined by the expression qn (z) = 11(zd -1)µ(n/d) din

where 1

r=1

µ(r) r = pl...ps Pi prime, Pi Pj 0 otherwise is the Mobius function. In general not all of the qd(z) are irreducible. As stated, considerations have been restricted to odd n. A simple induction argument, however, making use of the Z2 identity (zm-1)2 = z2m-1, extends this to n = 2mn0. Lemma 5.18 Let no >- 0 be odd, with no z 1= fj qd(z) ding

Then 2m

2m z2 no -1=

11 dino

The injectivity result given in Theorem 5.12 can also be stated by saying that an additive rule X:En- En is injective if and only if pX(z) does not split over Z2(w). The cyclotomic decomposition of pX(z) is used by Martin, et al [331 to count numbers and lengths of cycles. 5. Exercises for Chapter 1. Construct a table for 4-site rules similar to that of Figure 5.2.

102 2. Let the 3-site rule X be in additivity class 8 , and the 3-site rule Y be in additivity class 9. Prove that U(Y) consists of four copies of U(X), located in the four quadrants of the unit square (see Figures). 3. Prove the obstruction U(X) is symmetric about both diagonals of the unit square for 3-site rules X in additivity class 14. (See Figure) 4. The non-generative , non-additive 3-site rules partition into fifteen different additivity classes. These classes form a group when addition of two classes is defined in terms of the component-wise binary addition of their members. Construct the group table for this group. 5. Prove that all 3-site non-generative, non-additive rules X satisfying the condition X(iOili2) = i0 + Y(ili2), where Y is a 2-site rule, belong to the same additivity class. 6. Prove that all non- generative, non-additive 3-site rules X satisfying the condition X(iOili2) = Y(iOil) + i2 belong to the same additivity class. 7. Prove that if a 3-site rule X satisfies the condition given in exercise 5, then the map U(X) consists of four copies of the map for the nongenerative , non-additive 2-site rules , one copy located in each quadrant of the unit square. 8. How many additivity classes are there for the non-generative 4-site rules? 9. For what values of n, if any, are the following rules injective? a) I+(;4+a5

b) I+(y+a6 c) I+a2+a3+a4+(;6 d) I+a+a5+a6 10. If a configuration µ(w)e En then rule 150 acts on this configuration via multiplication by p(wn-1) = l+wn-2+vn-1 where w = e2161n. If n is not a multiple of 3 then rule 150 is injective, hence there must be a polynomial p-1(w) such that p-1(w)p(wn-1) = 1. Find this inverse polynomials for n = 5 and for n = 7. This shows that the inverse of an injective rule acting on En will in general depend on the value of n.

103

Non-Trivial Additivity Classes for 3-Site Rules Graphs are color coded on a 128x128 grid. Each grid site is coded by two 7-digit binary strings, which map to 5-digit strings under a 3-site rule. Each five digit string resulting from X(µ)+X(µ')+X(µ+µ') defines a particular color. Pure green stands for 0, pure blue for 15, and pure red for 31. Values in between are represented by combinations of colors. Green areas are regions in which rules in a given class are additive, at least to 7-digits. Red areas are regions in which they are immediatly non-additive.

104

Additivity Class 1

105

Additivity Class 2

106

Additivity Class 3

107

Additivity Class 4

108

Additivity Class 5

109

Additivity Class 6

110

Additivity Class 7

111

Additivity Class 8

112

Additivity Class 9

113

Additivity Class 10

114

Additivity Class 1 1

115

Additivity Class 12

116

Additivity Class 13

117

Additivity Class 14

118

Additivity Class 15

119

Chapter 6 Additive Rules: II. Cycle Structures and Entropy Major work on additive rules has been carried out to determine the structure of their state transition diagrams. This structure is specified in terms of such quantities as the in degree of verticies, maximum tree heights, cycle periods, and numbers of fixed points and cycles. After reviewing some of this work, some results on reachability are derived using the complex polynomial formalism developed in Chapter 5. Following this, the formalism is used to derive conditions for a configuration to lie on a cycle. Finally, two theorems are proved which show that the set of fixed points of an additive rule, and the set of all configurations on cycles, are sub-groups of the group {En,+}. 1. State Transition Diagrams Since the configuration space En is finite, every configuration maps eventually to either a cycle or fixed point. If a rule is injective, every configuration is on a cycle. Otherwise, there will be Garden-of-Eden configurations, which may serve as initial conditions but which are unreachable in the CA evolution. The state transition diagram (STD) for a k-site rule X:En-*En is a graph with 2n verticies labeled by the set {iO...in-1 I is = 0,11. An edge connects vertex i to vertex j if and only if X(i0...in-1) = X(j0...jn-1)• Since X:En-En is a function each configuration maps to a unique image, and the STD for X will consist of a set of trees rooted on cycles or fixed points. Each connected component of the STD will be a fixed point or cycle together with the trees rooted thereon. The configurations which label verticies at the tops of the trees are the Garden-of-Eden configurations. If h(X,n) is the maximum tree height for X:En-)En then states on trees at heights 0 < h' _< h(X,n) cannot appear in further evolution of the rule after at most h(X,n) - h' iterations. After h(X,n) iterations of the rule only configurations which are fixed points, or which lie on cycles remain. Thus, iteration of a non-injective rule on En necessairly decreases the availiable volumn of configuration space, resulting in a concomitant reduction of entropy. The maximum entropy reduction is achieved when only configurations on cycles remain. For a rule acting on E+ the reduction in entropy is related to the Hausdorff dimension of the Cantor set which is the image space of the rule,

120 as will be further discussed in Chapter 12. For additive rules acting on E+ there can be no entropy reduction [36]. On En, however, evolution according to a non-injective additive rule does decrease entropy. Thus, when the same rule is iterated on E+, it reduces entropy on the subset of all periodic sequences, but not on the entire configuration space E+. The reason behind this is that for additive rules acting on E+ there are no Garden-of-Eden configurations. The existance of such configurations in En is an artifact of the maximum period limit in En. For example, for the binary difference rule D the configuration 100 e E3 has no predecessors. It does, however, have two predecessors in E6: 100011 and 0111 00 . Figure 6.1 shows the STD for the binary difference rule, D:E6--*E6. The diagram is composed of four connected components, consisting of height two binary trees rooted respectively on a fixed point, a period 3 cycle, and two period 6 cycles.

Figure 6.1 STD for D:E6-.E6

121

2. Cycle Periods The basic studies of additive rules acting on En was carried out by O. Martin, A.M. Odlyzko, and S. Wolfram [33], and in several papers by E. Jen [35,55,62]. Contributions have also been made by a number of other researchers, including P. Guan & Y. He [66], D.A. Lind [66], and W. Li, N.H. Packard, & C. Langton [451. Much of what is presented in this and the following section is a recapitulation of this previous work. The major questions of interest to be dealt with here are the question of cycle periods, maximum tree heights, and reachability. The basic theorems on these questions are contained in the work of Martin, et al, and of Jen. Theorem 6.1 (Martin, et al, [33]) The periods of all cycles of an additive rule acting on En divide the period b(n) of the cycle obtained by starting from an initial state containing only a single 1. Theorem 6.2 (Martin, et al, [331) Let n = 2n0 with no odd. Then b(n) 12b(n/2). Further results on cycle periods are obtained in terms of multiplicative order and suborder functions. Definition 6.3 For n = no the multiplicative order and suborder functions are defined

by: o(n,m) = min(r) 12r = 2m mod(n0) s(n,m) = min(r) 12r = ±2m mod(n0) (6.1) Lemma 6.4 o(2n,m+1) = o(2m+ln0,m+1) = o(2mn0,m)+1 s(2n,m+l) = s(2m+ln0,m+1) = s(2mn0,m)+1 (6.2) Proof By definition, 2o(n,m) = 2111 mod(n). Hence there is a K > 0 such that 2o(n,m) = Kn+2m. Therefore 2o(n,m)+l = 2Kn+2m+l = 2m+1 mod(2n). Now suppose that there was an r < o(n,m)+1 such that 2r = 2m+1 mod(2n). Then there will be a K' such that 2r = 2K'n+2m+l so that 2r-1 = K'n+2m = 2111 mod(n) and r-1 < o(n,m) which is a contradiction. A similar argument shows that the suborder identity is valid. I Theorem 6.5 (Martin, et al, [331) Let n = 2mn0 with no odd, and let X:En-En be a k-site additive CA rule. Then the maximum cycle period b(n;X) satisfies

122 b(n;X) 12o(n,m)_2m

(6.3)

Further, if X is symmetric then b(n;X) 12s( n,m)_2m

(6.4)

Proof Martin, et al demonstrate that for n = no b(no;X) 12o(no,m)_1 while if, in addition, X is symmetric then b(no;X) 12s(n0,m)_1 By Lemma 6.3, however, o(n,m) = o(n0,0)+m s(n,m) = s(n0,0)+m Hence 2o(n,m)-2m = 2m2o(no,0)_2m = 2m(2o(n0,0)_1) 2o(n,m)-2m = 2m(2o(n0,0)_1) and the claim follows by application of Theorem 6.2.1 In many cases the maximum cycle period actually equals 2o(n,m)_2m or 2s(n,m)-2m. Jeri [551 has shown that when this is not the case it is a result of an "anomalous shift" which acts to reduce the cycle period. Because of this relation between shifts and mapping site, it is important to be aware that the cycle period may depend on the choice of mapping site. For example, rule 90 acting on E6 has a maximum cycle period of 2 when taken as a nearest neighbor rule, and a maximum cycle period of 3 when taken as left justified. More generally, if a rule X obeys the equation Xr(g) = a-s(µ) for all configurations g which lie on cycles then b(n;X) = rt where t is the minimum integer such that st = 0 mod(n). A change of mapping site is equivalent to multiplication by some power of the shift. Thus, on cycles (agX)r(g) = aq-s(µ), and the minimum t' such that t'(q-s) = 0 mod(n) need not be the same as the minimum t such that is = 0 mod(n).

3. Reachability A configuration fie En is reachable by a rule X if there is a configuration g such that X(µ) = B. Clearly all configurations are reachable if Xis injective. If this is not the case, then the unreachable configurations are just the Garden-of-Eden states. The next two theorems restate the reachability results of [33,551 in the language of Chapter 5.

123

A polynomial p(w) is irreducible if p(w) = q(w)r(w) implies that either q(w) or r(w) is a constant. Suppose that X:En-En is represented by the polynomial pX(w), and that this polynomial has a decomposition into irreducible factors given by r s PX (w) = fl Pi (w)[J Dj (w) = P(w)S2(w) (6.5) i=1 j=1 where the pi(w) represent injective rules and the S2j(w) represent noninjective rules . Then A(X) = A(p)A()) and A-1(p) exists, but A-1(i2) does not. Denote the nullity of A(f) by v(X). Theorem 6.6 A configuration 13(w) is reachable by a rule X if and only if S2(w) 113(w). Proof If 13(w) is reachable then there is a µ(w) such that 13(w) = pX(w)g(w). Since pX(w) = p(w)12(w) the first half of the claim is obvious. Suppose that S2(w) 113(w). Then there is a configuration µ'(w) such that 13(w) = S2(w)µ'(w). But pX(w) = p(w)S2(w) and p(w) represents an injective rule. Therefore there is a polynomial p-1(w) such that p-1(w)p(w) = 1. Thus 12(w) = PX(w)p-1(w) and 13(w) = PX(w)p-1(w)g'(w) and p-1(w)g'(w) is a predecessor for 13(w). I In terms of the matrix A(X) the reachability condition may be written as the requirement that there exist a g such that AW)F(µ) = A-1(p)F(6) (6.6) which, when coupled with standard theorems of linear algebra, leads to the next theorem. Theorem 6.7 The fraction of configurations reachable by an additive rule X:En-+En is 2-V(X)The in degree of a vertex in a directed graph is defined as the number of edges directed into that vertex. Similarly, the out degree of a vertex is the number of edges directed out of that vertex. Since CA rules are functions, the out degree of every vertex in the STD is 1. The in degree of all verticies will be 1, however, if and only if the rule is injective. Lemma 6.8 Let X:En-4En be an additive CA rule. The in degree of configurations ge En is given by

124 0 g is Garden - of - Eden P) (X, µ) = 12v( X) otherwise Proof If g is a Garden-of-Eden configuration then d(i)(X,g) = 0 by definition. Otherwise , suppose that X(µ) = X(µ') = 13. Since X is additive this means that X(µ+µ') = 0, or PX(w)(µ(w)+µ'(w)) = 0 = S2(w)(g(w)+4'(w)) = 0 and this has 2v(X) solutions, say yl, i = 1,...,2v(X). Then for µ(w) a particular solution of X(µ) =13, µ'(w) = µ(w) + yl( w), and this exhausts the possibilities. I The next lemma simplifies the study of STD structure for additive rules. Lemma 6 . 9 (Martin , et al, [331) Trees rooted at all verticies on cycles of an additive CA are isomorphic to the tree rooted at Q. The definitive result on STD structure for additive rules defined on En was also presented in [33] . Let X: En-4En be an additive rule represented by pX(w) = p(w)S2(w), and let n = 2mn0 with no odd. Take ci to denote the multiplicity of the i-th irreducible factor of 92(w) over the field Z2 extended to include the n-th roots of unity, with c = min(ci). Theorem 6 . 10 (Martin, et al, [331) The STD for an additive rule X:En-En consists of a set of cycles (including fixed points as cycles of period 1) at each vertex of which are rooted identical 2v(X)-ary trees . A fraction 2-2mv(X) of the possible configurations appear on cycles. For c > 0 (i.e., a non-injective rule) the tree height is I

2^" E

and the trees are balanced if and only if either (a) ei >- 2m for all i; or (b) for all i,j ci = ej and a 12m.

4. Conditions for States on Cycles As discussed in Chapter 3, many CA rules have shift cycles, that is, cycles on which the rule acts as a shift. In addition, Jen [55] has emphasized the fundamental role of shifts by showing that a configuration µe En lies on a cycle of a rule X (additive or not) if and only if there exist integers r and s such that Xr(g)

=

a-s(µ)

(6.7)

125 For additive rules, this fundamental shift relation leads to a formulation of the conditions for a configuration to lie on a cycle in terms of linear recurrence relations [35]. An alternate formulation of these conditions can be given in terms of the complex formalism developed in Chapter 5. Let h(n,X) be the maximum tree height for an additive rule X:En-En. Then Ah(n,X)(X)F(g) is always on a cycle. Further, since all trees for additive automata are isomorphic, a configuration Be En can be on a cycle if and only if there is a configuration g such that F(B) = Ah(n,X)(X)F( g) (6.8) Since A(X) = A(p)A(S2), equation (6.8) is equivalent to Ah(n,X)(Sl)F(g) = A-h(n,X)(p)F(g) (6.9) Since Ah (n,X)(i2) is diagonal , but not invertible , some of its diagonal elements must be 0 . Hence the conditions for B to be on a cycle are just the consistency conditions for equation (6.9). Likewise , the maximum cycle period is given as the smallest integer b(n,X) such that

Ab(n,X)+h(n,X)(X) = Ah(n,X)(X)

(6.10)

b(n,X) is also the smallest integer for which the non-zero diagonal elements of Ab(n,X)( X) are equal to 1 mod(2). In terms of A(X), equation (6.7) becomes n-s(a)F( g ) Ar(X)F( µ ) = A

(6. 11)

where P6(w) = wn-1 and 1 0 0 0 ... 0 0 O w 0 0 ... 0 0

A(a) = 0 0 w2 0 ... 0 0

(6.12)

^0 0 0 0 ... 0 wn-1) As shown by Jen [35], equation (6.7) defines a linear recurrence relation which must be satisfied by the components of all configurations on cycles . Equation (6.11) recasts this relation as a set of linear equations. As an example, consider the binary difference rule acting on E6. The STD for this rule has already been shown in Figure 6.1. On E6, w = e276'6 and powers of A(D) mod(2) are computed to be A(D) = diag(0,1+w,1+w2,1+w3,1+w4,1+w5 ) A2(D) = diag(0,1+w2,1+w4,0,1+w2,1+w4)

126 A3(D) = diag(0,1+w+w2 +w3,w2+w4,0,w2+w4,1+w3+w4+w5) A4(D) = diag(0,1+w4, 1+w2,0,1+w4,1+w2) A5(D) = diag(0,1+w+w4+w5,1+w4,0,1+w2 , 1+w+w2+w5 ) (6.13) A6(D) = diag(O,w2+w4,w2+w4 ,0,w2+w4,w2+w4) A7(D) = diag(O,w2+w3+w4+w5 , 1+w2,0 , 1+w4,w+w2+w3+w4) A8(D) = diag(0,1+w2 , 1+w4,0, 1+w2,1+w4) Thus A8(D) = A2(D) so by equation (6.10) the maximum tree height is 2 and the maximum cycle period is 6. Since for n = 6 , w2+w4 = - 1 = 1 mod(2), A6(D) = diag(O, 1, 1,0, 1, 1) so that if µ is to be on a cycle then the equation A6(D)F(g) = F(µ) implies that the conditions which µ must satisfy are given by [F(µ)10 = [F(9)13 = 0. Making use of equation (5.17), µ1+µ2+µ3+µ4+µ5+µ6 µ1+µ2W5 +µ3W4 + µ4W3 +µ5w2 +µ6w F(µ)=^1 µ1 + µ2W43 µ3W2+µ4w µ5W4+µ6w2 6 µ1+µ2W + µ3+µ4W +95+96W 91 +92w2 +µ3w4 + µ4 +µ5w2 +µ6W4 µl+µ2W+µ3W2 +µ4W3 +µ5W4 +µ6W5

Equating coefficients of like powers of w, the conditions for µ to lie on a cycle can be read off as µ 1+µ2+µ3+µ4 +µ5+µ6 = 0 91+93+95 = 0 (6.14) µ2+94+96 = 0 The second and third of these conditions already imply the first. Taken together, they give the necessary and sufficient form for a configuration µ which is on a cycle: ( 6.15) µ = (µ1,µ24µ3µ4,µ1+µ3µ2+µ4 ) indicating that 16 of the 64 possible configurations are on cycles, a result confirmed by Theorem 6.10 since v(D) = 1 for n = 6. The shift relation of equation (6.11) can also be verified for r = 2 , s = 2 (Or, n-s = 4). Computation of A4(a)F( g ) yields

127

µ1+µ2+µ3+µ4+µ5+µ6 91W 4 + µ2w3 + µ3w2 + 4w + µ5 + µ6w5 9 1 91W 2 +92+93W 4 + µ4w2 + 95+96W 4 A4(a)F(µ)= 91+42w3+93+ 94F3+95+96K'3 91W 4 +92+93W 2 + µ4W4 +95+46W 2 µ 1K'2 + µ2K'3 + µ3K'4 + µ4W5 + 95 + 96K' while A2(D)F(g) is given by 0 9 1+93 +( 42+96 ) w+(9 1+95 ) w2+(94 + 96)w3+ ( 93+95 ) w4+(92+44)w5 1 91 +93+44+96 + 4t2+93+45+µ6 ) w2+(µl+112+44+45)w4

0 µ1+µ3 + µ4 +µ6 + (µ1+µ2 +µ4 +µ5 ) w2 +(µ2 +µ3 +µ5 +µ6)w4 91+93+ ( 92+94 ) w+(93+95 ) w2+(94 + 96)w3+ ( µl+µ5)w4+(42+96)w5 Which, on comparison with A4(a)F(µ) again yields the conditions of equation (6.14), as well as verifying equation (6.11).

It is also possible , using this method , to determine configurations on cycles of less than maximum period. Theorem 6.11 Let X:En-4En be an additive CA rule , and let $ be a proper subset of {0,...,n-1} such that I gi # 1 i E 15 [Ar(X)].. = n l mod(2) i e 15 Then µ is on a cycle of X of period r if and only if [F(µ )]i = 0 for all ie 15. In the D :E6-.E6 example , w2+w4 = - 1 = 1 mod(2), hence A3(D) = diag(0,1+w+w2+w3 , 1,0,1,1+w3+w4 +w5) mod(2) so that a configuration µE E6 will be on a period 3 cycle of D if and only if in addition to [F(µ)] 0 = [F(9)]3 = 0 it also satisfies [F(µ)]1 = [F (9)]5 = 0. Making use of the n = 6 identities w3 = w2+w4 = -1, w4 = -w, and w5 = -w2, and taking coefficients mod(2 ) these last two conditions become ( 91+µ4 ) + ( 93+96 )w + (92+95)w2 = 0 ( 91+µ4 ) + ( 92+95)w + ( 93+96)w2 = 0 which, setting coefficients of distinct powers of w to 0 , yields the conditions µ4=µ1 µ5 = µ2 96 = µ3

128 Thus µE E3, and in addition , by equation (6.14) µl + µ2 + 93 = 0 so that µ must be shift equivalent to 11 0 . If D:E7-E7 is considered , the components of Ar(D), with coefficients reduced mod(2), are found to be A(D) = diag(0,1+w,1+w2 ,1+w3,1+w4, 1+w5,1+w6) A2(D) = diag(0,1+w2 ,1+w4,1+w6,1+w,1+w3,1+w5) A3(D) = diag(0,1+w+w2 +w3,1+w2+w4+w6 , 1+w2+w3+w6, 1+w+w4+w5,1+w+w3+w5 , 1+w4+w5+w6) A4(D) = diag(0,1+w4,1+w,1+w5, 1+w2,1+w6,1+w3) A5(D) = diag( 0,1+w+w4+w5 , 1+w+w2+w3,1+w+w3+w5, 1+w2+w4+w6 , 1+w4+w5+w6 , 1+w2+w3+w6) A6(D) = diag(0,1+w2+w4+w6 , 1+w+w4+w5 , 1+w4+w5+w6, 1+w+w2+w3,1+w2+w3+w6 , 1+w+w3+w5 ) A7(D) = diag(0,1,1,1, 1,1,1)

(6.16)

Thus the cycle period is 7 and the condition that a configuration g be on a cycle is obtained from [F (µ)]0 = 0, which requires that an even number of the µi be equal to one. Inspection of equations (6.16) indicates that the transition from A(D) to A2(D) to A4(D) and back to A(D) is given by repeated applications of the permutation

0

1234567) 1526374

to the diagonal elements of these matricies, while the transition from A3(D) to A5(D) to A6(D) and back to A3( D) is obtained by applying the inverse of this permutation: (1234567 1357246 A similar computation for rule 150 with n = 6 yields A(A) = diag(1,1+w+w2,1+w2+w4 , 1,1+w2+w4,1+w4+w5 ) A2(A) = diag( 1,1+w2+w4,1+w2+w4,1,1+w2+w4,1+w2+w4) A3(A) = diag(1,w+w3+w5 , 1+w2+w4,1 , 1+w2+w4,w+w3+w5) A2r+2(A) = A2(A)

A2r+3 (A) = A3(A) r >_ 1

(6.17)

The period 2 oscillation in the value of A(A) makes it clear that the maximum cycle period is 2. Thus, by inspection of A2(A), the conditions for

129 being on a cycle are [F(µ)]s = 0 for s = 1,2,4, and 5 . Refering to equation (5.17), these conditions require that 91+42w5+93w4+114w3+95w2+µ6w=0 9l+92w4 +93w2+µ4+45w4 + 96w2=0 91+92w2+93w4+µ4+µ5w2+96w4=0 91 + 42w + µ3w2 + µ4w3 + µ5w4 + 96w5 = 0 (6.18) Addition of the first and fourth equation , or of the second and third equations , together with the n = 6 identities w+w5 = w2+w4 = 1 mod(2), yields the condition µ2+93+95+96 = 0. Since 1+w2 +w4 = 0 the second and third of these equations are satisfied whenever µl+µ4 = 92+95 = µ3+46. In addition, simultaneous satisfaction of the first and fourth equations imposes the condition that either all µi are equal , or µ1 = µ3 = 95 and 92 = 94 = 96. This gives the only states on cycles as the two fixed points 000000 and 111111, and the period two cycle 010101-101010. The configuration space En is an additive group under component-wise binary addition. The next results relate to some properties of cycles, considered as subsets of this group. Theorem 6.12 Let X: En-En be an additive CA rule, and let PB = {µE En I X(µ) = B) be the set of all predecessors of B. If Xis not injective , then for all BE En X Y- µ =0 µEPp

(6.19)

Proof If X is injective then each configuration has exactly one predecessor and the quantity in equation (6.19) is clearly not Q. If X is not injective, however, then by Lemma 6.8 the in degree of any configuration µ is 0 if µ is a Garden-of-Eden configuration, and is 2v (X) otherwise. In the first case µ has no predecessors and the quantity is automatically Q. In the second case µ has an even number of predecessors so that the quantity in equation (6.19) is a sum over an even number of copies of B, and taken mod(2 ) this is Q. I Theorem 6.13 Let X:En-En be an additive CA rule , and let Cp be any cycle of X having period p. Then

130

X "Y, Y,µ = Iµ EC IAECP

That is, the sum over any cycle of X is a fixed point of X. Proof X(µ) and since g is on Cp so is X(µ). Since Xis additive , X Y,µ = µECp µECp

Hence the sum over X(µ) is the same as the sum over the cycle. I Finally, it is possible to consider the set of all configurations on cycles for an additive rule X. Theorem 6.14 Let X: En-En be an additive CA rule, let {fp} be the set of fixed points of X. Then {fp ,+} is a subgroup of the group {En,+}.

Erad Since Xis additive Q is a fixed point. The inverse of fp is just fp, and if fp and fq are two distinct fixed points then X (fp+fq) = X(fp)+X(fq) = fp+fq is also a fixed point. I Theorem 6.15 Let X:En-*En be an additive CA rule, let Cp be any cycle of X having period p , and let b be the maximum cycle period for X. Then b CX = UCplb,+ plb=1

is a sub-group of {En,+}. Proof Since Xis additive , CX contains .Q as a 1-cycle , and this is the additive identity element . Also, the inverse of µE CX is again µ. Thus it is only necessary to show that µ+µ'E CX whenever µ,µ'E CX. Suppose that µ is contained in a cycle of X having period p, and µ' is contained in a cycle having period q. Without loss of generality , assume that p:5 q. Then p I b and q I b, and in addition XP(R) = µ and Xq( µ') = µ'. Then XsP(µ+µ') = µ + XSP (µ'). But choice of s such that sp = 0 mod (q) then indicates that XSP (µ+µ') = µ+µ', hence the sum also lies on a cycle of X and so is contained in CX. I

131 The two preceeding theorems show that the configurations which are fixed points, and which lie on cycles, form subgroups of En, obviously satisfying the inclusion relations {fp,+} cCXcEn. 5. Entropy Reduction If an additive rule X:En- En is not injective unreachable configurations exist , and entropy decreases as an initially uniform ensemble of configurations is evolved under X. The normalized measure entropy as a function of time (iteration step) is given by Sn(t)=-n 1 Pt 40 1092 Pt (9)

(6.20)

gEEn

where pt(g) is the probability of configuration g after t iterations of the rule. The in degree of verticies in the STD for Xis given by Lemma 6.8 in terms of the nullity of A(X). Lemma 6.16 I d(') (X, 9) = 2n gEEn

(6.21)

Proof The out degree d(o)(X,g) of every vertex in the STD is equal to 1, hence the sum of out degrees over En is just the number of verticies which is 2n. But every edge directed out of a vertex must terminate on some other vertex, and every edge directed into a vertex must have originated at some other vertex. Therefore the sum of in degrees must equal the sum of out degrees. I Since the in degree is the same for all verticies with predecessors it is a characteristic of the rule X, and is denoted independently of configuration as d(i)(X). Let GE(X) be the subset of En consisting of configurations without predecessors, and take N(o) and N(P) as the number of states in GE(X) and En-GE(X) respectively. Then, by Lemmas 6.8 and 6.16 N(P) = 2nd(i)(X) = 2n-v(X) N(o) = 2n(1-2-V(X)) (6.22) Theorem 6.10 indicates that the total number of configurations on cycles for n = 2mnn0 (no odd) will be 2n-2v(X) and, for rules with balanced trees, the number of rules reachable after t iterations of X is given by 2n-tv(X) for t < h(n,X). Thus, if the STD for X has balanced trees, equation (6.20) gives the entropy as

132 1-tv(X) 05t5h(n,X) Sn (t) -

1 - h(n, X)v(X) t > h(n , X) n For these cases , the fractional change in Sn(t) per time step is

( 6.23)

(6 . 24) S„ (t + 1) - Sn (t))=nh(n X ) h(n,X) ( so that v(X) appears as the discrete analog of the rate of change of entropy. If trees are not balanced, however, this rate of change is not constant. The rule DA2 = I+a+a2+a3+a4+a5 acting on E6, for example, collapses to Q_ and 1 after one iteration, while on the second iteration 1-)Q. Thus, for this rule almost all of the entropy reduction occures in the first interation. 6. Exercises for Chapter 6 1. Draw the STD for the following left justified rules acting on E5 and on E6. a) I+a3 b) I+a+a2 2. Draw the STD for rule 90 acting on E6, first as a nearest neighbor rule and next as a left justified rule. 3. Find the maximum cycle period for the binary difference rule D acting on E37. 4. If X is the rule represented by pX(w) = (1+w+w3)(1+w)2, determine which of the following configurations in E8 are reachable: a) µ(w) = w b) µ(w) = 1+w3+w6+w7 c) µ(w) = 1+w4 d) µ(w) = w+w2+w3 5. What fraction of configurations in E 12 are reachable by the rule defined in exercise 4? 6. What is the maximum tree height for each of the following rules acting on E12: a) I+(;+a2 b) I+a2 c) I+(;2+a3 d) I+a

133 7. What are the conditions for configurations to be on cycles for each of the following rules acting on E 12: a) I+a+a2 b) I+a2 c) I+a3 d) I+a2+a4 8. What are the cycle periods, and conditions for configurations to be on cycles of periods which are divisors of the maximum cycle period, for the rules in exercise 7? 9. Compute the entropy as a function of number of iterations for rule 150 acting on E 12 10. Compute the entropy as a function of number of iterations for the rule DL2 = I+a+a2+a3+a4 +a5 acting on E12.

134

Chapter 7 Additive Rules: III. Computation of Predecessors If BE E+ and Xis a CA rule, the predecessor problem is to find those configurations g such that X(µ) = B. For additive rules it is possible to find a closed form solution for the predecessors of any given configuration. This solution involves a partitioning of configurations by shuffling out sub-configurations so as to cast the predecessor equation into the form of a set of coupled equations which can be solved. The solution of these equations often involves an operator B which plays a role of integration with respect to sequence index. Following several examples of solutions, the properties of this operator are derived. 1. Predecessors of 3-Site Rules One of the more interesting questions which can be asked, given an arbritrary CA rule X:E+->E+, is the question of predecessor configurations. That is, for any given BE E+ find all configurations µ such that X(µ) = B. For general CA rules, this is a difficult problem. If Xis additive, however, a closed form solution is possible. To establish this, it is useful to begin with consideration of the left justified 3-site rules. The additive 3-site rules are listed in nearest neighbor form in Table 1.1. Table 7.1 gives the left justified form of these rules. Rule

Symbol

Components

Shift Form

0

0

00000000

0

240 204

I a

00001111 00110011

I identity Cr left shift

170

a2

01010101

a2

60

D

00111100

I+a

90 102

S = D2 D* = aD

01011010 01100110

I+a2 a+a2

150

A

01101001

I+a+a2

Table 7.1 Left Justified 3-Site Additive Rules

All configurations except Q have no predecessors under rule 0. For the rules I , a, and a2 computation of predecessors is trivial. Let a(s)e E+ be the configuration with [a(s)1i = 8is. That is, the only non-zero entry in a(s) is a 1 in the s-th position.

135

Theorem 7.1 Let µe E+ . For the operators I, a, and a2 the solutions of X(µ) = B are given by

1. X=I:µ=B 2. X = a: µ = ala(1) + a-1(f3) 3. X = a2 : µ = ala ( 1) + a2a(2 ) + a-2(B) The leading terms involve boundary conditions specified at the left boundary of sequences in E+. Inspection of Table 7 . 1, together with Theorem 7.1, indicates that predecessors can be computed for all 3-site additive rules if then can be computed for the rules D and A. To compute predecessors for these rules, an operator B:E+->E+ is defined by i

[13(µ)]i = Ipj mod(2)

(7.1)

j=1

B is a parity operator in the sense that [B(µ)]i is 0 or 1 as the number of l's contained in the sequence µ, up to and including the i -th term, is even or odd. A simple computation yields a formula for powers of B. Lemma 7.2 n ' n+i-j

1(i-j+1)

mod(2) (7.2)

where n + i - jl is the i-th entry of the n-th row of the mod(2) Pascal ( i-j+l) triangle. Lemma 7.3 The operators B and a-1 commute. That is, [B,a-1] = 0. Theorem 7.4 The general solution of D(µ) = B is given by µ = alB(a(1)) + Ba-1(B) al = 0 ,1 (7.3) Proof D[alB(a(1)) + Ba- 1(B)] = a1DB(a(1)) + DBa- 1(3) = D(Ba- 1(B)) since B((x(1)) = 1 and D( 1) = 0. But [D(Ba-1(B))]i = [Ba-1(13)li+[Ba-1(B)li+l mod(2), and from equation ( 7.1) this is given by i

i+1

i

i+1

j[6 1(B)lj +I[a 1(B)lj = lBj- , + ^Bj-1=Bi

j=1 j=1 j=1 j=1

136 The boundary condition contains the parameter al which can be either 0 or 1, so equation (7.3) yields two solutions. But it is known that configurations have only two predecessors under D, so equation (7.3) gives all predecessors. I Iteration of equation (7.3) yields the next theorem. Theorem 7.5 The general solution of Dn(i) =13 is given by n a8B1 (a(s))+Bna n(13) as = 0,1 (7.4) S=1

This theorem gives an immediate formula for the predecessors of configurations evolved under the much studied rule 90. Corollary 7.6 Let 8 = D2 be the global operator for rule 90. The general solution to 8(i) =13 is given by i = B(ala(1 )+a2a(2 )) + B2a-2(B). An alternate method for solution of D2(µ) = 13 will be given, however, as an introduction to the method which will be used to compute predecessors for rule 150. Writing D2(µ) = 13 in component form yields the infinite system of equations ii+942 = 13i . Sequences in E+ are now subdivided into odd and even parts according to

[i(o)]i = 42i-1 [i(e)]i = µ2i i = 1,2,3,... (7.5) with a similar division defining 13(0) and 13(e). With this partitioning of sequences, D2(µ) =13 may be written as the pair of equations D(i(°)) = !3(0) D(i(e)) = B(e)

(7.6)

which can be solved use of equation (7.3). The solution i is then reconstructed by use of a "shuffle" operator S*: 90+ 1)/2 i odd (7.7) e i even ii/2 Solution of equations (7.6) is easy since they are uncoupled. This is not the case for rule 150 . Shuffling out odd and even components of sequences in ii = [S*(i( o),i(e))]i =

the same way for the equation A(i) = B yields the set of coupled equations (o) (o) _ (e) (o)

ii +N'i+1- Ili +131 (7.8) iie) + (e)1 = (o) + Bie) which can be written as

137 D(µ(o)) = µ(e)+13(0) D(µ(e)) = a(µ(o))+f3(e)

(7.9)

These equations may be formally solved by use of equation (7.3) to obtain µ(o) = alB(a(1)) + Ba-1(µ(e)+13(o)) µ(e) = a2B(a(1)) + Ba-1(a(µ(o))+13(e)) (7.10) Substitution of the second equation of (7.10) into the first then yields µ(o) = a1B(a(1)) + Bc-1{13(o)+a2B(a(1)) + Ba-l(a(µ(o))+f3(e))} Since a-1(a(s)) = a(s+l), and Cr2a(µ(o)) = a-1(µ(o)+µ(°)a(1)) , this may be rewritten, by absorbing the constant µ(°) term, as (I+B2(r1)(.(o)) = a1B(a(1))+a2B2(a(2))+B2a-2(13(e))+Ba-l(b(o)) (7.11) with al and a2 taking values 0 and 1. Equation (7.11) is solved by inverting the operator I+B2a-1 as indicated in the next theorem. Theorem 7.7 (I+B2(r1)-1 = I + C(2;1) where C(2;1) is defined by ['32]

[C(2;1)(µ)]i =

^(µi- 3s-2+µi-3s-1) S=O

with ia] indicating the greatest integer less than or equal to a. This theorem follows as a corollary to the more general result of Theorem 7.9 which will be proved when k-site rules are considered. Given the solution of equation (7.11) for µ(o), it is substituted into the second equation of (7.10) to find µ(e). The inversion operator C(2;1) has a representation as an infinite lower triangular matrix for which the j-th column (j = 1,2,...) is given by [a i+1 11)]T: 0 0 0 0 0 0 0

N

1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 C(2;1) =

1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1 0

(7.12)

138 2. k-Site Rules The predecessor computation for rule 150 is illustrative of a general technique for computation of predecessors for k -site additive rules. That is, sequences in E+ are partitioned into subsequences, yielding a set of coupled equations which are easier to solve than the original equation X(µ) = B. How this partitioning is done will generally be a matter of choice. There is an analogy to the difference between differentiation and integration: By comparison to Taylor's theorem , the binary difference rule D is the analogue of a derivative with respect to sequence index , since µi+1 = gi+[D(g )]i. The operator B then appears as the analogue of integration with respect to sequence index. Just as there is no general algorithm for computation of integrals, there is no general way to choose the optimal partition of sequences . It is, for example, always possible to choose a partition such that the resulting set of coupled equations involves only the operators I, as, and D: for a k-site rule all that is necessary for this is to choose the number of subsequences in the partition to be greater than or equal to k/2. The equations which result, however, may not be as easy to solve as those obtained from some other choice of partition. In the solution of coupled equations which involve D, or which reduce to equations involving D, the operator B is involved in the form (I+Br(rs). Thus it is useful to be able to invert this operator . For the case r = s = 1 this inversion is easy. Proof of the next theorem follows immediatly from the definitions of the operators B and a-1. Theorem 7.8 1. B = (I+Ba-1) 2. B-1 = (I+a-1) In addition, as shown in Lemma 7 .3, B and a- 1 commute. Thus Bn = :&

: l^ -^

B_n = 1 s+1 :&'': In order to deal with the more general case, matrix representations of the operators B and a - 1 acting on E+ are required . These are given by the infinite lower triangular matricies shown in Figure 7.1.

139 1 0

B=

1 1 1

0 0 0

0 0

..

0 0 0 0

0

0

0

..

1 0 0 0 1 1 0 0 1 1 1 0

0 0 0 0 0 0

.. .. ..

1 0 0 0 0 1 0 0 0 0 1 0

0 0 0 0 0 0

0 0 0

.. .. ..

_1

1 1 1 1 1 0 0 ..

0 0 0

1

1

1

1

1

0 0 0 0

1 0

1

1

1

1

1

0 0

0

1 0 .. 1 1 ...

1 0 0 0 ..

0 0

0

..

1 0

..

Figure 7.1 Matrix Representations of B and a-1

Theorem 7.9 All operators of the form (I+Bra-s) are invertible. The inverse has the form (I+Bra-s)-l = I + C(r;s) (7.13) where C(r;s) has a unique representation as an infinite lower triangular matrix with elements [C(r;s)hj determined by solution of the equation m.( r+1 ) m - j+ 1 [C(rs)lim +[C(rs) lij+s = Slj+s

(7.14)

Proof If an operatorC(r;s) exists which satisfies equation (7.13), then (I+C(r;s ))(I+Br(rs) = I, hence C(r;s)(I+Bra-s ) = Bras

(7.15)

By Theorem 7.8, B-r = (I+a'l)r. Multiplying equation (7.15) on the right by this expression yields

C(r;s)[(I+a-l)r+a-s] = as which becomes r r+l (r+ l) 1m -s 1 -r C(rs) a +a J = 6 (7.16) M=1 m

Making use of the matrix representation for a-l given in Figure 7.1 shows that the matrix elements of (I+(r 1)r+a-s are given by r+1 +S;,j +s j_i5j+r (7.17) [(I +a 1)r +a s]•= i-j+1)

'^

ft1

i ,j+s

otherwise

140

This combines with equation (7.16) to yield equation ( 7.14). Thus, any solution of (7.14) yields a C(r;s ) which satisfies (7.15) and hence (7.13). If this solution is to be unique , however, it is necessary to impose the additional requirement that it have a lower triangular matrix representation I The next theorem follows immediatly by use of equation (7.14) together with theorem 7.8. Theorem 7.10 The operators C(r;s) satisfy the non-linear recurrence relation C(r;s)[C ( r;s)+C(r;s+1)+C(r-l ; s)l = C(r-l;s)[I+C(r ; s+l)l (7.18) Computation of predecessors can also be used to determine configurations on cycles of given periods. If X is an additive rule, and µe E+ is a solution of Xn( g) = g then it is also a solution of (I+Xn )(4) = Q. Thus, the solution set of this equation yields all configurations g which lie on cycles of X having period a divisor of n.

3. Examples A. Consider the homogeneous equation A(µ) = 0, where A is the operator for rule 150 . Equations (7.11) and the second equation of (7.10) now reduce to (I+B2a- 1)(µ(o)) = all + a201 µ(e) = a21 + Bcr 1a(g(o)) The matrix representation of C(2 ; 1) is given in equation ( 7.12). Making use of this , the identity a-la(g(0)) = µ(O)+µ(°)a(1), again absorbing the constant term , and Theorem 7.7, g(0 ) and µ( e) are determined to be the period three sequences g(o) = (al , a2,al+a2,...) µ(e) = (a l+a2 ,a l,a2,... ) The possible solutions are Q (al=a2=0 ), 011 (al=0 ,a2=1), 11 (al=1,a2=0 ), and 1 1 (al=a2=1). Now, suppose that 13 = 10000 in the equation A(µ) =13. Again, the equations to be solved are (7 . 11) and the second equation of (7.10). The following quantities are easily computed: a- 1(13(0)) = 01000 Ba-1(13(0)) = 0111110000 13( 0) = 10000 a- 1(13(e)) = 00010 a-2(13(e)) = 00001 13(e) = 00100 Ba-2(f3(e )) = 0000111110 B2a-2(f3(e)) = 00001010111111010100 Ba-1(13( 0)) + B2a-2(13(e)) = 01110110111000100100

141

From these, using the infinite matrix expression for C(2;1), µ(o) and µ(e) can be computed for the case al = a2 = 0 as the period 15 sequences 4(o) = 010101000111100 µ(e) = 011110001010100 which shuffle together to give the solution for g as g= 001101110110000 This is the initial conditions specified by al = a2 = 0. Any other solution of the homogeneous equation A(µ) = 0 may be added to this to give another solution. Thus, the four possible solutions to 0(µ) = 10000 are given

by Table 7.2. 4 period

1 0

0

001101110110000 15

0 1 1

1

01011

0 1

5 111011000000110 15 100000011011101 15

Table 7.2 Solutions of b(µ) = 14040

Since A:E5-4E5 is injective there can be only one period 5 solution. The solutions given in Table 7.2, however, shows that there are also three period 15 solutions if E5 is embedded as a subspace of E+, again illustrating the fact that for additive rules, at least, configurations at the tops of STD trees in En lack predecessors only as a result of the finite configuration space. B. Rule 150 has already been shown to exhibit distinctive behavior for periods which are multiples of 3. Thus it is instructive to consider the following examples: a) 13 = 100000 b) 13 = 100011 c) 13 = 101010 These particular examples are choosen following a tree in the STD since A(100000) = 100011, and A(1000 11 = 101010. Further, 100000 has no predecessors in E6. Carrying out the computations, the resulting solutions of A(µ) =13 are listed in Table 7.3. For 13 = 100000 there are four period 12 solutions; for 13 = 100011 there are four period 6 solutions; and for 13 = iQ there are three period 6 solutions and one period 2 solution.

142

period

al

B2

0

0

001101100000

12

0

1

010110111011

12

1

0

111011010110

12

1

1

100000001101

12

0

001101

0 1

0 1

010110

6 6

0

111011

6

1

1

100000

6

IL B = 100000:

B=100011:

B = 101010: 0

0

001110

6

0

1

2

1

0 1

01 111000 100011

6

1

6

Table 7.3 Solutions of 0(µ) . B for Specified B

C. For a final example, consider the 7-site rule X = I+a+a3+(y4. For this rule a "3-dimensional" partition is made, defined by µi0) = µ3i-2 µi1) = µ3i-1 µi2) = µ3i

With these definitions, the equation (I+a+(y3+a4)(µ) = B yields the set of three coupled equations D(µ(0)+µ(1)) = 13(0) D(µ(1)+µ(2)) = B(1) D(µ(2)+aµ(0)) = 13(2) (7.19) Rather than immediatly computing the formal solutions of these equations by use of equation (7.3), they are added together to yield an equation involving only 9(0): D(µ(0)+a4(0)) = D2(µ(0)) = 13(0)+13(1)+B(2) (7.20) This equation can be solved immediatly by use of Theorem 7.5: µ(0) = all + a29-1 + B2a-2(B(0)+B(1)+B(2)) (7.21) Theorem 7.4 now gives the formal solutions for the first two equations of (7.19) as

143

g(0)+g(l) = a31 + B(-l(13(0)) g(1)+g(2) = a41 + Ba-l(13(1)) and the solutions for g(1) and g(2) can be immediatly read off from these by making use of equation (7.21): g(0) = all + a201 + B2a 2(13(0)+13(1)+13(2) ) g(l) = (al+a3)1 + a2Q1 + Ba-1(B(0)) +B2a-2(13(0)+13(1)+13(2)) (7.22) g(2) = (al+a3+a4)1 + a2U + B61(13(0)+B(1)) +B2r2(13(0)+13(1)+B(2) ) There are 16 homogeneous solutions, corresponding to the possible values for the boundary conditions al, a2, a3, and a4. These consist of the possible linear combinations of the four solutions given in Table 7.4. For each B #.Q there will be 16 solutions obtained by adding the 15 non-trivial homogeneous solutions to the solution for B with a1 = a2 = a3 = a4 = 0. period

^1- a2 0 0

0

1

001

3

0 0

1

0

Oil

3

0 1

0 0

0 0

000 111

6 1

1 0

4

LI

1 Table 7.4

Basic Homogeneous Solutions for I+a+63+a4

Computation of predecessors can also provide information about cycles. If ge E+ is on a cycle of an additive operator X, with period n, then Xn(g) = g. Turning this around, all configurations on cycles of X with period a divisor of n are solutions of (I+Xn)(g) = 0 (7.23) When coupled with the next lemma, this allows derivation of some results for cycles of period 2n. Lemma 7.11 k-1 Let X be the k-site additive operator X = I asas . Then s=0 X2n = 2°(k`-1)a a2ns L s s=0

Theorem 7.12 A configuration ge E+ lies on a cycle of the binary difference rule D with period a divisor of 2n if and only if gi = 0 for i > 2n.

144

Proof If g is on a cycle of D with period a divisor of 2n then by equation (7.23), (I+D2n)(g) = 0 (7.24) Since D = I+a, by Lemma 7. 11, D2n = I+a2n , hence equation (7.24) becomes a2n(µ) = 0, which completes the proof. I

4. The Operator B A major role in computing solutions to equations of the form X(µ) =13 is played by the operator B:E+->E+ defined in equation (7.1). As indicated at the beginning of section 2 of this chapter, operating on a configuration with B can be seen as integration with respect to sequence index. Lemma 7.2 gives a formula for [Bn(g)h, and Lemma 7.3 indicates that B commutes with the right shift. It is not true, however, that B commutes with the left shift. Lemma 7.13 r-1 [B,ar] = BPJ 1as (7.25) s=0

where P1 is the projection defined by P1(µ) = µ1a(1). Proof The claim is easily demonstrated for r = 1, and the remainder of the proof follows by induction. Assume that equation (7.25) is true for r. Then [Bar+l] = Bar+1 + ar+1B = Bara + ar+1B but r-1 Bar = arB + BPl as s=0

so that [B,ar+l] becomes r-1 arBa + BP 11 as+1+ ar+1B s=0 and on substitution of Ba = aB + BP1 this yields equation (7.25) with the summation from 0 to r. I

In the examples of section 3, it often occured that the predecessor of a periodic configuration µ had a period greater than that of its image. This is a consequence of the period doubling property of the operator B, described in the next two theorems. Theorem 7.14 Let µE E+ have period n. Then:

145

1. If [B(9)1n = 0 then B(µ) also has period n. 2. If [B(µ)]n = 1 then B(µ) has period 2n. Proof Let Itr be the first non-zero entry in the configuration g. By definition, [B(4)]n+1 = [B(9)1n + 9n+1• If g has period n then this equals [B(9)1n + µl. Hence, if [B(9)1n = 0 it follows that [B(g)]n+r = [B(µ)]r for 15 r< n and thus B(µ) has period n. Suppose that [B(µ)ln = 1. Then [B(µ)]n+r = 1+[B(µ)]n for 1<_ r:5 n, and in particular [B(9)]2n = 0. Since g has period n, it also has period 2n, and from the first part of the theorem B(µ) must also have period 2n. I This theorem shows that B doubles the period of configurations which contain an odd number of l's in one period, and leaves the period unchanged if this number is even. The next theorem shows that iteration of B will always result in period doubling. Theorem 7.15 Let µe E+ have period n, and be such that [B(4)1n = 0. Let µr be the first non-zero entry in g. Then there is an s<_ n-r+1 such that [Bs(R)1n = 1. Proof The proof will follow by showing that it is impossible to construct a non-zero period n configuration g for which [B5(µ)]n = 0 for all s. From the definition of B, [Bs(1)1n = [B8(9)]n-1 + [Bs-1(9)]n. Expanding this gives the expression

[B5(4)]n = [Bs(µ)]n-s + [Bs-1(9)]n-s+l + .. + [B3(9)]n-3 + [B24)]n-2 + [B(9)]n-1 + µn (7.26) Further, for all s it is true that [Bs(µ)]r = 1 since µr is the first non-zero entry in R. Also, by assumption, [B(µ)]n = B(µ)]n-l + µn = 0. Thus [B2(9)]n = [B2(9)]n-2, and if this is equal to 1 then s = 2 will satisfy the condition. Thus, suppose that [B2(9)]n-2 = 0. Then [B3(9)]n = [B3(µ)]n_3 and if this is equal to 1 then s = 3 is sufficient. Continuation along this line must, at the worst, eventually reach [Br(R)1n-r, which is necessairly equal to 1. The combination of Theorems 7.14 and 7.15 should make it clear that B can have no cycles other than Q. This result can also be demonstrated by making use of the expression for [Bn(µ)]i given in equation (7.2). Theorem 7.16 B:E+-.E+ has no cycles other than the trivial fixed point Q.

146 Proof Suppose there is an s > 0 and a non-zero µe E+ such that Bs(g) = µ. Then, for all i, µl + ' (s+i - jµ =0 mod ( 2) (7.27) i- +11 j=1 Let µr be the first non-zero entry of µ. Expansion of equation (7.27) yields the hierarchy of equations Cs1

C

Rr =0 22 ) s+1 (s+21 2 µr+1+ 3 Jµr = 0

+3 ) (s ) (s+1) (s +2 2 µr+2+I\3 µr+l+l\ 4 9r=0 etc.

l ( s 21) = 0. Since µr # 0 the first of these equations requires that I Substitution of this into the second equation of (7.28) then r equires that s+2 3 = 0. Repeated substitutions will lead to the general condition that s+j j+1

= 0 for j >- 1 . But these coefficients come from the s-th diagonal of the

mod(2) Pascal triangle , and no diagonal of this triangle consists of a leading 1 followed by only 0 's. Hence equation ( 7.27) can never be satisfied for µ * 0. The proof of Theorem 7.16 follows from the non -existence of a diagonal consisting only of 0 's following the leading 1 in the mod(2) Pascal triangle. This triangle does, however, contain diagonals which have arbritrairly large numbers of 0's following the leading 1, and this fact can be used to show that B is almost periodic , in the sense of being self-accumulating. That is, every iterate of B is an accumulation point for further iterates. Theorem 7.17 For all n,s >- 0, and for all µE E+ g(Bn(µ),Bn+2s (µ)) < [2(22s - 1)l-1 (7.29) where g( p,p') is the metric on [0,1] defined in section 4 of Chapter 1.

Proof It is sufficient to prove the theorem for n = 0. For this case, consider the sequence 4 = µ - B25 (µ), which can be shown to have components

147

i -1

2s +.1 Ij E j=1 i-j+1 - ) The coefficients in this sum are the first i entries of the 2s diagonal of the mod(2) Pascal triangle, with the first entry excluded since the sum is only up to i- 1. By Lemma A.5 of appendix 4, these coefficients are all 0 for i <- 2s. In fact, 2s+1-j _ 1 i-j=0, 2s,22s. (i - j+ 1) 0 otherwise so that the values of i giving non-zero contributions to ^ are i = 2sm+1 for m>1. But l g(µ,B 2s (µ) I ^_^' -2 - 21 i122i+1 2 i=1

The final sum in this expression is just (22s-1 )- 1, finishing the proof. I The content of this theorem is summarized by saying that the operator B is self-accumulating; i.e., with respect to the topology induced on E+ by the metric g, the sequence Bn(g) eventually returns to arbritrairly small neighborhoods of its previous iterates. It is also possible to show that B has no dense orbits . Define a partition of E+ by E+ = U ET where Er = WE E+ I µi=0, i
definition, B: Er -+ ET , so no orbit of B can be dense in E . If gE Er and g 'r= ES , with s < r, then g(g,Bn(g')) > 2-s-2-r. Theorem 7.18 No orbit of B: Es -* ES is dense in Eg for any s. Proof It is sufficient to prove the theorem for the case s = 1. Suppose that µE Ei is such that the orbit of B iterated on g is dense in E. Then, for any given 13,13'E Et, and for all N < oo there will be integers n and m (which may depend on N) such that g(13,Bn(g)) < 2-N and g(B',Bm(g)) < 2-N (7.30) Without loss of generality, assume that n and m are the smallest integers for which these inequalities hold. In terms of the components of g, 13, and B' equation ( 7.30) requires that 13i = [Bn(g)]i and 13'i = [Bm(g)]i, i<- N.

148 Thus, if such a µ exists, it must be possible to simultaneously satisfy the equations

n+l- j 13; = µ j i-j+1 j=l (

(7.31)

m+1j

13;= ^ µj j=1 i-j+1

for all i:5 N. Addition of th ese two eq uations yields

[ [ n±l

13; +13; =,

_i

m+ 1- j J + ( i-j+1)]µi

which can be written in matrix form as `P = fi n,m)

7.32)

(7.33)

where `P and n(n,m) are column vectors in Z2 having components

C

`Pj=13j+f3j n+j-1 nj(n,m) j Xm+j and F is the NxN matrix 0 0 0 µl

...

0

µ2

µl

0

0

...

0

µ3

92

91

0

•.•

0

(7.34)

0 µN 4N-1 9N-2 9N-3 ••• µl Since µe Ei , µl = 1 and Det(f) = 1. Hence I` 1 exists and equation (7.33) has a unique solution n(n,m) = r-1T. Further, this must be the case for all possible choices of 13 and 13'. Another choice would give `I1 l1*, and would involve different values of n and in, but it would not change the uniqueness of the new solution n*(n,m). And all solutions will be given as the sum of the first N entries in two specific diagonals of the mod (2) Pascal triangle. Let r be such that 2r-1 < N:5 2r. Then, for i:52r, the first N terms of the i+2r diagonal of the mod(2) Pascal triangle are equal to the first N terms of the i-th diagonal. Thus, taking only the first N terms, there are 2r distinct diagonals and hence 2r-1(2r- 1) possible sums of two diagonals . `P*, however, is an arbritrary N-vector, with the exception that its first component must be

149 equal to 0 since 131 =13'1 = 1 . Thus, there are 2N-1 possible choices for `Y*, and N can always be choosen large enough so that 2N - 1 > 2r-1( 2r-1). I 5. Exercises for Chapter 7 1. Use equations (7.10), (7 . 11), and Theorem 7.7 to find the general solution to A(µ) = 01001. 2. Give a proof of Theorem 7.10. 3. Find the general solution to the homogeneous equation X(µ) = 0 for the following operators X: a) a+a2 b) I+a3 c) I+a+a3 d) I+a2+a4 4. Find the general solution to X(µ) = 10010 for each of the operators in exercise 3. 5. Use equation (7.23), Theorem 7.5, and Lemma 7.11 to prove that the configurations on cycles of rule 150 which have period a divisor of 2n are given by 2n 2° l bsa(s ) + X asBsa(s+2° ) s=1 s=1

6. Use the result of exercise 5 to find all configurations on period 4 cycles for rule 150. 7. Find the spatial period of configurations on period 5 cycles of D. 8. Use Lemma 7.13 to compute [Br,as].

150

Chapter 8 The Binary Difference Rule This chapter presents an analysis of the geometrical and arithmatical properties of the binary difference rule D. The first part of the chapter contains derivations of some of the basic characteristics of this rule, and then gives an analysis of the graph of D:[0,1]- [0,1]. This is followed by some number theoretic relations contained in the graph, and a theorem which demonstrates that the set of all predecessors of all orders of any point in [0,1] is uniformly distributed in [0,1]. 1. Basic Properties of D After the right and left shifts, perhaps the most ubiquitous cellular automata rule is the binary difference rule. This is the rule D:E-*E defined by [D(i)]i = µi+µi+1 (8.1) which has frequently been mentioned to in previous chapters. This rule was particularly useful in Chapter 7, where it provided the basis for solution of the predecessor equation for additive rules in general. As indicated in Chapter 7, equation (8.1) can be seen as an analogue of Taylor's theorem, indicating that D can be viewed as a discrete derivative with respect to sequence index. Since D(µ) is uniquely defined for all ge E+, the map D:E+->E+ is a function. If D is taken as a map D:[0,1]-*[0,1], however, it is not a function. This is easily seen by consideration of the configurations g and g' defined by gi i < n µi i < n µi= 1 i=n µ;= 0 i=n 0 i>n 1 i>n Both of these configurations define the same point of [0,1], but _ [D(µ)]i i #n -1 [D(µ')], 1+[D(g)]i i=n-1 Every configuration of the specified form corresponds to a rational point of [0,1] having as denominator a power of 2. Hence D:[0,1]-4[0,1] is double valued at all points of the form I(/2s. Despite this, however, D is sufficiently function like that it can be treated as such in many cases.

151

This chapter presents some of the more elementary properties of the binary difference rule. The following lemma follows immediatly by iteration of equation (8.1). Lemma 8.1 [Dn(g)h = n+1(n+1 ) j=1 J

where the coefficients are drawn from the n+1 row of the mod(2) Pascal triangle. One question is whether or not D has cycles. Clearly it must when acting on the finite state space En . The next lemma, a version of Theorem 6.10, details the maximum number of iterations that are necessary to reach a cycle. Lemma 8.2 Let n = 2mn0 with no odd. Under iteration of D, every element of the state space En maps to a cycle or fixed point in at most 2m iterations. Proof If no = 1 then by equation (8.1) and the property of the mod (2) Pascal triangle [D2m(µ)]i = µi + 9i+2m = 0 since i+2m = i mod(n). If no > 1, let k be the smallest integer such that 2k = 2m mod(n). Then [D2k(g)]i = gi + g42k = 11i+2m = [D2m(g)]i which may be written as D2k-2m(D2m(µ)) = D2m(g)]i indicating that D2m(g) is on a cycle. I Returning considerations to the space E+, a configuration ge E+ will be said to terminate at n if there is an n > 0 such that gn = 1, while gi = 0 for all i > n. Likewise, a configuration will be called eventually periodic if there are integers n and m such that for all i, gi+n+m = gi+n• If m is the smallest integer such that this is true then g will be said to be eventually periodic with period m. If no such m exists, then g is non-periodic. If D is treated as a map of the interval [0,11 then eventually periodic sequences correspond to rationals while non-periodic sequences correspond to irrationals. The next few lemmas and theorems establish the invariance of the sets of rationals, and of irrationals, under the mapping defined by D. This, in turn, relates to the fact, mentioned at the end of Chapter 6, that D acting on En reduces entropy, but acting on E+ does not do so. This topic will be taken up further in Chapter 12.

152

Lemma 8.3 Let gE E+ terminate at n. Then µ lies on a cycle of D having period 2r where r is the smallest integer such that n:5 2r. Proof Since the 2r-th row of the mod(2) Pascal triangle contains all 0's, excepting the initial and final 1, application of equation (8.2) yields the following expression [1)2r(g)]i = µi + µi+2r Since g terminates at n:5 2r this means that [D2r(µ)]i = µi for all i. Suppose that there is an s > 0 such that D21-s(g) = g. By equation (8.2) this can be written out as 0=(2' -s+1) 2 1) µ2+ +(2r s+1 µi

Il J 0 = (2r -2s+ 1

)µ3+...+ 1 2r, -s+1 lµj + 2r + 1 µj+1

J

0 = [2r -s + 11µn-1+...+(2r 3 + 1 2 0=

[

)

un

(8.3)

2r -s+ll 2 )

un

where J n 2r-s+1- n = 2r-s+l 2r-s+1
C

2r-s+1 =0. 2 )

Substitution of this condition into the penultimate equation then requires that 2r-s+ll =0. Il 3 J Continuation of this process indicates that the full set of equations in (8.3) will be satisfied if and only if 2r-s+1 >- n, and in addition

153

[

2r_ s^1J_o 2

:5 j
The 2r-s+l row of the mod(2 ) Pascal triangle has 2r-s+l entries, and r is the smallest integer such that z i <- 2r. Hence 2r- 1 < n, so that satisfaction of equation (8.4) requires that more than the first half of this row must consist of 0's following the leading 1. But this will occur if and only ifs = 0. Let µe E+ be eventually periodic with period in. Then g will have the form g = g'+B where 41=1 i=n g1=0 i>n Bi = 0 i- n Bi+m 1>n

Additivity of D, combined with Lemmas 8.2 and 8.3 then yields a theorem on the action of D on eventually periodic sequences. Theorem 8.4 Let g be as above , let nm(B) be the period of the cycle to which B goes under iteration of D, and let r be the smallest integer such that n <_ 2r. Then D interated on g yields a cycle of period at most 2rnm(B). This theorem indicates that all eventually periodic configurations iterate to cycles under the action of D. The next theorems indicates that this property is only true for eventually periodic configurations. Theorem 8.5 Let µe E+ lie on a cycle of D. Then g is eventually periodic. Proof If µ is on a cycle then there is an integer n such that Dn(µ) = µ. Equation (8.2) and the identity n+ 1

n+1

1 )=(n+l)=1 then yields the set of equations Ni+n =

(n 1

1)µi +1 +•••

+(nn 1) µi+n-1

(8.5)

Thus, for all r > n, µr is determined in terms of 92 ,...,µn by a finite recurrence relation, and g must therefore be eventually periodic. I Theorem 8.6 Let µe E+ be non-periodic . Then D (µ) is non-periodic. Proof

154

The proof is by contrapositive, making use of the formula given in Theorem 7.4 for the solution of the equation D(µ) =13. Suppose that D(µ) is eventually periodic with period n. By Theorem 7.4, and the period doubling property of the operator B, µ must also be eventually periodic, with period either n or 2n. I From the above theorems and lemmas it is apparent that D maps eventually periodic configurations to eventually periodic configurations, and non-periodic configurations to non-periodic configurations. Thus, as a map of the interval , D maps rationals to rationals and irrationals to irrationals. In Chapter 12 this property will be shown to characterize all surjective rules. 2. The Graph of D:10.11-40.11 The graph of D:[0,1]-+[0,1] is shown in Figure 8.1 at the end of this chapter. Some of the general properties of this graph are derived in this section. Lemma 8.7 The graph of D:[0,1]-*[0,1] is reflection symmetric about 1/2. Proof D(1/2) = D(1Q) = D(01) = 1/2. Hence 1/2 is a fixed point. Suppose that XE [0,1 /2) and take two configurations µ, µ'E E+ such that µ = 2 - x and µ' = 2 + x. Then µ+µ' = 1 so that D( µ+µ') = 0. Since D is additive this means that D(µ) = D(µ'). I Without loss of generality, further considerations of the graph in Figure 8 . 1 will be restricted to the interval [0,1/2]. Coordinates can be introduced labeling points on this graph, based on the sequence {S(al)} = {(2

-al , 2-ai I 1<_ a1 < ^{ (8.6) )

Inspection of Figure 8.1 suggests that every point of this sequence is the limit point of a second sequence {S(al,a2)}, every point of which is, in turn, the limit point of a third sequence {S(a1,a2,a3)}, and so on, ad infinitum. To make this intuitive idea precise, define a subset of the graph of D consisting of all points having rational x-coordinate in [0,1 /2] with denominator a power of 2. Recall that these are precisely the points at which D is double valued. Points of this subset will be labeled S(a 1,...,an) for some

155

finite n, in a manner described below. The x and y coordinates of S(al,...,an) will be denoted x(al,...,an) and y(al,...,an) respectively. The binary sequence corresponding to x(al,...,an) is defined by the following algorithm: Algorithm 8.8 1. x(1,...,1 (n times)) is the configuration having the first n terms alternating 0 and 1, starting with 0. If n is odd the remaining terms in the configuration are 1 and if n is even the remaining terms are 0. For example, x(1,1)->01Q while x(1,1,1)-0101. 2. For 1<_ i<_ n the configuration for x(al,...,an) is generated from that for x(1,...,1 (n times)) as follows: a. If i is odd, replace the 0 in the i-th position by a block of 0's of length ai. b. If i is even, replace the 1 in the i-th position by a block of 1's of length ai. For example, x(2,3,3)-W0 111000 1 and x(2,3,1,2)-00111011Q. Lemma 8.9 Every number in [0,1/2] of the form K/2s has an expression x(al,...,an) with n finite, and every number which is not of this form has a unique expression lim x(al,...,an). n->-

Proof If x has the form K/2s then x can be expressed as a finite sum of powers of 1/2, and the form of x(al,...,an) can be directly read off from this expression. The second part of the lemma asserts that every number in [0,1/2] can be expressed as the limit of an infinite sequence in powers of 1/2. Points on the graph of D will be characterized in terms of two structural lemmas. Lemma 8.10 x(al,...,an,l) = x(al,...,an+1) y(al,...,an,l) = y(al,...,an+1) + 2-a(l,n)

(8.7)

where n

a(k,n) = Y as s=k

Proof Suppose that n is even. Then x(al,...,an+1) has the form

(8.8)

156 an+l

bal,...,ban_1 ,1,...,1,0 i in which each hai is a block of ai 1's (i even) or 0's (i odd). The sequence x(a1,...,an , 1) will have the form an

ba1,...,ban- 11 and the numerical values of these two sequences are the same. A similar argument for n odd finishes the proof of the first identity. To demonstrate the y identity note that regardless of whether n is odd or even, when the x sequences are written out as members of E+ Dx(al,...,an,1) = Dx ( a1,...,an+1) + (0,...,O,l,Q) where the 1 in the second term on the right is located in the a ( 1,n)-th position. Mapping this equation to [0,1/21 then yields the second identity. I Lemma 8.11 x(al,...,an+1+1) = (1/2)[x(al ,...,an+1) + x(al,...,an)] y(al,...,an+1+1) = (1/2)[y(al ,..., an+1) + y( al,...,an)l Proof Assume that n is odd. Then an+1

x(al,...,an ) = ba,...,ban ,1,...,1,1 x(al,...,an+1 ) = ( ba1,...,ban , han+1'O) (8.9) x(al,...,an+1+1) = (ba1,...,ban,ban + 171'0From these equations,

x(al,...,an ) = x(al,...,an+1+1 ) + 2-a(1,n+1)-1 = x(al ,...,an) + 2-a ( 1,n+1), which proves the first equation . The second then follows since y(al,...,an ) = Dx(al,...,an) and D is an additive map. Similar arguments hold when n is even. Lemma 8.12 The (x,y) coordinates of S(al ,...,an) are given by

157 n x(al,...,an ) = 1- 2-a(4 n) I (-1)n + J (- 1)s-12a(s,n) IL s=1 y(al,...,an) = 2-a(l,n )L1+ 1:2a(s,n) S=1

(8.10)

Proof For n = 1 the claim is true by inspection. For greater values of n the proof proceeds by induction. Assume that equations (8.10) are valid for n. By equation (8.8) and Lemma 8.10 x(a1,...,an,1) = 1-2-a(l n)-il (-1)n + 1(-l)s-12a(s,n)+11 (8.11) L S=1 JI Repeated application of the first equation of Lemma 8.11, starting with x(al,...,an,l) yields an+1-1 1 x(a1,..•,an , an+1) = 2an+1_1 x(ai,...,an, l)+x(ai ,...,an) I 2-s (8.12) S=1 The claimed result then follows by making use of the identity r r 2 -1 Y 2-s = 2r s=1

substituting from (8.11), and making use of the induction hypothesis. A similar argument yields the results for y(ai,...,an). I The set of points s(i,n) = (S(al,...,an) I ar fixed for r # i, 1<- ai < oo} can be shown to lie along a straight line. The proceedure is to pick any two values of ai and use equations (8.10) to compute the slope of the line joining them. The result is independent of ai, hence all points of s(i,n) are connected by straight lines having the same slope. This proves Theorem 8.13 All points of the set s(i,n) lie on a straight line with slope given by n 1+ 2a(s,n)

_ s=i+1 n

(-1)n+ J(-1)s-12a(s,n) s=i+1 The slope of the line defined by s(n,n) is (-1)n-1, while the slope of the an line defined by s(n-1,n) is given by (-1)n-12a 2n + - 11 which is ±3 when an = 1.

158

Since the slope of the line defined by s(i,n) depends only on ar with r > i, any two sets s(i,n) and s(j,m) such that n-i = m-j, and having all ar equal for the final n-i terms, define straight lines having the same slopes. The graph of Figure 8.1 is composed of infinite sequences grouped into triangular patterns. Any given triangle has verticies defined by S(al,...,an), S(al,•••,an,an+1), and S(al,...,an+1,1) for some a1,•••,an+1• By Theorem 8.13 the slopes of the sides of these triangles will be: m(S(al,...,an),S(ai,...,an+1)) = ±1 m(S(ai,...,an),S(al,...,an+1,1)) = ±3 m(S(ai,...,an+l),S(ai,...,an+1,1)) = -(±3) Thus, all of these triangles are similar. Further, the nature of the infinite sequences composing these triangles, as determined in Lemmas 8.10 and 8.11, insures that they are pointwise identical up to isometries and scaling. When reflections are included, the entire graph of D:[0,1]-[0,1] is covered. Since self-similarity is defined by the property that all components of a figure map identically onto all other components by a similarity transformation [711, this proves Theorem 8.14 The map of D:[0,1]-,[0,1] is strictly self-similar. 3. Some Numerical Relations Let µe E+ be periodic with period 2s. Then g defines the point of [0,1] given by 25-1 x(µ) = µ2s-j2J

(8.13)

(22s -1) j=0 If gi is 1 when i = 2sj, and 0 otherwise then x(µ) = 231 , and (2 -1) iterates to 0 in 28 steps. For in < 2s Dm(x(g))= 31 [DI"(µ)l23_j2j (22-1) j=0

which, on substitution from equation (8.2) becomes Dm (x(µ)) = l [m+1)2J (8.14) (22$ -1)jOJ The sum in equation (8.14) can be evaulated by making use of a result of Hewgill [72]:

159

Lemma 8.15 1JF^m+11 F(m+11

J

+1 1_ 1JJ 2 J m+

m 2 l(8.15) Jj= - F( m+1 m 0

j=o where Fr = 22r+1 is the r-th Fermat number. The denominator in equation (8.14) is given by the expansion 22,9-1 = Fs-1Fs-2 ... FO More generally, if µe E+ has period n = 28n0 with no odd then either µ is on a cycle, or it iterates to a cycle. For such a µ x(µ)= 1 ^,µ2sn (22$np - 1) j=0j

-2i

(8.16)

and the denominator is given by s

s-1

(22 no -1) = 1 [IFr(no) (8.17) 2n0 - l r=0

where Fr(n0) = (Fr-1)n0+1. If µi is 1 when i = 2snoj and is 0 otherwise then x(µ) iterates to a cycle in exactly 2s steps and, for m<- 28 Dm(x(µ)) = s 1 m (rn4+lJi (8.18) (22 no -1) j0 1 which can be numerically evaulated by making use of equations (8.15) and (8.17). 4. A Density Result Theorem 7.5 gives the general solution to the equation Dn(µ) =13. The first n terms in this solution are free parameters. Hence, for any µ'e E+ they can be choosen so that µi = µ'i for i< n. This proves Theorem 8.16 Let 13 and µ' be arbritrary elements of E+. Then 13 has an n-th order predecessor µ such that I x(µ)-x(µ') I <_ 2-n. Corollary 8.17 The set P(x(13)) of all predecessors of 13 is dense in [0,1]. An even stronger result is possible. Consider a partition of [0,1] into equal segments of length 2-n, so that the left end point of the r-th segment is given by n for0
160

where µ* represents the initial conditions. Then x(µ) = x(µ*) + x(µ') and x(µ*) takes on the values 2n . Denote these as x(µ*,r), and the corresponding value of x(µ) by x(g,r). It is clear that x(µ*,r) <_ x(g,r) <_ x(µ*,r+l). Thus, each segment of the partition contains exactly one n-th order predecessor of B. Theorem 8.18 Let x(B)r= [0,11. Then for all n the set Pn(x(13)) of n-th order predecessors of x(13) is uniformly distributed with respect to the partition of [0,11 into 2n equal segments. 5. Exercises for Chapter 8 1. Find the cycle to which 13 = 1001011 maps under D. 2. Verify that Lemma 8.12 holds for S(3,1,2,2,1). 3. Find the slope of the straight line connecting S(1,2, 1) and S(2,1,2). 4. Compute the numerical value of the predecessors of Q out to fourth order. 5. Compute the STD out to tree heights of 3 for the STD containing the cycle -Qll-^1Q1-11Q-*. 6. Use the results of exercise 5 to verify Theorem 8.18 for this case.

161

Figure 8.1 Graph of D:[0,1]-*[0,1]

163

4

-Y

164

Chapter 9 Computation of Pre-Images In Chapter 7, attention was focused on computation of predecessors for configurations evolved under additive rules. The additivity condition allowed closed form solutions to be found for the predecessor equation X(µ) = B. In the case of nonadditive rules the situation is more difficult, and no general method for computation of predecessors is known. For this reason, the goal of this chapter is more modest. A method is presented for finding the pre-image of any finite sequence B1...Bn for a given rule X. That is, given such a sequence this method yields another sequence µ1...µn such that if X is a k-site rule then there is a sequence sl...sk-1 such that X(µl...µnsl..•sk1) = B1...Bn. In addition, it provides a method for counting how many pre-images any given sequence has. Throughout this chapter the configuration space is taken as E+. 1. Pre-Images and Predecessors Jen [34] has carried out a careful study of pre-images, obtaining the result that the total number of pre-images of certain well defined sequences scales with sequence length, and deriving a set of recurrence relations for the number N(S) of pre-images of any given finite sequence S. Her relations will be rederived in section 4 of this chapter. A question closely related to computation of pre-images is whether or not a given rule X has a "Garden-of-Eden"; i.e., whether or not there exists a subset GE(X) of E+ consisting of configurations having no predecessor. For a given rule X, define GE(X) = (Be E+ I X(µ) = B has no solution) (9.1) Since the equation X(µ) = B is difficult to handle for non-linear rules, and nonperiodic elements of E+, however, the Garden-of-Eden is best expressed in terms of finite sequences. Thus it is useful to work in terms of the set GE*(X), defined by the conditions 3 has no pre - image Be GE *(X) if 11. ( 2. every sub -sequenc e of B has a pre - image From this definition, the elements of GE*(X) are the minimal sequences without pre-image. They will be called seeds for GE(X) since, as indicated in the next lemma, they generate GE(X) in the sense that every element of GE(X) must contain at least one element of GE*(X) as a subsequence.

165

Lemma 9.1 (Hedlund, [49]) Let S*(µ) be the set of all finite sub-sequences of µ. Then µe GE(X) if and only if S*(g)nGE*(X) # 0. As discussed in Chapter 6, additive rules acting on E+ do not reduce entropy, even though they do reduce entropy when acting on En for all n. More generally, a rule X acting on E+ will reduce entropy if and only if GE*(X) is non-empty. If this is the case, the image of X in [0,1] is a Cantor set whose dimension is related to the amount of entropy reduction. This will be taken up further in Chapters 11 and 12. 2. The Rule Graph and Basic Matrix Let (X,E+) be a k-site cellular automaton. The rule graph for X is defined to be the directed graph Gk-1(X) = {vi,eijl where the vertex set is the set of all k-1 digit binary strings vi = iO...ik-2, and there is an edge eij directed from vertex i to vertex j if there is at least one sequence s 1...sk-1 such that X(iO...ik-2s1...sk-1) =jO...jk-2. The adjacency matrix Ak-1(X) for this graph will be called the basic matrix for the rule X. Now, consider the set of graphs G(X) = {Gn(X) I k- 1<_n<-) and the corresponding set of adjacency matricies A(X), where each Gn(X) is defined in the same way as the rule graph, except that the vertex set consists of n-digit binary strings. These sets are highly redundant since all information about the rule Xis already contained in the edge labled version of Gk-1(X). What is significant, however, is the structure of this redundancy, which contains a self-similar decomposition of the rule X. To see how this decomposition works, it is useful to take X as a map of the interval. The method for doing this is given in section 4 of Chapter 1. Denote the ordinary graph of X:[0,1]-[0,1] by G(X). For each An(X)eA(X) define a matrix An(X) by ij = a2n -j,i. That is, An(X) is the upsidedown transpose of An(X). Let Q(X) be a grid which subdivides [0,1]x[0,1] into 2nx2n squares, labeled with the ij indicies ofAn(X), and let the ij cell in this grid be white or black as ij is zero or non-zero respectively. Each of the -Gn(X) gives a decomposition of X:[0,11-40,11 at a certain scale. As this scale becomes more and more refined, the picture given by -Qn(X) gets closer and closer to the graph of X. The viewpoint which this provides is different from the usual view of functions as point mappings. Instead, the value of a map at a point is seen as

1 66

being the limit of a family of interval maps containing that point, as the length of the intervals shrinks to 0. This intuitive statement is formalized in the next theorem. Theorem 9.2

lim Gn(X) = G(X) n->-

Proof G(X) = {(y,X(y)) I ye [0,1]). The point y corresponds to an element µe E+ which can be obtained by the following iterative proceedure : 1) If y is of the form K2-r for some r define two elements µ and µ' of E+, with g ending in an infinite string of 0's, and g' ending in an infinite string of 1's, such that the map f defined in equation ( 1.13) satisfies f(µ) = flµ') = y. 2) If y is not an inverse power of 2, divide [0,1] into the segments [0,1/2) and ( 1/2,1]. If y is in the first of these segments, write µ1 = 0 and if y is in the second write µ1 = 1; 3) Divide the segment which contains y in half again . If y is in the first half write 92 = 0 , and if y is in the second half write 92 = 1. This proceedure is continued, and after n steps has generated the first n digits µ 1...µn of g. By definition, the column of Gn(X) which is labeled by µ1...µn (or, the columns labeled by the first n digits of g and µ') contains black squares at only those grid locations which are images of µ1... µn under X. Hence, in the limit as n becomes infinite this must converge to G(X). I Figures 9 . 1 illustrates this theorem by showing Gn(a) and G(a) for 1<_n<_4, where 6:[0,1]-4[0, 1] is defined by a(y) = 2y mod( 1). Figures 9.2 at the end of this chapter, show the A7 (X) matricies for several 3-site rules. If Xis a k-site rule, then the maximum value for an entry in any of the matricies An(X) will be 2k-1. Further, this will also be the value for the sum of each row of each of the An(X). If Ak_ 1(X) consists entirely of 1's then the rule graph Gk-1(X) is completely connected --every vertex of this graph can be reached from every other vertex in one iteration of the rule X. Definition 9.3 Let X be a k-site rule such that X(iO...ir...ik-1) = it + Y( iO...ir -1ir+1... ik-1) (9.2) where Y: E+->E+ is a k- 1 site rule. Then X is said to be linear in the r-th variable . If r = 0, Xis linear in the initial variable (LIV), and if r = k-1, Xis linear in the final variable (LFV). If X is either LIV or LFV then it is said to be linear in an extreme variable (LEV).

167

n=1

n=2

n=3

n=4

G(a) Figure 9.1 Lin(a) and G(a) for n = 1,2,3,4

Lemma 9.4 A rule Xis linear in the r-th variable if and only if

168 X(i0...ir-10ir+1...ik-1) = 1 + X(i0...ir-llir+1...ik-1) for all

(9.3)

ik-1-

Pr Proof The claim is obviously true if X is linear in the r-th variable. Suppose, then, that equation (9.3) is satisfied, and set Y(iO...ir-lir+1...ik-1) = X(i0...ir-10ir+1...ik-1) Then X(i0...ir-lirir+l...ik-1) = Y(i0...ir-14+1...ik-1) it =0 (9.4) 1+Y(ip...ir_11r+1...ik-1) 1r = 1

which is the definition of linearity in the r-th variable. I Lemma 9.5 Let X:E+-E+ be a k-site rule which is linear in the r-th variable. Then the component expression (x0,...,x2k - 1) contains an equal number of 0's and 1's. Remark Equivalently, Langton's ? parameter [42] is equal to 1/2 for rules linear in the r-th variable. Remark The 3-site rule 172, (00110101) provides an immediate counterexample to the converse of this theorem. Proof Suppose that a k-site rule X is linear in the r-th variable. Application of this rule to the k-site neighborhood list yields the component expression for X, but X must also satisfy equation (9.2), and the structure of the numerically ordered neighborhood list is such that for every s the block i0...is remains fixed while is+1...ik-1 ranges through all possible values. In particular, for each fixed value of iO...ir-1, it will first take the value 0 while it+1...ik-1 ranges through all values, and then will take the value 1 while it+1...ik-1 again ranges through all possible values. Thus, for each fixed block i0...ir-1, equation (9.2) yields the following two equations X(i0...ir-10ir+1...ik-1) = Y(iO...ir-fir+1...ik-1) X(iO...ir-11ir+1...ik-1) = 1 + Y(iO...ir-1ir+1...ik-1) corresponding to the condition on the components of X xi+2r = 1+xi (9.5) and this is satisfied if the number of 0's and 1's in the rule table are equal. I Theorem 9.6 The basic matrix Ak-l(X) of a k-site rule X:E+-*E+ consists entirely of ones if and only if X is linear in the final variable.

169

Proof If every entry in Ak-1(X) equals 1 then the equation X(iO...ik-2s l...sk-1) = j0...jk-2 must have a unique solution for all values of j = jO•••jn-1. That is, it must be possible to find sr with 15r-
(9.6) X(ik-2s 1...sk-1) = ik-2 have a solution for all j, and all blocks i0...ik-2. In the first of these equations it must be possible to choose sl such that there is a unique solution for all iO...ik-2. This is only possible, however, if X(iO...ik-20) = 1 + X(iO ... ik-2 1) for all iO...ik-2 since it would otherwise be impossible to simultaneously satisfy both of the equations X(i0...ik-2s1) =j1 X(iO...ik-2s1) = 1 +jl ragardless of the choice of sl. This is equivalent to X being linear in the final variable. Suppose, now, that X is LFV. Examination of equations (9.6) then indicates that the final sr term in each equation may always be choosen so that a unique solution exists. Hence Ak- 1(X) must consist entirely of 1's. I 3. Computation of Pre- Images From the Basic Matrix Suppose that an n-digit pre-image is desired for a sequence jO...jn-1. This can be done immediatly by checking the row labels of those rows for which [An(X)]ij # 0. Each such row label i = iO...in-1 corresponds to a preimage ofjO...jn-1. Further, it is easy to obtain An(X) from Ak-1(X). Two algorithms for doing so are presented in this section. Algorithm 9.7 Derivation of An+1 from An: 1. Given the 2nx2n matrix An(X) form the 2n+1x2n+1 matrix A"n+l(X) as indicated An (X) 0 A"n+1(X)= (An(X) 0 where the 0 entries indicate 2nx2n blocks of 0's.

170

2. Let X+ indicate the right extension of X to n+1 sites, as defined in section 2 of Chapter 1. If n = k-1 set X+ = X. Let xr denote the r-th component of X+. 3. For 0:5r<2n+1: a. If xr = 0 leave the r-th row of A"n+1(X) unchanged. b. If xr = 1 interchange the first and second halves of the r-th row of A''n+l(X). It is easy to show that the matrix generated by Algorithm 9.7 equals An+1(X). In order to do this, consider the aij element of An(X). By definition, there are exactly aij blocks sl...sk-1 such that X(iO...in-1sl...sk-1) =j0...jn-1 (9.7) The rows of A"n+l(X) are labeled by i = iO...in-lin. Consider the equation X(iO...insl ...sk-1) =jOj1•• Jn where jl..jnis equal to thejO...jn-1 of equation (9.7), and J0 = X(iO...ik-1). By definition of the right extension, however,

X(iO...ik-1) = X+(iO...ik-1sl... sk-1) (9.8) so that j0- 10 x; =0 1 xi=1 where xi is the i-th component of X+. Thus, for fixed i, J0 = 0 implies that [A"n+1(X)lij = [An(X)Iij while J0 = 1 implies that[A"n+l(X)lij = [An(X)lij+2n, and this is equivalent to the interchange of the first and second halves of the i-th row. I By making use of a similar construction, it is also possible to derive An(X) directly from Ak- 1(X): Algorithm 9.8 Derivation of An(X) from Ak-1(X): 1. Construct the 2nx2n matrix A'n(X) in which the first 2k-1 columns consist of 2n-k+1 copies of Ak-1(X) while the remainder of the matrix consists of O's. Partition the columns of this matrix into blocks of 2k-1 columns each, and label each block with an index 0:5 j < 2n-k+ 1. 2. Construct the list L(k,n-k;X) defined in equation (2.1).

171

3. For 0<_ i < 2n shift the entries in the i-th row and j = 0 column block to entries in the i-th row and the j = Li(k,n-k;X) column block. Set all other entries in the i-th row equal to 0. This algorithm incorporates the proceedure of algorithm 9.8, but applies it in an immediate rather than iterative form . The next theorem follows as a consequence of Theorem 9.7. Theorem 9.9 The matrix generated by Algorithm 9.8 equals An(X). 4. Pre-Images and the Jen Recurrence Relations The maps which generate L(k,n-k;X) can be used in another way to compute pre-images, as well as to derive Jen's recurrence relations for the number of pre-images of a given sequence . Write equation (2.1) in the form X(iO...in-1 ) = j0•••jn-k These maps can be represented by a set of matricies indexed by iO...ik-2. Divide elements of the list iO...in - 1 into two parts: (i0...ik-2)(ik-1 ...in-1). For fixed iO ...ik-2 the rule X defines a map ik - 1...in-1-j0 •••jn-k which is represented by a 2n11- k+l^n-k+1 matrix defined by X)] _ J0 X(iO...ik-2ik - 1...in-1 ) * JO...Jn (P(n-k+2 )( k (9.9) Ll 10...ik -2 ij ll1 X (io...ik-21k-1...in-1 ) = JO... Jn-k

where the ij indices on P are given by i = ik - 1...in-k, andj = j0...jn- k. In the case n = k, iO...in- 1 is just the neighborhood list for X and j = jo is just the i-th component of X. Thus the matricies P^1) - (X) are just the 2x2 matricies 10...1k_2 P(1) (X) x2i x2i 1 10...ik-2 x2i+1 x2i+1

(9.10)

where i = iO ...ik-2 and x' = 1+x mod(2). There are only four possibilities for the matricies defined by equation (9.10). These are: (

1

0) `0 11 1 (1 00)

(0

1)

(

9.11)

As with the An(X) matricies , the matrix Pln-)kk 2) (X) can be obtained from 0) . (X) and P(n-k+1) (X) 10...1k - 2 10...ik-2 Algorithm 9.10

A+2) (X) from p(1) • (X) and p(n-k+1) (X): Derivation of & 10 ...1 k 2 10 'k-2 10 ...1 k-2 -

172 1. For each 0 in P^1) . ( X), substitute a 2nx2n block of 0's. 10...ik-2 2. Replace the 1 in the first row of P^1 ) (X) by P(n-k+1) (X), and 10... ik-2 11...ik-20 the 1 in the second row by P(n-k+l) il...ik-21(X)

Theorem 9.11 The matrix generated by Algorithm 9.10 equals P(n-k+2) (X). 10 ik-2

Proof The matrix P(n-k+1) 10...ik-2 (X) is generated by the mapping

X(iO...ik-2ik-1...in-1)- *jO...J n-k while the matrix p(n-k+2) 10...ik-2 (X) comes from the map X(iO...ik-2ik-1... in- lin)-4JO...J n-kin-k+l The block listing for P(n-k+2) 10...'k-2 (X) may be represented as indicated in

Figure 9.2.

i0 lk-2 ik-1 ik

In-1 in

P11 (nkik-20(X)

(n-k+1) Pi1... ik-21(X)

Figure 9.2 Construction for Theorem 9.11 As indicated in this figure, the upper block, for which ik-1 = 0, gives rise to P(n-k+l) (X), ,while the block for which ik-1 = 1 gives rise to the 11...ik-20 matrix P(n-k+1) i 1...^k-21(X). But, the block iO...ik-20 determines the first row of

173 P^1) (X) while the block ip...ik-21 determines the second row of this same 10... ik-2

matrix. I Corollary 9.12 For all X, the matricies P(n-k+1) (X), 0...0 <_ ip...ik 2 <_ 1...1, are lo ... 'k-2 permutation matricies if and only if the 10 P^1) i (X) matricies are ...k-2 permutation matricies for all iO...ik-2•

Implimentation of Algorithm 9.10, making use of equation (9.10), yields x2.p'n-k+1) 0(X) x2.p(n-k+1) 0(X) P(n-k+2) (X) = 1 k-2 1 k-2 (9.12) i0...ik-2 x2i+1 il.. kk_21(X) x2i+1 il. kk_21(X

The number of pre-images of a sequence j = jo...jn-1, beginning with the sequence ip...ik-2 is denoted by Lj (n; X) with i = ip...ik-2 and j = jO...jn-1. This number is obtained as the sum of the j-th column of &-k+ 1) (X). 10...'k-2 Hence, from equation (9.12),

L'.(n;X)=[P(1) . (X)] 0j, L'.,(n-1;X)+[P^1) - (X)] L'.,(n-1;X) (9.13) J i0...ik-2 10..^k2 where j' = j mod(2n-1), i' = i1...ik-20, and i" = ip...ik-21. This is Jen's recurrence relation, restricted to rules defined over a binary alphabet. Making use of equation (9.10), this formula can be written as (n; X) = x2iL (n-1;X)+x2i+1(n-1;X) (9.14) where xa = jxs j00 S xs j0=1 Summation of equation (9.14) with respect to the index i = ip...ik-2 they gives a formula for the number of preimages of the sequence j =j0..in-1: 2k-1_1r N(j0...jn-1)= ^, I xa+x.+2k-1)Lj (n-1;X) (9.15) i=0 \ (Note the j ' indices in equation (9.14), and the j index in equation (9.15).) -k+1) (X) relatively The next lemma makes direct computation of P(n 10...'k-2

easy. Its proof follows immediatly from the fact that if the set of all il...ir is listed in ascending numerical order then the list for ipil...ir is given by I lpl l...ir Oil ...ir = {l

174

and [X(ip ... ir)]s =

X(iOil...ik_1) S = 0 [X(il...ir)]S otherwise

Lemma 9.13 Let X be a k-site rule with component representation X = (xO...x2k-1), and let the i-th n+k-1 site block be denoted i = iO...in+k-2. Then X(iO...in+k-2) = xrl...xrn

where rs = 11mo2') Equations (9.12) and (9.14) or (9.15) can be used to obtain expressions for the number of pre-images of given sequences in terms of the components of the rule X. For example, suppose that X is a 3-site rule. Formula for the number of pre-images of the four two digit sequences are obtained by use of equation (9.12) in order to compute P^o^1(X). Summation of columns then yields the set of equations N(00)=(x'O+x'4)(x'O+x'1)+(x'1+x'5)(x'2+x'3)+(x'2+x 6)(x'4+x'5)+x('3+x'7)(x'g+x'7) N(01)=(x'0+x'4)(x0+x 1)+(x' 1+x'5)(x2+x3)+(x'2+x'g)(x4+x5)+(x'3+x'7)(xg+x7 ) N(10)=(xO+x4)(x'O+x' 1)+(x 1+x5)(x'2+x'3)+(x2+xg)(x'4+x'5)+(x3+x7)(x'g+x'7) N(11) = (xO+x4)(xp+x1)+(xl+x5)(x2+x3)+(x2+x6)(x4+x5)+(x3+x7)(x6+x7) (The sums in these equations are not taken mod(2).) If X = (01001000) is rule 18, for example, substitution into these equations gives N(00) = 9, N(O1) = N(10) = 3, and N(11) = 1. In writing out the equations for N(ipil) terms have not been simplified, and the order of factors has been preserved. The reason for this is that in general, the equation for the number of predecessors of a sequence i0•••in-1 can be obtained from the equation for the number of predecessors of the length n sequence consisting of all l's by putting a prime on each term corresponding to a 0 entry in iO...in-1. Inspection of the equations for N(00), N(01), and N(10) illustrates this point. Another method, defined in terms of the An(X) matrix, also allows computation of the number of pre-images for specified sequences, or for blocks of sequences. For a given value of n, define the equivalence classes Js(n,X) = {i mod(2k-1) 12k-ls < ji < 2k-1(s+l), 0<_ s< 2n-k+1-1} By the construction used in Algorithm 9.8, each equivalence class contains only those i values, reduced mod(2k-l), for which the i-th row of

175

An(X) has been shifted by 2k- ls places to the right. For each s , the rows of Ak-i(X) indexed by the i values in Js(n,X) can be checked. If this particular combination of rows contains a position r (0<_r<2k-1) with 0 as the only column entry, then column 2k-is+r of An(X) will consist entirely of O's. An example is provided by the 3 -site rule 54 , which has the basic matrix 1 1 1 1 A2(54) = 0 0 2 2

(9.16)

2 2 0 0 Table 9.1 lists the Js classes for this rule, for n = 3, 4, and 5. n=3: JO(3) = {000,011,110,111) = (0,2,32) mod(4) Ji(3) = {001,010,100,101) = {12,2,0) n=4: JO(4) = {0000,0110,0111,1110,1111) = {0,22,32} J1(4) = {0001,1100,1101} = {0,12) J2(4) = {0011,1000,1011} = {0,32} J3(4) = {0010,0100,0101,1001,1010} = (0,12,22} n=5: JO(5) = (00000,01110,01111,11110,11111) = (0,22,32) J1(5) = {00001,01100,01101,11100,11101) = (02,13) J2(5) = (00011,11000,11011} = {0,32} J3(5) = {00010,11001,11010) = {1,22} J4(5) = {00110,00111,10000,10110,10111} = {0,22,32) J5(5) = {10001) = (1) J6(5) = (01000,01011,10011) = {0,32} J7(5) = {00100,00101,01001,01010,10010,10100,10101} = {02,13,22} Table 9.1 Js Equivalence Classes for Rule 54, s=3,4,5

The notation in Table 9.1 first lists the elements of Js in binary form, then lists them mod(4), with multiplicities superscripted. The number of preimages for a sequence corresponding to the column 2k-1s+r given by 2k-1_1 1mr[Ak-1(X)]ir i=0

176 where mr is the multiplicity of r in Js. Examination of A2 (54), for example, indicates that a class Js will allow all 0's in a given column position if it does not contain i = 0, or both of i = 2 and i = 3. The first classes which satisfy this condition are J3 (5) and J5(5). The first of these corresponds to a column of 0's located at j = 4x3+1 = 13 (01101, column 13 in A5 (54)), while the second corresponds to columns of O's located at j = 4x5+1 = 21 and 4x5+2 = 22 ( 11001 and 11010, columns 21 and 22.) Examination of J7 (5), on the other hand, indicates that columns 4x7+r _r<4, corresponding to sequences 11100 , 11101, 11110 , and 11111 have for 0< 8, 2, 5, and 12 pre -images respectively. 5. Exercises for Chapter 9 1. Construct the matrix A4(X) for rules 18, 30, 54 , 90, and 150. 2. Find all pre-images of the sequence 1011 for the rules of exercise 1. 3. Give a proof of Theorem 9.9. 4. Sketch the first four approximations to the graph of a2. 5. How many pre-images does the sequence 10100 have under the following rules: a) Rule 90 b) Rule 30 c) Rule 54 d) Rule 116 6. Compute the Js(n,X) classes for n = 1,2,3 for the following rules: a) Rule 30 b) Rule 108 c) Rule 18 d) Rule 126

177

Figures 9.2 A7(X) Matricies for Selected 3-Site Rules The A7(X) matricies for 3-site rules are 128x128 . Only non-zero elements are shown. Matrix element values are color coded: Red = 1 Green = 2 Blue=3 These figures can be compared to the graphs shown in the Introduction, or in the case of rule 60 in Chapter 8.

178

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193

Chapter 10 The Garden of Eden The Garden-of-Eden, GE(X), for a rule Xis the set of all configurations without predecessors under X. This set is most easily studied via its countable set of generators, GE*(X). A rule is surjective if and only if GE(X) = 0. If rules X and Y are related by any of the symmetry transformations T1, T2, and T3 then GE*(X) a GE*(Y). The formalism introduced in Chapter 9 allows computation of GE*. In some cases the elements of GE* can be simply characterized. A first order classification of rules into four catagories can be made on the basis of the set GE*, and these catagories can be compared to those defined in other phenomenological classifications.

1. GE(X) and GE*(X) The Garden-of-Eden for a CA rule X was defined in equation (9.1) as the set GE(X) consisting of all configurations without predecessors; i.e., all unreachable configurations. If the configuration space is En, then as discussed in section 1 of Chapter 6, there will be unreachable configurations in En which do have predecessors in E+. The existence of these configurations for non-injective rules is the reason that certain rules which do not reduce entropy when acting on E+, do so when defined on En. The unreachable nature of these configurations is due to the finite configuration space rather than an intrinsic property of the evolution rule. On the other hand, if a configuration in E+ is unreachable by a rule X, then this unreachability is intrinsically characteristic of X. By definition, a rule X is surjective if and only if every configuration has a predecessor. If the configuration space is finite, surjectivity is equivalent to injectivity. This is not true for infinite configurations spaces, since infinite sets may be in one-toone correspondence with proper subsets. The definitions of GE(X) and of surjectivity immediatly yield the next lemma. Lemma 10.1 A rule X:E+- *E+ is surjective if and only if GE(X) = 0. In Chapter 9 the set GE*(X) was defined as the set of all finite sequences without pre-image, for which all proper subsequences do have preimages. Lemma 9.1 showed that a configuration was in GE(X) if and only if it contained at least one element of GE*(X) as a subsequence. That is, GE*(X)

194

is the generating set for GE (X). Thus, elements of GE*(X) will be called seeds for the Garden-of-Eden. Lemma 10.2 If GE( X) is not empty then it is uncountable. Proof IfjO ...jn-1 is in GE*( X), then every configuration which contains it as a subsequence is in GE(X), and the number of such configurations is uncountable. I Since GE*(X) is always countable , it is easier to work with than GE(X). Taking X:[ 0,1]-[0,1], the range of X is a Cantor set [36] with fractal dimension d(X) <- 1. Equality occurs if and only if GE*(X) = 0. The dimension d(X) is related to the spatial measure entropy , which is defined for sequences of length n by Sn(X)=-n 1, pn(s)1og2Pn(s) (10.1) sEEn

where pn (s) is the probability of the length n sequence s occuring in the evolution of rule X.

The fractal dimension d(X) is then given by d(X) = lim Sn (X) n- oo

(10.2)

The map X:[0,1]-x[0,1] will, in general , be no where continuous. Nevertheless , it may still satisfy the intermediate value property (IVP), that for all intervals [a,b]E [0,1] such that X(a) # x(b), and for all y between X(a) and X(b) there is a cE [a,b] such that y = X( c). Those discontinuous functions which still have the intermediate value property (IVP) are called Darboux functions [731. Lemma 10.3 If GE*(X) ;& 9 then X does not have the intermediate value property. Proof LetjO••in-lE GE*(X) and take [ a,b] as any sub-interval of [0,1] for which X(a) # X(b). Let the elements of E+ corresponding to a and b be 13(a) and 13 (b). Choose any point y between X(a) and X(b). If there is not an element 13(c)E E+ such that X(13(c)) =13(y) then X does not have IVP. Suppose, then, that such a 13( c) exists. By inserting the sequence jo.. jn -1 into 13(y) far enough to the right a new point, y' is obtained and y' is still between X(a) and X(b). Nevertheless, no 13(c) exists such that X(13(c)) = y'. I

195 Lemma 10.4 Let X:[0,1]-4[0 , 1] have the intermediate value property . Then the range of X is [0,11.

Proof Let c and c' be respectively the maximum and minimum values of X on [0,11. Suppose that c < 1 . Then there is a yE (c,1] for which 13(y)E GE(X). Therefore, by Lemma 10.3, X cannot have the intermediate value property. Thus c = 1, and a similar argument demonstrates that c'= 0.1 Lemma 10.5 GE*(X) = 0 does not imply IVP. Proof Rule 90 (X = S) provides a simple counter example. Choose 13(a) = 101100101111111Q 13(b) = 1011001100000002

(10.3)

Then 8(13(a)) = 0111100100000112 6(13(b )) = 0 111111 10000000.Q and the point y corresponding to

13(y) = 011110010000100Q satisfies the condition S(a) < y < S(b) when S is considered as a map on [0,1]. But the predecessors of 13 (y) can be computed from Corollary 7.6 and they are found to be 130 =000110000101011 131 = 010011010000041 132 = 101100101111114 133 = 111001111010102 Comparison of these with equation ( 10.3) shows that 130 <131 <132 < 13(a) <13(b) < 133 hence no predecessor of y is in the interval [a,b]. I Since GE *(X) = 0 is equivalent to surjectivity, this shows that the intermediate value property is a more restrictive condition than surjectivity. 2. Computation of GE*(X) Equations (9.12), (9 . 14), and (9.15) allow computation of the number of pre-images for any finite sequence, for any given k-site rule X:E+->E+.

196 Sequences in GE*(X) will be those for which this number is 0. It is also possible to turn this around. Since these equations were written in terms of the components of X, a sequence jO...jn-1 can be given and the question asked: for what k-site rules X does is this sequence contained in GE*(X)? Phrased in this way, the question becomes a satisfaction problem, and its solution will in general be computationally hard. In fact, not only is it necessary to find those rules for which the sequence has no pre-image, but it is also necessary to exclude from this solution set those rules for which there are proper sub-sequences without pre-images. The number of rules involved can be computed via the principle of inclusion-exclusion, but the computation is difficult if the sequence is long. A related question can be posed: Given a finite, or algorithmically generated infinite set S of sequences satisfying the condition that none is a sub-sequence of any of the others, when is there a k < 00 such that there is a k-site rule X for which this set is GE*(X). For example, if X is a 3-site rule, expansion of equation (9.15) gives N(111) = xOxOxO + xOx2x4 + xOxOx4 + xOx4x6 + xOxOxl + xlx2x4 + xOxlx4 + xlx4x6 + xOxlx2 + x2x2x5 + xlx2x4 + x2x5x6 + xOxlx3 + x2x3x5 + xlx3x4 + x3x5x6 + xlx2x4 + x3x4x6 + x25 + x4x6x7 + xlx2x5 + x3x5x6 + x2x5x5 + x5x6x7 + xlx3x6 + x3x6x7 + x3x5x6 + x6x7x7 + xlx3x7 + x3x7x7 + x3x5x7 + x7x7x7 Since xi >: 0 each term in this expansion must be separately equal to 0 if N(111) = 0. The solution set to these equations consists of rules 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 32, 34, 40, 42, 48, 50, 56, 64, 66, 68, 70, 72, 76, 80, 84, 96, 98, and 112. The single digit sequence 1 has no pre-image only for the rule 0. The rules for which the sequence 11 has no pre -image are computed to be 0, 2, 4, 8, 12, 16, 24, 32, 34, 48, 64, 66, and 68. Thus 111E GE*(X) for rules 6, 10, 14, 18, 20, 28, 40, 42, 50, 56, 70, 76, 80, 84, 96, 98, and 112. If rules for which sequences 111 and 00 both have no pre-image are sought, a quick computation shows that no such 3-site rules exist. The equations to be solved are those for N(111) = 0, and in addition those for N(00) = 0. These latter are:

197

x'Ox'0 + x'Ox' 1 + x' lx'2 + x 1'x 3 = 0 x'2x'4 + x'2x'5 + x'3x'6 + x'3x'7 = 0 x'4x'O + x'4x' 1 + X'5x'2 + x'5x'3 = 0 x'6x'4 + x'6x'5 + x'7x'6 + X'7x'7 = 0 Since all terms must separately equal 0, in particular x'O = x'7 = 0 which implies that xO = x7 = 1, while for 111 to have no pre-image it was necessary for xO = x7 = 0. If the rule Xis specified, computation of GE*(X) can be accomplished by use of one of the algorithms given in section 3 of Chapter 9. With these the set of matricies An(X) can be computed, and checked for columns of 0's. When such a column, sayjO..jn-1 first shows up, all columns which contain this sequence in An+r(X) will automatically be all 0's, and can be ignored. In addition, it is only necessary to compute GE*(X) for equivalence classes of rules under the transformations T1, T2, and T3 of Chapter 1. If GE*(X) is known for one rule in an equivalence class then it is known for all rules in the class . This is a result of the next theorem. Theorem 10.6 1. jO...Jn-lE GE*(X) rz jn-l...JOE GE*(T1X) 2. jO..in-lE GE*(X) $-->jO...jn-lE GE*(T2X)

3. jO...jn-lE GE*(X) a 1...1 + jO...jn-l€ GE*(T3X) Proof The first part of the theorem follows directly from left/right symmetry. In order to prove the second suppose that jo..in-lE GE*(X), but there is a sequence iO...in_1 such that T2X(iO...in-1sl...sk) =jO..jn-1 for some sl...sk. Taking in+k-1 = sk for notational convenience, this means that for all r< n, T2X(ir...ir+k-1) =Jr. Since jO...jn-le GE*(X), however, there must be at least one r such that X(ir...ir+k-1) #Jr. But by the definition of the transformation T2, T2X(ir...ir+k-1) = X(ir...ir+k-1 + 1...1) for all r. Hence T2X(ir...ir+k-1) = jr implies that X(ir...ir+k-1 + 1...1) = jr as well, contradicting the assumption that jO...j n- lE GE *(X). Finally, suppose thatjO...jn-lE GE*(X) but 1...1 +jO...jn-1 GE*(T3X). Then there is an iO...in-1 and an s1...sk such that T3X(iO...in-1sl...sk) = 1...1+ JO.-in-1• But T3X(9) = (1 + X)(µ) = 1(µ) + X(µ), and i(µ) = 1 for all R. Therefore, T3X(iO...in-lsl...sk) = 1...1+ X(iO...in-lsl...sk) = 1...1+ JO...Jn-1, so that X(i0...in-lsl...sk) =JO•••Jn-1, contradicting the initial assumption. I

198

3. GE*(X) for 3-Site Rules The space of 256 3-site rules is partitioned into 47 equivalence classes by the transformations T1, T2, and T3. Appendix 5 gives either GE*(X), if it is finite, or the initial portions of GE*(X) otherwise, for particular exemplars of each of these equivalence classes . The T-invariant equivalence classes are listed in Table 10.1. (0,255) (1,127,128,254)

(2,8,16,64,191,239,247,253)

(4,32,223,251)

(3,17,63,119,136,192,238,252)

(5,95,160,250)

(6,20,40,96,159,215,235,249)

(10,80,175,245)

(7,21,31,87,168,224,234,248)

(15,85,170,240)

(9,65,111,125,130,144,190,246)

(18,72,183,237)

(11,47,81,117,138,174,208,244)

(19,55,200,236) (22,104,151,233)

(12,34,48,68,187,207,221,243)

(23,232)

(14,42,84,112,143,171,213,241)

(24,66,189,231)

(25,61,67,103,152,188,194,230)

(29,71,184,226)

(26,74,82,88,167,173,181,229)

(33,123,132,222)

(27,39,53,83,172,202,216,228)

(36,126,129,219)

(28,56, 70,98,157,185,199,227)

(37,91,164,218) (43,113,142,212)

(30,86,106,120,135,149,169,225)

(46,116,139,209)

(38,44,52,100,155,203,211,217)

(50,76,179,205)

(41,97,107,121,134,148,158,214)

(51,204)

(45,75,89,101,154,166,180,210)

(54,108,147,201)

(58, 78,92,114,141,163,177,197)

(57,99,156,198)

(62,110,118,124,131,137,145,193)

(13,69,79,93,162,176,186,242)

(35,49,59,115,140,196,206,220)

(60,102,153,195) (73,109,146,182) (77,178) (90,165) (94,122,133,161) (105,150) Table 10.1 T-Equivalence Classes of 3-Site Rules

199

If the elements of GE*(X) are computed out to sequences of length eighteen. Four first order catagories are obtained. These are defined as: EM: For rules in this catagory GE*(X) is empty. The rules in EM are listed in Table 10.2 FI: For rules in this catagory GE*(X) is finite. SI: For rules in this catagory, the number of elements in GE*(X) of length <_ n scales linearly with n. LI: For rules in this catagory, the number of elements in GE*(X) of length <_ n grows at a rate greater than linear in n. Of the 256 3-site rules a total of 30 are in the EM catagory (Table 10.2), 146 are in FI, 28 are in SI, and 52 are in LI. As already indicated, GE*(X) is given in Appendix 5 for exemplars of each of the 47 equivalence classes of rules. Examination of these exemplars leads to several observations. Homogeneous and Inhomogeneous Linear Rules 15,51,60,85,90,102,105,150,153,165,170,195,204,240 Non-Linear Rules 30,45,75,86,89,101,106,120,135,149,154,166,169,180,210,225 Table 10.2 3-Site Rules With GE* = 0

1. The maximum number of seeds for rules in FI is six. 2. For rules in the catagory SI it is possible to identify conditions for a sequence to be in GE*, and to determine the way in which the number of seeds scales with n. For the four exemplars of this catagory the results are: Rule 38 For every even n >_ 4 there is a single seed of the form 01...101 where the block of 1's has length n-3. There are no seeds of odd length. The number of seeds scales with n according to N(n) n23I (10.4) where Lx J is the smallest integer Lgreater than or equal to x. Rule 46 For every odd n >_ 3 there is a single seed of the form 01...10 where the block of 1's has length n-2. There are no seeds of even length. The number of seeds scales with n according to

N(n) n221 (10.5)

200 Rule 58 1111 is a seed. For all n > 7 there are two seeds, having the forms 010...0100, and 010...0111 with the central block of 0's having length n-5 in each case. The number of seeds scales with n according to N(n) = 2n-11 (10.6) Rule 172 1010 is a seed. For all n >_ 7 there is a single seed with the form 001...1001 if n is odd, and 101...1..1 if n is even. The central block of 1's has length n-5 and the number of seeds scales with n according to N(n) = n-5

(10.7)

3. The LI catagory includes several rules which have been extensively studied, in particular rules 18 [52,64], 22, and 110 [74,75]. For a rule X denote the number of seeds of length n by K(n,X). Table 10.3 gives the value of K(n,X), n<_ 18, for the exemplar rules in this catagory. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

18 0 0 1 0 0 0 1 2 3 4 6 10 17 28 45 72 116 188

22 0 0 0 0 0 0 0 2 4 3 11 17 23 47 84 118 223 394

26 0 0 0 1 0 0 0 2 1 4 3 9 8 21 23 52 63 128

94 0 0 0 0 1 0 0 2 5 5 6 7 13 25 43 72 109 176

110 0 0 0 0 1 1 1 1 2 3 4 5 7 10 14 19 26 36

134 0 0 0 0 1 1 0 1 1 2 6 7 9 16 24 35 59 90

146 0 0 0 0 0 2 1 4 1 5 6 4 15 12 25 36 41 78

152 0 0 0 0 1 0 1 1 2 2 4 5 8 11 17 24 36 52

164 0 0 0 0 0 0 0 0 5 6 14 23 39 56 103 135 248 337

Table 10.3 K(n,X) for LI Exemplars

In most cases in LI there is not an easily discernable pattern for predicting K(n,X). For rules 18, 110, and 152, however, some phenomenological results can be stated. Rule 18 In [33], conditions for an n digit sequence s = sO...sn-1 to be in GE*(18) are given as: a) s = 111; b) s has the form 110s3...sn-4011 with a3...an-4

201

containing an odd number of isolated 1's. On this basis it is possible to compute K(n,18) by counting the number of sequences of length n-6 which contain only an odd number of isolated l's. By Corollary 3.7 the number of length r sequences with only isolated 1's is given by Lr - 1, where Lr is the r-th Lucas number. The Lucas numbers satisfy the Fibonacci relation L0=1 L1 = 3 Lr+1 = Lr + Lr-1 It is essential to recall , however, that Corollary 3.7 refers to sequences with periodic boundary conditions whereas the sequences in condition (b) above refer to sequences with open boundary conditons . For example, a sequence 10101 is acceptable with open boundary conditions , but not with periodic boundary conditions. Lemma 10.7 Let Kr be the number of length r sequences with periodic boundary conditions containing only an odd number of isolated 1's. Then Kr satisfies: K1=1 K2 = 2 K3=3 Kr+1 = Kr + ( Lr-1- Kr-1) r >_ 3 (10.8) Proof The set Sr + 1(1;1) of length r+1 sequences containing only isolated 1's can be constructed as follows: a) Add a 0 to the right end point of each sequence in Sr(1;1) to obtain a set Sr+1 ( 0). The elements of this set have only isolated 1 's, and in addition, they terminate in 0; b) Add a 1 to the right end point of each sequence in the set Sr (0) of elements of Sr( 1;1) which terminate in 0. Call this new set Sr + 1(1). Clearly this set consists of length r+1 sequences containing only isolated 1's, and terminating in 1. Then, Sr+1(1 ; 1) = Sr+l (0)uSr+1(1),){0...011 since the final sequence accounts for the only remaining element of Sr+1(1;1). By construction , the elements of Sr+1 ( 0) will have an odd or even number of 1 's if the corresponding elements of Sr(1;1) do, hence the number of sequences with an odd number of isolated 1's contained in Sr+l ( 0) is given by Kr. The elements of Sr+l(1), on the other hand, will have an odd number of l's if the corresponding elements of Sr(O) have an even number of l's. But

202 Sr(0) is just Sr-1(1;1) with a 0 appended to the right end point of each member. Hence the number of sequences in Sr+1 ( 1) having an odd number of isolated 1 's is just the number of sequences in Sr-1(1;1) with an even number of 1's. But this is just the number of length r-1 sequences with only isolated 1's minus the number of length r- 1 sequences with an odd number of isolated 1's. Finally, the term { 0...01} is counted by adding a final 1. Thus, recalling Corollary 3.7, Kr+1 (1;1) = Sr(1;1) + (Lr-1 - 1 - Kr-1) + 1, and this proves the lemma. Theorem 10.8 Excluding the anamolous seed 111 , K(n,18) satisfies 0 n<7 K(n,18 )={ >7 (10.9) K(n-4, 8 )+ Kn_6 n Proof Excepting the seed 111, all elements of GE *( 18) are of the form 110s3 ... sn-4011 where s3...sn -4 has an odd number of isolated 1's. This sequence has length n-6, so if n < 7 no such sequences can exist . The number of such sequences will be Kn-6 plus a term which adds in those sequences which are allowed by open boundary conditions , but forbidden by periodic boundary conditions . All such additional sequences will have the form 10s5...sn-701 where the n- 10 digit sequence s5...sn-7 contains an odd number of isolated l's. But this is exactly the condition on the interior sequence contained in K(n-4,18).1 Equation ( 10.9) can be expanded to yield In461

K(n,18 ) = JK(n-6) mod(4)+4j j=0

(10.10)

Rule 110 This rule has been studied as a possible example of a rule which carries out computations at the edge of chaos [75,76]. Examination of the pattern of seeds suggests that a sequences is in GE *(110) if and only if it has the form 010s3... sn-4010 where s3...sn -4 consists either of all O's, or of blocks of 111 separated by one or more 0 's. The values listed in Table 10.3 follow the recursion relation K(n,110) = 0 n<44

203

K(5,110) = 1 K(n+1,110) = K(n,110) + K(n-4,110) n >- 5 (10.11) The number of sequences consisting of blocks of 111's separated by one or more 0's can be computed from Lemmas 3.5 and 3.6. Using these lemmas, the number of seeds of length n is given by K(n,110) = Kn-6(1;3) + 1 in which the final 1 accounts for the s3...sn-4 = 0... 0 sequence. Rule 152 Explicit conditions for membership in GE*(152) appear to be rather complex. Inspection of the list of seeds for n<- 18, however, suggest that these conditions include: a) All seeds have the form 110s3...sn-4011 with the sequence s3•••sn-4 having only isolated 1's. b) If n is odd , s3...sn-4 contains an even number of 1's. c) If n is even, s3...sn-4 contains an odd number of 1's. Further, if there is only a single 1 it must appear in a position s5+j for 0 <- j <- n-10. d) For n > 5, sn-6 = 0. Phenomenologically, the following relations hold: K(n,152)=0 n<<-4 K(n+1,152)- K(n,152)+K(n-2,152)-1 n-odd (10.12) 1 K (n,152) + K (n - 2,152 )-+ 1 n - even For rule 94, elements of GE*(94) appear to have the form Ols2...sn-310 with the sequence s2...sn-3 having no isolated 1's. For rule 146, with n > 6, seeds have the form 11s2...sn-311 with the internal sequence having only isolated 1's. For rule 22 the seed form is 10s2...sn-301. 4. Classification With GE* Four catagories of 3-site rules have been distinguished, based on the behavior of the set GE*: either it is empty (EM), finite (FI), grows linearly with n (SI), or grows at a rate faster than linear with n (LI). Since, roughly speaking, the elements of GE* define how many "holes" are cut out of [0,1] to form the Cantor set X([0,1]), it is to be expected that the fractal dimension of this set will satisfy d(LI) < d(SI) < d(FI) < d(EM) = 1. Thus, the greatest entropy reduction is expected for rules in the LI catagory. The best known classification of CA rules has been made by Wolfram based on limiting behavior. The four classes distinguished are defined as [28]: Class I: rules which evolve after a finite number of steps, from almost all

204

initial states , to a unique homogeneous state in which all sites have the same value ; Class II : Rules which evolve to simple structures , either fixed points or cycles . The set of these structures corresponds to the words generated by a regular grammer; Class III: These rules evolve to aperiodic , chaotic patterns. These patterns appear as asymtotically self-similar fractals ; Class IV: Rules for which the persisting structures exhibit no simple pattern, and appear to be essentially unpredictable. This classification has been refined by Li and Packard [58], who distinguish five catagories: Null Rules (Wolfram Class I); Fixed Point Rules (Wolfram Class Ila); Periodic Rules (Wolfram Class IIb); Locally Chaotic Rules (chaotic dynamics confined by domain walls); and Globally Chaotic Rules (Wolfram Class III). Wolframs Class IV is difficult to fit into the Li & Packard scheme, falling under either the chaotic heading (based on spatial response to perturbations ), or the periodic heading (based on possible periodicity of limit configurations). The classification on the basis of GE * does not respect the Wolfram, or Li & Packard classes since , although the transformation T2 leaves GE* invariant, it crosses the Wolfram/Li&Packard boundaries . For example, both rule 18 and rule 72 are in the LI catagory, yet in terms of asymtotic behavior rule 18 is Class 3 (or Chaotic in the Li & Packard scheme) while rule 72 is Class I (Null). Rules 18 and 72 are related by T2, however , so by Theorem 10.6, GE *( 18) = GE *( 72). Table 10 .4 lists the members of the Li & Packard classes which belong to each of the GE * catagories . Note that exemplars are shown only for the 88 equivalence classes distinguished under the transformations Ti and T2T3. Several points are worthy of note: 1. GE* is finite for all null rules; 2. With two exceptions , all rules for which GE* is finite are either null, fixed point, or periodic. That is, in Wolfram Classes I and II. 3. With the same two exceptions , chaotic rules are found either in the EM or LI catagories. 4. The only rules in EM which are not chaotic are the identity, powers of the shift, and 1 plus these rules. The exceptions mentioned in points (2) and (3) are rules 54 and 129. These rules are in Wolfram Class IV, and are assigned to the chaotic class by Li & Packard , although they suggest that other considerations might well place them in the periodic class. Rule 54 is related to rule 108 by both T 1 and

205 T2, and rule 108 is in the periodic class, so it seems reasonable to place rule 54 in this class as well. Likewise, rule 129 is related to rule 126 by the transformation T2T3 so an argument could also be made for placing this rule in the periodic class.

Null Fixed Point

Periodic

Locally Chaotic Chaotic

EM FI SI 0,8,32,40,128, 136,160,168 44,46,58,78, 2,4,10,12,13, 170,204 172 24,34,36,42, 56,57,76,77, 130,132,138, 140,162,184, 200,232 1,3,5,6,7,9,11, 27,38 15,51 14,19,23,28, 29,33,35,43, 108,142,156, 178 154 30,45,60,90, 105 106,150

54,129

LI

72,104,152, 164

25,37,41,74, 131,133,134

26,73 18,22,137, 146,164

Table 10.4 GE* Catagories for Li & Packard Classes

One final point is that membership in the Wolfram catagories has been shown to be formally undecidable [53], and results have been obtained which indicate that class membership may be undecidable in any scheme based on infinite time behvior [77]. The GE* catagories, on the other hand, are decidable, and the classes which they define are invariant under the transformations Ti, T2, and T3. The conceptual difficulty with GE* classification is that members of the same catagory may have very different long term behaviors, as indicated already in the example of rules 18 and 72. What is interesting about these two rules in particular is that they are defined on exactly the same Cantor set, yet have very different behaviors on this set. Rule 72 has the left justified decomposition a+x+t (or, the nearest neighbor decomposition I+x+t), while rule 18 has the decompositions 6+B++Band 4+x+t. Under rule 72, all states map to states which contain only isolated pairs of 1's, and these shift to the right with each iteration (left justified) or remain fixed (nearest neighbor). Rule 18, on the other hand, imitates rule 150 on these configurations, and imitates rule 90 on domains

206 which consist of regions of isolated l's separated by an odd number of 0's, sandwiched between domain walls consisting of 11 's, which preform anihilating random walks [64]. This suggests that GE* classification is useful when the interest is in the nature of the Cantor set which is the domain of a rule, and that a classification based on long term behavior is more of interest from a dynamical perspective. Another classification of 3-site rules has been given by Jen [35], based on the recurrence relations for counting pre-images. Since GE* is defined as the minimal set of sequences having no pre-images , it is the generator of the null set of these relations. That is, for a given rule X, the elements of GE*(X) are generators of the set of sequences which yield 0 when substituted into the recurrence relation for X. The catagories distinguished by Jen are defined in terms of the number of pre -images of a sequence s as: A. For all s the number of pre -images is constant; B. The number of pre-images of s is a product of integers representing block lengths (of 0's and 1' s) in s; C. The number of pre-images of s is a product of integers representing specific terms in s; D. The number of pre-images of s satisfies a telescoping recurrence relation; E. The number of pre -images of s is given by terms in a sequence whose values vary periodically; F. Linear recurrence relations with non-constant coefficients which do not obviously simplify (default catagory). Table 10 . 5 indicates the GE * catagories occupied by the rules in the six different classes defined by Jen. With the exception of rule 0, Jen's Class A corresponds exactly to the catagory EM, corresponding to an empty Garden-of-Eden. GE* is finite for rules in Classes B, C, and F with the exception of rules 38, 44, and 46 in Class B; rules 18 and 72 in Class C; and rules 58, 74, and 78 in Class F. All rules in Class E are in the LI catagory except rules 27 and 172 which are in SI. No rules in SI are in Classes C or D, and no rules in LI are in Class B. In addition, almost all of Jen's classes are preserved under T-transformations, the exceptions falling into the default class F. With further analysis it might be possible to show that rules in her class F can fit into other classes.

207 EM FI SI A 15,30,45,51, 0,255 60,90,105, 150,170,204 B 11,12,14,24, 38,44,46 34,35,42,138, 140 C 1,2,3,4,7,8,16, 17,19,28,32, 36,50,56,57, 76,126,128, 136,156,162, 168,184,200 D 5,6,9,23,33, 40,54,108, 130,132,160, 232 E 27,172 F

10,13,29,43, 77,142,178

58,78

LI

18,72

37,62,73,94, 110,122,146

22,25,26,41, 104,134,152, 164 74

Table 10.5 GE* Catagories for Jen Classes

5. Exercises for Chapter 10 1. In section 1 of this chapter, a counter-example is given to the conjecture that GE*(X) = 0 implies that X has the intermediate value property. Find another counter-example to this conjecture. 2. Find all 3-site rules X for which the specified sets of sequences are in GE*(X): a) 101 b) 101, 111 c) 1100 d) 101, 1001 e) 01110 3. Suppose that Xis a rule for which GE*(X) = {111}. Use Lemmas 3.5 and 3.6 to compute a formula for the fractal dimension of the Cantor set which is the image of X on [0,1].

208

Chapter 11 Time Series Simulation In this chapter the possibility of finding CA rules and initial conditions which generate specified time series is considered. The strongest general result is that no k-site rule can generate arbritrary time series with entries of k or more digits, while a k- site rule can generate arbritrary time series with k-1 digit entries if and only if it is linear in the final variable. A matrix technique is derived which allows determination of whether or not a given rule could have generated a specified time series , and if so what are the possible initial conditions. This allows estimation of probabilities for specified series , but these estimates are unbiased only for rules with empty Gardens-of-Eden. 1. Cellular Automata Generating Time Series Measurements carried out on a dynamical system yield discrete time series of finite accuracy. The theoretical problem is to reconstruct the underlying dynamics from this data set. This reconstruction is generally carried out within a particular theoretical framework which specifies the kinds of modelling paradigms which are to be used. For example, in classical physics it is assumed that modelling will be carried out in terms of either ordinary or partial differential equations. Cellular automata form one of the simplest model classes for systems which exhibit spatially and temporally discrete pattern generation. Several researchers have proposed them as models of pattern formation [78,791, and they have been employed in a variety of applications for theoretical modelling. The suggestion has also been made that CA equations of motion could be infered from time series data [80,811. That is, given a time series, that it would be possible to determine a CA rule which generated that series from an initial condition. This last claim has been questioned on the basis that if distortions introduced by the measuring apparatus are included, CAs do not form a wide enough modelling class to accurately reproduce underlying dynamics [571. Despite this negative result, the question of when a CA dqg serve as a good model for the dynamics which generate a time series is still of interest. In fact, if it turns out that the output time series of a particular measuring instrument can be represented by a CA, this conveys more information than if it were the case that all time series could be so simulated.

209

If the effect of the measuring instrument itself can be modelled by a CA rule, say T, then the condition that the dynamics be modelled by a CA rule is that there exist rules X and Y such that Y generates the output series, and X is Ito related to Y via T. In this case, the series observed is YnT(µ(0)) where µ(0) is the initial configuration, so that the rule sought is Y acting on an initial state T(µ(0)). The methods discussed in Chapter 4 can then be used to compute X. Let S(n,t) be a time series of n digit binary numbers containing t+1 elements: S(n,t) = µ1(0)...µn(0) µ1(1)...µn(1) 41(t)...9n(t) The two questions to be considered are: 1) Could a given CA rule X:E+-*E+ have generated this series, and if so, what are the possible initial states? 2) What CA rules, if any, could have possible generated this series? The second of these questions is the more difficult, although a partial answer is easy to obtain. Theorem 11.1 Let X:E+_E+ be a k- site rule . Then X cannot generate an arbritrary series S(n,t) for any n > k. Proof By definition , if i = io ...ik-1 then X cannot generate any series which contains the sequence i0...ik-1 x'i since X(i0...ik_ 1) = xi. I One approach to finding an answer to the first question is through the process of backward reconstruction . This method begins with the last entry in the time series and looks for a sequence s1...sk-1 such that when this sequence is appended to the penultimate entry in the time series , the final entry is generated by the given rule. That is X4t1(t-1)...9n(t- 1)s1...sk-1) = µl(t)...µn(t) (11.1) If this is possible , then the process is repeated with the new sequences µ1(t-l) ...µn(t-l)sl...sk-1, for all sl...sk-1 such that equation (11.1) is satisfied, now moved to the right side of this equation , while the argument on the left

210

side is now µ1(t-2)...gn(t-2)rl...r2k-2. An algorithmic proceedure for carrying out these computations has been given by Wuensche & Lesser[43]. So long as n < k there is no a priori restriction on the possibility of success in this process, but it is computationally intensive, and there are often cases in which it will fail, although this failure may not become apparent until the process has continued for some time. For example, Rule 22 cannot generate the time series 01,10,11,01,11, as indicated by the following attempted reconstruction: 01011 # 1 100111 0 01 110000 0100 11 There is one case, however, in which the backward reconstruction process will always succeed. Theorem 11.2 Let X:E+- *E+ be a k-site rule which is linear in the final variable. Then X can generate arbritrary sequences S(n,t) for all n < k. Proof Since Xis LFV, X(iO...ik-20) = 1 + X(iO...ik-21). Since n < k this means that there will always be the freedom to flip the final bit from 0 to 1 or vice versa, in order to escape an impass. I The next theorem shows that this result is the best possible. Theorem 11.3 There exist time series S(k-1,t) which cannot be generated by any ksite rule which is not LFV. Proof If a rule X is not LFV then there will be at least two neighborhoods iO...ik-20 and iO...ik-21 which map to the same value , say 1+a = a', under X. Let S(k-1,t) be any time series which contains the conbinations iO...ik-2 a.......... i0...ik-2 a..........

211

for all possible choices of iO...ik-2. Clearly, no X which is not LFV can generate such a series. I To summarize the conclusions of the preceeding paragraphs, if X:E+_*E+ is linear in the final variable then X can generate all time series S(n,t) for which n < k, but there will be time series with n > k which X cannot generate. If X is not LFV, then there will be time series S(k-l,t) which cannot be generated. The more general results for k-site rules are best introduced through the example of single digit time series S(1,t) and 3-site rules. The question is now, given a 3-site rule X, and a time series S(l,t), is it possible that the rule X could have generated this series, and if so, what are the initial conditions? It would also be useful to know the probabilities that one rule X generated the series as compared to some other rule X'. If a 3-site rule X can generate a time series S(1,t) = {41(0),...,µ1(t)} then a backward reconstruction as indicated below is possible. µ1(0)42(0).93(0)... µ 1(t-3)µ2(t-3)µ3(t-3)µ4(t-3)µ5(t-3)µ6(t-3)µ7(t-3) µ 1(t-2)µ2(t-2)µ3(t-2)µ4(t-2)µ5(t-2) µ1(t-1)µ2(t-1)93(t-1) µ1(t) That is, there must be at least one neighborhood starting with µ1(t-1) which maps to µ1(t); there must be at least one five digit string starting with µ1(t-2) which maps to one of these neighborhoods, and etc. This situation can be expressed in terms of the matricies X(k,m) defined in Chapter 2. Recall that X(k,m) is a 2k+mx2m+1 matrix with ij element given by the j-th element of the complete Boolean product of Li(k,m;X). To formalize the backward reconstruction process for one digit time series, these matricies are partitioned into quadrants: X(k,m) = (PQO(m+ 1) P0,1(m+ l)1 p10(m+ 1) P1,1(m+ 1)) (11.2) In the case under consideration, the reconstruction 91(t-1412(t-1)43(t-1) µ1(t) will be possible for arbritrary µ1(t) and µ1(t-1) if and only if the 8x2 matrix

21 2

P0,0(1) P0,1(1) (p10(1) P1,1(1)

X(3,0) =

(11.3)

xg X6 X'7 X7

has non -zero entries in each quadrant pb,c(1). Likewise, the reconstruction µ 1(t-2)42(t-2)µ3(t-2)94(t-2)95(t-2) 91(t-1)µ2(t-1)µ3(t-1) will be possible for all µ1(t-1) and µ1(t-2) if and only if the 32x8 matrix X(3,2) = po,0(3) P0,1(3) (P1,0(3) p1,1(3) has at least one non-zero entry in each quadrant. By construction, if the 16x1 column matrix pa,b(3)Pb,c(1) has any entry equal to 1 then there is a five digit sequence aµ2(t-2)93(t-2)94(t-2)µ5(t-2) which maps to the three digit sequence bµ1(t-1)92(t-1), which in turn maps to c under the rule X. That is, the rule X can generate the time sequence a, b, c if and only if pa,b(3)Pb,c(1) # 0. The k-site generalization of this is given in the next theorem. Theorem 11.4 Let X:E+-4E+ be a k-site rule and let S(1,t) = {µ1(0),...,µ1(t)} be a single digit time series. Then X can generate S(1,t) if and only if the column matrix defined by the product Pµ(0),µ(1) ((k - 1)t - k + 2)Pµ(1),µ(2) ((k - 1)t - 2k + 3)... P4(t-1),µ(t) (1) 0 _ [IPµ(t-s-1),µ(t-s)((k-1)s+ 1) (11.4) s=t-1

has at least one non-zero entry. The binary form of the row index for nonzero entries of this matrix are the initial conditions for the time series. Since the above theorems give not only the possibility of a rule X generating a given time series, but also the initial conditions from which this series might have been generated, it allows comparison between different rules . That is, if X and X' are two rules both of which could have generated a time series , then the relative probabilities of these two rules as the actual

213

generators of the series can be taken as related to the relative numbers of initial conditions from which each could in fact have generated the series. For example , if X generates the series from a single initial condition, while X' could have generated the series from four distinct initial conditions , then all other things being equal the probabilities for X and X ' to have generated the series would be 20% and 80% respectively. For a time series S (n,t) with n < k, a similar result holds. Let a(s,n) represent the sequence 91(t-s )...µn(t-s), and define pa(s-1 ,n),s(s,n)( r) by a partition of the matrix X(k,r-1 ) into 22n blocks of equal size. Theorem 11.5 Let S(n,t) be as above , with terms denoted a(s,n). A k- site rule X will generate this series if and only if 0 X7 1 1Pa(s-1,n),a(s,n ) (( k - 1)s + n) # 0 (11.5) s=t-1 Further, the binary forms of the indices for rows having non -zero entries are

the initial conditions for the series. Since the p-matricies are defined in terms of the components of X, this gives an immediate test for whether or not a given rule can generate a given time series . If a rule is not given , on the other hand, it gives a set of necessary and sufficient constraints on the components of any rule which could possible generate a given time series. A simple argument utilizing the properties of matrix multiplication yields a count of the number of distinct time series which a given rule could generate. Theorem 11.6 The number of distinct time series S(n,t) beginning with a (t,n) and ending with a(0,n) is given by the number of 1's in the block of the product 0 11 X(k, (k - 1)s + n - 1) S=t-1

which has row indices with binary form starting with a(t,n). 2. Statistics of Time Series In analogue with the spatial measure entropy defined in equation (10.1) there is a temporal measure entropy defined for time series S(1,t) of length n by

214

Sn(X)=- 1 lim I pn(s) log2P. ( s) n t-*°° SEE n

(11.6)

where pn( s) is the probability of the length n sequence s occuring in the temporal output series S(1,t). An equivalent entropy can be defined for series S(n,t). In the following, however , only sequences S(1,t) will be considered. For these series, the sum of the column matricies defined in equation (11.4) gives the total number of initial configurations from which a rule X can generate the specified series. Division of this number by the total number of possible initial configurations gives the probability that a particular series will appear. The catch in this argument , however, is that some configurations may be forbidden for a rule X because they contain sequences from GE*(X). These configurations may appear only once , at the beginning , and thereafter will never appear in the later evolution of the rule . This can act to skew the naive probabilities computed from counts of the column entries in equation (11.4), and corrections will need to be made. An infinite sequence p is called infinitely distributed if for all n, every possible combination of n digits appears with equal probability 2-n. The probabilities computed from the products of equation ( 11.4) are based on the assumption that all spatial sequences are infinitely distributed at each step in the iteration of the rule X. As indicated , this assumption will be false if there are Garden-of-Eden configurations . On the other hand, if GE*(X) = 0 then a theorem of Hedlund [49] states that for all n, every n digit sequence has an equal number of pre -images. Since each pre-image will be an n+k-1 digit sequence , each such possible sequence must occur with equal frequency so if a sequence is infinitely distributed then its successor sequences, under a rule for which GE*(X) = 0 , will also be infinitely distributed. Theorem 11.7

Let µe E+ be infinitely distributed and let X:E+-*E+ be a k - site rule. Then for r > 0, Xr(t) is infinitely distributed if and only if GE *(X) = 0. Table 11 . 1 shows probabilities for 1 , 2, and 3 digit sequences in S(1,t) for several representative 3-site rules, computed from the matrix product introduced in Theorem 11.4. These are estimates of probabilities for the appearence of the given 1, 2, and 3 digit sequences as the specified rules evolve from an infinitely distributed initial condition.

215

Rule 18 .75 0 1 .25

22 .625 .375

30 .5 .5

54 .5 .5

90 .5 .5

108 .5 .5

110 .375 .625

114 .5 .5

120 .5 .5

150 .5 .5

178 .5 .5

.25 .25 .25 .25

.125 .375 .25 .25

.375 .125 .125 .375

.375 .125 .125 .375

.25 .25 .25 .25

.375 .125 .125 .375

.0312 .0938 .2188 .1562 .0625 .1875 .1562 .0938

.3125 .0625 0 .125 .0312 .0312 .0312 .3438

.3125 .0625 .0625 .0625 .0625 .0625 .0625 .3125

.125 .125 .125 .125 .125 .125 .125 .125

.2812 .0938 0 .125 .125 0 .0938 .2812

00 01 10 11

.375 .125 .375 .125

.25 .25 .375 .125

.125 .375 .375 .125

.25 .25 .25 .25

.25 .25 .25 .25

000 001 010 011 100 101 110 111

.3125 .0625 .0312 .0938 .3125 .0625 .0938 .0312

.1562 .0938 .1562 .0938 .25 .125 .0938 .0312

.0312 .0938 .2812 .0938 .0938 .2812 .0938 .0312

.0938 .1562 .1875 .0625 .125 .125 .125 .125

.125 .125 .125 .125 .125 .1875 .125 .0625 .125 .1875 .125 .0625 .125 .1875 .125 .0625 Table 11.1

Predicted Probabilities for 1, 2, and 3 digit Time Series for Some 3-Site Rules Based on Preservation of Randomness in Automata Evolution (Although listed horizontally, sequences are temporal outputs)

Table 11.2 gives probabilities computed empirically by evolution of sixteen different 1023 digit pseudo-random sequences for 511 iterations and averaged over the sixteen outputs. In both of these tables, the rules are taken as left justified. There are differences which arise if the mapping site is changed which will be discussed later in this chapter. Rule 0 1

18 .751 .249

22 .648 .352

30 .501 .499

54 .530 .470

90 .513 .487

108 .689 .311

110 .427 .573

114 120 .438* .511 .562* .489

150 .501 .499

178 .511 .489

00 01 10 11

.627 .126 .126 .122

.429 .219 .219 .133

.118 .383 .383 .116

.292 .239 .239 .229

.265 .247 .247 .241

.454 .236 .236 .075

.090 .337 .337 .235

.439* .000 .000 .561*

.390 .120 .120 .371

.252 .247 .247 .254

.425 .084 .084 .406

000 001 010 011 100 101 110 111

.560 .065 .063 .063 .065 .061 .063 .059

.295 .133 .155 .065 .133 .086 .063 .070

.027 .092 .294 .088 .092 .290 .088 .027

.239 .057 .208 .031 .055 .183 .031 .198

.136 .248 .010 .130 .206 .081 .120 .191 .187 .126 .045 .152 .130 .206 .079 .118 .029 .258 .126 .045 .152 .114 .029 .083 Table 11.2

.439* .000 .000 .000 .000 .000 .000 .561*

.332 .059 .059 .061 .059 .061 .059 .310

.128 .126 .120 .126 .126 .122 .126 .130

.341 .085 0 .085 .085 0 .085 .321

Empirical Probabilities for 1 , 2, and 3 digit Time Series for Some 3-Site Rules (Although listed horizontally, sequences are temporal outputs)

216 Comparison of these two tables gives some indication of the degree to which a rule fails to satisfy the criterion that each successive iteration preserve the random characteristic of the initial condition. As expected, empirical results for the two additive rules, 90 and 150, match the computed values of Table 11.1 closely, while the results for the non-linear rules do not. For the additive rules the expected temporal entropy for the three values of n used is equal to 1, which is maximal. For other rules it will be less than this. Table 11.3 gives the p-matrix estimates of entropies, and the entropies computed from the empirical probabilities of Table 11.2, together with the difference dS between these two numbers. 90 1.000 .998 .002

108 1.000 .894 .106

110 .955 .984 -.029

178 114 120 150 1.000 1.000 1.000 1.000 .989 1.000 1.000 1.000 0 0 0 .011

Rule 18 Si(p) .812 Si(e) .811 dS .001

22 .955 .936 .019

30 1.000 1.000 0

54 1.000 .997 .003

S2 (p) .906 S2 (e) .773 dS .133

.953 .936 .017

.906 .893 .013

1.000 1.000 1.000 .953 .996 .999 .890 .931 .004 .001 .110 .022

.906 .495 .411

.906 .897 .009

1.000 .906 1.000 .826 0 .080

S3 (p) .834 S3(e) .743 dS .101

.943 .930 .013

.874 .856 .018

.980 .883 .097

.716 .330 .386

.850 .836 .014

1.000 .807 1.000 .755 0 .052

1.000 .953 .938 .999 .864 .911 .001 .089 .027 Table 11.3

Sn(X) Entropies for Time Series for n = 1,2,3 (Sn(p) is from p-matricies, Sn(e) is empirical)

The values computed from the p-matrix method can be adjusted for those rules with GE* # 0 by excluding contributions from the row index in equation (11.4) which contain a sequence from GE*. For rule 18, for example, the first element of GE*(18) is 111. There are 24 of the row indices in the product Pa,b(3)Pb,c(l) which do not contain this sequence. When all other indicies are excluded the new probability estimates for 2 and 3 digit series for rule 18 are: 00 01 10 11 000 001 010 011 100 101 110 111 .542 .177 .167 .125 .292 .125 .042 .083 .25 .042 .125 .083 For greater accuracy it would be necessary to compute products involving longer initial sequences, excluding those containing a 111 block, and computing statistics over those which remain. Wolfram [281 has pointed out that for n digit sequences, substantially more than 2n samples need to be considered to obtain accurate estimates for length n block probabilities, those

217 estimates based on small samples will systematically underestimate the entropies computed.

The value S1(p) is just the percentage of l's in the rule table. This quantity is Langton' s ? parameter. Rule classes can be roughly defined by choosing values of ? between 0 and 1 /2. Maximum chaos in the space-time patterns generated tends to occur at A. = 1 /2. Class IV behavior in Wolfram's classification occurs as a phase transition between periodic class II and chaotic class III behaviors . In terms of temporal series, A. is just the probability that the symbol at time t in the series is a 1 , given an equal probability of k-site neighborhoods occuring at time t-1. The entries for rule 114 in Table 11 . 2 are starred because the empirical output from this rule is not well represented by the average values given in the table. In the actual output from this rule, either 0 , 00, and 000; or 1, 11, and 111 showed up almost exclusively . There were seven cases in which the value 0 appeared an average of 510 of 511 . times, the series 00 appeared an average of 508 of 510 times, and the series 000 appeared an average of 507 of 509 times, all other values being essentially 0. In the remaining 9 cases the value 1 appeared an average of 508 of 511 times, the series 11 appeared an average of 507 of 510 times, and the series 111 appeared an average of 506 of 509 times. In seven of the sixteen runs there were flips from 0 ' s to l 's. This gives an example of a bifurcation between two possible behaviors, with the evolution rule being very sensitive to the initial conditions choosen. This can be seen by consideration of the rule table. The rule table for rule 114 is 000 001 010 011 100 101 110 111 0 1 0 0 1 1 1 0 Thus, if the initial condition is a sequence starting with a 0 the only way that a 1 can appear in future iterations is if the neighborhood 001 appears , while if the initial sequence starts with a 1 then the only way that a 0 can appear in later iterations is if the neighborhood 111 appears first. Rule 178 gives an example of another behavior. Here the series 010 and 101 can never occur, as can be seen from the attempted backward reconstructions . The rule table for rule 178 is 000 001 010 011 100 101 110 111 0 1 0 0 1 1 0 1 so the attempted reconstruction goes as indicated below.

218 001#

110#

110

001

0 1 One point of interest relates to rule 30. The empirical and p-matrix generated probabilities for series match within accepted errors, which is to be expected since the Garden-of-Eden for rule 30 is empty. Examination of Tables 11.1 and 11.2 indicates that the probabilities for sequences to appear in S(l ,t) peaks about sequences of alternating 0's and 1's. It has been suggested, however, that this rule can serve as a random sequence generator [50]. Wolfram claims that the sequence obtained by taking successive time values at a fixed site, or at sites lying along any diagonal (i.e., gi(t) or Ri±t(t)) through the spacetime pattern generated by iteration of this rule will be random . The difference is that Wolfram uses a nearest neighbor realization of rule 30 while that used here is left justified. The rule table for rule 30 is 000 001 010 011 100 101 110 ill 0 1 1 1 1 0 0 0 In the left justified form of the rule, the probability that a 0 value of gl (t) will be followed by a 1 value of µi(t+1) is .75, as is the probability that a 1 value will be followed by a 0 value . In nearest neighbor form, however, each of these probabilities is .5. Likewise, computation of the products pa,b(3 )Pb,c(1) indicates that in nearest neighbor form the probabilities of all three digit series are equal to .125. Since these two forms of the rule are related by a shift, the sequence gl(t) in this paper becomes the diagonal sequence gl+t(t) in nearest neighbor form. Right diagonal sequences in the nearest neighbor form, then, will have sequence distributions peaked about alternating sequences of 0's and l's, while nearest neighbor vertical series can be expected to be random. 3. Exercises for Chapter 11 1. Use backward reconstruction to determine which of the following rules could have generated the time series 10, 01, 00 , 10, 00: a) Rule 18 b) Rule 22 c) Rule 54 d) Rule 110

219

2. Find a 2 digit time sequence which could not have been generated by rule 30. 3. Use Theorem 11.5 to prove that no 3-site rule having xg = x7 = 0 can generate a 2 digit time series containing the sequence 11, 11. 4. Find the initial conditions from which the following rules generate the single digit time series 1, 0, 0, 1. a) Rule 54 b) Rule 18 c) Rule 22 d) Rule 110 5. Let X be a k-site rule which is linear in the k-1 variable. CanX generate all k-2 digit time series? Why?

220

Chapter 12 Surjectivity of Cellular Automata Rules A CA rule is surjective if every configuration has a predecessor. There are a number of important properties of CAs which are equivalent to surjectivity, a number of which are listed in a "kitchen sink" theorem in section 1. There are also graph theoretic techniques for the study of surjectivity, centered on the de Bruijn diagram which is defined in section 2. Another diagram, derived from the de Bruijn diagram, is the subset diagram. Both diagrams provide surjectivity criteria. Analysis of the adjacency matrix for the de Bruijn diagram allows definition of a semi-group associated to each CA rule, and several theorems on the structure of this semi-group are proved. Finally, a replacement diagram is associated to each CA rule. 1. A Kitchen Sink Theorem As defined in Chapter 10, a CA rule is surjective if every configuration has a predecessor, and injective if this predecessor is unique. The interest in surjectivity was first stimulated by Moore [82], who introduced the idea of Garden-of-Eden configurations. As indicated already, a rule is surjective if and only if it has an empty Garden-of-Eden. In Chapters 9 and 10 the Garden-of-Eden set GE(X) was defined in terms of the set of seeds GE*(X), and lemma 9.1 showed that a configuration is in GE(X) if and only if it contains at least one sequence from GE*(X). Whether or not specific rules are surjective is a major question. No general test for surjectivity is known, and for dimensions greater than 1, no such test can exist. This is a consequence of the following theorem proved by Kari [68] on the basis of the undecidability of tilings of the plane. Theorem 12.1 (Kari, [681) For any dimension d >_ 2 it is undecidable whether or not an arbritrary d-dimensional CA rule is injective, or surjective. For dimension 1 there are proceedures for determining whether or not specified rules are surjective. Finding an effective proceedure for all 1 dimensional rules, however, remains an open problem, and forms the focus of interest in this chapter. Theorem 12.2 (Amoroso & Patt, [831) Let X be a 1-dimensional k-site CA rule. Then the injectivity and surjectivity of X are decidable.

221

The classic study of surjectivity is contained in the paper by Hedlund [49]. Several of the results of this paper are synthesized in the following three theorems: Theorem 12.3 Let X be a k-site CA rule . Xis surjective if and only if every finite sequence sl...sn has exactly 2k-1 pre-images, and every infinite configuration has at most 2k-1 predecessors. Corollary 12.4 If a rule Xis surjective then it has an equal number of 0's and 1 's in its rule table. Proof Every finite sequence has exactly 2k-1 pre -images. In particular, therefore, 0 and 1 each have 2k- 1 pre-images, and these are the k-site neighborhoods. I Theorem 12.5 Let X be a k-site surjective CA rule . If gE E+ is periodic or eventually periodic, then all predecessors of.t under X are also periodic or eventually periodic. Proof Suppose to the contrary that X is surjective, and that B has period n in E+, but that B has a non-periodic predecessor g. But arn(B) = 13 for all r >_ 1. Since g is not periodic, however, arn(µ) * asn(g) for any r,s > 0, r,*s , otherwise it would be the case that a(r-s)n ( g) = g and g would be periodic . For all r >_ 1, however, X(arn(g)) = arnX(g) = am(13) = B . Thus B must have at least a countable infinity of predecessors , in contradiction to Theorem 12.3, and the claim is demonstrated. I Theorem 12.6 _ 1 and suppose that X:E+_3E+ is not surjective. Then there Let n > exists a finite sequence sl...sr having more than n pre -images . There also exist spatially periodic configurations having an uncountable number of predecessors. In terms of the map X :[0,1]-9[0,1] periodic and eventually periodic configurations correspond to rationals . Since the rationals are countable, Theorem 12 . 6 implies that if X is not surjective then there will be rationals in [0,1] having uncountably many irrational predecessors . If X is surjective, on the other hand, Theorem 12.5 requires that all rationals have only a finite

222 number of rational predecessors. Use of this in a contrapositive argument proves the next theorem. Theorem 12.7 X:E+-E+ is surjective if and only if X:[0,1]-[0,1] maps rationals to rationals and irrationals to irrationals. The next lemma shows that surjectivity is preserved under composition of rules. Lemma 12.8 Let X and Y be CA rules defined on E+. The composition XY is surjective if and only if X and Y are each surjective. Proof Clearly XY will be surjective if X and Y are surjective. Suppose then that XY is surjective, but X is not. Then there will be a configuration B which does not have a predecessor under X. But it does have one under XY, say g. Thus XY(µ) = B, hence Y(µ) is a predecessor to B under X, contradicting the assumption. Suppose that Y is not surjective. By theorem 12.5 there will be a periodic configuration, say B, with an uncountable number of predecessors under Y. Let these predecessors be denoted µ(s). Then XY(t(s)) = X(B), hence the configuration X(B) has uncountably many predecessors under XY, but this cannot be since by theorem 12.3 and the surjectivity of XY this rule can have at most 2k+r-1 predecessors where Xis a k-site rule and Y is an r-site rule. I The spatial measure entropy of a rule X was defined for sequences of length n in equation (10.1), and was related to the fractal dimension of the Cantor set d(X) defined by X in equation (10.2). The topological entropies for sequences g are defined by [28]: 1 2n St (n; t)=-nlog2 H(pn(i;µ)) i-1

(12.1)

where pn(i;µ) is the probability of occurence in g of the i-th length n sequence and O H(p)={0 Pp=0 Comparison of equations (10.1) and (12.1) indicate that the topological and measure entropies satisfy the inequality Sn(X) S St(n;X(-t)) < 1 (12.2)

223 Since elements of GE*(X) have length greater than or equal to some critical length nc the topological entropy will be 1 for values of n less than this critical length. If a configuration µ is infinitely distributed and Xis a k-site surjective rule then, as indicated by Theorem 11.7, the configuration X(µ) will also be infinitely distributed since each n digit sequence in X(µ) has 2k-1 pre-images, and each of the 2n+k-1 digit pre-images occurs in g with equal probability. Thus Sn(X) = 1 if X is surjective. In fact, if g is infinitely distributed and Sn(X(4)) = 1 then X is surjective since maximum spatial measure entropy requires that all probabilities be equal, implying that for all n, all length n sequences have the same number of pre-images. In addition, if Sn(X(g)) = 1 for all n, then by equation (10.2) the image of X has dimension d(X) = 1. The temporal probabilities for occurence of length n series were discussed in Chapter 11, and their temporal measure entropy is defined in equation (11.6). It was pointed out that the value of this entropy for a rule X, as estimated via the p-matrix method, would be expected to correspond to the empirically determined entropy if and only if GE*(X) = 0. This follows since the p-matrix technique assumes that the property of infinite distribution is preserved under interation of X, and this is true if and only if X is surjective. The various properties equivalent to surjectivity which have been discussed can be combined into a "kitchen sink" theorem, although it must be understood that this theorem makes no claim to be exhaustive. Theorem 12.9 Let X be a k-site CA rule. Then the following statements are equivalent: 1. X is surjective. 2. GE*(X) = GE(X) = 0. 3. Every finite sequence sl...sn has exactly 2k-1 pre-images. 4. Every element of E+ has at most 2k-1 predecessors. 5. X maps eventually periodic configurations to eventually periodic configurations, and non-periodic configurations to non-periodic configurations. 6. X:[0,1]-x[0,1] maps rationals to rationale and irrationals to irrationals. 7. X has maximal spatial measure entropy (Sn(X) = 1 for all n). 8. The Cantor set defined by X has dimension d(X) = 1.

224 9. X preserves the infinite distribution property of configurations. 10. The p-matrix method is an accurate estimator for temporal measure entropies. In Chapter 6 it was pointed out that while all non -injective rules defined on En led to a decrease in entropy , a similar result was not true for some of these rules when defined on E+. Indeed, the question might arise as to why any rule can have maximum entropy since it is known that periodic configurations map to fixed points or cycles . For a finite configuration space this will necessairly lead to a reduction in entropy unless a rule is injective. For the configuration space E +, however, the situation is different . Property 6 in theorem 12.9 provides the explanation for the entropy reducing properties of CA rules . The set of eventually periodic configurations, corresponding to the set of rationals in [0,1], has measure 0 with respect to the full configuration space . Hence, if a rule is surjective , it has an entropy reducing effect only on a set of measure 0 . On the otherhand, if a rule is not surjective, then by Theorem 12.6 there will be rational elements of [0,11 having uncountably many irrational predecessors . In these cases, entropy reduction is only to be expected. 2. The de Bruijn Diagram Applications of graph theory have played a major role in much of the research on cellular automata . For present purposes the two graphs which are important are the de Bruijn diagram and the subset diagram . Reviews detailing many of the possible applications of these, and other graphs to the study of cellular automata have been published by McIntosh [38,39,84]. The de Bruijn diagram plays a central role in Jen 's work on computation of preimages [34,85], and in Wolframs studies of the relation between cellular automata and formal languages [29]. These diagrams can be considered one of the essential tools of the trade. For a k- site CA rule X, the de Bruijn diagram is defined as a labeled directed graph with 2k- 1 verticies and 2k edges . The verticies are labeled with the binary digits of the integers ranging from 0 to 2k-1-1. An edge is directed from vertex i = i0...ik-2 to vertex j = j0•••jk - 2 if and only ifjs = is+1 for all s such that 0:5 s:5 k-3 . That is , if and only if these two digit strings can be overlapped to form a k-site neighborhood, which will be denoted i•j. When

225 this is the case, the edge connecting vertex i to vertex j is labeled with the value X(i *j). The de Bruijn diagram without edge labels tells which k-1 digit strings can be overlapped to form a k-site neighborhood. When the edge labels are included it gives the same information about a rule as does the rule table. As will be seen, however, it also provides a means of continuing the analysis of a rule, unpacking the information in the rule table. Figure 12.1 shows the de Bruijn diagram for the generic 3-site rule, labeled in terms of components. An immediate property of this diagram is that it allows determination of pre- images. KO

00

7 Figure 12.1 Generic de Bruijn Diagram for 3-Site Rules

If sl...sn is any binary sequence, the pre-images for this sequence under a rule X are found by finding all length n paths in the de Bruijn diagram for X which have edge labels sl...sn, and then constructing the length n+k-1 sequences generated by overlapping the k-1 site blocks which label the verticies of these paths. If no such path exists, then sl...sn has no pre-image [38,391. For example, if X is the 3-site rule 30 and the sequence 10011 is given, the pre-images of this sequence can be read off from the de Bruijn diagram of Figure 12.2.

226 Examination of this figure shows that there are four paths with the label 10011. These connect the verticies listed, together with the pre- images constructed from their overlap, in Table 12.1. Connected Verticies

Pre-Image

10, 00, 00, 00, 01, 10 10, 00, 00, 00, 01, 11

1000011

01, 11, 11, 10, 00, 01

0111001

01, 11, 10, 01, 10, 00

0110100

1000010

Table 12.1 Pre-Images for 10011 Under Rule 30

0

11

0 Figure 12.2 de Bruijn Diagram for Rule 30

In Definition 9.3 linearity in the r-th variable was introduced, and the particular cases of linearity in the initial variable (LW), and linearity in the final variable (LFV) were distinguished, and a rule possessing one or the other of these properties was called linear in an extreme variable (LEV). It will turn out that these properties are closely related to surjectivity. Lemma 12.10 Let X be a k-site CA rule. Then:

227

1. If X is LIV the incoming edges at each vertex in the de Bruijn diagram for X have distinct labels. 2. If X is LFV the out going edges at each vertex of the de Bruijn diagram for X have distinct labels. Proof If X is LIV then, by Lemma 9.4 (equation (9.3)) xi+2k_1 = 1 + xi for all i between 0 and 2k- 1-1. But each vertex il...ik-2 of the de Bruijn diagram for X has incoming edges from verticies OiO...ik -3 and 1i0 ...ik-3, and hence overlaps with these verticies to form the neighborhoods 0i0...ik -2 and 1iO ...ik-2 which map to distinct symbols under X. Likewise, if X is LFV then x2i+1 = 1 + x2i for 0<_ i:5 2k-1-1. Each vertex il ...ik-2 also has out going edges to verticies il...ik-20 and il ...ik_21, overlapping with these to form neighborhoods iOil ...ik-20 and iOil ...ik-21. Again, these will map to distinct symbols under X. I The adjacency matrix for the unlabeled de Bruijn diagram for all k-site rules has the form d;j _

1 is = Js-1 1!- s<_ 2k-2 _ 1 0 otherwise

which defines a "staircase" structure . For example, for 3- site rules this matrix is 1 1 0 0 d= 0 0 1 1 1 1 0 0 0 0 1 1 Since this matrix is the same for all rules, some means of identifying particular rules is required. This is accomplished by including the edge labels and defining two matricies, d0 and dl, such that d0 counts only the de Bruijn edges labeled 0 and dl counts only those edges labeled 1. These matricies, called de Bruijn fragments by McIntosh [38,39,84], are most easily given in terms of the V matrix introduced in equation (3.3) of Chapter 3. Lemma 12.11 For a given k-site CA rule X the de Bruijn fragments are d0(X) = V(1+X) = V(T3X) di(X) = V(X)

Remark By definition, d = d0(X) + dl(X).

228 Use of the de Bruijn fragment matricies for analysis of CA properties has been extensively advocated by McIntosh. These matricies have also been discussed by Jen [85] , and they have been used in other applications by Backhouse & Carr [86], and by Conway [87]. Theorem 12.12 Let X be a k-site CA rule. Then: 1. d0(X) and dl(X) are permutation matricies if and only if X is both LIV and LFV. 2. If X is LIV but not LFV then both d0(X) and dl(X) have at least one row of 0's, but no columns of all 0's. 3. If X is LFV but not LIV then both d0(X) and dl(X) have at least one column of 0's, but no rows of all 0's. Proof By definition, if d0(X) is a permutation matrix then so is dl(X), and vice versa. Thus, for proof of the first statement it is only necessary to show that dl(X) is a permutation if and only if X is both LIV and LFV. Suppose that dl(X) is a permutation matrix. Then each row and each column contains a single 1 . Hence, by the structure of dl (X) = V(X), reading down each column yields the condition xi+2k_1 = 1+xi for all i <_ 2k-1-1, and so Xis LIV. Reading across each row yields x2i+1 = 1+x2i for 0!5 i!5 2k-1-1, so X is also LFV. On the other hand, if Xis both LIV and LFV this same argument implies that dl (X) can have only a single 1 in each row and column so that the matrix is a permutation. Now suppose that Xis LIV but not LFV. Then there will be at least one value of i such that x2i+1 = x2i and hence the row in dl(X) which contains these values will have either two 1 ' s, or two 0 's. In Theorem 12.18 it is shown that LEV implies surjective. Thus, Corollary 12.4 implies that each of the matricies d0(X) and dl(X) contains an equal number of 1's. Thus a row of all 0 's in one of these matricies must be compensated by a row of all 0's in the other. Since Xis LIV, however, xi+2k-1= 1+xi for all i<_ 2k-1-1 and so neither d0(X) nor dl(X) can contain a column of 0's. A similar argument will demonstrate that if Xis LFV but not LIV then d0(X) and dl(X) must each contain at least one column of 0's , but will have no rows of 0's. I By definition, [ds(X)]ij = 1 if and only if the neighborhood i ♦j maps to s under the rule X. But [dr(X)ds(X)]ij = 1 if and only if there is at least one m

229 such that [dr(X)]im[ds(X)Imj = 1 , meaning that the neighborhood i ♦ m maps to r and the neighborhood m•j maps to s . If this is the case , then the sequence i•m•j maps to the sequence rs under X . Thus, the value of the matrix product element [dr(X)]im[ds (X)]mj counts the number of distinct length 2 paths labeled rs which connect vertex i to vertex j of the de Bruijn diagram. Theorem 12.13 The number of pre-images of a sequence s1...sn under a k-site rule X is given by the sum of elements in the product 7n ^ l ldsi ( X) i=1

(12.3)

Corollary 12.14 A sequence s1...sn has no pre-images under a rule X if and only if the product of ( 12.3) equals the zero matrix. Another way of stating this result is given in the next theorem. Theorem 12.15 Let X be a given k-site CA rule . GE*(X) = 0 , hence Xis surjective, if and only if the free semi-group G(X) with generators d0(X) and dl(X) does not contain the zero matrix. The d0 (X) and dl (X) matricies may be used as another means of generating the set GE*(X). This is done by making use of the formal expansion do = [d0(X)+dl(X)]n, bearing in mind that these matricies will not, in general, commute. The smallest value of n for which a term in this expansion is 0 will be the critical sequence length nc at which the topological entropy begins to decrease . The shortest sequences in GE *(X) will then be the sequences sl...snc such that nc

fldsi (X) = 0 i=1

Each of these products is dropped from the expansion, and the remaining terms are multiplied by d0 (X)+dl(X). Again, some terms may turn out to be equal to 0. The corresponding nc+1 digit sequences not already containing one of the sequences already in GE*(X) as a sub -sequence are added to GE*(X), and all zero terms are again dropped . The process continues, either until it terminates with no new sequences in GE *(X) being

230 generated, or until GE *(X) has been computed to whatever length sequences are desired. 3. The Subset Diagram Closely related to the de Bruijn diagram for a rule Xis the subset diagram. This diagram has two forms which will be called the full , and the pruned subset diagrams . The second of these is the more useful since it is substantially smaller and requires correspondingly less computation. The pruned subset diagram is generated from the de Bruijn diagram by the following algorithm: Algorithm 12.16 1. Define the full subset S as the vertex set for the de Bruijn diagram. Take S as the initial vertex. 2. For a given rule X let SO be the set of verticies of the de Bruijn diagram which can be reached from some element of S by an edge labeled 0, and let S l be the set of verticies which can be reached from some element of S by following an edge labeled 1 . Take SO and Sl as the next level verticies Draw an edge labeled 0 from S to SO and an edge labeled 1 from S to Sl. 3. Let S00 be the subset of verticies which can be reached by following an edge labeled 0 from some element of SO , with equivalent definitions for SO1, S10, and Sll. Add each of these subsets to the subset diagram vertex set, and draw edges to them from SO and Sl, labeled with their defining symbol from the de Bruijn diagram. 4. Continue this process, now taking S00, Sol , S10, and Sll as the initial subsets . Continue in this way until no new subsets are generated. 5. If any subset generated in this process has edges emerging from its elements in the de Bruijn diagram which are labeled with only a single symbol, then an edge labeled with the missing symbol is drawn to the empty set 0.

In the full subset diagram this process is carried out for all sets of de Bruijn verticies rather than starting with the full subset. Figure 12 .3 shows the pruned subset diagram for the 3 -site rule 54 (01101100). Sequences without pre-images are identified in the subset diagram as those corresponding to labeled paths which begin at the full subset and terminate at the empty subset.

231

1

0 V 0011)

(0000) 1

0

Figure 12.3 Pruned Subset Diagram for Rule 54 (Subset elements are 00, 01, 10, and 11. Verticies are labeled to indicate whether or not a particular one of these elements is present or not in the denoted subset.) As with the de Bruijn diagram, the LEV properties are closely

connected with the properties of the subset diagram. Lemma 12.17 Let X be a k-site CA rule. Then in the full subset diagram: 1. If X is LIV the out going edges of the full subset S are loops. 2. If X is LFV the incoming edges of the empty subset are loops. Proof If Xis LIV then by Lemma 12 . 10 the incoming edges at each vertex of the de Bruijn diagram have distinct labels . Thus, every vertex i0...ik-2 of this

232 diagram has an incoming edge labeled 0 and an incoming edge labeled 1. Hence the full subset , which contains all verticies , maps to itself with both 0 and 1 edges. Again, if Xis LFV then the out going edges at each vertex of the de Bruijn diagram have distinct labels . Hence there is no vertex lacking an out going edge with a label of either 0 or 1, and so no edge is directed to the empty subset except those originating at the empty subset. I Since each case in Lemma 12.17 excludes the possibility of a path from the full subset to the empty subset , this proves the next theorem. Theorem 12.18 Let X be a k-site CA rule. If X is LEV then X is surjective. This theorem indicates that the LEV property is sufficient for surjectivity. In addition, all 2-site surjective rules are additive , while all 3site surjective rules are LEV. The LEV property is not , however, necessary for surjectivity, as is indicated by the example of the 5 - site rule which is obained as the composition of rules 30 and 86 . Rule 30 is LIV while rule 86 is LFV, hence their composition is surjective. It is easy to verify , however, that their composition is not linear in any variable. It is tempting to conjecture that surjectivity is equivalent to being either LEV, or composed of rules which are LEV. Since all surjective 3-site rules are LEV, an induction argument shows that this conjecture would be true if there were no prime k-site rules with k > 3 which are surjective but not LEV. This last condition, however, is not true . Patt [88] has shown that there are exactly eight non trivial 4-site rules which are injective (hence surjective) and prime, but not LEV; and Amoroso & Patt [83] prove the following theorem , which indicates that further such rules will appear for all values of k. Theorem 12.19 For all n >_ 3 there exist non-trivial injective global maps which define a k-site CA rule for k > n, but are not the global map of any CA rule with k = n. Proof Let jr denote a block of r 1's. Then , for r _> 2, let s = 1r00 , s' = 0jr-k00, and define an r+2 site rule X such that for i = i0...ir+1 0 i=s X(i) = 1 i = s'

ip otherwise

233 With this definition, for all µe E+, X2(µ) = g, hence X is injective. If X is defined for more than 3-sites, it need not be LEV. Further, if X is defined for 4 sites then, if it were composed it would have to be a composition of a 2site and a 3-site rule, but it is easy to check that this is not the case. I The proof of this theorem is accomplished by exhibiting rules which satisfy the desired conditions. The way in which these rules were constructed, however, leads to the possiblity of other constructions as well. For example, in the proof given, the rules produced were taken as left justified. They could equally well be taken as defined for other mapping sites , and in fact Amoroso & Patt defined them for i 1 as the mapping site. Of the eight injective 4-site rules which are prime but not LEV, four are non-generative. The component expressions for these rules are given in Table 12.2. (0010110100001111) (0011100100110011) (0011001101100011) (0000111101001011) Table 12.2 Component Forms for Prime 4-Site non-LEV Injective Rules

The determination of exactly what conditions the components of a CA rule must satisfy in order for the rule to be surjective remains an outstanding problem. The matricies d0 and d1 can also be used to construct the subset diagram, a construction which McIntosh calls the vector subset diagram [38,391. As in Figure 12.3, let each of the verticies of the de Bruijn diagram for a rule X be taken as a coordinate for a binary vector in 2k-1 dimensions. Then each of the possible 22k-1 vectors c in this space corresponds uniquely to a subset C of the full subset via (0 vertex i e C

C: =

Sl 1 vertex i 4< C The pruned subset diagram is then generated by the following algorithm:

Aleorithm 12.20 1. Let c(1) be the vector with all 1 components. Take c(1) as the initial vertex.

234 2. Compute c(1)d0 and c ( 1)dl. Set all terms greater than 1 equal to 1. Take these products as the next two verticies , with an edge labeled 0 directed from c(1) to c(1)d0 and an edge labeled 1 directed from c( 1) to c(1)dl. 3. Repeat this proceedure , using c( 1)d0 and c( 1)dl as the new initial vectors . Continue until no new vectors are generated. 4. The vector c(0) consisting of all 0 entries corresponds to the empty subset. GE*(X) is empty if and only if the vector c(0) is not generated in this proceedure. This is the method used to generate the subset diagram for rule 54 given in Figure 12 . 3. Note that if a rule Xis LIV then by Lemma 12.17 c(1)d0 (X) = c(1)dl(X) = c(1), hence the pruned subset diagram consists of only a single vertex, c(1), with two loops. If a rule is not surjective then the subset diagram provides yet another means of computing GE*. If M is the adjacency matrix of the subset diagram (either the full diagram , or the pruned diagram will do) then the ij entry of Mn counts the number of length n paths which start at vertex i and end at vertex j . In particular, the entry with the label c(1),c(0 ) will count the number of length n paths which begin at c(1 ) and terminate at c(0), and the labeling of these paths gives all length n sequences without pre -images. The edges in the subset diagram as defined are labeled with either 0 or 1, but they could equally well be labeled with d0 and dl . With this labeling, the entries in the adjacency matrix of this diagram will be 0, d0, and dl. For example , for rule 54 this form of the matrix M(54) is given in Figure 12.4. do + d1

0

0

0

0

0

0

0

0

0 0

d1

0

0

do

0

0

0

0

0

0 0

do

0

0

0

0

0

0

0

d1

0 0

0

0

0

do

0

0

0

0

d1

0 0

0

do

d1

0

0

0

0

0

0

0 0

0

do

0

0

0

0

0

0

0

d1 0

0

0

0

0

d1

0

0

do

0

0 0 0 0

0

0

0

0

0

0

0

do

d1

0

0

0

0

0

d1

do

0

0

0 0

0

0

0

0

0

0

do

0

0

d1 0

0

0

0

0

0

0

0

do

0

d1 0

235 In this form the c( 1),c(0) entry of Mn not only counts the number of length n sequences without pre -images, but lists them, as products of d0 and dl. In this way, the difference between the cases in which GE* is finite or infinite is seen to depend on the way in which the matrix M behaves when raised to powers . If after some finite n the c(1),c(0 ) element always contains d0 and d1 products which have already appeared for lower values of n, then GE* will be finite. In the rule 54 example, computation yields the first non-zero [Mn(54 )lc(1),c(0) as

[M5(54) lc(1),c(0 ) = dldpoldpd1 + d1dpdidp +dpdidpd1 indicating that the first entries in GE* (54) are the five digit sequences 10101, 10110, and 01101 . No new seeds appear until n = 8, where the sequence 10111101 shows up, after which no further seeds appear. 4. The Semi-Group OX) The free semi-group generated by d0(X) and di(X) was introduced in Theorem 12.15. In this section some of the properties of this semi -group are determined. Theorem 12.21 G(X) is a group if and only if Xis both LIV and LFV. Proof If X is both LIV and LFV then by Theorem 12.12(1) G(X) is a permutation group. The proof in the other direction will proceed by contrapositive. Suppose that Xis not LW. Then, by Theorem 12.12(2) both d0 and di will have at least one row of zeros, say r0 and rl . Further, these cannot be the same row since d0+dl = d. Assume now that G(X) is a group. Then it will contain an identity element e, and all elements will have inverses . In particular, d0 and dl will have inverses . Let v0 be a row vector which has a 1 in the r0 position only, and vl a row vector with a 1 only in the rl position . Then v0 (d0d-10) =0 since vodo = 0 . Likewise , vl(dld- 1l) = 0. Thus, v0(edl ) = vl(ed0 ) = 0. But these last products cannot be zero since d 1 must have a pair of l 's in row r0, and d0 must have a pair of l 's in row 1. Hence not all elements of G(X) can have inverses , and G(X) cannot be a group. Likewise , if X is not LFV then each of

236 d0 and dl will have a column of 0's, and these cannot be the same column, so a similar argument applies to show again that G(X) cannot be a group. I As might be expected, the semi-groups defined by rules which are related by the transformations Ti, T2, and T3 are also closely related. The action of these transformations on a rule X can be expressed in terms of their action on the components of X--through their actions on neighborhoods in the case of Ti and T2, and in terms of complimentation in the case of T3. Thus [Tl(X)]j = xT10), [T2(X)]j = xT2(j), [T3(X)]j = 1+xj. Proof of the next theorem follows from the observation that if j and m are k-1 site partial neighborhoods which can be overlapped to form a k-site neighborhood then T1(j ♦ m) = T1(m),T1(j) and T2(j ♦ m) = T2(j),T2(m). Theorem 12.22 Let X be a k-site CA rule. The groups G(X), G(Tl(X)), G(T2(X)), and G(T3(X)) are isomorphic. The isomorphisms relating these groups are defined in terms of their generators by: [ds(Tl(X)]jm = [ds(X)1T1(m),T1(j) [ds(T2(X)]jm = [ds(X)1T2(j),T2(m) [ds(T3(X)Ijm = [ds'(X))]jm s' = 1+s mod(2) Appendix 6 lists the semi-group G(X) for some rules of interest. 5. The Subset Matrix and Some Replacement Diagrams In construction of the subset diagram via Algorithm 12.20, the d0 and dl matricies were multiplied on the left by vectors c which described subsets of the full subset of de Bruijn verticies. In addition, the "topological set" condition that terms greater than 1 be set to 1 was imposed. If the i-th component of the product cds is non-zero it means that there is an edge labeled s from one of the verticies in the subset characterized by c to vertex i. Similarly, if the i-th component of n

c[Tdsj j=1

is non-zero it means that there is a path labeled sl...sn from at least one of the verticies in the subset characterized by c to vertex i. The matricies d0 and dl can also be used to multiply column vectors on the right. If c is defined in the same way as before, but now as a column

237 vector, then if the i-th component of dsc is non-zero it means that there is an edge in the de Bruijn diagram labeled s from vertex i to one of the verticies in the subset characterized by c. Without the topological set condition, the i-th component of n r1dsj j=1

c

(12.4)

counts the number of length n paths labeled sn...sl which originate at vertex i and terminate at a vertex in the subset characterized by c. (Note the reverse order of indices!) Thus, if c is taken as a unit vector with r component equal to 1 and all other components equal to 0, then the product in (12.4) counts the number of length n paths labeled sn...s1 which originate at vertex i and terminate at vertex r. In addition, by a process similar to that of Algorithm 12.20, a diagram is generated which in general will be infinite, but which can, at least in some cases , be characterized by a relatively simple recurrence diagram, which shown how CA rules can generate specific number sequences such as the Fibonacci sequence. The following examples will serve to illustrate the idea: 1. For the 2-site binary difference rule D = (01110) do(D)=I0 1 d1(D)=C1 Ol 0) 0 1 The diagram generated, taking c = 10 as the starting point, is shown in Figure 12.4.

d 10 01 d1 Figure 12.4 2. For rule 90 , S = (01011010) 1 0 0 0 0 1 0 0 do(8) = 0 0 O1 00 d1(8) = 0 1 00 0 0 0 0 1 0 0 1 0 and the diagram obtained is shown in Figure 12.5.

238

0010

d1

1000

Figure 12.5 In both of these first two examples the diagrams are compact. There are no positive feedback loops and no numbers larger than 1 appear in any of the vertex nodes . Thus, the number of paths connecting pairs of verticies will be a constant, independent of the path length. Note also that only unit vectors appear as vertex labels in these diagrams, a consequence of the fact that the d0 and dl matricies are permutation matricies. The next set of examples consider more general cases, in which the rules considered are not additive, and GE* is not empty. In these examples the diagrams will be infinite, but it will be seen that they can be easily represented by finite replacement diagrams. That is, diagrams which start with the vector c = (abcd), with variable components, and indicate how these components map at each multiplication by d0 or dl. The replacement diagrams for the rules given in the two previous examples represent simple permutations. For the rules D and 8 respectively, these are a-^b b->d

a-a a b^c c->b

C->a C1-4C

d-,d

D

S

3. Let X be the 2-site rule (0111). Then

do (X) _ (0 0) d1(X) _

(0

1

These matricies generate the infinite diagram which is shown in Figure 12.6.

239

d0 01 d^ 00 'd1l 11 d

d 23 -20

.d1 35 >30 d1i

Figure 12.6 Although infinite, inspection shows that this diagram can be represented by the finite recurrence diagram with replacements shown below. C11

4. For the 3-site rule 18 (1 0 0 0 0 0 1 11

dp(18)=

I0

1 00

0 0

1 1

d1(18) =

0

1 0 0

0

0

0 0

1 0 0 0 0

0 0 0

240 The diagram generated with these matricies from an initial 0001 is shown in Figure 12.7.

d1 10000

d0

0101

d1

d1 ->1000

0010

d0 J d1- >220000 d j; 0020 d0`J 0323

dl 3000 d

d0 0535

d0^

0030

d0o 5000 d

d

0050

d Figure 12.7

Again the diagram is non-compact, and can be seen to be generating Fibonacci sequences in its output. The numerical relations which it contains are generated by the recurrence diagram

d0 b-+s

241 In reading off pre-image numbers from this diagram the additional information dO(000a) = (OaOa), dl(000a) = (0000), dO(Oa00) = (00a0), and d1(OaO0) = (a000) is useful. As an example, suppose that the sequence 000110001 is given. The number of paths with this label in the de Bruijn diagram, which terminate at 00 is determined by starting at a000 with the initial condition a = 1, and following the arrows labeled in the order 100011000 (recall the reverse ordering in equation (12.4)), making the indicated substitutions along the way. This gives 1000-0010--)0101->0111-0212---)2000->0020-)0202->0222-->0424 indicating that there are four paths labeled 000110001 which start at 01 and terminate at 00, four which start at 11, and two which start at 10 and terminate at 00. The replacement diagrams can most compactly be specified in terms of the mappings of the full subset vector, represented abstractly for 3- site rules as abcd. For rule 18, the diagram becomes b -^cf dd0 c-,b d-+ cid

abcd

d1a-4b ^b-40 C -4 a

d-40

Likewise, for the 2-site rule of example 3, the diagram is d0_^, A-d1 a-4b b -^ O b-a+b With c( 1) the vector consisting of all 1 entries, the i-th component of the product n

fj dsj c(1) j=1

counts the number of length n paths labeled sn...sl in the de Bruijn diagram which start at vertex i = iO...ik-2. This counts the number of pre-images of the sequence sn...sl which start with iO...ik-2. This is the number LS (n; X) used in the Jen recurrence relations for numbers of pre-images [33,34]. Thus the total number of pre-images of sn...sl is given by the expression 2k-1 -1

N(sn...si ) =

E

[[ ft ds.]c(1)]

i=0 j=1

(12.5)

242 which gives a connection of the replacement diagram to Jens recurrence relations. 5. In the rule 18 example, the number series which were generated satisfied Fibonacci recurrence relations . Rule 12 provides an example in which the number series is a simple counting relation. For rule 12

(1 1 0 0' (0 0 0 0' d1(12) =

d0(12)= 0 0 0 0 0 0 1 1

0 0 1 1 0 0 0 0 0 0 0 0

generating the diagram of Figure 12.8.

1000 d1 0000 ----

1010^•'0100,• d1.

10 1 d 1 ..

d1

0200 ,.

d>2020 d-1 } 0400

0300 , •"0d>3030 -}>

2022

Figure 12.8 The replacement diagram for this rule is CIO dl b-+0 abcdb->cid c -* a+b c -00 C1-+ c 4d d ->0 In the first two of these examples, the number of paths determined from the diagrams generated remained constant. In the remaining examples this number grew with path length as the result of the appearence of a positive feedback loop in the diagram. The disadvantage of the compact form of the replacement diagram is that it does not make the presence of feedback loops manifestly obvious, while the presence of such loops is directly related to surjectivity since all finite sequences have the same number of pre -images for surjective rules . On the other hand, the compact form does present an

243

iterated replacement scheme on 2k-1 letters which is easy to implement on a computer. Theorem 12.23 Let X be a k-site CA rule . Xis surjective if and only if a positive feedback loop does not appear in the replacement diagram for X. Proof If a positive feedback loop appears then the number of paths with a given label will grow with increasing path length. It will always be possible to choose a path long enough that this number exceeds 2k-1 so by Theorem 12.3, X cannot be surjective. Suppose that Xis not surjective. Then , by Theorem 12.6, for any given n there will be a finite sequence having more than n pre -images . This can only be the case if there are more than n paths with this sequence as label in the de Bruijn diagram for X and hence the number of paths increases with path length for at least some paths . But this can only occur if there is a positive feedback loop in the dsc diagram. I 6. Exercises for Chapter 12 1. Prove that LIV and LFV are sufficient conditions for the matrix products VT(X)V(X), and V(X)VT(X) respectively, to be diagonal. 2. Find a prime 5 -site injective rule which is not LEV. 3. Construct the de Bruijn diagrams for the following rules: a) (01011000) b) (01111100) c) (0010110100001111) d) (01001000) e) (01011010) 4. Use the de Bruijn diagrams to find all pre-images for the following sequences for each rule in exercise 3: a) 0110 b) 011100 c) 1001011 d) 010101 e) 101001000 5. Use Theorem 12.13 to compute the number of pre -images for each of the sequences in exercise 4, under each rule from exercise 3.

244 6. Construct the pruned subset diagram for each rule in exercise 3. 7. Refering to Figure 12.3, find all 6 digit sequences without pre-image under rule 54. 8. Construct the replacement diagrams for the following 3-site rules: a) Rule 36 b) Rule 164 c) Rule 108 d) Rule 22 e) Rule 178 9. Compute the number of pre-images of each of the following sequences for each rule in exercise 8: a) 10 b) 11111 c) 011011 d) 01010111 e) 10110110101 10. For rule 18 prove that ldo+1(18)d1(18 )]0 = Fn+2 where Fn is the n-th Fibonacci number (FO = F1 = 0, F2 = 1).

245

Appendix 1 Boolean Expressions for 2 and 3-Site Rules Expressions are given only for rules with x0 = 0. The Boolean expression for a rule (1xl...x2k_1) is obtained by adding 1 to the expression for the rule (0x1 ...x2k_1). The rule is applied to variables x,y (2-site) or x,y,z (3site ). All sums are mod(2). The Boolean expressions in these tables are given in terms of the AND and XOR (exclusive or) operations. They differ from the Boolean entries given by Wolfram in his Table 1, pages 516-521 of Cellular Automata and Complexity since he uses the AND and OR operations. 1. Two Site Rules: Boolean Expression 0 xy xy X,

Rule 0000 0001 0010 0011 0100 0101 0110 0111

xy

y x+y x+ +x

2. Three Site Rules: Rule

A

Y

00000000 00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000 00001001 00001010 00001011 00001100 00001101 00001110 00001111 00010000 00010001 00010010 00010011 00010100

0000 0001 0001 0000 0010 0011 0011 0010 0010 0011 0011 0010 0000 0001 0001 0000 0100 0101 0101 0100 0110

0000 0000 0001 0001 0000 0000 0001 0001 0010 0010 0011 0011 0010 0010 0011 0011 0000 0000 0001 0001 0000

Boolean Expression 0 xyz xyz' xy xy'z xz x(y+z) x(y+z+yz) xyz' x(1+y+z) xz' x(1+y'z) xy' x(1+yz') x(l+yz) x x'yz yz y(x+z) y(x+z+xz) z(x+ )

z( ,x+T) z A Ax+z,Ax Ax+z(A+x+T) z(A+x+T) zA x xx+^+x zAx+Ax+A+x zAx+A+x

A+x zAx+,zx+A,x ,zx+Ax A+,zx zAx+A+x z,Ax+A zx+A zx+Ax z,Ax+, x A zAx+A

zAx+A,x Ax zAx+,zA+A ,zA+Ax zA+x zd x+A+x Ax+,z(A+x) z(Ax+A+x) zAx+,z(A+x) z(A+x) zA+zx z,Ax+zA zx+z^X

zA+zx+Ax A + (z x+T )A zA (z+x)A

,z,^x zA,x+x zA+x zA+Ax ;fx+zAx

zA+ zx zAx+ ,Ax z^fx+(z+^f+T)x zA+(z+X+T )x zA+zx+Ax zAx+zA+zx+Ax (Ax+A+x)z

TTTTT000 01111000 TOT1T000 00111000 11011000 OTOTT000 TOOTT000 OOOTT000 TTTOT000 OTT01000 TOTOT000

0010 TOT0 1010 OOTO OTTO TITO TITO OTTO OTTO TITO TTTO

TT00 TT00 OT00 OT00 TT00 TT00 OT00 0100 T000 T000 0000 ,

00000100

OOTO

OOTO

TOTOOOTO OOTOOOTO 110000TO 010000TO 100000TO 000000TO TTTTTTOO 01111100 TOT1T100 OOTTTTOO TTOTTTOO 0101TT00 TOOI1100 OOOITT00 TTTOTT00 01101100 10101100 OOTOTT00 TTOOTT00 OT001T00 10001100 OOOOTT00 TTTTOT00 OTTIOT00 10110100 OOTTOT00 11010100 OTOTOT00 TOOTOT00 00010100 11100100 OT100T00 TOTOOT00 00100100 11000100 OTOOOT00 TOOOOT00

TTOT OTOT OOOT TOOT TOOT OOOT 0000 T 000 T000 0000 OT00 TTOO TT00 OT00 OT00 TT00 TT00 OT00 0000 T000 1 000 0000 OOTO OOTO TOT0 OOTO 0110 TITO TI10 OTTO OTTO T1TO TITO OTT O OOT O T010 TOT0

0000 0000 T000 T000 0000 0000 TTTO TTTO OTTO OTTO TITO TTTO OTTO OTTO TOT0 TOT0 OOTO OOTO TOT0 TOT0 0010 OOTO TITO TITO OTTO OTTO 1110 TITO OTTO OTTO TOM TOT0 OOTO OOT 0 TOT O TOT0 0010

9bZ

247 (1+x'y)z+xy y'z+xy (x+z)y' xy'+(l+x'y)z x+(1+x'y)z x+y'z xy'+x'y'z xy'+(1+x+y)z x+(1+x+y)z x+x'y'z x'z ( 1+xy')z xy+(l+xy')z xy+x'z ( 1+xy)z z xy+z z+xyz' xy'+(l+xy)z xy'+z x+z x+z+xyz xy'+x'z z+xy'z' x+(1+xy')z x+z+xz x'(y+z) x y+(l+xy')z y+(1+xy')z y+x'z x y+(l+xy)z x'y+z y+z y+(1+xy)z x+y+(1+xy)z x+y+z x+y+z+xy x+y+z+xyz' x+y+z+xz x+y+z+xy'z x+y+z+xz+xyz' x+y+z+xy+xz x'y+x'y'z x y+(l+x+y)z y+(1+x+y)z y+x'y'z x y+y'z

01000110 01000111 01001000 01001001 01001010 01001011 01001100 01001101 01001110 01001111 01010000 01010001 01010010 01010011 01010100 01010101 01010110 01010111 01011000 01011001 01011010 01011011 01011100 01011101 01011110 01011111 01100000 01100001 01100010 01100011 01100100 01100101 01100110 01100111 01101000 01101001 01101010 01101011 01101100 01101101 01101110 01101111 01110000 01110001 01110010 01110011 01110100

1011 1010 1010 1011 1011 1010 1000 1001 1001 1000 1100 1101 1101 1100 1110 1111 1111 1110 1110 1111 1111 1110 1100 1101 1101 1100 1100 1101 1101 1100 1110 1111 1111 1110 1110 1111 1111 1110 1100 1101 1101 1100 1000 1001 1001 1000 1010

0001 0001 0010 0010 0011 0011 0010 0010 0011 0011 0000 0000 0001 0001 0000 0000 0001 0001 0010 0010 0011 0011 0010 0010 0011 0011 0100 0100 0101 0101 0100 0100 0101 0101 0110 0110 0111 0111 0110 0110 0111 0111 0100 0100 0101 0101 0100

01110101

1011

0100

z+x'yz'

01110110

1011

0101

y+(1+x' )z

248 01110111 01111000 01111001 01111010 01111011 01111100 01111101 01111110 01111111

1010 1010 1011 1011 1010 1000 1001 1001 1000

0101 0110 0110 0111 0111 0110 0110 0111 0111

y+y'z x+y+y'z x+y+(1+x'y)z y+(x+z)y'+xyz y+(x+z)y' x+y+x'y'z z+(x+y)z' x+y+z+xy+xz+yz x+y+z+xy+xz+yz+ xyz

249

Appendix 2 Canonical Forms and Decompositions of 3-Site Rules The first column of this table lists the decimal label of the rule. The second column gives the label of its conjugate rule under the transformation T1. If this column is empty then the rule is self-conjugate. The third column gives the expression of the rule in terms of the canonical basis operators, and the fourth column lists the strongly legal decompositions of the rule. Recall that if a rule Xis decomposed as X = A+F where A is an additive rule then this decomposition is strongly legal if the kernel of F in En-{Q,j is large. What this requires is that F be one of the following combinations of basis operators, listed togeather with the configurations which they map to Q, in Table A2.1. The full listing of conditions for a configuration to map to Q under various combinations of basis operators is given in Table 3.3. F 13+,13±+X T1+, T1±+K

Configurations Manned to 0 Contain only isolated 1's Contain only isolated 0's

13++8,13±+X+8

Isolated 1's separated by two or more 0's

11±+t, Tl±+t+K x 8 t K

Isolated 0's separated by two or more 1's Contains no 111 blocks Contains no isolated 0's Contains no isolated 1's Contains no 000 blocks

Table A2.1 Note that this table lists forms involving K. If this basis operator is involved in the canonical form of a rule, then the rule is generative. In terms of the listing given below, canonical forms for generative rules with numerical label 2m+ 1 are obtained by adding the basis operator K to the form for the non-generative rule with label 2m. Rule 0 2 16 4 20 6 8 64 10 80

Canonical Form

Strongly Legal Decompositions

11+ t it++t 13-+T1+

a2+13-+X+B a+13++13-+x aD+13++8 none a2+X+8

12 14

13-+t 13-+Tl++t

a+f3++X a2+X+8+t, aD+13++f3-+8

68 84

3-

250 16 18 20 22 24 26 28 30 32 34

2

I+B+ +x +9

66 82 70 86

iiTl++rl71-+t rl++tj-+t 13-+,nB-+il++iiB-+i1 +t B-+il++Tl-+t

48

il++9

none a2+B- +x

9+t

a+B++B-+x+9

6

8

36

5+13++B-, A+X+t D+13-+9

e+x I+B++B-+x+9, D+9+t S+B+ D+9

e +B- + x

38 40 42

52 96 112

t1++9+t B-+9 B- +Tl ++8

aD+B+ D+il-+t a2+x, D+il++il-+t

44

100

B-+0+t

a+B+ +x+9, D +T--

46 48 50 52 54 56 58 60

116 34

B-+Tl++9+t Ty-+9 Tl++rl-+9 iI-+8+t

a2+x+t, aD+B++B-, D+tl++,qI+B++x

8+B++B-+9 , 4+x+9+t D+B-

Ti++il-+8+t

o+x+9

98 114 102

B-+il-+9

I+B++B- +x, D+t 8+B++9, D+il++t D

62 64 66 68

118 8 24 12

8- +Tl ++Ti- +9+t 13+

70 72

28

B++il++t B++B-

74

88

38

76 78 80 82 84 86 88

90

92 10 26 14 30 74

B-+,q++tl-+9 B-+Tl-+8+t

B++fl+ B++t

D+,q+, 4+B-+x+9 none a2+B++B-+x+9, aD+9+t

a+B- +x (;D+9

a+x +t, S+ti + + l-

B++B-+il+

a2+B++x+9, S+,q-

13++B-+t

a+x, S+tl++ij-+t

B++B-+71++t B++qB++tl++rlB++rl-+t

aD+B-+9 , S+il -+t

B++1l++Tl-+t

B++B-+rl-

B++B-+ij++rl-

I+x+9

S+BI+x+9+t , D+B++B-+9

4+B+ +x I+B- +x +9, S +Ti+ S

251

92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166

78

6++B-+Tl-+t

D+B++O, S+11++t

B++B-+T1++T1-+t

S+t, a+B++B-+x

40 56 44 60

B++e

aD+Tl++t

8++T1++e B+ +e +t B++T1+ +e +t

a+B- +x +e, aD+il+ aD

B++B-+e B++B-+Tl++e

a+x+e+t

120

B++B-+e+t

a+ x+e aD+BI+xaD+il++r1-+t aD+Tl-+t, 8+B-+O I+x+t, aD+Tl++Tl -, D+B++BaD+TI-, A+B+ +x+e

124 42 58 46 62 106 110

144

B++B-+T1++e+t B++T1- +e B++11++Tl- +e B++Ti- +e +t B++Tl++TI-+e+t

134 194 210 198 214 176 180

a2+B+ +x

B++B-+TI-+e

I+B-+x

B++B-+Tl++Tl-+e

8+e

B++B-+T1-+e+t B++B-+TI++Tl-+e+t

D+B+

x 11++x

A+Tl++TI-+t

x+t

148 192 208 196 212 130

a2+B++B-+x, aD+t

8+e+t, a+B++B-+x+8

a2+B-+e , A+Tl-+t

B-+Tl+ + x

a+B++B-, 4+Tl++ilaD+B++x+e, a+rlnone a2+e

B-+x+t

a+B+

B-+T1++x+t n-+x Tl++T1-+x

a2+e+t, aD+B++B-+x+e I+B+ +e, 4 +T1 ++t

Tl++x+t B-+x

T1-+x+t Tl ++T1-+x+t

B-+Tl-+x B-+Tl++,n-+x

S+B++B-+x, A+t D+B- +x +e, A +Tl+

A I+B++B- +e, D + x + e +t S+B+ + x

B-+rl-+x+t

D + x+e

8-+Tl++T1-+x+t

4+B-

x+e Tl++x+e

none a2+B-

x+e +t

a+B++B-+e aD+B++x

T1+ + x+ e +t

252 168 170 172 174 176 178 180

224 240 228 244 162 166

182 184 186 188 190 192

226 242 230 246 136

194 196 198 200 202 204

152 140 156

206 208 210 212 214 216 218 220 222 224 226 228 230

216 220 138 154 142 158 202 206

a2+11+

13- +Tl+ +x + e

a2

B-+x+e+t 13- +il + +x + e +t

a2 +1+ +t,

rl- + x +e Tl++rl- + x +9 rl - + x+e +t

I+13+

Tl++TI -+x+9+t 13-+r1-+x +e 13-+T1 ++rl -+x+e B-+Tl- +x + e +t 13-+Tl++TI-+x+9+t 13++x

168 184 172 188

252 170

a+13++e aD+13++13- + x

8+I3+ +B- +x +e, A +e+t D+B- + x A+e a2+11++rl-, I+B++13-, D+x+t a2+T1a2+rl++T1-+t, D+x a2+rl-+t, M+13-+e none a2+B++B-+9 , aD+x+e+t a+B-

B++B-+T1+ +x +t B++T1-+x B++71++71-+x B++T1-+x+t B++Tl ++rl-+x+t 8++B-+T1- +x 8++B-+Tl++T1-+x B++B-+r1- +x +t

a+rl+, aD+B-+x+9 I+9

B++x+9 B++r1++x+9 B+ +x +e +t B++r1 + +x +9+t

8++B- +x+e 248

a2 +t,

B++Tl++x B++x+t B++71+ +x +t B++B-+x B++B-+r1++x B++B- +x +t

13++13-+r1++T1-+x+t

232 234 236 238 240

B - +x +e

8++B-+il++x+e B++B-+x+9 +t B++B-+T1+ +x+e +t B++il-+x+e

aD+x+9 a+t a2+6++9 , a+71++1 a

8+B-+x I+e +1, D+B++B-+x +e O+B+ I+B-+e , a +Tl -+t a+Tl++T1-+t,

8+x

a+T1-, D+B++x+9 a+T1++r1-, 8+4+t, O+B++BI+11a2+B++B-, I+rl++rl-, aD +x +t I+11- +4 a+B- +e I+r1++rl-+t, aD+x a+e+t

a2 +B+

a+e aD+B- + x I

253 242 244 246 248 250 252 254

186 174 190 234 238

B++il++il-+X+8 B++qq -+X+0+t

I+11+ I+t, D+B++B-+X

8++ Ti++1-+X+8+t

B++B-+il-+X+8

I+tl++t, A+B++O I+6-

B++B-+tI++,I-+x+0

8+x+0

B++B-+11-+X+0+t B++B-+rl++tl-+x+0+t

D+B++x

S+x+O+t, a+B++B-+0

254

Appendix 3 Strongly Legal 3-Site Non-Generative Rules This appendix lists the strongly legal non-generative 3-site rules which reduce to the identity or to shifts. The forms given are for nearest neighbor rules. The expressions given can be changed to their left justified form by the substitutions a-1-I, I->a, and a->a2. Table A3.1 lists rules which are have strongly legal fixed points. Table A3.2 lists rules with strongly legal shift cycles. Note that the same rule may have both strongly legal fixed points, and strongly legal shift cycles. 1. Configurations with only isolated 1's are fixed

140

1+13+

196 1+13- 132 I+B++13-

12 I +B + +x 68 I+B-+x 4 I+B++B-+x 2. Configurations with only isolated 0's are fixed 206 I+il+ 220 I+rl- 222 I+il++il3. Configurations with isolated 1's separated by two or more 0's are fixed 172 I+13++0 228 I+B-+O 164 I+B++13-+0 44 I+B+ +x +O 100 I+B-+x+O 36 I+B++B-+x+9 4. Configurations with isolated 0' separated by two or more 1's are fixed 202 I+rl++t 216 I+rl-+t 218 I+il++il-+t 5. Other cases 236 1+0 Fixed points have no isolated 0's 200 I+t Fixed points have no isolated 1's 76 I+x Fixed points have only 1 and 11 blocks 108 I +x +O Fixed points have 1 and 11 blocks, no isolated 0's 232 I+O+t Fixed points have no isolated 0's or 1's 72 I +x +t Fixed points have only 11 blocks 104 I +X +O+t Fixed points have 11 blocks and no isolated 0's Table AM Strongly Legal Fixed Point Rules Complimentary rules are indicated by pairing them within parentheses. In nearest neighbor form, the right and left shifts are complimentary. Hence if X and Y are complimentary rules, and X has a strongly legal right shift decomposition then Y will have a strongly legal left shift decomposition, and vice versa.

The comments made in section 1 of Chapter 3 describe some cases of rules which have both strongly legal right and left shift decompositions, acting on different configuration subsets. Decompositions in which both a shift and the identity occur are also possible.

255 1. Configurations with isolated l's on shift cycles (234,248) (a+B+,^l+B ) (162,176) ((rl+B+,a+B-) (106,120) (a+13++x,(r1+13-+x) (a+B'+x,a+B-+x) (34,48) (98,56) (a+B++B-+x,(rl+B++B'+x) 2. Configurations with isolated 0's on shift cycles (168,224) (a+il+,(rl+rl-) (186,242) ((rl+rl-,a+rl+) 3. Configurations with isolated 1's and 0's on shift cycles (226,184) (a+B++B' = a-1+71++il-,(Y'1+B++B' = a+rl++ij-) Rule 226 reduces to a left shift on configurations with only isolated 1's, and a right shift on configurations with only isolated 0's, and vice versa for rule 184. 4. Configurations with isolated l's separated by two or more 0's (202,216) (a+B++e,a-l+B-+8) (130,144) (a+B-+O,a-1+B++9) (74,88) (a+B++x+9,(Y'l+B-+x+8) (2,16) (a+B'+x+8,(rl+B++x+9) (66,24) (a+B++B'+x+A,(rl+B++B'+x+9) 5. Configurations with isolated 0's separated by two or more l's (168,224) (a+Tl++t,a'l+Tl-+t) (186,242) (a+rl-+t,a-l+Tl++t) (230,188) (a+rl++rl-+t,a-1+Tl++rl-+t) 6. Other Cases (138,208) (a+9,(rl+8) No isolated 0's Only 1 and 11 blocks (42,112) (a+x,(-l+x) (174,244) (a+t,a-1+t) No isolated 1's (10,80) (a+x+9,a-1+x+8) No isolated 0's, only 1's and 11's (142,212) (a+0+t,a'1+0+t) No isolated 0's or isolated 1's (46,116) (a+x+t,(rl+x+t) Only 11 blocks (14,84) (a+x+O+t,a-l+x+O+t) Only 11 blocks, no isolated 0's Table A3.2 Rules With Strongly Legal Shift Cycles

256

Appendix 4 The Mod (2) Pascal Triangle Many of the properties of the additive cellular automata depend on properties of Pascal's triangle taken modulo 2. In this appendix some of the more important properties of this triangle are given. Most of the results presented here are due to Calvin Long [891, who proved them for the more general case of Pascal's triangle taken modulo any prime p. Figure A. 1 shows the first 33 rows of this triangle. 1 11 101 1111 10001 110011 1010101 11111111 100000001 1100000011 10100000101 111100001111 1000100010001 11001100110011 101010101010101 1111111111111111 10000000000000001 110000000000000011 1010000000000000101 11110000000000001111 100010000000000010001 1100110000000000110011 10101010000000001010101 111111110000000011111111 1000000010000000100000001 11000000110000001100000011 101000001010000010100000101 1111000011110000111100001111 10001000100010001000100010001 110011001100110011001100110011 1010101010101010101010101010101 11111111111111111111111111111111 100000000000000000000000000000001 , Figure Al First 33 Rows of Mod (2) Pascal Triangle

257 This triangle is generated by the binary difference operator D acting on the space of double infinite binary sequences with an initial sequence containing only a single 1. Two immediately obvious properties are the presence of self similar copies of the Pascal triangle at different size scales, and the separation of these copies by inverted triangles of 0's. This structure is explained by an elegant structural theorem. Let An,k denote the triangle 2m n 2m k

2mn+2m -1l 2°'n+2m-1 ( 2mk ) ..................(2mk+2m - 1 / Theorem A4.1 (Long, [ 89]) above is the triangle An,k (0<_k<<-n) defined

(k)(0) (k)(0) (k)(1)

(n)(2m -1 ) ......... (n)(2m -1) k 0 k 2m-1 with all products taken mod(2). Further, An,k + An,k+1 = An+1,k+1 where the addition is element-wise mod(2). Finally, every element in Pascal's triangle which is not contained in one of the An,k equals 0 mod(2). Long also demonstrates that the triangle of triangles A0,0 A1,0AL1 A2,0A2,1A2,2

is isomorphic to the mod(2) Pascal triangle, explaining the basic self-similar property of this triangle: "If we repeatedly interate this process by mapping the triangles An,k onto the residues it follows that, modulo p, Pascal's triangle is a triangle that contains a Pascal's triangle of triangles, that in turn contains a Pascal's triangle of triangles,..., ad infinitum." [89]

258 It also turns out that the 0 entries in the mod(2) Pascal triangle form inverted triangles of the form 2mn 2"'n 2mk+1 2mk+2r"+1

2mn+2"'-2 2mk+2m-1 A question of particular interest is the location of these inverted triangles of 0's , and the self-similarity property . Two immediate results are that the inverted triangles of 0's are based on rows k such that 2 I (k-1), while if k = 2m+1 then there will be only one inverted triangle of 0's based on this row, with base length 2111 - 1. That is , the 2m+1 row of the mod (2) Pascal triangle consists of an initial and final 1 separated by 2m- 1 zeros. Lemma A4.2 Let As be the mod(2) Pascal triangle truncated at row 2s. Then As consists of 3 upright triangles isomorphic to As-1 and 3 inverted triangles of 0's having base length 2s-1-1 . These are the largest inverted triangles of 0's contained in As. Proof The upright triangles are the An , k defined by Long . Since the triangle of these triangles is isomorphic to Pascal 's triangle , the number of upright triangles isomorphic to As-1 is the same as the number of non -zero elements in the first two rows of this triangle , and this equals 3. The inverted triangles fall between the upright triangles so the number of these is just 1+2.1 This lemma makes it possible to count the number of inverted triangles of 0's of any size which are contained in As. There is one with the maximum base length 2s-1 - 1, while each of the upright triangles is isomorphic to As-1. But As- 1 contains 3 upright triangles isomorphic to As-2 , and 3 inverted triangles of 0's with base length 2s-2-1. Thus As contains 9 upright triangles isomorphic to As-2, and 3 inverted triangles of 0's of base length 2s-2-1. Continuation of this line yields the lext lemma. Lemma A4.3 Let 1 <- r<- s-1. As contains 3r upright triangles isomorphic to As-r and 3r-1 inverted triangles of 0's with base length 2s-r-1.

259 In Chapter 7 much use was made of the properties of the diagonals of the mod(2) Pascal triangle which are given in the next two lemmas. Lemma A4.4 The 2s diagonal of the mod(2) Pascal triangle is periodic with period 2s. The first 2s terms of this diagonal consist of a leading 1 followed by 2s-1 0's. Proof The first term of every diagonal in Pascal's triangle is 1. However, there is an inverted triangle of 0's with base length 2s-1 based on row 2s+1, and the next 2s-1 entries of the 2s diagonal lie along a side of this triangle of 0's. Periodicity follows from self-similarity. I Proof of the next lemma follows from an almost identical argument. Lemma A4.5 The 2s-1 diagonal of Pascal's triangle mod(2) has period 2s and consists of two leading l's followed by 2s-2 0's. It is also possible to express every element of Pascal's triangle as the sum of elements along a diagonal. Lemma A4.6

C +i n

+l,^ (jn+j)

(

Proof An analytic proof follows directly from the addition rule for Pascal's k) triangle , noting that 1= 1 for all k. The essence of the proof, however, is easily grasped through figure A.2. x x x

Figure A2 Illustration of Lemma A6 The Element at the Bottom of the Line is the Sum of Elements Indicated With Open Circles

260

Appendix 5 GE*(X) for 3-Site T-Equivalence Class Exemplars GE* shown for underlined rule. Class GE* Case 1 : GE* Finite (23,232)

01001 01101 10010 10110

Class--

GE*

(77,178)

00011 00111 11000 11100 0001000 1110111

(4,32,223,251)

11

0100010 1011101 (1,127,1,254)

101 1001

(5,95,160,250)

10001 10011 11001 11011

(10,80,175,245)

101 111

(29,71,.] ,226)

1100

(33,123,132,222)

1011 1101 110011

(,126,129,219)

101

(43,113,142,212)

00010 11101

(Q,76,179,205)

111

(,108,147,201)

01101 10101 10110 10111101

(57,99,150,198)

001000 111000 111011 1110101000

(2,8,16,64, 191,239,247,253)

11 101

(3,17,63,119, 1M,192,238,252)

101

(,20,40,96, 159,215,235,349)

111 100101

(7,21,31,87, 168,224,234,248)

1001

1001001 11000101 110001001

24 66,189,231)

11

261 (9,65,111,125, 1101 130,144,190,246) 10011 10101 11000011 101000011

(11,47,81,117, 101 13$,174,208,244)

(12,34,48,68, 11 (187,207,221,243)

(13,69,79,93, 0011 132,176,186,242) 11011

14,42,84,112, 111 143,171,213,241)

(2,56,70,98, 111 157,185,195,227)

(35,49,59,115, 1101 --Q,196 ,206,220) (14

Case 2 : GE* Scales With n (SI) (4,116, 139,209) 010 01110 0111110 011111110 01111111110 0111111111110

(27,39,53,83, 1010 172,202,216,228) 0011001 10111001 001111001 1011111001 00111111001

(,78,92,114, 1111 (.$,44,52,100, 0101 141,163,177,197) 0100100 155,203,211,217) 011101 0100111 01111101 01000100 0111111101 01000111 011111111101 010000100 01111111111101 010000111 0111111111111101

TOTOTTTTTTOT TOTOOTTTTTOT TOTTTOTTTTOT TOTTOOTTTTOT TOTTTTOTTTOT TOTITTOOTTOT TOTOTOOOTTOT TOT TTTTOOTOT TOOTTOTTOTOT 101000TTOTOT 101100010101 TOTIT110010T 101000TOOTOT 101011000TO1 TOTOOT000TOT TOTOTTOTTOOT IOOTOTTOTOOT TOOTOTTTIOT T001101TTOT T001100110T 100111TOTOT 100111OOI01 101001OOTOT TOIOT11T00T 101001TTOOT TOT 11011001 101100TTOOT 101111OTOOT TOTOTTOTOT 100101TOOT TOOTTOTOOT TOTTOTTOT TOTOOTTOT TOTTTOTOT TOTTOOTOT TOOTOTOT TOTOTOOT (999'T9T`f7OT`-ZM uut JUOUVI

1T00010TOTOTI TTOOTOOTOTOTT TTOT000TOTOTT TIOOTO100TOTT TTOIOOOOOTOTT TTOTOT000TOTT TT0000000TOTT TTOOTOTOTOOTT 110100101001I 1TOT0100100TT TT000000TOOTT TTOIOTOT00011 TT00000T000TT 110000TOOOOTT TI000T00000TT ITOOT000000TT TTOT0000000IT TTOOTOTOIOTT 110100TOTOTT TTOTOTOOTOTT TT000000TOTT TTOTOTOTOOTI TT00000TOOTT 1100001OOOTT ITOOOTOOOOTT TTOOT00000TI TIOTOOOOOOIT ITOTOTOTOTT TT00000TOTT TI0000TOOTT 1100010OOII ITOOTOOOOTT TTOT00000TT 1T000OTOT1 110001OOTI TTOOT000IT 1101000OTT TTOOOTOTT TTOOTOOTT T10T000TT TTOOTOTT TTOTOOTT TTOTOTT TTT (L8ti`E8T`ZL`-FT) qj i 3 Al 1 Z9Z

101101TITOT 101001TTTOT 101100TTTOT 101000TTT01 TOTTTTOTTOT TOOTOTOTIOT 101110OTIOT TOTT1100101 TOTTT000TOT 100101TTOOT 100110TTOOT 100111OTOOT IOTT010100T 100010TOOOT TOOOTOTTOT TOOOTOOIOT TOOTOTTOOT 100110TOOT TOTTOT000T TOTOOT000I TOTTOTTOT TOTOOTTOT IOTTOOIOT TOIOOOIOT 100101001 (8TZ`T`T6`LE)

1011111111107 1010001111107 1011000111107 1011100011107 TOTITT000TTOT TOIT1110001OT 101000T00010T TOTTT1TTTTOT TOTOOO11110T 1011000T1TOI T011100OTTOT 101111000TOT 101111TTTOT 101000TTTOT 10110001101 10111000TO1 1011111101 TOI0001101 1011000101 TOTTT1TOI 101000101 101111OT TOTI101 TOTTOT TOI01 (Z8T`^T ` 60T`EL)

010000OTTTO 010000TTOTO 010001TOOTO 010011000TO OTOI10000T0 0111000OOTO 010000TTTO OTOOOTTOTO 01011000TO OITTOOOOTO 010001ITO 010111OTO OTOOTTOIO OIOTTOOTO OTTTOOOTO 010011TO 011100TO OOT00 (T9I`EET `ZZI`-f9)

ITT000TOOTTT TOTOT000011T IOTOTOOITOTT TOOT000TTOTT TOTTOTOOTOTT TTTOOTOOTOTT TTT0000000TT 1001000OTTT TTT00000TTT TIT000TTOTT TOTOTOOTOIT TTTOOT000TT TTT000TOTT TOTOT000TT TTTOOOOTT 111001TT TITOII TOTTT (fiTZ`89T ` 8J71 `FT`TZ1` LOT`L6`TI7)

E9Z

TTTOT000TTOTT TT000000TTOTT TTOOTTOTOTOTT IT000010OTOTT TTOOT0000TOTT ITOOOOIIOOOIT TTOOTT00000TT TTOOTTOTOOTOT 111010TOT011 110000TOTOTT 1100100OTO11 TTI00000TOTT TT000011OOTT TTOOT10000TT TTTOTOTOOTOT 111001000TOI T1100000OTOT 110000TTOTT 1100TOOTOTT 110011OOOTT TTOOTOTOTT TTTOOOTOTT ITOOTTOOTT 111000OTOT 110011011 TT TOT OTT 11100101 TTTT (6ZZ`T 8T `ELT `L9T `88`Z8`TL`SZ)

TTOTOOOOTOIOTT TT000TOTOTOOTT TTOTOOTOOTOOTT TTOTOT000TOOTT TT0000000T00TT TTOTOOTOT000TT TTOTOTOT0000TT TT00000T0000TT TT000T000000TT 110100000000TT TTOOOOOTOTOTT TT000TOOTOOIT TTOT0000TOOIT TTOOOIOTOOOIT TTOIOOT0000TT TTOTOT00000TT TT000000000TI 110100TOTOTT TTOTOTOTOOTT IT00000I00T1 TT00010000TT 1101000OOOTT TT000TOTOTT TTOTOOTOOTT 110TOT000TT 110000OOOTT TT000TOOTT 1101000017 TTOTOTOTT 110000011 ITOIOOT1 1100017 TTOIT

(089`i,6T `887 `-Z-gT`EOT `L9`T 9`9Z) b9Z

265 (62,x,118,124 , 131,137 , 145,193) 01010 010010 0100010 01000010 010000010 010111010 0100000010 0100111010 0101110010 01000000010 01000111010 01001110010 01011100010 010000000010 010000111010 010001110010 010011100010 010111000010 0100000000010 0100000111010 0100001110010 0100011100010 0100111000010 0101110000010 0101110111010 01000000000010 01000000111010 01000001110010 01000011100010 01000111000010 01001110000010 01001110111010 01011100000010 01011100111010 01011101110010 continued next column

cont. 010000000000010 010000000111010 010000001110010 010000011100010 010000111000010 010001110000010 010001110111010 010011100000010 010011100111010 010011101110010 010111000000010 010111000111010 010111001110010 010111011100010

266

Appendix 6

G(X) for Selected Rules The semi-group G(X) is listed in terms of element labels. Thus, the table entry sl...sn indicates the product ds1...dsn, and also any path labeled Si. ..sn in the de Bruijn diagram. The product of two group elements is formed by concatanation of their labeling strings. If a string labeled si...sn can be reduced via group operations to a string rl...rm with m < n this means that every path in the de Bruijn diagram labeled s1...sn can be replaced by a path labeled ri...rm which starts and ends at the same vertex. The table lists semi-group elements together with the elements to which they map when multiplied by d0 and di. If the semi-group contains a zero matrix this is indicated by Z, while if it contains an identity matrix it is indicated by I. A star in the table indicates that the corresponding element maps to itself. 1. Rule 18 (01001000) The semi-group defined by this rule contains 37 elements, including the zero matrix Z.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Element Maps by d Maps by d1 Element Maps by d Maps by d 0 00 10 1 0101 00101 1 1 01 11 1 1101 01101 Z 00 100 000 2 0011 * 1011 10 010 110 2 1011 * 11 01 001 2 00100 101 * 10100 11 011 Z 2 10100 010100 1101 000 * 100 2 01100 001100 101100 100 0100 1100 2 01010 001010 10 010 1010 0010 2 00110 * 10110 110 0110 Z 2 10110 * 110 * 001 101 2 00101 * 1 101 0101 1101 2 01101 001101 101101 011 0011 1011 3 010100 000 100 0100 00100 10100 3 001100 * 101100 1100 01100 Z 3 101100 * 1100 0010 * 1010 3 001010 * 10 1010 01010 1101 * 34 001101 101101 0110 00110 10110 * 3 101101 1101 0101 00101 1 Table A6.1 Semi-Group Table for Rule 18

267 2. Rule 30 (01111000) No Z or I element . The semi-group contains 40 elements. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0 1 00 10 01 11 010 110 001 101 011 111 0010 1010 0110 1110 0101 1101 0011 1011

00 01 * 010 001 011 0010 0110 001 0101 0011 0 * 01010 00110 00 0010 01101 * 01011

- --- _ 20 01010 10 21 11010 11 22 00110 10 110 23 10110 24 01101 101 111 25 11101 26 01011 1010 27 11011 1110 28 010110 101 1101 29 110110 1011 30 001101 31 101101 1 32 011011 1010 11010 33 111011 10110 34 0101101 35 1101101 10 36 0011011 1010 37 1011011 11101 1011 38 01011011 11011 39 11011011

0010 * * 010110 001101 001 0010 011011 0010 1110 * 0101101 0011011 0011 0010 11101 * 01011011 0010 111011

Table A6.2 Semi-Group Table for Rule 30

3. Rule 46 (01110100) The semi-group contains 14 elements, including Z. Element Maps by d0 Maps by dl10 00 0 0 11 01 1 1 10 00 2 00 110 Z 10 3 101 01 4 01 1 011 11 5 10 0110 110 6 1101 Z 101 7 1011 011 011 8 10 0110 0110 9 101 01 10 1101 11011 Z 11 1011 1011 Oil 12 11011 Table A6.3 Semi-Group Table for Rule 46

1010 01010 10110 110110 101101 101 1010 111011 1010 0110 101101 1101101 1011011 1011 1010 01101 1011011 11011011 1010 011011

268 4. Rule 84 (00101010)

0 1 2 3 4 5 6 7 8

The semi- group contains 24 elements, including Z. Element Maps by a Mans by al li;lement Maps by cl Mans by C11 Z 12 1100 01100 00 10 0 11010 11 13 1010 010 1 01 10110 14 0110 100 * 00 00 Z 110 15 1101 01 10 010 011 11 101 * 16 1011 01 110100 0100 11 011 Z 17 10100 101100 1100 18 01100 * 100 0100 Z 010 1010 19 11010 010 * 110 0110 110 0110 Z 20 10110

9

101

10 011 11 0100

01

1101

21 110100

0100

Z

* *

1011 10100

22 101100

01100

1100

Table A6.4 Semi-Group Table for Rule 84

In the first four examples, a zero matrix Z always appears. The shortest product sequence resulting in the zero matrix can be determined from the tables by taking the first time Z occurs. For example, in the table for rule 84 given above, the first occurence of Z is 111 (i.e., 11 multiplied by di). Thus di is the first sequence to result in Z. Note, however, that for many rules there will be several "shortest" sequences which result in a zero matrix. In the next two examples, rules 90 and 150, the semi-group is a permutation group. 5. Rule 90 (01011010)

0 1 2 3 4 5 6

The semi- group is a group with 8 elements, including the identity I. Element Mans by d0 Maps by di 10 I 0 11 1 01 110 010 10 1 0 01 110 010 11 I 010 10 11 01 110 Table A6.5 Semi-Group Table for Rule 90

269 6. Rule 150 (01101001) The semi-group is a group with 12 elements, including the identity I. 0 1 2 3 4 5 6 7 8 9 10

01 11 100 110 010 I 1100 0100 0 01 00

a 00 01 I 010 110 100 0100 1100 1 11 10

0 1 00 10 01 11 100 010 110 0100 1100

Table A6.6 Semi-Group for Rule 150 7. Rule 61 ,620 (0010110100001111) This is one of the prime non-LEV 4- site rules . The semi-group has 31 elements with no Z or I. 0 1 2

00 01 000

0 1 00

10 11 100

16 0010 17 0110 18 1110

0110 * 0010

10 10

3

10

010

110

19 0001

*

101

4 5 6 7 8 9 10 11 12 13 14 15

01 11 000 100 010 110 001 101 011 111 0100 1100

001 011 * 0100 0010 0110 0001 0101 0011 0111 00100 000

101 111 100 1100 10 1110 101 1101 1011 * 100 11100

20 21 22 23 24 25 26 27 28 29 30

00101 0001 * 01011 0011 000 00100 0001 00101 0111 0011

101 11101 1011 11011 1011 100

0101 1101 0011 1011 0111 00100 11100 00101 11101 01011 11011

101 1011 111

Table A6.7 Semi-Group for 4 -Site Rule 61,620

In Table A6.8 the statistics on exemplars of each of the T-Invariant equivalence classes listed in Table 10.1 are given. By Theorem 10.6, these statistics will be valid for all rules in each equivalence class. The additive rules 60, 170, and 204 do not show an identity because they are really defined for fewer than 3-sites.

270 Note the exceptionally large numbers of elements in the semi -groups defined by rules 22 (2374) 26 (838), 94 (1245) 146 (734), and 164 (1007). $,W_@ Semi-Group Comment $WI@ Semi-Group Comment

Size

Size 0 2 6 10 18 22 26 30 32 38

3 13 124 37 37 2374 838 40 19 42

Z present Z present Z present Z present Z present Z present Z present No Z, or I (LIV) Z present Z present

132 134 136 138 140 142 144 146 150 152

70 974 15 26 24 73 121 734 12 83

Z present Z present Z present Z present Z present Z present Z present Z present Group Z present

46

14

Z present

154

20

No Z, or I (LFV)

48 50 54 56 58 60 66 84 90 94 110 126 128

9 22 108 43 88 4 26 24 8 1245 74 44 19

Z present Z present Z present Z present Z present No Z, or I Z present Z present group Z present Z present Z present Z present

156 160 164 168 170 172 178 184 186 204 232 236

125 91 1007 36 6 82 119 33 43 4 119 20

Z present Z present Z present Z present No Z, or I Z present Z present Z present Z present No Z, or I Z present Z present

Table A6.8 Semi -Group Statistics for T-Invariance Classes

271

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