Logic and the Foundations of Game and Decision Theory - LOFT 8: 8th International Conference, Amsterdam, The Netherlands, July 3-5, 2008, Revised ... Lecture Notes in Artificial Intelligence)

Lecture Notes in Artificial Intelligence

6006

Edited by R. Goebel, J. Siekmann, and W. Wahlster

Subseries of Lecture Notes in Computer Science FoLLI Publications on Logic, Language and Information Editors-in-Chief Luigia Carlucci Aiello, University of Rome "La Sapienza", Italy Michael Moortgat, University of Utrecht, The Netherlands Maarten de Rijke, University of Amsterdam, The Netherlands

Editorial Board Carlos Areces, INRIA Lorraine, France Nicholas Asher, University of Texas at Austin, TX, USA Johan van Benthem, University of Amsterdam, The Netherlands Raffaella Bernardi, Free University of Bozen-Bolzano, Italy Antal van den Bosch, Tilburg University, The Netherlands Paul Buitelaar, DFKI, Saarbrücken, Germany Diego Calvanese, Free University of Bozen-Bolzano, Italy Ann Copestake, University of Cambridge, United Kingdom Robert Dale, Macquarie University, Sydney, Australia Luis Fariñas, IRIT, Toulouse, France Claire Gardent, INRIA Lorraine, France Rajeev Goré, Australian National University, Canberra, Australia Reiner Hähnle, Chalmers University of Technology, Göteborg, Sweden Wilfrid Hodges, Queen Mary, University of London, United Kingdom Carsten Lutz, Dresden University of Technology, Germany Christopher Manning, Stanford University, CA, USA Valeria de Paiva, Palo Alto Research Center, CA, USA Martha Palmer, University of Pennsylvania, PA, USA Alberto Policriti, University of Udine, Italy James Rogers, Earlham College, Richmond, IN, USA Francesca Rossi, University of Padua, Italy Yde Venema, University of Amsterdam, The Netherlands Bonnie Webber, University of Edinburgh, Scotland, United Kingdom Ian H. Witten, University of Waikato, New Zealand

T· L· G

Texts in Logic and Games

Texts in Logic and Games (TLG) was a book series created in 2007 and published by Amsterdam University Press from 2007 to 2009. The Editor-in-Chief of the series was Johan van Benthem (Amsterdam & Stanford), the Managing Editors were Wiebe van der Hoek (Liverpool), Benedikt Löwe (Amsterdam), Robert van Rooij (Amsterdam), and Bernhard von Stengel (London). The following volumes were published by Amsterdam University Press: Volume 1 Johan van Benthem, Benedikt Löwe, Dov Gabbay (eds.) Interactive Logic Selected Papers from the 7th Augustus de Morgan Workshop, London, December 2007 Volume 2 Jörg Flum, Erich Grädel, Thomas Wilke (eds.) Logic and Automata: History and Perspectives, December 2007 Volume 3 Giacomo Bonanno, Wiebe van der Hoek, Michael Wooldridge (eds.) Logic and the Foundations of Game and Decision Theory (LOFT 7), June 2008 Volume 4 Krzysztof R. Apt, Robert van Rooij (eds.) New Perspectives on Games and Interaction, December 2008 Volume 5 Jan van Eijck, Rineke Verbrugge (eds.) Discourses on Social Software, March 2009 From 2010 onwards, TLG will become part of the Springer FoLLI series and continue to publish books at the interface between logic, computer science, and game theory in the area of “intelligent interaction” linking into economics, linguistics, mathematics, philosophy, social sciences, and other congenial areas.

Giacomo Bonanno Benedikt Löwe Wiebe van der Hoek (Eds.)

Logic and the Foundations of Game and Decision Theory – LOFT 8 8th International Conference Amsterdam, The Netherlands, July 3-5, 2008 Revised Selected Papers

13

Series Editors Randy Goebel, University of Alberta, Edmonton, Canada Jörg Siekmann, University of Saarland, Saarbrücken, Germany Wolfgang Wahlster, DFKI and University of Saarland, Saarbrücken, Germany Volume Editors Giacomo Bonanno University of California, Department of Economics One Shields Avenue, Davis, CA 95616-8578, USA E-mail: [email protected] Benedikt Löwe Universiteit van Amsterdam Institute for Logic, Language and Computation (ILLC) P.O. Box 94242, 1090 GE Amsterdam, The Netherlands E-mail: [email protected] Wiebe van der Hoek University of Liverpool, Department of Computer Science Liverpool L69 7ZF, UK E-mail: [email protected]

Library of Congress Control Number: 2010932134 CR Subject Classification (1998): F.3, F.4.1, G.2, F.1, H.3, I.2.3

LNCS Sublibrary: SL 7 – Artificial Intelligence

ISSN ISBN-10 ISBN-13

0302-9743 3-642-15163-9 Springer Berlin Heidelberg New York 978-3-642-15163-7 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2010 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper 06/3180

Preface

This volume is conceived as a ramification of the eighth conference on Logic and the Foundations of the Theory of Games and Decisions (LOFT8), which took place in Amsterdam, in July 2008. In all, 61 papers were submitted to this event, of which 48 were presented. This volume is based on a selection of the presented papers and the invited talks: they survived a thorough and lengthy reviewing process. The LOFT conferences have been a regular biannual event since 1994. The first conference was hosted by the Centre International de Recherches Mathematiques in Marseille (France), the next four took place at the International Centre for Economic Research in Turin (Italy), the sixth conference was hosted by the Graduate School of Management in Leipzig (Germany), the seventh took place at the University of Liverpool (UK) and LOFT8 was organized at the ILLC in Amsterdam.1 The LOFT conferences are interdisciplinary events that bring together researchers from a variety of fields: computer science, economics, game theory, linguistics, logic, multi-agent systems, psychology, philosophy, social choice and statistics. Its focus is on the general issue of rationality and agency. In the last decade, there has been a floroushing activity in exactly the kind of areas that LOFT aims to address, which is demonstrated by not only the number of submissions to LOFT8, but also the fact that the conference is becoming more and more a home for a large group of researchers, rather than a host for passersby from different disciplines. The papers collected in this volume reflect the contemporary interests and interdisciplinary scope of the LOFT conferences. Dietmar Berwanger opens this volume with a paper on “Infinite Coordination Games.” He investigates the prescriptive power of sequential iterated

1

Collections of papers from previous LOFT conferences can be found in a special issue of Theory and Decision (Vol. 37, 1994, edited by M. Bacharach and P. Mongin), the volume Epistemic logic and the theory of games and decisions (edited by M. Bacharach, L.-A. G´erard-Varet, P. Mongin and H. Shin and published by Kluwer Academic, 1997), two special issues of Mathematical Social Sciences (Vols. 36 and 38, 1998, edited by G. Bonanno, M. Kaneko and P. Mongin), two special issues of Bulletin of Economic Research (Vol. 53, 2001 and Vol. 54, 2002, edited by G. Bonanno and W. van der Hoek), a special issue of Research in Economics, (Vol. 57, 2003, edited by G. Bonanno and W. van der Hoek), a special issue of Knowledge, Rationality and Action (part of Synthese, Vol. 147, 2005, edited by G. Bonanno) and the volume Logic and the Foundations of Game and Decision Theory (LOFT 7) edited by G. Bonanno, W. van der Hoek and M. Wooldridge, Texts in Logic and Games No 3, AUP, 2007.

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Preface

admissibility in coordination games of the Gale-Stewart style, i.e., perfect information games of infinite duration with only two payoffs. Such infinite games are interesting, because they provide a natural model for non-terminating interactive computation, for instance. Berwanger shows that, in the kind of games described, the procedure of eliminating weakly dominated strategies is independent of the elimination order and that, under so-called maximal simultaneous elimination, this procedure converges after at most ω many steps. In the paper “Conditioning by Minimizing Accessibility,” Konstantinos Georgatos presents an axiomatization of a class of set-theoretic conditional operators. Conditionals use a notion of ‘closeness to a world,’ often based on a local notion of distance, parameterized by worlds. Georgatos, however, is interested in a global distance notion, of which the hamming distance between valuations is an example. The purpose of the paper is to introduce and characterize a class of conditional operators that are modelled by a global distance. The idea here is to assume a binary relation of indistinguishability between states, and associate the global distance with the number of steps it takes to go from one state to the other. Using this global map of indistinguishability, one can define and characterize belief revision and update operators. The paper “Correspondences in the Theory of Aggregation,” reports on an analysis by Davide Grossi of the interrelationship between the social-theoretic problems of preferences and judgment aggregation from a formal logic perspective. The study reveals that on the one hand, preference aggregation on total pre-orders is equivalent to the aggregation of specific types of judgments. On the other hand, judgment aggregation on propositional logic is proven equivalent to the aggregation of specific types of preferences. This indicates that aggregation of preferences and of judgments can be viewed as two faces of the same coin. In “Logic and Data Exchange: Which Solutions Are “Good” Solutions?” Andr´e Hernich and Nicole Schweikhardt introduce the area of data exchange, which is concerned with translating data that are structured in one format to data structured in a new format. A solution is then a database that does this translation, given a database and some conditions that should hold between the source and target database. They focus on good solutions for query answering on the target database, in particular universal solutions, the core of the universal solutions, and CWA solutions. They also present a game theoretic characterization of the CWA solutions. As the title suggests, “Preference Change Triggered by Belief Change: A Principled Approach” by J´erˆome Lang and Leendert van der Torre is about preference change: the authors introduce eight properties for preferences evolving after some new fact has been learned. Four properties are concerned with persistence of preferences when something that was preferred is (partly) satisfied or dissatisfied, four other properties regard situations where the agent learns something she already suspected to hold. They then define a family of preference change operators, parameterized by a revision function on epistemic states, and they give semantic conditions on those change operators and revision function for each of those eight conditions to hold.

Preface

VII

Riccardo Pucella and Mehrnoosh Sadrzadeh present a ‘runs-and-systems semantics for logics of announcements’ in a paper with exactly that title. Rather than using updateable Kripke models as a semantics for logics of announcements, they provide a semantics based on runs in a system, a semantics known from the interpreted systems literature. This alternative semantics is more concrete and better suited to accurately model a given scenario. They then study notions like honesty of an announcement (meaning its content is true) and belief of the agents when announcements are not public, i.e., need not be broadcast to all agents in the system. In “From Nash to Dependency Equilibria” Wolfgang Spohn challenges the assumptions underlying the Nash equilibrium concerning the causal independence of the decisions and the actions of the players. Although Spohn acknowledges that independence of the actions is constitutive of normal form games, he argues that the independence of the decisions may and should be given up. This then leads to the wider and quite different notion of a dependency equilibrium, of which cooperation in the single-shot prisoners’ dilemma is an example. The paper argues that this notion of dependency equilibrium is meaningful and significant, and it sketches some of its consequences. Where Pucella and Sadrzadeh relate dynamic epistemic logic with a temporal logic for knowledge in their paper, Johan van Benthem and C´edric D´egremont try to connect logics for the dynamics of belief with existing temporal logic approaches for belief. In their paper “Multi-agent Belief Dynamics: Bridges Between Dynamic Doxastic and Doxastic Temporal Logics” they focus on dynamic logics for changing beliefs using plausibility relations between worlds to represent agents’ beliefs and conditional beliefs and try to relate them to models of doxastic temporal logic. They identify crucial agent features behind dynamic doxastic belief revision, and position them inside the broader temporal setting. In “Multi-agent Belief Revision with Linked Preferences,” Jan van Eijck and Floor Sietsma establish a connection between dynamic epistemic logics of belief revision and approaches to collective judgment and multi-agent preference change. They first define an abstract propositional dynamic logic (PDL) style framework in which one can reason about preferences. They then focus on a central property of preferences, called linkedness, that roughly says that if two alternatives are each ranked with respect to a third alternative, it should be possible to rank (or compare) the two as well. They give a criterion under which linkedness is preserved under belief change. The paper then demonstrates how the logic can be used to model consensus-seeking procedures, focusing on socalled plenary Dutch meetings. In their “Note on Assumption-Completeness in Modal Logic,” Johathan Zvesper and Eric Pacuit come up with a positive result in the following question. Since in a game, players reason about (others’) beliefs, the question arises whether there exists such a thing as the space of all possible beliefs. For their analysis, they identify a belief with an ‘assumption,’ which is the characterization of what an agent exactly believes in the sense that it implies all its (other) beliefs. Now, roughly, a belief model is assumption complete for a language of

VIII

Preface

agent Ann, if for every sentence of this language, it is possible that Bob exactly believes that sentence. Now, a negative result in the first-order language literature says that there are no assumption-complete models for this language. However, the result of Zvesper and Pacuit says that for the modal logic language, there do exist assumption-complete models. We thank all the authors for their input and their patience, and we thank the reviewers for all their carefully written reports and suggestions. The General Chairs would like to thank the sponsors for their generous support and the Program Committee for their input during the refereeing of the submissions. Special thanks go to Peter van Ormondt and Sanchit Saraf for their support during the organization of the conference. October 2009

Giacomo Bonanno Benedikt L¨ owe Wiebe van der Hoek

Organization

LOFT 2008 was organised by the Institute for Logic, Language and Computation (ILLC), University of Amsterdam.

General Chairs Giacomo Bonanno Benedikt L¨ owe Wiebe van der Hoek

University of California at Davis, USA Universiteit van Amsterdam, The Netherlands The University of Liverpool, UK

Program Committee Krzysztof Apt Alexandru Baltag Johan van Benthem Adam Brandenburger Hans van Ditmarsch Ulle Endriss Erich Gr¨ adel Joe Halpern Wojtek Jamroga J´erˆome Lang Hannes Leitgeb Larry Moss Rohit Parikh Marc Pauly Andres Perea Ramanujam Robert van Rooij Hans Rott Robert Stalnaker Moshe Tennenholtz Rineke Verbrugge

Centrum voor Wiskunde en Informatica, The Netherlands Oxford University, UK Universiteit van Amsterdam, The Netherlands New York University, USA University of Otago, New Zealand Universiteit van Amsterdam, The Netherlands Rheinisch-Westf¨ alische Technische Hochschule Aachen, Germany Cornell University, USA Technische Universitt Clausthal, Germany Universit´e Paul Sabatier, France University of Bristol, UK Indiana University, USA City University of New York, USA Stanford University, USA Universiteit Maastricht, The Netherlands Institute of Mathematical Sciences, India Universiteit van Amsterdam, The Netherlands Universit¨ at Regensburg, Germany Massachusetts Institute of Technology, USA Technion, Israel Rijksuniversiteit Groningen, The Netherlands

X

Organization

Sponsoring Institutions LOFT 2008 was hosted by the Institute for Logic, Language and Computation (ILLC) of the Universiteit van Amsterdam, and was funded by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO, DN 612.080.000 CN 2008/05570/EW) and by the Marie Curie Research Training Site GLoRiClass (“Games in Logic Reaching out to Classical game theory”; MEST-CT-2005020841).

Table of Contents

Infinite Coordination Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dietmar Berwanger

1

Conditioning by Minimizing Accessibility . . . . . . . . . . . . . . . . . . . . . . . . . . . Konstantinos Georgatos

20

Correspondences in the Theory of Aggregation . . . . . . . . . . . . . . . . . . . . . . . Davide Grossi

34

Logic and Data Exchange: Which Solutions Are “Good” Solutions? . . . . Andr´e Hernich and Nicole Schweikardt

61

Preference Change Triggered by Belief Change: A Principled Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J´erˆ ome Lang and Leendert van der Torre

86

A Runs-and-Systems Semantics for Logics of Announcements . . . . . . . . . . Riccardo Pucella and Mehrnoosh Sadrzadeh

112

From Nash to Dependency Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wolfgang Spohn

135

Bridges between Dynamic Doxastic and Doxastic Temporal Logics . . . . . Johan van Benthem and C´edric D´egremont

151

Multi-agent Belief Revision with Linked Preferences . . . . . . . . . . . . . . . . . . Jan van Eijck and Floor Sietsma

174

A Note on Assumption-Completeness in Modal Logic . . . . . . . . . . . . . . . . . Jonathan A. Zvesper and Eric Pacuit

190

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

Infinite Coordination Games Dietmar Berwanger Laboratoire Sp´ecification et V´erification, Centre National de la Recherche Scientifique ´ (CNRS) & Ecole Normale Sup´erieure de Cachan, 61, avenue du Pr´esident Wilson, 94235 Cachan, France [email protected]

Abstract. We investigate the prescriptive power of sequential iterated admissibility in coordination games of the Gale-Stewart style, i.e., perfect-information games of infinite duration with only two payoffs. We show that, on this kind of games, the procedure of eliminating weakly dominated strategies is independent of the elimination order and that, under maximal simultaneous elimination, the procedure converges after at most ω many stages.

1

Introduction

Modern computing systems should interact successfully with the environment and never break. As a natural model for non-terminating interactive computation, extensive games of infinite duration have proved to be a suitable analytic framework. For such games, a vast and effective theory has been developed over the past fifty years at the intersection between logic and game theory (for a survey, see [9]). The fundamental model at the basis of this development are Gale-Stewart games [6]: perfect-information games between two strictly competing players with two possible payoff values: win or lose. This basic model has been successfully extended into various directions, including multi-valued payoffs, stochastic effects, partial information, player aggregation, etc. As a common feature most of these extensions postulate a strictly competitive setting. One major challenge for the analysis of interactive systems consists in handling multiple components that are designed and controlled independently. One can interpret the transition structure of such a system as a game form for several players, each identified with a component, and derive the utility function of each player from the specification of the corresponding component. Via this interpretation, rational strategies in the game correspond to sound designs for components. However, this translation gives rise to infinite non-zero sum games, the theory of which is yet in an initial phase of development. (See [10], for a recent study on Nash equilibrium and refinements in this framework.) Taking a point of view diametrically opposed to pure conflict models, we investigate extensive games of infinite duration where all participating players receive a common payoff. The players, there may be two or more, thus aim at coordinating their behaviour towards achieving a mutually beneficial outcome. For our analysis, we preserve the remaining aspects of the Gale-Stewart model G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 1–19, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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D. Berwanger

and restrict our attention to infinite coordination games of perfect information with only two possible payoffs. Our focus on coordination is motivated by a recurring pattern in the analysis of open systems, in which several components are conceived as a team acting against an adverse environment [1,16,12]. Traditionally, such systems are modelled as two-player zero-sum games, and the problem is to construct a strategy for each team member so that the interplay of these distributed strategies guarantees an optimal outcome against the environment. In general, however, the profile of distributed strategies is synthesised by a centralised instance, the designer of the open system, who effectively acts as an external coordinator. As a far-range objective, we aim at developing an alternative approach to synthesising interaction within a team of players, where the members are themselves responsible for constructing optimal strategies, without involving an external coordinator. Here is the motivating scenario for our investigation. To build a multi-component system, the system designer distributes to different agents a game form representing the possible transitions within a system, and a utility function specifying the desired behaviour of the global system. Each agent is in charge for one component. Independently of the other agents, he should provide an implementation that restricts the behaviour of this particular component in such a way that the composed system satisfies the specification. It is common knowledge among the agents that they all seek to fulfill the same specification, but they are not able to communicate on implementation details, nor to rely on the way in which the game model is represented; this is because they may have different input formats which allow them to reconstruct the original model only up to isomorphism. To accomplish their task, the agents obviously need to share some basic principle of rationality. Our aim is to find principles that are possibly simple and efficient. In game-theoretic terms, proposing a procedure for resolving this problem amounts to defining a solution concept for coordination games. The concept should prescribe, individually to each player, a set of strategies. Hence, the global solution should be a rectangular set: any profile composed of strategies that appear as part of a solution should also constitute a solution. On finite game trees, coordination games with perfect information and binary payoffs are disconcertingly simple. They can be solved by backwards induction yielding subgame-perfect equilibria, all of which are Pareto efficient, i.e., they attain the maximum available payoff. An equivalent solution is obtained through iterated elimination of weakly dominated strategies. In the infinite setting, it is a-priori less clear which solution concept would be appropriate. Subgame-perfect equilibria always exist, but they may not form a rectangular set, and prescribing the players to choose a subgame-perfect equilibrium independently could thus lead to coordination failure. The binary payoff scheme induces wide-ranging indifference among the outcomes, offering no grip to refinements based on payoff perturbations. For the same reason, forwardinduction arguments do not apply either.

Infinite Coordination Games

3

We analyse iterated admissibility, i.e., elimination of weakly dominated strategies, as a solution concept for infinite coordination games. The procedure has been shown to be sound for infinite perfect-information games with two payoffs [4]. Here, we consider a sequential variant of admissibility and show that, on coordination games, it enjoys two desirable properties, that do not hold in the general case. (i) For any game, the procedure of maximal elimination of dominated strategies converges in at most ω many stages to a non-empty set. (ii) The outcome of the procedure does not depend on the order of elimination (up to renaming of strategies and deletion of duplicates). Besides constituting a meta-theoretical criterion for the stability of the proposed solution, order independence is crucial for our application area. If the solution was sensitive to the elimination order, the system designer would need to optimise over different orders, which is a very difficult task. Applying the procedure towards solving infinite coordination games, we shall prove, on the positive side, that games with an essentially winning subgame are solvable, i.e., iterated admissibility delivers a rectangular set of strategies, the combination of which always yields the maximal payoff. On the negative side, we show that this classification is tight: if no player has a winning strategy that does not involve the cooperation of other players, admissibility cannot avoid coordination failure. Our proof is based on a potential characterisation of coordination games. This characterisation also implies that, on infinite coordination with binary payoffs, iterated admissibility provides a refinement of subgame-perfect equilibrium which favours secure equilibria, where a player’s payoff cannot decrease under any deviation of other players. To justify the restrictions assumed for our present model, we point out that the most straightforward relaxations lead to complications that raise doubts on whether admissibility can serve as a meaningful solution concept for more general classes of infinite games. Nevertheless, the question whether the good properties of infinite coordination games with two payoffs can be extended to games with finitely many payoffs remains open. We show that, unlike the case of finite coordination games with perfect information, or infinite non-zero games with two payoffs, already a few payoffs are sufficient to generate forward-induction effects in infinite coordination games, which appear to take the analysis out of the reach of our present methods.

2

Formalities

In situations that involve n players, we refer to a list of elements x = (xi )i
4

2.1

D. Berwanger

Coordination Games

  An extensive coordination game for n players is a structure Γ = T , u , where T is a directed tree over a domain T of elements called positions, equipped with a partition (T i )i
Sequential Admissibility

We define a sequential version of iterated admissibility based on simultaneous maximal elimination of weakly dominated strategies, following Bicchieri and Schulte [5].

Infinite Coordination Games

5

Given two strategies si , ri ∈ S i and a set Q−i ⊆ S −i , we say that si dominates r on Q−i at a position p ∈ T i , if i

up (si , t−i ) ≥ up (ri , t−i ) for all t−i ∈ Q−i , and up (si , t−i ) > up (ri , t−i ) for some t−i ∈ Q−i .1 For a set Q ⊆ S and a position p ∈ T i , we say that a strategy si ∈ S i is admissible on Q at p, if no strategy in Qi dominates si on Q−i at p. A strategy si ∈ S i is sequentially admissible on Q, if it is admissible on Q at every position p ∈ T i. For a game Γ , we define simultaneously for all players i: - Qi0 := S i ; - Qiα+1 := {si ∈ Qiα : si is admissible on Qα }, for every successor ordinal α + 1, and  - Qiλ := α<λ Qiα , for every limit ordinal λ. A strategy si ∈ Qiα is called α-admissible. As the stages are decreasing, the iteration reaches a fixed point (Qi∞ )i
3

Solvable and Unsolvable Games

In the following two sections, we restrict our attention to infinite coordination games with only two payoffs. A coordination game Γ is solvable, if the utility function is constant on Qi∞ , that is, if all iteratively admissible profiles yield the same payoff. In any solvable game where the winning set is nonempty, iterated admissibility thus guarantees a winning outcome. In contrast to this, if a game is not solvable then, by the definition of admissibility, for every iteratively admissible strategy of a player there exist iteratively admissible strategies of other players such that the resulting profile is losing; hence, in this case, iterated admissibility cannot avoid coordination failure. 3.1

Examples

We begin with some examples to illustrate how the elimination procedure works. In the graphical representations throughout the paper, there will be only two players. We draw the positions of Player 0 in circles and those of Player 1 in squares; binary payoffs in {−1, 1} are represented by their sign. 1

Whenever we refer to dominance throughout the paper, we actually mean weak dominance, in the classical sense. There is no risk of confusion, as the notion of strict dominance plays no role when we are concerned about pure strategies in games with only two payoffs.

6

D. Berwanger 1

1 r

l 2 r

l 4

2 r

l

5

r

l 3

r

l

+

6

3

4

5

6

l

r

l

r

l

r

l

l

r

l

+

•

•

•

+

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+

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(a) Game tree

+

(b) Admissible strategies

Fig. 1. Finite coordination game

The case of finite games is very simple in our setting. In the game of Figure 1, for instance, any strategy of Player 0 which at position 6 chooses r is sequentially dominated by the one which instead chooses l. Likewise, on the full set of strategies of Player 0, any strategy of Player 1 which at 3 chooses l is dominated by the one that instead chooses r. However, the two strategies are incomparable on a set from which Player 1 has previously eliminated the strategies choosing r at 6. In general, for finite games, one round of simultaneous elimination of dominated strategies yields a restriction where losing outcomes are reachable only from positions from which there are no reachable winning ones. Moreover if, for some subgame, a player i has a strategy that guarantees a win regardless of how the other players move, (as it is the case at position 3 in the example), then the admissibility procedure eliminates any strategy of i that is not winning for this subgame. As a consequence, every finite game is solvable, and the number of rounds needed until the iteration stabilises is bounded by the depth of the game tree. Figure 2 shows a game with one infinite play which is losing, whereas all finite plays are winning. In the first round, the elimination procedure removes all strategies that persist in playing in from some position onwards, which readily solves the game and ensures coordination.

1 out

in

2

out +

in

3

out +

in

4

in

•

out +

+

Fig. 2. Game of infinite duration

However, unlike the finite case, there are infinite games which are not solvable. Consider, for instance the two-player game on the infinite binary tree, where the players strictly alternate in choosing either the left or the right successor of the current position, and the winning condition requires that the sequence of choices be finally constant. In this game, let us call it Δ, no player has a winning strategy.

Infinite Coordination Games

7

The first round of elimination removes all strategies where a player would keep shifting between left and right, regardless of how his partner moves. However, in the obtained restriction, the players can still not guarantee a win, and none of the surviving strategies are dominated. Thus, the iteration terminates leaving the game unsolved. We can use the game Δ to build a game Δk that requires k iterations as shown in Figure 3. Clearly, the only iteratively admissible strategy in Δk is the one that always chooses in. However, the strategies that choose out at position  can be eliminated only at the stage at which all strategies that choose out at some later position  >  are eliminated. To be solved, the game thus needs k rounds to propagate the winning payoff. Figure 4 extends this example to obtain a game that requires ω many iterations.

1

in

out

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in

out

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in out

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k

in

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out

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Fig. 3. Game Δk requiring k iterations

1 out

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in

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in out

out 1

in

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k

in

out k• 1

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k

Fig. 4. Game Δω requiring ω iterations

3.2

Characterisation

Before we proceed to a formal discussion, let us describe the procedure step-bystep for an arbitrary game Γ with a non-empty winning condition. In the first stage, there may exist subgames Γp in which a player has a winning strategy (with respect to S). We call such strategies strongly winning in Γp . For any subgame Γp in which a player i has a strongly winning strategy, all strategies of S i that are not winning in Γp are eliminated. All strategies that are losing (with respect to S) are also eliminated. This yields Q1 . Notice that no strategy in Q1 can be losing with respect to Q1 . However, there may exist subgames Γp in which a player has a winning strategy with respect to Q1 . We call such strategies weakly winning. In the second stage, if no weakly winning strategy exists, the procedure terminates, concluding that the game is not solvable. Otherwise, for any subgame Γp where a player i has a weakly winning strategy, all strategies of Qi that are not winning in Γp with respect to Q1 are eliminated, thus yielding Q2 . It may happen that no strategies are eliminated in this stage (note that strongly winning

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strategies are also weakly winning). Then, the procedure terminates, concluding that the game is solvable. In the following stages α ≥ 2, strategies are eliminated only due to the discovery of new subgames where a player has a winning strategy with respect to the previous stage. These subgames arise by backwards propagation of winning subgames discovered in the first two stages. The propagation continues until it reaches the root of the initial game or of a previously solved subgame. Every subgame where a player has a (weakly) winning strategy will thus propagate for a finite number of stages. Since winning subgames may occur arbitrarily deep in the tree, the number of stages to propagate them cannot be bounded. Therefore, the iteration may require ω many steps to stabilise. Based on the insights from the above description, we advance the following statement which will be proved in the remaining part of the section. Theorem 1 (Constructiveness). For any coordination game with binary payoffs, the procedure of simultaneous maximal elimination of dominated strategies converges in at most ω many steps. In case the game is not solvable, the procedure terminates at the second stage. It becomes apparent that, for coordination games, the elimination procedure is considerably simpler than in the case when payoffs are not common. Indeed, losing strategies can occur only at the initial stage. Accordingly, each subgame that contains no subgames where a player has a weakly winning strategy freezes from the second stage onwards: every strategy of a player in such a subgame can lead to a winning or a losing outcome, depending on the choice of the other players. None of the players is able to avoid this interdependence and, therefore, no strategy can be eliminated. This leads us to the following characterisation of solvable games as being, essentially, those games where coordination effort is required only for finitely many steps at the beginning of a play. Theorem 2 (Solvability). A game Γ with binary payoffs is solvable if, and only if, it contains a subgame where one of the players has a weakly winning strategy, i.e., a strategy that is winning with respect to the set of 1-admissible strategies. 3.3

Proof

Our justification of Theorems 1 and 2, and the analysis of order dependency in the following section, relies on the notion of potential of (a restriction of) a coordination game. Intuitively, the potential of a restricted game describes the payoff that the players can achieve when confining themselves to playing within the restriction The notion of potential is based on the value function defined in [4] for characterising the dominance relation. Let us fix an extensive coordination game Γ for the rest of the section. The value of a restriction Q ⊆ S of Γ for player i is a function χi that assigns to every position p ∈ T a value χip (Q) ∈ {−1, 0, 1}, as follows:

Infinite Coordination Games

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– χip (Q) = 1 (winning) if there exists a strategy si ∈ Qi such that up (si , t−i ) = 1, for all t−i ∈ Q−i , – χip (Q) = −1 (losing) if up (s) = −1, for every profile s ∈ Q, and – χip (Q) = 0 (undetermined), otherwise. The following lemma is an adaptation of the Value Characterisation from [4, Lemmata 8 and 9] for the sequential variant of dominance. Lemma 3 (Value characterisation [4]). Let Q ⊆ S be an iteration stage of the admissibility procedure. (i) A strategy si ∈ S i dominates ri ∈ S i on Q if, and only if, • χi (si , Q−i ) ≥ χi (ri , Q−i ), and • χip (si , Q−i ) > χip (ri , Q−i ), for some p ∈ T . (ii) A strategy si ∈ S i is sequentially admissible on Q if, and only if, we have χi (si , Q−i ) ≥ χi (Q). In the special case of coordination games, the value functions of the different players are intimately connected. Obviously, if χip (Q) = −1 for one player i, then χjp (Q) = −1 for all players j. Moreover, if χip (Q) = 1 for a player i, then for the set Q obtained from Q by maximal simultaneous elimination of dominated strategies, we have χjp (Q ) = 1 for all players j. We define the potential χp (Q) of a restriction Q ⊆ S at position p to be maximum value χip (Q) over all players i < n. The potential function induces a partial order over restrictions of Γ : for Q, Q ⊆ S, we write χ(Q) ≤ χ(Q ) if χp (Q) ≤ χp (Q ), for all p ∈ T . Likewise, we denote by max(χ(Q), χ(Q )) the function that assigns to every p ∈ T the value max{χp (Q), χp (Q )}. When one of the sets Qi is a singleton {si } we write χ(si , Q−i ) rather than χ({si }, Q−i ). According to Lemma 3, the restrictions Q where all strategies are admissible on Q itself are characterised by χ(si , Q−i ) = χ(Q) for all players i < n and all si ∈ Qi . The fixed points of the elimination procedure are examples of such restrictions. In contrast to the general case, in which the value function may increase and decrease along the elimination stages, the value—and thus the potential—of coordination games is monotonic. Indeed, the potential observed at a particular position p during the iteration can change only from 0 (undetermined) to 1 (winning). Lemma 4 (Monotonicity, stationary values). For any two iteration stages Q ⊇ Q  ∅ of the admissibility procedure, the following properties hold: (i) χ(Q) ≤ χ(Q ), and (ii) if χp (Q)  = 0 then χp (Q ) = χp (Q), for any position p ∈ T . Proof. The statement is a consequence of the Lemma 3. Here, we shall only sketch the argument. In case χp (Q) = −1, every strategy profile in Q ⊆ Q is losing, hence χp (Q ) = −1. In case χp (Q) = 1, there exists a strategy si ∈ Qi

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which is winning in the subgame Γp restricted to Q. Then, any later stage Q contains a strategy ri that agrees with si on Γp , hence χp (Q ) = 1. Finally, in case χp (Q) = 0, there exists a profile s ∈ Q with up (s) = 1. To derive a contradiction, suppose that χp (Q) = −1. Then, there exists a player i for which all strategies in Qi that agree with si on the play π determined by s in Γp are eliminated. This can happen only if, at some stage Q between Q and Q , player i has a winning strategy ri for a subgame Γp rooted at a position p on π; let us choose i and p such that p is of minimal distance from p. Then, any play in Q that proceeds along π up to p and then follows the strategy ri is winning, witnessing that χp (Q ) ≥ 0. When reiterating the argument, the profitable deviations propagate towards the root, and we obtain χp (Q ) = 1, a contradiction.

The potential function satisfies the following consistency property along the edges of the game tree. Lemma 5 (Propagation). Let Q ⊇ Q be two consecutive iteration stages of the admissibility procedure on Γ , and let p ∈ T . For p ranging over the set of direct successors of p in T , we have: χp (Q) ≤ max χp (Q)  p

and

χp (Q ) ≥ min χp (Q).  p

Proof. For an arbitrary player i, the value of Q at p is bounded by min χip (Q) ≤ χip (Q) ≤ max χip (Q)   p

p

On the one hand, as the potential is the maximum value, this implies that χp (Q) ≤ maxp χp (Q). On the other hand, χjp (Q ) ≥ χip (Q) for all players j and all positions p (eliminating dominated strategies from Qi is profitable for all players). This implies χp (Q ) ≥ χip (Q) and, as i was chosen arbitrarily, we can conclude χp (Q ) ≥ minp χp (Q).

In particular, within a restriction Q, we have χp (Q) = −1 if, and only if, χp (Q) = −1 for all p  p. Also, it follows that, for any position p, either p is a leaf with χp (Q) corresponding to its payoff, or there exists a direct successor p with χp (Q) ≥ χp (Q). From Lemmas 4 and 5, we can immediately derive the following properties of individual elimination steps. Lemma 6 (Potential transformation). For any two consecutive iteration stages Q ⊇ Q of the admissibility procedure, the following properties hold: (i) χ(Q) = χ(Q ), if, and only if, all strategies in Q are admissible on Q . (ii) For any position p, if χp (Q) ≤ 0, for all p  p, then χp (Q ) = χp (Q) for all p  p. (iii) If χp (Q) = 0 and χp (Q) = 1 for some p  p, then there exists a position p with p  p ≺ p such that χp (Q) = 0 and χp (Q ) = 1.

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On the basis of these properties we can describe, in terms of potential, how the strategy elimination proceeds and how it terminates. First, if no winning strategy for a subgame Γp is encountered at stages Q0 or Q1 , that is, if χp (Q1 ) ≤ 0 for all p, then the iteration terminates with Q1 , by Lemma 6 (i) and (ii). Otherwise, consider the upwards closure X := {p ∈ T : χp (Q1 ) = 1 for some p  p } of the positions that are winning in Q1 . On the one hand, for all positions p ∈ T \X, the potential χp is stationary from stage Q1 onwards, by Lemma 6 (ii). On the other hand, Lemma 6 (iii) implies that every position p ∈ X will have χp (Qα ) = 1 for some finite ordinal α (winning positions travel upwards). Hence, after at most ω many steps χ(Qα ) will become stationary and the iteration terminates with χp (Q∞ ) = 1 if, and only if, p ∈ X. This concludes the proof of Theorems 1 and 2.

4

Order Independence

One way to argue why iterated admissibility is a meaningful solution concept is by illustrating that the players may identify dominated strategies from previous or fictive plays of a game and learn to avoid them henceforth. By taking different parts in the game, they would also understand that other players will avoid dominated strategies, which leads to iterating the elimination procedure until the evolution reaches a fixed point [14]. This view, however, raises the criticism that the solution concept would depend on the order in which strategies are eliminated. The choice of eliminating all dominated strategies at once for all players may seem arbitrary. Indeed, there are simple examples of games with payoffs that are not common to the players, where a procedure in which the players take turns while eliminating strategies would lead to a different outcome [18]. Over the past thirty years, several conditions to characterise finite extensive games that are immune against this criticism have been proposed [11,17,7,15]. Typically, these conditions feature a form of transference of decisionmaker indifference requiring that no player should be indifferent between choices which affect other player’s payoffs. In the kind of games we consider here, this condition obviously holds. Nevertheless, there are coordination games with infinitely many payoffs, where the outcome of iterated elimination of dominated strategies depends of the elimination order, as we shall see in 5. It turns out that, for games with binary payoffs, the order in which dominated strategies are eliminated does not matter when solving infinite coordination games via iterated admissibility. To make this statement more precise, we introduce the necessary notions following Marx and Swinkels [15]. A reduction sequence for a game Γ is a descending sequence (Rα )α∈On of rectangular sets of strategy profiles satisfying the following conditions: - R0 := S; i i - Rα+1 ⊆ Rα for every successor ordinal α + 1, and each si ∈ Rα+1 \ Rα is dominated on R , and α  - Rλ := α<λ Rα , for every limit ordinal λ.

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We denote the fixed point of the sequence with R∞ . A reduction sequence is full, if no strategy in R∞ is dominated. Notice that the stages of maximal simultaneous elimination of dominated strategies used to define admissibility form a particular (full) reduction sequence. However, arbitrary reduction sequences may decrease over uncountably many stages before reaching R∞ . Consider, for instance, a 2-player game on the infinite binary tree with only one winning play. Then, the full reduction sequence which removes the dominated strategies one by one is of length at least ω1 . We remark here, that the Value-Characterisation Lemma 3 and the Monotonicity Lemma 4 still hold when we consider stages of arbitrary reduction sequences, instead of stages of simultaneous maximal elimination. This can be shown following the corresponding proofs in [4]. Any full reduction sequence of a game Γ induces a restriction of the game i )i
Infinite Coordination Games 1

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(b) 4r, 2l, 1r, 3r

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Fig. 5. Elimination orders with different outcomes

Our aim is to prove that, in coordination games with two payoffs, all procedures of iterative elimination of dominated strategies lead to a unique outcome, up to equivalence. Theorem 8 (Order independence). Any two reductions of an infinite coordination game with binary payoffs are equivalent. 4.1

Proof

Following the unifying approach of Apt [2] for proving order-invariance of finite games, we formulate our proof of Theorem 8 in terms of abstract reduction systems. As we deal with strategy spaces of infinite cardinality, we cannot rely on Newman’s Lemma which is the basic principle of this framework. Instead, we establish that the reduction sequences which we consider here, satisfy a transfinite confluence property. For the sequel of this section, let us fix a game Γ . We use the notation R  R to express that there exists a (possibly transfinite) reduction sequence with R0 = R and Rα = R for some α ∈ On. First, we introduce a notion of game equivalence which is weaker than ≡, but easier to handle. Given two restrictions of Γ , we define R ≡χ R to hold if χ(R) = χ(R ). Clearly, R ≡ R implies R ≡χ R , but the converse is in general not true. For fixed points of full reduction sequences, however, it turns out that the two notions coincide. Lemma 9. Let R, R ⊆ S be reductions of Γ . Then R ≡ R if, and only if, R ≡χ R  . Proof. Let X ⊆ T be the set of positions with χp (R) = χp (R ) = 1. As all strategies in R are admissible on R, this set is upwards closed with respect to , i.e., if p ∈ X then p ∈ X for all p  p. To define a suitable embedding f of R into R , consider any strategy ri ∈ Ri for a player i, and pick an arbitrary strategy ri ∈ Ri . Let f (ri ) be the strategy that agrees with ri on X and with ri on T \ X.

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We claim that f (ri ) ∼S ri , that is up (f (ri ), t−i ) = up (ri , t−i ) for all t−i ∈ S . This is sufficient to show that the image f (ri ) belongs to Ri , since for any restriction Q ⊆ S, and in particular for the stages of S  R , the set of strategies that are admissible on Q is closed under ∼S . To prove the claim note that, for any position p ∈ T \ X, the strategies f (ri ) and ri agree on {p : p  p}, by construction. As T \ X is downwards closed, we have up (f (ri ), t−i ) = up (ri , t−i ). For positions in T \ X, we argue as follows. Since ri and ri are admissible on R and R , respectively, we have, by Lemma 3, χ(ri , R−i ) = χ(R−i ) and χ(ri , R−i ) = χ(R−i ). By hypothesis, these potentials are equal. In particular, for any p ∈ X, we have χp (ri , R−i ) = χp (ri , R−i ) = 1. On the other hand, both ri and ri are image-closed on X, i.e., ri (p) ∈ X for any p ∈ X and likewise for ri . Therefore, we have for all p ∈ X and every t−i ∈ S −i , −i

up (f (ri ), t−i ) = up (ri , t−i ) = 1 = up (ri , t−i ). Next, let us verify that f is an embedding, that is, that up (s0 , . . . , sn ) equals up (f (s0 ), . . . , f (sn )) for all p and all s ∈ R. At any p ∈ T \ X, the condition holds because each strategy si agrees with its image f (si ) on T \ X; at p ∈ X all profiles in R and R yield the same payoff 1. To justify that f respects payoff equivalence, the argument is similar. Consider a pair of strategies ri ∼R si , that is up (ri , t−i ) = up (si , t−i ) for all p ∈ T and t−i ∈ R−i . For p ∈ T \ X, the strategies ri and si agree with their f -image on T \ X, and hence up (f (ri ), t−i ) = up (f (si ), t−i ), for every p ∈ T \ X and all t−i ∈ R−i . On the other hand, for p ∈ X, we have, up (f (ri ), t−i ) = up (f (si ), t−i ), because f (ri ) and f (si ) are in R . Symmetrically, we can construct an embedding of R into R that preserves payoff equivalence, thus concluding the proof that R ≡ R .

In the next step, we relate reduction sequences obtained by arbitrary elimination of dominated strategies with the sequence (Qα )α∈On obtained by maximal simultaneous elimination of dominated strategies. We define the rank |p| of a position p ∈ T to be the least ordinal α such that χp (Qα ) < χp (Qα+1 ) if such an ordinal exists, and 0 otherwise. Notice that for all positions p ∈ T with χp (S) ≤ 0, we have |p| = 0, according to Lemma 6(ii). Likewise, any position p such that there exists a strongly or weakly winning strategy in Γ |p has rank 0 or 1, respectively. Essentially, the rank of a position represents the stage at which it is discovered as being winning, losing, or undetermined during the process of maximal simultaneous elimination. Intuitively, the following lemma points out that any set of winning positions discovered through arbitrary elimination is also discovered through maximal simultaneous elimination. Lemma 10 (Covering). If S  R, then χ(R) ≤ χ(Qκ ), for the stage κ = sup{|p| : p ∈ T with χp (R) = 1}. Proof. If there are no positions p ∈ T with χp (R) = 1, we simply have χ(R) = χ(S) = χ(Q0 ). Otherwise, for any position p with χp (Q∞ ) = −1, we have

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χp (R) = −1 ≤ χp (Qκ ). For positions p with χp (Q∞ ) = 1, we have χp (Q0 ) ≥ 0 and thus, by monotonicity χp (Qκ ) ≥ 0; due to the way k was chosen, if |p| ≤ κ then χp (Qκ ) = 1 and, if |p| > κ, then χp (R) = 0. Thus, χp (Qκ ) ≥ χp (R), in both cases. Finally, for positions p with χp (Q∞ ) = 0 we argue by transfinite induction along the reduction sequence, that no undetermined strategy is ever eliminated in the subgame starting at p, that is, {si |p : si ∈ R1i } ⊆ {si |p : si ∈ Qiα }, for any stage α and every player i < n. Hence, we can conclude that χp (R) = 0 = χp (Qκ ).

The next lemma, illustrated in Figure 6(a), shows that, when we set out on a particular reduction sequence S  B, there is no risk to miss a position p which would have been discovered following a different sequence; it is always possible to continue the reduction from B onwards such that p will be discovered in at most ω many steps. Lemma 11 (Catch up). If B  S  C, then (i) there exists an index δ such that χ(Qδ ) ≥ max{χ(B), χ(C)}, and (ii) for the least such δ, there exists D  B such that χ(D) = χ(Qδ ). Proof. Let δ := sup{|p| : p ∈ T with χp (B) = 1 or χp (C) = 1}. According to Lemma 10, χ(Qδ ) ≥ max{χ(B), χ(C)} and there is no index smaller than δ with this property. To prove the second point, let us construct a sequence (Bα )α<δ with B0 := B and Bα+1 consisting of those strategies si ∈ Bα that are admissible  on Bα at all positions of rank at most α; for the case δ = ω, we set Bω := α<ω Bα . We claim that the set D := Bδ has the required properties. Clearly, D  B via a reduction sequence of length δ ≤ ω. For every position p of rank up to δ and every index α ≤ δ, by construction, χp (Bα ) = χp (Qα ). Thus, for all these positions, χp (D) = χp (Qδ ). On the other hand, for each position p with rank greater than δ, we have χp (D) = χp (B) = 0, again by Lemma 10. Hence, overall χ(D) = χ(Qδ ).

Finally, we are ready to state our confluence property for reductions of extensive coordination games. S

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Fig. 6. Reduction patterns

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D. Berwanger

Lemma 12 (Confluence). If B  S  C, there exist restrictions D  B and E  C such that D ≡χ E. Proof. It is sufficient to apply the construction of 11 on the two reduction paths as illustrated in Figure 6(b). The restrictions D and E obtained in this way then have the same potential as the least stage Qδ with χ(Qδ ) ≥ max{χ(B), χ(C)}.

By Lemma 10, the potential of any reduction is bounded above by χ(Q∞ ). Hence, for any fixed point R of a full reduction sequence, we have R ≡χ Q∞ . According to Lemma 9, it follows that every extensive coordination game has a unique normal form, which concludes our proof of Theorem 8.

5

Beyond Binary Payoffs

From the perspective of the more classical theory of extensive games, the restriction to binary payoffs assumed in Gale-Stewart games may appear too limiting. However, it seems doubtful whether admissibility can yield a sound solution concept for games of infinite duration with a more general payoff structure.

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Fig. 7. Conflicting interests

One problem is that, already when there are three different payoffs, one round of maximal simultaneous elimination can remove infinite dominance chains without letting any dominating strategy survive. As an example, consider the game in Figure 7 which represents a coordination situation with conflicting interests. Notice that, on the full set of strategies, any strategy of Player 0 that chooses in in the first k − 1 rounds, that is, at positions 1, 3, . . . , 2k − 1, and out in round k is dominated by one that keeps choosing in until round k, and only afterwards, in round k + 1 chooses out. Thus, maximal simultaneous elimination, will eliminate all strategies of Player 0 except the one that chooses in forever; the same reasoning applies to Player 1. Consequently, the only 1-admissible strategy profile consists in choosing in forever yielding payoff (0, 0), which is less than any combination of the eliminated strategies would yield for any player. With maximal elimination in sequential order, where the dominated strategies of Player 0 are removed first, Player 1 would retain all strategies that choose out after a finite number of rounds, which yields a solution with payoff (2, 1). On the other hand, if we start by removing all dominated strategies of Player 1 first, we obtain a solution with payoff (1, 2).

Infinite Coordination Games

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Hence, in the general case of infinite non-zero sum games, the elimination of weakly dominated strategies in infinite games with more than two payoffs may be order dependent and lead to implausible predictions, even if it is not iterated. Similar phenomena occur in coordination games with infinitely many payoffs. Consider, for instance, the game in Figure 8, where the play ends as soon as one of the players chooses out with a payoff corresponding to the number of moves taken in the play. Clearly, any strategy to choose out after k many in-steps is dominated by any strategy that chooses out after at least k + 1 many steps, whenever the current reduction stage contains a strategy of the other player to choose in for at least k many steps. Therefore, maximal simultaneous elimination would lead both players to choose in forever and earn payoff 0. Worse than this, a reduction sequence may lead to empty strategy sets. For instance, if we first eliminate all dominated strategies of Player 0, only the one playing in forever survives. With respect to this single strategy, the strategy of Player 1 to play in forever is dominated by the one that moves out, e.g., at position 2. On the other hand, any strategy to choose out at position k (after a sequence of in) is in turn dominated by the strategy to choose out at k + 2. Thus, none of the strategies available to Player 1 is admissible at this stage of the reduction.

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Fig. 8. Infinitely many common payoffs

As a larger class of infinte games for which iterated admissibilty may still constitute a meaningful solution concept, the above examples single out coordination games with an arbitrary finite number of payoffs. However, as we shall see, the structure of admissible strategy sets changes substantially when the number of possible payoffs is increased. Intuitively, for games with two payoffs, the elimination of dominated strategies is backwards inductive, in the sense that any combination of solutions for independent subgames extends to a solution of the entire game, according to the value characterisation in Lemma 3. This is no longer the case in games with three or more payoffs. Here, admissibility incorporates a form of forwardinductive reasoning due to which a strategy for a subgame may be dominated or not depending on choices at off-play positions, i.e., positions that are never reached in a play that reaches the subgame. (See [8], for a recent and very thorough account on forwards induction). As a consequence, even though we set out with a game of perfect information, simultaneous maximal elimination of dominated strategies may yield a game restriction which cannot be understood any more as a game with perfect information. Accordingly, partial solutions for independent subgames do not necessarily combine to a solution of the entire game.

18

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Fig. 9. Loss of perfect-information character

To illustrate the effect of forward-inductive reasoning in an infinite coordination game, we adapt an example from [3]. Recall the basic game Δ described in Subsection 3.1 as an example where admissibility cannot ensure coordination, and let us denote by Δa,b the modified version where, instead of payoff +1 and −1, the players receive payoff a and b, respectively. When considered in isolation, Δa,b should of course not be different of Δ−1,+1 , for any a < b. Let us now look at the game Γ depicted in Figure 9. Here, any strategy of player to choose in at the root, and out both at positions 3 and 4, is dominated by the strategy to choose out already at the root. Also, any strategy that is losing on Γ3 or Γ4 is (sequentially) dominated. All the other strategies are undominated, because the choices (1in, 2out, 3in), (1in, 2in, 3out) and (1in, 2in, 3in), combined with any strategy that is admissible in Γ3 and Γ4 , lead to mutually incomparable strategies. Hence, it is admissible for Player 0 to play out when reaching position 3, only under the counterfactual condition that he would have played in if the play would have reached position 4. Phrased as a forward-induction argument, this is to say that, upon reaching position 2, Player 1 knows that Player 0 will choose in either at 3 or at 4, which he could not have derived—by means of admissibility— by considering just the subgames Γ3 and Γ4 . Thus, in admissible strategies the choices at position 3 and 4 are not any more strategically independent in the sense of [13], although reaching one position would exclude reaching the other one. In fact, the restriction induced by the set of undominated strategies is not equivalent to any game with perfect information. The loss of perfect information encountered here is in sharp contrast to the case of games with only two payoffs. For infinite games with two—not necessarily common—payoffs it was shown in [4], that strategy sets encountered during simultaneous maximal elimination of dominated strategies satisfy a particular well-formedness criterion: they allow shifting, in the phrasing of the paper. Informally, the criterion requires that at any decision point reached during a play, the player i in turn to move at p can deviate from the strategy he used so far and switch to any strategy that is compatible with reaching p. The property that α-admissible strategy sets allow shifting is crucial for most proofs in [4]. As games with more than two payoffs fail to satisfy this property, we cannot rely on our previous results about admissibility in infinite non-zero sum games.

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We can neither confirm nor reject the hypothesis that the good properties of admissibility on infinite coordination games with two payoffs are preserved in games with an arbitrary number payoffs. The presence of forward-inductive effects, however, suggests that sharper analytic tools will be needed to settle this question.

References 1. Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time Temporal Logic. J. ACM 49, 672–713 (2002) 2. Apt, K.R.: Uniform Proofs of Order Independence for Various Strategy Elimination Procedures Contrib. Theoret. Econ. 4(1), article 5 (2004) 3. Battigalli, P.: On Rationalizability in Extensive Games. J. Econ. Theory 74, 40–61 (1997) 4. Berwanger, D.: Admissibility in infinite games. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 188–199. Springer, Heidelberg (2007) 5. Bicchieri, C., Schulte, O.: Common Reasoning About Admissibility. Erkenntnis 45, 299–325 (1997) 6. Gale, D., Stewart, F.M.: Infinite Games with Perfect Information Kuhn. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, Princeton. Annals of Mathematics Studies, vol. 28, pp. 245–266 (1953) 7. Gilboa, I., Kalai, E., Zemel, E.: On the Order of Eliminating Dominated Strategies. Oper. Res. Letters 9, 85–89 (1988) 8. Govindan, S., Wilson, P.: On Forward Induction. Econometrica 77, 1–28 (2009) 9. Gr¨ adel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002) 10. Gr¨ adel, E., Ummels, M.: Solution Concepts and Algorithms for Infinite Multiplayer Games. In: Apt, K., van Rooij, R. (eds.) New Perspectives on Games and Interaction. Texts in Logic and Games, vol. 4, pp. 151–178. Amsterdam University Press, Amsterdam (2008) 11. Gretlein, R.J.: Dominance Elimination Procedures on Finite Alternative Games. Int. J. Game Theory 12, 107–113 (1983) 12. Henzinger, T.A.: Games in System Design and Verification. In: van der Meijden, R. (ed.) TARK 2005, pp. 1–4. National University of Singapore (2005) 13. Mailath, G.J., Samuelson, L., Swinkels, J.M.: Extensive Form Reasoning in Normal Form Games. Econometrica 61, 273–302 (1993) 14. Marx, L.: Adaptive Learning and Iterated Weak Dominance. Games Econ. Behav. 26, 254–278 (1999) 15. Marx, L.M., Swinkels, J.M.: Order Ondependence for Iterated Weak Dominance. Games Econ. Behav. 18, 219–245 (1997) 16. Mohalik, S., Walukiewicz, I.: Distributed games. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 338–351. Springer, Heidelberg (2003) 17. Moulin, H.: Dominance Solvability and Cournot Stability. Math. Soc. Sci. 7, 83–102 (1984) 18. Samuelson, L.: Dominated Strategies and Common Knowledge. Games Econ. Behav. 4, 284–313 (1992)

Conditioning by Minimizing Accessibility Konstantinos Georgatos

2

1 Department of Mathematics and Computer Science, John Jay College, City University of New York, 445 West 59th Street, New York, NY 10019, United States of America The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016, United States of America [email protected]

Abstract. This paper presents an axiomatization of a class of set-theoretic conditional operators using minimization of the geodesic distance defined as the shortest path generated by the accessibility relation on a frame. The objective of this modeling is to define conditioning based on a notion of similarity generated by degrees of indistinguishability.

1

Introduction

Stalnaker [27] developed semantics for counterfactual conditionals by using the following principle A > B is true at a state s iff the closest A-world to s is a B-world. By “closest” one understands the most similar; closeness is identified with similarity. For example, one may reason that the counterfactual: If it had rained, then I would have arrived later is true because in the most similar world to the present where it had rained, I would have driven more carefully, and thus more slowly. However, it is a curious fact that neither Stalnaker nor Lewis, who relaxed the uniqueness assumption of the closest world [20], considered a standard notion of closeness because closeness is not judged with a global criterion but rather it is parameterized by worlds. Stalnaker made use of selection functions f (x, A) where the variable x is a world in the model, while Lewis made use of a class of closeness orderings ≤x , one for each world x. Although such modelings are justifiable in theory, in practice similarity is modeled with global criteria. In Information Retrieval, for example, documents may be thought of as vectors in a metric space. The closer in space the more similar those documents are [25]. Similarly, a popular global measure for similarity among propositional valuations is the number of variab! les they differ, their hamming distance (see [4]). Note that assuming a global distance does not exclude parameterization. In fact, a single global distance d generates a closeness ordering ≤x for each epistemic state x: y ≤x z defined by d(x, y) ≤ d(x, z). Therefore, one wonders why global distance semantics for conditionals have not been studied or developed further. This is even more surprising in view of [26], G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 20–33, 2010. c Springer-Verlag Berlin Heidelberg 2010 

Conditioning by Minimizing Accessibility

21

a

e d

u

u @ @ @ u b1

u @ @ @u

u b2

c Fig. 1.

where it is shown that, for each parameterized ordering model, we can always find an equivalent model based on a global distance. In other words, we gain no advantage by parameterizing because an invariant distance relation can encode any set of parameterized orderings. The purpose of the present paper is to introduce and characterize a class of conditional operators that are modeled by a global distance. We consider frames, that is, pairs (T, R) where T is a set of states and R is an accessibility relation. Every frame comes equipped with a distance map called geodesic which is simply defined as the times we need to apply R to reach a state. Minimization over a geodesic is straightforward. We illustrate the process with the following example where the accessibility relation R, represented by the edges of the graph, is reflexive and symmetric, in Figure 1. Let A = {a, b1 , b2 }, and B = {b1 , b2 }. We denote conditioning with >. Then A > B = {b2 , c} because the closest A-worlds to b2 and c are {b1 , b2 } and {b2 } respectively. However, d ∈ A > B because its closest A-worlds are {a, b2 }. Similarly, we have A > ¬B = {e}. We use the accessibility relation to represent indistinguishability. This idea is not new. Indeed, this is the idea behind the use of frames as semantics for epistemic modal logics. When a possible world w2 is accessible from a word w1 means that, for an agent in w1 , the world w2 is (epistemically) indistinguishable from w1 . What are the properties of accessibility as indistinguishability? Many authors have argued that indistinguishability is better expressed through equivalence [1,17,5]; many have also dropped transitivity as early as [24] (see also [13] for similarity), while others have weakened transitivity, the most notable example being the t-norm transitivity of the similarity relation in Fuzzy sets [28]. A set equipped with a reflexive symmetric relation has been called a proximity space in [2]. In the framework of Rough Sets [23], a reflexive, symmetric, and transitive relation of indistinguishability is called indiscernibility, and if it is only reflexive and symmetric it is called a tolerance relation [29,21]. In addition, their logical status has been well studied and give rise to orthologic [12], and the modal logic system B (see [18]). Further, in the usual modal logics of belief reflexivity is dropped, as the agent may distinguish her present state from the one she believes (falsely) that she is in. We shall make no assumptions on

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K. Georgatos

indistinguishability although we show that properties such as reflexivity and symmetry can be added. Examples are easily generated, as the accessibility relation needs no quantitative information. Metrics is perhaps the easiest way to generate indistinguishability; for example, let xRy if and only if d(x, y) < e for some appropriate fixed metric d and non-negative real number e. Other examples follow: Example 1. Let T be the set of finite binary strings of finite length n. We can say that two strings are indistinguishable when they have the same length and differ in at most one digit. Otherwise, they are distinguishable. Example 2. Let T be a set of documents (sets of terms) and n a positive integer. Two documents d1 and d2 are indistinguishable when they have at least n common terms. That is, d1 Rd2

iff |d1 ∩ d2 | ≥ n.

The above examples are important and will be discussed in more detail in the next section. We use the geodesic distance to express similarity. In [7], we argued that it is possible to view similarity (the geodesic distance) as a derived notion, the more basic concept being indistinguishability (the accessibility relation): Our idea rests on the following maxim: two objects are similar when there is a context under which they are indistinguishable. Therefore, similarity can be measured with degrees of distinguishability. For example, although two similar houses might appear different in various details when we stand in front of them, they might appear identical if we observe them from an appropriate distance x. Thus, indistinguishability at distance x implies similarity. The smaller the distance x, the less similar the objects are. This way our approach follows closely the Stalnaker idea of conditioning by picking the most similar worlds [27]. In our framework, the closest interpretations (according to the geodesic distance) are the most similar, as long as similarity is measured by degrees of distinguishability. First, we discuss frames and their geodesics and present examples. In Section 3, we define geodesic conditional operators that correspond to update, and present their axioms and the associated characterization result (Theorem 9). In the next section, similar results are presented for a conditional operator based on revision. The form of Ramsey tests that allow us to go back and forth between the conditional and update/revision operators is also discussed.

2

Frames and Their Geodesics

In the following, we shall consider frames with a single accessibility relation, that is, pairs F = (T, R) where T is a set of states and R ⊆ T × T an accessibility

Conditioning by Minimizing Accessibility

23

relation between them. We shall write xRy when (x, y) ∈ R. Given such a pair F we can define its geodesic distance dF , or simply d, as a map from T × T to N ∪ {∞}, where N = {1, 2, 3, . . .} is the set of natural numbers, defined by  min{n | xRn y} if {n | xRn y} = ∅ dF (x, y) = ∞ otherwise. Note that dF is not a topological metric. First, its range does not include 0. Although this could be mended, for example we could subtract 1 in the above definition, dF may fail identity. Nevertheless, if R is assumed to be reflexive (see 3.1 and the range is extended to contain 0, then R is a pseudometric. If R is connected, i.e., for all x, y ∈ T there is n ∈ N such that xRn y, then the range of the geodesic restricts to N . In addition, (an extension of) the triangle inequality holds. In particular, we have the following Lemma 3. Let F = (T, R). Then: 1. 2. 3. 4. 5.

For all x ∈ T , dF (x, x) = 1 iff R is reflexive R is symmetric iff dF (x, y) = dF (y, x) If R is transitive then dF (x, y) = 1 or ∞ dF (x, y  ) ≤ dF (x, y) + dF (y, y  ) (assuming n + ∞ = ∞ + ∞ = ∞) For all x, y ∈ T such that dF (x, y) = n with 2 < n < ∞ there is z ∈ T with z  = x, y such that dF (x, y) = dF (x, z) + dF (z, y). In particular, we can choose z so that dF (x, z) = 1.

We shall now discuss an interesting construction of a geodesic space given an arbitrary metric space. This construction is a straightforward modeling of the concept of threshold which was the original motivation of the present work. Let (X, d) be a a metric space and let e > 0 be a real number representing a threshold. Then (X, de ) is called the e-threshold space of (X, d), with de (x, y) = dF (x, y), where F = (T, R) is a frame with (x, y) ∈ R iff d(x, y) < e. Every e-threshold space is geodesic. A more interesting space arises, if we let dm e (x, y) = n, where n = min{k : d(x, y) ≤ ke, k > 1}. Call (X, dm e ) the multiple e-threshold space of (X, d). Unfortunately, we have dm = dF (x, y), that is the multiple e-threshold space is e (x, y)  not geodesic although dm e (x, y) is an integer valued map. However, an arbitrary integer distance map can be embedded into a geodesic space if we add enough elements as the following lemma shows. Lemma 4. Let (X, d) where d is a map from X × X to N ∪ {∞} satisfying the extended triangle inequality (as in Lemma 3.4). Then there exists a frame F = (X, R) such that dF = d. Proof. Let X  = X ∪ Y , where Y = {y((x,x ),k) : d(x, x ) = n < ∞, and k = 1, 2, . . . , n − 1}, that is, we add n − 1 elements for all x, x ∈ X such that d(x, x ) = n. Also, let R be defined by

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– – – –

K. Georgatos

xRx iff d(x, x ) = 1, xRy({x,x },1) y({x,x },n−1) Rx , y({x,x },m) Ry({x,x },m+1) , where 0 < m < n − 1.

Consider the frame F = (X  , R). Observe that dF (x, x ) ≤ d(x, x ), for d(x, x ) = n < ∞ implies the existence of sequence of R-related elements of Y of lenth n−1. We shall show that d(x, x ) = dF (x, x ) by induction on dF (x, x ). Let dF (x, x ) = 1. Then xRx so by the above definition d(x, x ) = 1. Assume d(x, x ) = dF (x, x ) for all x, x ∈ X such that dF (x, x ) = m with m < n. Now, suppose dF (x, x ) = n. This means there is a (n − 1)-length sequence of R-related elements between x, x . If this sequence contains no x ∈ X then the elements are all of the form y({x,x },k) and necessarily d(x, x ) = n by the definition of Y . If this sequence contains x ∈ X, then dF (x, x )+dF (x , x ) ≤ n. By induction hypothesis, as 1 < dF (x, x ), dF (x , x ) < n, dF (x, x ) = d(x, x ) and dF (x , x ) = d(x , x ), and therefore d(x, x )+d(x , x ) ≤ n. By the triangle inequality d(x, x ) ≤ n. Since n = dF (x, x ) ≤ d(x, x ), we have d(x, x ) = n. Finally, suppose dF (x, x ) = ∞, then d(x, x ) = ∞ as dF (x, x ) ≤ d(x, x ). 

3

Geodesic Conditioning

Not surprisingly, conditional reasoning has been linked to belief change very early [22,6,19]. Although the relation between conditionals and the operators of update and revision is straightforward when the conditionals are not nested, there are results that extend this link to full conditional logics (see [14,16] for update, and [3,10,11] for revision). Our approach is set theoretic and the results of this paper do not compare directly with these works. Nevertheless, the conditional operators we present should be viewed as an extension of those conditional logic systems to geodesic spaces. We follow the presentation of conditioning of [15]. Halpern observed that an alternative to the usual syntax-semantics approach is to consider modalities and conditionals as set-theoretic operators. The justification is that eventually we interpret those operators set-theoretically, so there is no harm in doing this from the start. Mathematicians and economists have used this approach for a long time, the former by using probability theory, the latter by using events. The main advantage of this approach is that it gives easier completeness theorems because we skip limiting cases. Additionally, it is still possible to turn such completeness theorems into usual logic completeness theorems in a uniform way (see [15] for details). Given a frame, there is a straightforward notion of conditioning based on its geodesic. As soon as we have a notion of distance, then we can choose the closest states. Let  {y ∈ A | dF (x, y) = dF (x, A)} dF (x, A) = ∞ fF (x, A) = ∅ otherwise where x ∈ T and A ⊆ T .

Conditioning by Minimizing Accessibility

25

Observe that fF is a selection function in the sense of Lewis, that is, it returns a subset of states rather than a unique state. Also note that fF extends to an operation between subsets with  B•A= fF (x, A), x∈B

where A, B ⊆ T . In [8], B • A was called the geodesic update operation. Definition 5. Given a frame F = (T, R) and x ∈ T , A, B ⊆ T then we can define the geodesic conditional operator >d : 2T × 2T → 2T by A >d B = {x : fF (x, A) ⊆ B}. The definition of A >d B readily translates to the following C ⊆ A >d B

iff

C • A ⊆ B,

which is a form of the Ramsey Test for the geodesic update operation. The fact that conditioning is defined through the Ramsey Test should not come as a surprise. As early as [14], it is known that the update is compatible with the Ramsey test. We shall attempt to formulate a similar identity for revision in the next section. The axioms that characterize the geodesic conditional operator appear in Table 1. It is not difficult to show that >d satisfies all axioms apart from REFL and SYMM. Axiom NORM has appeared in [15] while ID, CV, MP, CN are straightforward translations of conditional logic axioms (for example see [22]). Axioms INDB, IND and CN* axiomatize the behavior of T > A that acts as the belief modality. Axioms INDB controls the base case, IND the induction case, while CN* the iteration limit case. Axioms REFL and SYMM stipulate that the accessibility relation is reflexive and symmetric, respectively, and were not assumed for the results of this section. Now, let a GC space be a pair (T, >), where T is a set of worlds and > an operator on the subsets of T , that is, >: 2T ×2T → 2T , which satisfies the axioms of Table 1. A GC space gives rise to a frame (T, R), where R is defined by xRy

iff

x∈T >A⇒y∈A

for all x, y ∈ T and A ⊆ T . Denote its associated geodesic conditional operator with >d . Lemma 6. Given a GC space (T, >) then, for all A ⊆ T we have T >A

=

T >d A.

Proof. First, suppose x ∈ T > A. Let y ∈ T be such that xRy. We need to show that y ∈ A but this is immediate from the definition of R.

26

K. Georgatos Table 1. System GC 

j∈J (A

> Bj ) = A >



Bj

NORM

A>A=T

ID

¬(A > ¬B) ∩ (A > C) ⊆ (A ∩ B) > C

CV

(A > B) ⊆ (T > (A → B))

MP

If C ⊆ ¬(T > ¬A), then (T > A → B) ∩ C = (A > B) ∩ C

INDB

If C ⊆ T > ¬A, then (¬(T > ¬A) > (T > A → B)) ∩ C = (A > B) ∩ C IND If A ⊆ B, then B > ∅ ⊆ A > ∅

CN

If A ⊆ T > A, then A ⊆ ¬A > ∅

CN*

T >A⊆A

REFL

¬A ⊆ T > ¬(T > A)

SYMM

Next, suppose x ∈ T >d A. Suppose towards a contradiction that x ∈ T > A and  B=∅ ¬A ∩ x∈T >d B

which implies



B ⊆ A.

x∈T >d B

Using NORM we have



T >

B ⊆ T > A.

x∈T >d B

Applying again NORM we get 

T > B ⊆ T > A.

x∈T >d B

This shows that x ∈ T > B which is a contradiction. Therefore,  ¬A ∩ B =∅ x∈T >d B

and let z ∈ T belong to the above set. Then we have that xRz and z ∈ ¬B which contradicts the initial hypothesis.  The following two lemmas are useful for the main result Lemma 7. For all x ∈ T , A ⊆ T and n > 1, dF (x, A) = n

iff

dF (x, ¬(T >d ¬A))) = n − 1.

Conditioning by Minimizing Accessibility

27

Proof. Suppose that dF (x, ¬(T >d ¬A)) = n − 1. Then there exists y ∈ ¬(T >d ¬A) and z ∈ A such that dF (x, y) = n − 1 and dF (y, z) = 1. By Lemma 3.4, we have dF (x, z) ≤ n. Therefore, dF (x, A) ≤ n. Suppose towards a contradiction that dF (x, A) = k < n. Then there exists w ∈ A such that dF (x, w) = k. By Lemma 3.5, there exists y ∈ T such that dF (x, y) = k − 1 and dF (y, w) = 1. The latter implies that y ∈ ¬(T >d ¬A) and therefore, dF (x, ¬(T >d ¬A)) = k − 1 < n − 1, a contradiction. The other direction is similar.   For the following, let x∗ = ¬ x∈A>d ∅ A, that is the set of all states accessible from x. Lemma 8. For all x ∈ T we have x∗

⊆

x∗ > ∅.

Proof. Suppose y ∈ x∗ , so there is n < ∞ such that xRn y. Let yRz for a z ∈ T . Obviously xRn+1 z and therefore z ∈ x∗ . This shows that y ∈ T >d x∗ so by Lemma 6 y ∈ T > x∗ . Therefore, x∗ ⊆ T > x∗ . By CN*, we have y ∈ x∗ > ∅.  The main result of this section is the following Theorem 9. Given a GC space, then, for all A, B ⊆ T A>B

=

A >d B.

Proof. We shall show that for all x ∈ T and A, B ⊆ T , x∈A>B

iff

x ∈ A >d B,

by induction of the distance of x from A. – Let d(x, A) = 1. Suppose x ∈ A > B. Let y ∈ fF (x, A). This implies that xRy. By MP, we have that x ∈ T > A → B, so by the definition of R we have that y ∈ A → B. Since y ∈ A, we have y ∈ B. This shows that x ∈ A >d B. For the other direction, suppose x ∈ A >d B. This implies x ∈ T >d A → B so, by Lemma 6, we have x ∈ T > A → B. Observe that d(x, A) = 1 implies that x ∈ ¬(T > ¬A). Applying INDB we have x ∈ A > B. – Suppose the hypothesis is true for d(x, A) ≤ n and assume d(x, A) = n + 1. Observe that since d(x, A) > 1, we have x ∈ T >d ¬A and, by Lemma 6, x ∈ T > ¬A. Now, suppose x ∈ A > B which is equivalent to x ∈ ¬(T > A) > (T > A → B) given that x ∈ T > ¬A and using IND. Observe that ¬(T > A) > (T > A → B) = ¬(T >d A) > (T >d A → B)

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K. Georgatos

using Lemma 6. We have that d(x, ¬(T > ¬A)) = n using Lemmas 7 and 6. By induction hypothesis ¬(T > A) > (T > A → B) = ¬(T >d A) >d (T >d A → B). So x ∈ A > B is equivalent to x ∈ ¬(T >d A) >d (T >d A → B) which is equivalent x ∈ A >d B, given that x ∈ T >d ¬A and using IND. – Finally, let d(x, A) = ∞. In this case, fF (x, A) = ∅ so x ∈ A >d ∅ and x ∈ A >d B for all B ⊆ T . We have A ⊆ ¬x∗ . By CN we get ¬x∗ > ∅ ⊆ A > ∅. By Lemma 8 x∗ ⊆ A > ∅. So x ∈ A > ∅, and therefore x ∈ A > B, for all B ⊆ T. 

4

Geodesic Conditioning Using Revision

Recall that the selection function we used in the previous section could be extended to a selection function between two subsets which was subsequently used to define the update operator. There is an alternative way to extend the selection function fF to pairs of subsets: instead of pointwise minimization defined through union, we use global minimization. Let  {x ∈ B : d(A, x) = d(A, B)} d(A, B) < ∞ A∗B = ∅ otherwise As in the case of update, we called A ∗ B the (non-centered) geodesic revision of A with B in [9]. Observe that the second part of the definition could be eliminated for an alternative definition of the revision operator that does not satisfy consistency preservation. Definition 10. Given a frame F = (T, R) and A, B ⊆ T then we can define T the revision geodesic conditional operator >∗d : 2T × 2T → 22 by A >∗d B = {C : C ∗ A ⊆ B}. Again as in the case of update, the above definition translates to the following Ramsey test: A ∗ B ⊆ C iff A ∈ B >∗d C RMT which is akin to the Ramsey test of the previous section in the sense that RMT forces conditioning to operate on a metalevel, that is, conditioning returns a set of subsets rather than a single subset. This difference should not be understood as a different mechanism. Indeed, we essentially minimize the induced accesibility R∗ relation on subsets defined by AR∗ B if B = {y : xRy, x ∈ A}. Since the induced accesibility is a function, this opens up the possibility of a simpler representation of the revision conditional based on the previous section. For example, consider the graph of Figure 2. Let C = {c1 , c2 } and D = {c1 }. Then C >∗d D contains all the subsets whose distance from c1 is (strictly) less than the distance from c2 . Those are {b1 }, {b1 , a}, {b1 , c2 }, {c2 }, and {c2 , a}.

Conditioning by Minimizing Accessibility

29

a

b1 u

u @ @ @ u b2

c1 u

u c2 Fig. 2.

Table 2. System GCR 

j∈J (A

> Bj ) = A >

 j∈J

Bj

A > A = P(T ) ¬(T > ¬A) ∩ (T > A → B) = A > B

NORM ID INDB

If X ∈ T > ¬A, then X ∈ A > ¬B iff X ◦ ∈ A > B IND If X ⊆ Y , then Y ∈ T > A implies X ∈ T > A

SB

If A ⊆ B, then B > ∅ ⊆ A > ∅

CN1

If X ⊆ Y , then Y ∈ A > ∅ implies X ∈ A > ∅

CN2

If A ∈ T > A, then A ∈ ¬A > ∅

CN3

If A ∈ T > B, then A ⊆ B

REFL

If A ∈ T > ¬B, then B ∈ T > ¬A

SYMM

The set of axioms that characterizes this class are those of Table 2 excluding REFL and SYMM. Axioms REFL and SYMM stipulate that the accessibility relation is reflexive and symmetric, respectively, and, as in the previous section, were not assumed for the results of this section. Notice that Axioms INDB and IND control induction on the frame. Axiom INDB is stronger than the Axiom MP of the preceding section. Axiom SB stipulates that beliefs persist when there is more information about the present state, i.e., monotonicity of beliefs. Axioms CN1, CN2, and CN3 stipulate the maximal expansion of our beliefs. We proceed as in the previous section. Call a pair (T, >), a GCR space if > is an operation on the subsets of T that returns a set of subsets of T , that is, T >: 2T × 2T → 22 , which satisfies the axioms of Table 2. Similarly, a GCR space gives rise to a frame (T, R), where R is defined by xRy

iff {x} ∈ T > A ⇒ y ∈ A

for all x, y ∈ T and A ⊆ T . Denote its associated geodesic conditional operator with >∗d .

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K. Georgatos

As in the previous section, we shall also make use of the following sets. The first set is the set of states the agent believes she is in when she is in A:  B. A◦ = A∈T >d B

The second set is the set of states accessible from A:  A∗ = T − B. A∈B>d ∅

Lemma 11. If (T, >) is a GCR space, then for all A ⊆ T we have A ∈ T > A◦ . Proof. Axiom NORM implies  A∈ (T > B) = T > A∈T >B



B = T > A◦ .

A∈T >B

 Lemma 12. Given a GCR space (T, >) then, for all A ⊆ T we have T >A

T >∗d A.

=

Proof. First, suppose C ∈ T > A. We shall show that C ∈ T >∗d A. By the definition of >∗d , it is enough to show that C ∗ T ⊆ A. Let y ∈ C ∗ T then by the definition of ∗ we have d(C, y) = d(C, T ) = 1 and, therefore, there exists x ∈ C such that d(x, y) = 1, i.e. xRy, which, by the definition of R implies that if {x} ∈ T > A then y ∈ A. Notice that C ∈ T > A and therefore {x} ∈ T > A by Axiom SB. So, we have y ∈ A. For the opposite inclusion, suppose C ∈ T >∗d A. If ¬A ∩ C ◦ = ∅ then there exists x ∈ C and y ∈ ¬A such that xRy which is a contradiction to our assumption. If ¬A ∩ C ◦ = ∅ then C ◦ ⊆ A, so T > C ◦ ⊆ T > A by Axiom NORM. So by Lemma 11 C ⊆ T > A.  Observe that the above Lemma implies  B= A◦ = A∈T >∗ dB



B,

A∈T >B

that allow us to make use of the operator −∗ in the axioms. We omit the proof of the following lemma which is similar to the proof Lemma 7. Lemma 13. For all A, B ⊆ T and n > 1, dF (A, B) = n

iff

dF (A◦ , B) = n − 1.

Conditioning by Minimizing Accessibility

31

We have the following Lemma 14. For all A ⊆ T we have C ∗ ∈ T > C ∗ . Proof. Suppose C ∗  ∈ T > C ∗ so C ∗  ∈ T >∗d C ∗ by Lemma 12. Therefore ∗ ∗ C ∗T  ⊆ C , which implies that there exist x ∈ C ∗ and y ∈ T such that xRy  and y ∈ ¬C ∗ , which contradicts that dF (C ∗ , ¬C ∗ ) = ∞. We may now show the following theorem Theorem 15. Given a GCR space, then, for all A, B ⊆ T A>B

=

A >∗d B.

Proof. We shall show that for all C ⊆ T and A, B ⊆ T , C ∈A>B

iff

C ∈ A >d B,

by induction of the distance of C from A. – Let d(C, A) = 1. Suppose C ∈ A > B. So C ∈ T > A → B, so C ∈ T > A >∗d B by Lemma 12. Therefore C ∗ A ⊆ A → B, so C ∗ A ⊆ B and so C ∈ A >∗d B. For the other direction, suppose C ∈ A >d B. This implies that C ∈ T >d A → B by Axiom MP, so by Lemma 12 we have C ∈ T > A → B. We have that d(C, A) = 1 implies that C ∈ ¬(T >∗d ¬A) and again by Lemma 12 we get C ∈ ¬(T > ¬A). Applying Axiom MP we have that C ∈ A > B. – Suppose the hypothesis is true for d(C, A) ≤ n and let d(x, A) = n + 1. Observe that since d(C, A) > 1 then we have C ∈ T >d ¬A and, by Lemma 12, C ∈ T > ¬A. Suppose C ∈ A > B which is equivalent to C ◦ ∈ A > B by Axiom IND. By Lemma 13, we d(C ◦ , A) ≤ n and therefore we can apply the induction hypothesis to infer that C ◦ > A = C ◦ >∗d A. Again by applying Axiom IND we have C ∈ A >∗d B. – Finally, let d(C, A) = ∞. We have C ∈ A >∗d ∞ and we shall show C ∈ A > ∅. By Lemma 14, we have C ∗ ∈ T > C ∗ which implies C ∗ ∈ ¬C ∗ > ∅ using Axiom CN3. By Axiom CN2, we get C ∗ ∈ A > ∅, because A ⊆ ¬C ∗ using Axiom CN1. Applying Axiom CN2, we get C ∈ A > ∅. 

5

Conclusion

We introduced a class of set-theoretic conditional operators based on a global distance map of indistinguishability, and characterized this class by a finite set of axiom schemas. This result extends to a variety of properties we can assume for the indistinguishability relation, such as reflexivity and symmetry. These results belong to a larger study of the reasoning based on the minimization of a geodesic map. In particular, we have studied and characterized belief revision and update operators [9,8].

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K. Georgatos

Although our approach is part of the wider field of distance-based reasoning, its most salient feature is that it provides a bridge between qualitative and quantitative reasoning. The reason for this is twofold: first, quantitative information prone to error is not continuous. The threshold of error limits the range of values into integers which in turn can be embedded into a geodesic space. Second, limited observational powers introduce vagueness phenomena in human reasoning and we have argued [7] that such phenomena can be modeled satisfactorily using a binary relation of indistinguishability. The use of indistinguishability may not be applicable to all facets of formal epistemology but it seems particularly suitable when a metric space is already in place.

References 1. Aumann, R.: Agreeing to Disagree. Ann. Stat. 4, 1236–1239 (1976) 2. Bell, J.L.: A New Approach to Quantum Logic. Brit. J. Philos. Sci. 37, 83–99 (1986) 3. Boutilier, C.: Revision Sequences and Nested Conditionals. In: Bajcsy, R. (ed.) IJCAI 1993, pp. 519–525. Morgan Kaufmann, California (1993) 4. Dalal, M.: Investigations into a Theory of Knowledge Base Bevision: Preliminary Report. In: Rosenbloom, P., Szolovits, P. (eds.) AAAI 1988, vol. 2, pp. 475–479. AAAI Press, Menlo Park (1988) 5. Fagin, R., Halpern, J.Y., Vardi, M.Y.: A model-theoretic Analysis of Knowledge. J. ACM 38(2), 382–428 (1991) 6. G¨ ardenfors, P.: Knowledge in Flux. MIT Press, Cambridge (1985) 7. Georgatos, K.: On Indistinguishability and Prototypes. Log. J. IGPL 11(5), 531– 545 (2003) 8. Georgatos, K.: Belief Update Using Graphs. In: Wilson, D., Lane, H.C. (eds.) FLAIRS 2008, pp. 649–654. AAAI Press, Menlo Park (2008) 9. Georgatos, K.: Geodesic Revision. J. Log. Comput. 19, 447–459 (2008) 10. Giordano, L., Gliozzi, V., Olivetti, N.: Iterated Belief Revision and Conditional Logic. Stud. Log. 70(1), 23–47 (2002) 11. Giordano, L., Gliozzi, V., Olivetti, N.: Weak AGM Postulates and Strong Ramsey Test: A Logical Formalization. Artif. Intell. 168(1-2), 1–37 (2005) 12. Goldblatt, R.I.: Semantic Analysis of Orthologic. J. Philos. Log. 3, 19–35 (1974) 13. Goodman, N.: The Structure of Appearance, 3rd edn. Boston Studies in the Philosphy of Science, vol. 53. Reidel, Dordrecht (1977) 14. Grahne, G.: Updates and Counterfactuals. In: Allen, J.F., Fikes, R., Sandewall, E. (eds.) KR 1991, pp. 269–276. Morgan Kaufmann, San Francisco (1991) 15. Halpern, J.Y.: Set-theoretic Completeness for Epistemic and Conditional Logic. Ann. Math. Artif. Intell. 26(1-4), 1–27 (1999) 16. Herzig, A.: Logics for Belief Base Updating. In: Gabbay, D., Smets, P. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management, vol. 3, pp. 189–231. Kluwer Academic Publishers, Dordrecht (1998) 17. Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithaca (1962) 18. Hughes, G.E., Cresswell, M.J.: A Companion to Modal Logic. Methuen, London (1984) 19. Katsuno, H., Satoh, D.: A Unified View of Consequence Relation, Belief Revision and Conditional Logic. In: Mylopolous, J., Reiter, R. (eds.) IJCAI 1991, pp. 406– 412. Morgan Kaufmann, San Francisco (1991)

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20. Lewis, D.: Counterfactuals. Harvard University Press, Cambridge (1973) 21. Nieminen, J.: Rough tolerance equality. Fund. Inform. 11, 288–294 (1988) 22. Nute, D.: Conditional Logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 387–439. Reidel, Dordrecht (1984) 23. Pawlak, Z.: Rough Sets — Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991) 24. Poincar´e, H.: La Valeur de la Science. Flammarion, Paris (1905) 25. Salton, G.: Automatic Text Processing: The Transformation, Analysis, and Retrieval of Information by Computer. Addison-Wesley, Reading (1989) 26. Schlechta, K., Makinson, D.: Local and Global Metrics for the Semantics of Counterfactual Conditionals. J. Appl. Non-Class. Log. 4(2), 129–140 (1994) 27. Stalnaker, R.: A Theory of Conditionals. In: Rescher, N. (ed.) Studies in Logical Theory. American Philosophical Quarterly Monograph Series, vol. 52, pp. 98–112. Blackwell, Oxford (1968) 28. Zadeh, L.A.: Similarity Relations and Fuzzy Orderings. Inform. Sciences 3, 177–200 (1971) 29. Zeeman, E.C.: The Topology of the Brain and Visual Perception. In: Fort, M.K. (ed.) The Topology of 3-Manifolds, pp. 240–256. Prentice Hall, Englewood Cliffs (1962)

Correspondences in the Theory of Aggregation Davide Grossi Institute of Logic, Language and Computation, Universiteit van Amsterdam, Postbus 94242, 1090 GE Amsterdam, The Netherlands [email protected]

Abstract. The paper studies the interrelationships between the socialtheoretic problems of preference and judgment aggregation from the perspective of formal logic. The result of the paper is twofold. On the one hand, preference aggregation on total preorders is proven equivalent to the aggregation of specific types of judgments. On the other hand, judgment aggregation on propositional logic is proven equivalent to the aggregation of specific types of preferences. This shows how the aggregation of preferences and judgments can be viewed, in fact, as the two faces of a same coin.

1

Introduction

Preference aggregation (PA) and judgment aggregation (JA), are two related sub-disciplines of social choice theory [9] broadly concerned with the problem of collective decision-making. While PA, which reached maturity with the work of K. Arrow [1,2], studies the aggregation of a profile of agents’ preferences into one collective preference, the younger research field of JA (see [16] for an overview of the field) studies the aggregation of a profile of agents’ judgments concerning the acceptance/rejection of a set of logically interconnected issues. Both these aggregation problems have a common manifestation, “each individual thinking himself intelligent and yet all together feeling themselves barren”, to say it with [18, Ch. 99]. The problem is for the aggregation process to preserve, in a non-trivial way, some characteristic ‘rational’ aspects of the individual tobe-aggregated stances, e.g., transitivity in the case of preferences, and logical consistency in the case of judgments. Recent results [19,15,6] have shown how PA theorems, such as Arrow’s impossibility [1], can be obtained as corollaries of impossibility theorems concerning the aggregation of judgments. The idea behind this reduction consists in viewing preferences between issues as special kind of judgments, i.e., formulae to which a 

I am very grateful to Gabriella Pigozzi, Leon van der Torre, Paul Harrenstein, Franz Dietrich, Christian List and the reviewers of the postproceedings of LOFT 2008. Their comments have compelled me to think through a number of important details, helped me rectify several mistakes, and helped me improving the presentation. This study was made possible thanks to the Minist`ere de la Culture, de L’Enseignement Sup´erieur et de la Recherche, Grand-Duch´e de Luxembourg (BFR07/123) and the Nederlanse Organisatie voor Wetenschappelijk Onderzoek (VENI grant 639.021.816).

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 34–60, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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truth-value is attached. In [15,6] the formulae used for representing preferences are first-order formulae of the type xP y (“x is strictly preferred to y”) where x, y are variables for the elements in the set of issues of the PA problem. Then, in order for the judgments concerning such formulae to behave like a strict preference relation, the three axioms of asymmetry, transitivity and connectedness1 are added to the judgment aggregation framework. The first result of the present paper consists in a different approach to obtain the same kind of reduction. More precisely, preferences will be studied as implications y → x (“y is at most as preferred as x”) in a many-value logic setting [12,11]. Such reduction does not improve on [15,6]—except for considering the more general class of weak preferences instead of the strict ones—but it paves the way for the second result of the paper, which consists in an inverse reduction of judgments to special kind of preferences. To the best of our knowledge, this is the first work advancing a proposal on how to reduce JA on standard propositional logic to PA. All in all, the paper provides a study of two correspondences holding between PA and JA. The correspondence between the aggregation of weak preferences and the aggregation of implications in many-valued logics (Theorem 11), and the correspondence between the aggregation of propositional formulae and the aggregation of a special class of dichotomous preferences (Theorem 28). The paper is structured as follows. In Section 2 the frameworks of PA and JA are briefly exposed and some basic terminology is introduced. In Section 3 a correspondence is proven between the class of PA problems and a specific subclass of the JA problems. Arrow’s impossibility, as well as Sen’s impossibility of a Paretian liberal [24] are thereby imported from PA to JA. Then, along the lines of the aforementioned [15,6], an impossibility result similar to Arrow’s theorem is imported from JA to PA. Such transfer of results will highlight interesting discrepancies between the PA and the JA frameworks. The same program is pursued in Section 4, although in the different direction. There, a correspondence is established between the class of JA problems based on standard propositional logic, and a subclass of the PA problems. Also in this case impossibility results are transfered on the ground of such correspondence, and their relative strength is compared. Conclusions follow in Section 5. Proofs are provided in the Appendix.

2

Preliminaries

The present section is devoted to the introduction of the two frameworks of preference and judgment aggregation.2 2.1

Preference Aggregation

Preference Aggregation (PA) concerns the aggregation of the preferences of several agents into one collective preference. A preference relation  on a set of 1 2

Asymmetry: ∀x, y(xP y → ¬yP x); Transitivity: ∀x, y, z((xP y ∧ yP z) → xP z); Connectedness: ∀x, y(x  = y → (xP y ∨ yP x)) Our presentation is inspired by [17].

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issues IssP is a total preorder, i.e., a binary relation which is reflexive, transitive, and connected. The set Tp(IssP ) denotes the set of all total preorders of a set IssP . As usual, on the ground of  we can define its asymmetric and symmetric parts: x ≺ y iff (x, y) ∈ & (y, x)  ∈; x ≈ y iff (x, y) ∈ & (y, x) ∈. Sometimes we shall use PATp to make explicit that we are referring to PA on total preorders. The notion of PA structure can now be defined. Definition 1. (Preference aggregation structure) A PA structure is a quadruple SP = AgnP , IssP , PrfP , AggP where: AgnP is a finite set of agents such that 1 ≤ |AgnP |; IssP is a countable set of issues such that 3 ≤ |IssP |; PrfP is the set of all preference profiles, i.e., |AgnP |-tuples p = (i )i∈AgnP where each i is a total preorder over IssP ; AggP is a function taking each p ∈ PrfP to a total preorder over IssP , i.e., AggP : PrfP −→ Tp(IssP ). The value of AggP is denoted by . Typically, PA studies the aggregation function AggP under the assumption that it satisfies specific conditions. The most common of such conditions are the following ones: Unanimity (U). If all agents strictly prefer x over y, so does the aggregated preference: (∀x, y ∈ IssP )(∀p ∈ PrfP )[(∀i ∈ AgnP )[y ≺i x]] ⇒ [y ≺ x]]. Independence (I).3 The strict part of the aggregated preference over x and y depends only on the agents’ strict preferences over x and y: (∀x, y ∈ IssP )(∀p, p ∈ PrfP )[(∀i ∈ AgnP )[y ≺i x ⇔ y ≺i x]] ⇒ [y ≺ x ⇔ y ≺ x]]. Non-dictatorship (NoDict). There is no agent i such that for all profiles the aggregated preference relation always agrees with i’s strict preferences: (i ∈ AgnP )(∀x, y ∈ IssP )(∀p ∈ PrfP )[y ≺i x ⇒ y ≺ x]. The following less common condition will also be considered: Systematicity (Sys).4 If agents’ strict preferences over the pair (x, y) and the pair (x , y  ) are interdependent, so are the aggregated strict preferences: (∀x, y, x , y  ∈ IssP )(∀p, p ∈ PrfP )[[[(∀i ∈ AgnP )[y ≺i x ⇔ y  ≺i x ]] ⇒ [y ≺ x ⇔ y  ≺ x ]]. Notice that the definition of AggP in Definition 1 directly incorporates the aggregation conditions usually referred to as Universal domain and Collective rationality. In the rest of the paper the superscript P will be often omitted when this does not give rise to confusion. 2.2

Judgment Aggregation

Judgment aggregation (JA) concerns the aggregation of judgments about the acceptance or rejection of a set of interrelated issues (i.e., logical formulae) into 3 4

This condition is more commonly named IIA (Independence of Irrelevant Alternatives). This condition is equivalent to Independence plus the condition that all issues are treated in the same way (Neutrality) [25]. It is also known as Independence of NonWelfare Characteristics. In [10] it is called Strict Neutrality.

Correspondences in the Theory of Aggregation

37

one collective judgment. This section introduces a framework for JA built on the language of propositional logic. The way the issues to be judged upon are interrelated with one another depends on the logic underlying the aggregation problem. Logic. The logic underlying the aggregation problem defines the notion of “rationality” by which the individual and collective judgments should abide. To make the standard discursive dilemma5 example [22], which presupposes propositional logic, if p and q are accepted as true, so should p ∧ q and vice versa. In this paper we assume the semantics of the propositional language to be based on many-valued propositional valuation functions f on the [0, 1] interval. Let L be a propositional language built on the set of propositional atoms P. The meaning of {¬, ∧, ∨} is given by the standard algebraic operations of complementation, max, and min: f ∗ () = 1 f ∗ (¬φ) = 1 − f ∗ (φ) f ∗ (φ ∧ ψ) = min(f ∗ (φ), f ∗ (ψ)) f ∗ (φ ∨ ψ) = max(f ∗ (φ), f ∗ (ψ)) where f is a valuation function and f ∗ is its inductive extension.6 The designated value for the notion of satisfaction is 1: f |= φ iff f ∗ (φ) = 1. To put it in the JA jargon, issue φ is accepted iff it is attributed value 1. Similarly, if F is a set of valuation functions, by writing F |= φ we mean that for all f ∈ F , f ∗ (φ) = 1. Note that it does not hold in general that f  |= φ iff f |= ¬φ, or equivalently that = 1 iff f ∗ (φ) = 0, where f is a valuation function for the propositional f ∗ (φ)  language and f ∗ its inductive extension. This setting slightly generalizes the JA framework built on classical propositional logic [15].7 We can introduce now the central notion of JA, the one of judgment set. Judgment Sets. As anticipated above, the issues IssJ of a JA problem are propositional formulae. The set IssJ0 denotes the set of propositional atoms in Iss. Note that it is not necessarily the case that IssJ0 ⊆ IssJ . In other words, the issues might all be complex formulae. In what follows we shall often refrain to use the superscript J when possible. 5

6

7

The dilemma is often illustrated as follows. A three-member committee has to make a collective judgment on three interconnected formulae: p (“CO2 emissions are above the threshold”); p → q (“if CO2 emissions are above the threshold then there will be global warming”); q (“there will be global warming”). Propositionwise majority voting leads to an inconsistent collective judgment. The inductive extension of a valuation function f : P −→ [0, 1] is the function f ∗ : L −→ [0, 1] which assigns a value to all formulae in the language L on the ground of the value assigned by f to the atoms in P. Recall that, in propositional logic, a formula φ is assigned value 0 (i.e., it is false) iff φ is assigned a value different from 1 (i.e., it is not true).

38

D. Grossi

The position of an individual agent about the issues in Iss is represented by a judgment set. A judgment set accepts and rejects elements of Iss consistently with the underlying many-valued semantics introduced above. Definition 2 (Judgment sets). A judgment set J is a subset of Iss such that there exists a valuation function f : Iss0 −→ [0, 1] for which it holds that: (1) ∈ J implies f ∗ (φ) < 1. The ∀φ ∈ Iss: φ ∈ J implies f ∗ (φ) = 1; (2) ∀φ ∈ Iss: φ  set of all judgment sets for Iss is denoted J(Iss). In other words, a judgment set J states all and only the issues of Iss which are accepted, by keeping consistency with the underlying semantics. This means that, while choosing the accepted issues, it should be possible to give at the same time value 1 to all issues that J accepts and a value different from 1 to all issues that J does not accept. Notice that, from Definition 2, judgment sets are complete (any issue either belongs to J or not), non-contradictory (by item 1), and deductively closed (by item 2). The standard literature on JA also introduces a notion of agenda. Normally, an agenda is viewed as a set of acceptance-rejection elements φ and ¬φ which both belong to Iss. According to this view, the judgment themselves can be seen as formulae of the propositional language from which the issues are drawn. In our setting instead, we see judgments as meta-formulae, which do not belong to Iss, but which state whether a given issue in Iss is accepted (satisfied) or rejected (not satisfied), i.e., as properties or constraints of valuation functions (as it is evident from Definition 2). We do not introduce the notion of agenda here, since it would make the exposition of the results in the next sections slightly more involved. However, it will be of use later in Section 3.5, and we shall then introduce it formally in Definition 15. The following simple observations are relevant for our purposes. Given a set of issues Iss and a valuation f of Iss0 , the set TIss (f ) := {φ ∈ Iss | f |= φ} is clearly a judgment set. Now, two valuations can be said to be equivalent with respect to Iss when they give rise to the same judgment sets: f ∼Iss f  iff TIss (f ) = TIss (f  )

(1)

Let FIss0 be the set of all valuation functions under consideration. A set of issues Iss partitions FIss0 into equivalence classes: |f |Iss = {f  | f ∼Iss f  }. On the other hand, by Definition 2.2, a judgment set J always determines a set of valuations J for the set of issues Iss, which is so defined: J := {f | ∀φ ∈ Iss : f ∗ (φ) = 1 iff φ ∈ J}

(2)

The following semantic characterization of judgment sets follows from Definition 2, the definition of J in Formula 2, and of ∼Iss in Formula 1. Fact 3 (Semantic content of a judgment set). Let Iss be a set of issues, J ∈ J(Iss) and f ∈ FIss0 such that TIss (f ) = J. It holds that: J = |f |Iss

(3)

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39

In other words, each judgment set for a set of issues corresponds to an equivalence class in the partition of the valuation space yielded by that set of issues.8 On the ground of these considerations, in what follows we shall often use the semantically connoted “J |= φ”, instead of “φ ∈ J”. JA Structures. We can now precisely define the structure of the JA problem, mirroring Definition 1. Definition 4. (Judgment aggregation structure) A judgment aggregation structure is a quadruple SJ = AgnJ , IssJ , PrfJ , AggJ where: AgnJ is a finite set of agents such that 1 ≤ |AgnJ |; IssJ is a countable set of issues consisting of propositional formulae from a language L and containing at least two atoms, i.e., IssJ ⊆ L s.t. 2 ≤ |IssJ0 ∩ Iss|; PrfJ is the set of all judgment profiles, i.e., |AgnJ |-tuples j = (Ji )i∈AgnJ where each Ji is a judgment set for the set of issues IssJ ; AggJ is a function taking each j ∈ PrfJ to a judgment set for the set of issues IssJ ), i.e., AggJ : PrfJ −→ J(IssJ ). J denotes the value of AggJ . Just like PA, JA studies aggregation functions under specific conditions. The following conditions are the JA variants of the ones presented for PA in the previous section:9 Unanimity (U ). If all agents accept (or reject) x, then so does the aggregated judgment: (∀x ∈ IssJ )(∀j ∈ PrfJ ) [[[(∀i ∈ AgnJ )Ji |= x] ⇒ J |= x]] & [[(∀i ∈ AgnJ )Ji  |= x] ⇒ J  |= x]]. Independence (I ). The aggregated judgment on x depends only on the individual judgments on x: (∀x ∈ IssJ )(∀j, j ∈ PrfJ ) [(∀i ∈ AgnJ )[Ji |= x ⇔ J i |= x] ⇒ [J |= x ⇔ J |= x]]. Systematicity (Sys ). If the agents’ judgments on x are interdependent on the agents’ judgments on y, then so are the aggregated judgments: (∀x, y ∈ IssJ )(∀j, j ∈ PrfJ ) [[[(∀i ∈ AgnJ )[Ji |= x ⇔ J i |= y]] ⇒ [J |= x ⇔ J |= |= y]] ⇒ [J |= x ⇔ J  |= y]]]. y]] & [[(∀i ∈ AgnJ )[Ji |= x ⇔ J i   Non-dictatorship (NoDict ). There is no agent i such that the value of the aggregation function is always the ith -projection of its argument: (i ∈ AgnJ )(∀x ∈ IssJ )(∀j ∈ PrfJ )[Ji |= x ⇔ J |= x]. Notice that Definition 4 incorporates the aggregation conditions usually referred to as Universal domain and Collective rationality. 2.3

Setting the Stage

The investigations presented here build on three main considerations. 8

9

Notice that if Iss is closed under atoms, i.e. Iss0 ⊆ Iss, and we assume the standard semantics of propositional logic, then J corresponds exactly to one propositional valuation. To avoid confusion, the names of the JA conditions will contain  as a superscript.

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D. Grossi

1. The first one consists in the simple observation that preferences are actually defined by sets of statements—judgments—of the type: (x, y) ∈ and (x, y)  ∈. 2. The second one, which is a platitude in economic theory, is that (ordinal) preferences can be studied in terms of numerical functions, such as the functions on the [0, 1] interval [4]. 3. Finally, the third is borrowed from logic and consists in the recognition that numerical functions can be used as basis for logical semantics, like it happens in many-valued logic [12,11]. In such logics, as well as in propositional logic, the semantic clause u(y) ≤ u(x) typically defines the satisfaction by u of the implication x → y where u is a valuation: u |= x → y iff u(x) ≤ u(y).

(4)

Intuitively, implication x → y is true (or accepted, or satisfied) iff the rank of x is at most as high as the rank of y.10 These observations suggest that preferences can be viewed as special kind of judgments in many-valued logic and that, in turn, judgments in standard propositional logic, can be viewed as special sorts of preferences. The paper systematically explores this idea and the light it sheds on the theory of aggregation.

3

Preferences as Judgments

This section establishes a correspondence between PA structures as introduced in Definition 1 and a subclass of the JA structures introduced in Definition 4. We shall proceed as follows. First of all, in Section 3.1, the simple fact is noted that every total preorder specifies a set of order-isomorphic ranking functions. In Sections 3.2 and 3.3, it is shown that the judgment sets obtained by appropriately translating a total preorder, specify the very same set of ranking functions which is specified by the total preorder they translate. This leads us to the desired correspondence. Finally, in Section 3.5, PA impossibility results are imported to JA and vice versa, and their relative strength is compared. 3.1

Preferences and Ranking Functions

Let us first briefly recall the following well-known fact (see, for instance [4]), which follows from a simple argument based on the quotient yielded by a total preorder. Fact 5 (Representation of  by u). Let X be a countable set and ∈ Tp(X). There exists a ranking function u : X −→ [0, 1] such that ∀x, y ∈ X: x  y iff u(x) ≤ u(y). 10

(5)

Notice that we do not commit to any precise semantics for →, the only requirement on it being that u  |= x → y iff u∗ (x → y) < 1. See also Footnote 12.

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Such a function is unique up to ordinal transformations.11 This fact plays a central role in the present section. Notice that each ranking function u on a countable set X determines a linear order u(X), ≤ , where u(X) is the set of values of u for X. In other words, Fact 5 states that each total preorder  specifies a non-empty set of ranking functions all determining isomorphic linear orders. Given a ranking function u, let us denote |u|u(X),≤ the set of functions which are ≤-isomorphic to u for X. However, each total preorder also specifies a set of ranking functions in a natural way. Definition 6 (Semantic content of ). Let  be a total preorder on a countable set X. The semantic content of  is defined as follows: u() = {u | ∀x, y ∈ X, u(x) ≤ u(y) iff x  y}

(6)

where u : X −→ [0, 1]. In other words u() is nothing but the set of all ranking functions which embed  on the natural order of the set of their images. The following simple fact holds. Fact 7. Let ∈ Tp(X) with X countable, u() be the semantic content of , and u a ranking function on X preserving . It holds that: u() = |u|u(X),≤ Proof. It follows directly from Definition 6 and Fact 5.

(7)  

To sum up, any total preorder can be associated to a non-empty set of ranking functions which expresses exactly the same ordinal information. As a consequence, the set Tp(X) of all total preorders over X yields a partition of the set of all ranking functions u of X. 3.2

Condorcet’s Paradox as a JA Paradox

In Condorcet’s paradox, pairwise majority voting on issues generates a collective preference which is not transitive. From Fact 5 we know that any preference relation which is a total preorder can be represented by an appropriate ranking function u with codomain [0, 1]. Table 1 depicts the standard version of the paradox in relational notation, and the corresponding version which makes use of a ranking function u. The grey line displays the outcome obtained by pairwise majority. The basic intuition underlying this section consists in reading the right-hand part of Table 1 as if u was an interpretation function of propositions x, y, z on the real interval [0, 1], like Formula 4 suggests. It is then just a matter of closing the circle drawn by Formulae 4 and 5. Given a total preorder , there always 11

We recall that an ordinal transformation t is a function such that for all utilities m and n, t(m) ≤ t(n) iff m ≤ n.

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D. Grossi Table 1. Condorcet’s paradox {x, y} y≺x y≺x x≺y y≺x

{y, z} z≺y y≺z z≺y z≺y

{x, z} z≺x x≺z x≺z x≺z

{x, y} u(y) < u(x) u(y) < u(x) u(x) < u(y) u(y) < u(x)

{y, z} u(z) < u(y) u(y) < u(z) u(z) < u(y) u(z) < u(y)

{x, z} u(z) < u(x) u(x) < u(z) u(x) < u(z) u(x) < u(z)

Table 2. Condorcet’s paradox as a JA paradox {x, y} y≺x y≺x x≺y y≺x

{y, z} z≺y y≺z z≺y z≺y

{x, z} z≺x x≺z x≺z x≺z

{x, y} |= x → y |= x → y  |= y → x  |= x → y 

{y, z} |= y → z |= z → y  |= y → z  |= y → z 

{x, z} |= x → z |= z → x  |= z → x  |= z → x 

exists a ranking function u, unique up to order-preserving transformations, such that: xy

iff u(x) ≤ u(y) iff

u |= x → y.

(8)

We thus obtain a direct bridge between preferences and judgments via ranking functions. So, by exploiting Formula 8 we obtain the equivalent reformulation of Condorcet’s paradox depicted in Table 2. Notice that, since all preferences in the paradox are strict, by Formula 8, they all correspond to the falsity of the related implications: that y is strictly preferred to x (x ≺ y) means that it is not true that y gets a truth-value at most as high as x’s ( |= y → x). The type of JA paradox we obtain from Condorcet’s is not just a mathematical diversion, since it relates to the aggregation of judgments in the context of fuzzy classifications. As a matter of fact, ranking functions can be viewed as fuzzy interpretation functions, and fuzzy implications12 lie at the ground of the semantics of concept subsumption statements in fuzzy description logics [13]. In fuzzy logic, an implication denotes the relative strength of the truth-degrees of antecedent and consequent. The following example illustrates a fuzzy reading of a variant of Condorcet’s paradox. 12

Formula 4 sets a constraint for the semantics of implication which is satisfied by several fuzzy semantics for implication such as: ∗

f (x → y) =

⎧ ⎨ 1 if f ∗ (x) ≤ f ∗ (y) ⎩

0 if f ∗ (y) < f ∗ (x)

∗

f (x → y) =

⎧ ⎨ ⎩

1

if f ∗ (x) ≤ f ∗ (y)

f ∗ (y) if f ∗ (y) < f ∗ (x)

which are the standard strict implication and, respectively, the G¨ odel implication. We refer the reader to [12,11] for further details.

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43

Table 3. Weak Condorcet’s paradox as a JA paradox {x, y} y x y x x≺y y x

{y, z} z y y≺z z y z y

{x, z} z x x≺z x≺z x≺z

{x, y} |= y → x |= y → x |= y → x  |= y → x

{y, r} |= z → y |= z → y  |= z → y |= z → y

{x, r} |= z → x |= z → x  |= z → x  |= z → x 

Example 8 (Classifying Platypus). It is the end of the 18th century. A specimen of Platypus (Ornithorhynchus anatinus) has just been shipped from Australia to England. Three specialists have to provide a zoologic classification of the animal. However, the three zoologist cannot make up their mind about the species under which the animal should be classified. In fact, it looks part-bird, part-reptile, and part-mammal. An exact answer seems impossible, but they do want to provide some guidelines for the classification, and to solve the impasse they decide to vote by majority about accepting or rejecting the following satements: “it is a mammal at least as much as a reptile” (rpt → mml); “it is a reptile at least as much as a bird” (brd → rpt); and “it is a mammal at least as much as a bird” (brd → mml). Now if the first zoologist accepts all three statements, the second accepts only the first one, and the third accepts only the second statement, then they would conclude that if the animal looks like a bird then it does not look like a mammal, even though it logically should. We thus obtain Table 3, which is a variant of Table 2 with weak preferences.13 To sum up, by first reading the Condorcet’s paradox in terms of ranking functions (Fact 5), and then interpreting such functions from the point of view of logical semantics (Formula 8), we can show the equivalence between a concrete PA problem and a JA one. This finding is generalized in the next section. 3.3

PATp = JA→ [0,1]

What we are after is to show that any total preorder can be translated to a judgment set in such a way that the semantic content of the total preorder is preserved by its translation. This directly yields a translation of PA structures to JA structures. As illustrated in the previous section, preferences can be viewed as implications in many-valued semantics. Now, consider a countable set of propositional atoms P, and the set im(P) = {x → y | x, y ∈ P}. If IssJ = im(P), then it is a set of issues consisting of implications alone, or a set of implicative issues. The desired translation function can now be defined. 13

Notice that, in this case, the paradox works with underspecified preference relations since each row in Table 3 describes a set of total preorders. Notice also that this version of the paradox is sensitive to the way pairwise comparisons are framed. For instance, in the example it is asked for each pair whether the first element in the pair is weakly preferred to the second one or not. Reversing the order of the weak comparison might end up giving different results under pairwise majority voting.

44

D. Grossi Table 4. Correspondences between preferences and judgments

preferences

judgments

x y x≺y

 |= x → y iff J( )   iff J( ) |= y → x

x≈y

 |= x → y, J( )  |= y → x iff J( )

Definition 9 (Translating ). Let IssP be a countable set of issues. The P function J : Tp(IssP ) −→ 2im(Iss ) is defined as follows: J() := {x → y | (x, y) ∈} Informally, J sends total preorders to subsets of the set of implicative issues built out of the PA issues, i.e., where IssJ = im(IssP ).14 The point is now to show that J does actually better, sending total preorders exactly to judgment sets. As the following fact shows, it is so that the set of functions satisfying the constraints specified by J() consists exactly of the set of functions preserving . The next fact follows from Definition 9 and Fact 5. Fact 10. Let ∈ Tp(X) with X countable and u : X −→ [0, 1] preserving :  = |u|u(X),≤ J()

(9)

 = u() (by Fact 7) and As an immediate consequence we also obtain that J() hence that J turns out to be a bijection between Tp(IssP ) and J(im(IssP )). It is now possible to prove the correspondence result. Theorem 11 (Correspondence between Tp(IssP ) and J(Tp(IssP )). Let ∈ Tp(IssP ) with IssP countable. It holds that:  |= x → y. x  y iff J()

(10)

Leaving technicalities aside, Theorem 11 states that each total preorder can be translated to a judgment set which has exactly the same semantic content, that is to say, which orders the alternatives in IssP in exactly the same way. Table 4 spells out the judgments corresponding to the most common preference statements about a total preorder . Theorem 11, in addition to the fact that function J is bijective, guarantees that the set of all PA structures can be mapped into the set of all JA structures in such a way that each PA structure corresponds exactly to one many-valued 14

It might be instructive to notice that, as a consequence of Definition 9, given a set of PA issues, for each pairs of elements x, y, there correspond four possible implicative judgments, i.e.: |= ( |=)x → y and |= ( |=)y → x.

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JA structure on implicative issues. Given a PA structure SP , this can be easily done by constructing the corresponding JA structure J(SP ) as follows: we pose AgnJ := AgnP ; IssJ := im(IssP ); PrfJ := J(PrfP ) (where J over sets of profiles is defined in the natural way); and finally, AggJ (J(p)) := J(AggP (p)). Let us call such JA structures many-valued implicative JA structures, and let us denote the JA problem they formalize as JA→ [0,1] . We thus find a surjective map of PA into JA→ [0,1] . 3.4

JA and Many-Valued Logics

Before moving to the next section it is worth spending a few words about the role that many-valued logic has played in our analysis. It is not the first time many-valued logics are considered within JA. In particular, [21] studies impossibility results within Post’s many-valued logic [23] which, however, presupposes a different semantics from the one considered here. An identical semantics to ours (although on a finite set of truth-values) is assumed in [26]. Both these papers, however, are concerned with the study of a somewhat generalized setting for the arising of paradoxes of the type of the discursive dilemma, where besides ‘acceptance’ and ‘rejection’ further intermediate values are allowed. This is not the case in our paper, where a formula can only be accepted or rejected (there are no intermediate values!) since acceptance and rejection are assumed to correspond to the meta-level notions of satisfaction (|=) and falsification ( |=). It is important to stress this aspect, since it lies at the heart of the proposed correspondence. In a way, we use many-valued logic only as a way to simulate preferences—exploiting essentially Fact 5—as sets of implications at the object-level, while retaining the acceptance-rejection dichotomy of standard JA at the meta-level. This allows us to represent a preference statement (x, y) ∈ as  |= x → y where J()  is the translation of  (Theorem the logical statement J() 11). Such representation lies beyond the scope of both [21] and [26]. 3.5

Importing and Comparing Impossibilities

We have obtained a match between the standard PA structures and a specific subset of all JA structures. It becomes therefore possible to import impossibility results between the two frameworks. Results for PATp structures transfer directly → to JA→ [0,1] structures via J, and results for JA[0,1] structures, transfer directly to Tp −1 PA structures via J . Before we do that, it is worth observing the following. In the literature on JA, the import of impossibility results typically follows this pattern: an impossibility theorem is proven in JA, and then it is used to derive known PA theorems as corollaries via a suitable translation, such as the one based on first-order logic sketched in Section 1. In particular, this has been done in [19,15,6] for Arrow’s theorem, and in [7] for Sen’s impossibility of a Paretian liberal [24]. In this section we follow a similar pattern but, instead of trying to prove PA results as special

46

D. Grossi

cases of more general JA ones, we show how results in PA obtain a logically equivalent reformulation as JA results and vice versa. In other words, our aim is not to prove old results as special cases of new ones, thereby generalizing the formers, but rather to prove new equivalent reformulations of old results and thus to show that each impossibility theorem, as well as its proof, has both a PA and a JA side. In this section, we first do that by reformulating Arrow’s and Sen’s theorem in JA (Section 3.5) and then by reformulating a JA theorem in PA (Section 3.5). From PATp to JA→ [0,1] . To see how results from PA can be imported to manyvalued JA on implicative issues it suffices to notice that function J (Definition 9) yields corresponding JA versions of the PA aggregation conditions (see Section 2.1) in the natural way. We denote the translation of a condition by prefixing J to the name of the condition. For example, J(U) denotes the following condition: (∀x, y ∈ im(IssP ))(∀j ∈ J(PrfP )[[(∀i ∈ AgnP )[Ji  |= x → y]] ⇒ [J  |= x → y]].15 As an example of the results that can be imported we provide the JA→ [0,1] formulation of Arrow’s theorem. J Corollary 12 (Arrow in JA). For any JA→ [0,1] structure S , there exists no aggregation function which satisfies J(U), J(I) and J(NoDict).

Proof. Follows directly from Arrow’s theorem [1]on total preorders by Theorem 11 and Definition 9.   A preference aggregation theorem which acquires an interesting interpretation in the JA setting under our translation is the so-called impossibility of a Paretian liberal [24], whose JA version in first-order logic has already been studied in [7]. Here, we provide a translation of the theorem using our function J. Such theorem makes use of the following PA aggregation condition: Minimal liberalism (ML). There are at least two agents who always dictate the ordering of two pairs of issues: (∃i  = j ∈ AgnP )(∀x, y, w, z ∈ IssP )(∀p ∈ P Prf )[[y ≺i x ⇒ y ≺ x] & [z ≺j w ⇒ z ≺ w]]. Under our translation, the JA version of the theorem runs as follows. J Corollary 13 (Paretian liberal in JA). For any JA→ [0,1] structure S , there exists no aggregation function which satisfies J(U) and J(ML).

Proof. Follows directly from the theorem of the impossibility of a Paretian liberal [24] on total preorders by Theorem 11 and Definition 9.   That is to say, there is no way of aggregating the judgments of different agents preserving unanimity if there are at least two agents who have the authority to impose the acceptance or rejection of at least one implication each. The impossibility can be illustrated by expanding on on Example 8. 15

Recall Table 4.

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47

Example 14 (Classifying platypus: conflicts of expertise). The three zoologists decide to ask two eminent authorities—Prof. A and Prof. B—for help. Prof. A is a celebrated expert on fishes and reptiles and so, they think, his opinion should settle the question whether the animal is a reptile rather than a fish or vice versa. Similarly, Prof. B is an expert on mammals and reptiles and, they think, his opinion will also settle the question whether the animal is a mammal rather than a reptile or vice versa. Finally, what the two experts agree upon will also be taken as settled. So they let Prof. A and Prof. B cast their opinions:16 A  Prof. |= mml → brd,  |= brd → rpt,  |= mml → rpt  Prof. B  |= mml → brd,  |= rpt → brd,  |= rpt → mml The following judgment set Prof. AB would result:  Prof. AB  |= mml → brd,  |= brd → rpt,  |= rpt → mml which, however, no ranking function u can satisfy since u should be s.t. u(mml) < u(rpt) and u(rpt) < u(mml), which is impossible.17 Tp From JA→ . The transfer of impossibilities can take the other [0,1] to PA direction too. Building on results presented in [6], this section shows how to transfer a JA→ [0,1] impossibility into a PA setting. In [5,6], an approach to JA is developed which studies the minimal structural properties an agenda has to exhibit in order to give rise to Arrow-like impossibility results. In a nutshell, such structural perspective allows for the proof of impossibility results which are based only on how tightly the issues of the aggregation problem are interconnected, abstracting from the type of logic involved. In particular, [6] proves that two specific structural conditions on agendas are sufficient to yield Arrow’s theorem. In this section we briefly recapitulate those conditions and show that agendas on implicative issues in JA→ [0,1] enjoy them, thus proving a result very similar (but not identical!) to Corollary 12. Let us first formally introduce the notion of JA agenda in a many-valued setting.

Definition 15 (Agenda). The agenda for the set of issues Iss is defined as follows: ag(Iss) = {φ |φ ∈ Iss} ∪ {φ |φ ∈ Iss} where φ denotes that the issue gets value 1 (accepted) and φ denotes that the issue gets a value different from 1 (rejected). Intuitively, The fact that an agent accepts an issue φ is denoted by φ belonging to the agenda, and the fact that it can reject an issue φ is denoted by the fact that φ belongs to the agenda. Put it yet otherwise, an agenda is a set of properties of valuation functions, which is closed under negation: φ is the property enjoyed by those functions f s.t. f |= φ; and φ is the property enjoyed by those functions f s.t. f  |= φ. An agenda can have the following relevant structural properties. 16 17

We do not represent the whole judgment sets, but just their salient parts. This is a reformulation of Sen’s famous example about Lady Chatterly’s Lover [24].

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D. Grossi

Definition 16 (Structural properties of agendas). Let α, β, . . . denote the elements of an agenda ag(Iss), and let α := φ if α = φ and α := φ if α = φ for some φ ∈ Iss. Consider a subset X of an agenda ag(Iss). Such set is minimally inconsistent if it is inconsistent while each of its proper subsets is consistent.18 Now, an agenda ag(Iss) is minimally connected iff: 1. There exists X ⊆ ag(Iss) s.t. 2 < |X| and X is minimally inconsistent; 2. There exists X ⊆ ag(Iss) s.t. X is minimally inconsistent and for some Y ⊆ X of even size the set X\Y ∪ {α | α ∈ Y } is consistent.19 We say that an element α of an agenda is contingent if both α and α denote non-empty sets of valuation functions in the underlying semantics. Given two X elements α, β ∈ ag(Iss), we write α −→ β if there exists X ⊆ ag(Iss) s.t. {α, β} ∪ X is inconsistent, but X is consistent with both α and β. An agenda ag(Iss) is path-connected iff for any two contingent α, β there exist α1 , . . . , αn ∈ X

X

Xn−2

Xn−1

1 2 ag(Iss) (with α = α1 and β = αn ) s.t. α1 −−→ α2 −−→ . . . −−−→ αn−1 −−−→ αn . An agenda is said to be strongly connected iff it is both minimally connected and path-connected.20

It is worth providing some of the intuitions that back Definition 16. Item 1 of minimal connectedness guarantees that inconsistent sets of judgments can result by picking elements from consistent ones. Item 2 requires that judgment sets do not necessarily contain one of the consistent maximal subsets of each minimally inconsistent set. From the point of view of aggregation, they mean altogether that judgment sets can be aggregated by taking care that from each small set of conflicting issues (i.e., the minimally inconsistent sets) either at least one of their elements is removed, or an even number of them. Finally, pathconnectedness guarantees that each two elements in the agenda are linked via a path of minimally inconsistent sets in such a way that if the first element of the pair (i.e., α) is in the set (i.e., {α, β} ∪ X), then the second (i.e., β) is out. In other words, it holds for any two judgments α and β that, in the presence of some X, accepting α compels to accept β too. The following result for strongly connected agendas is proven in [6]. Proposition 17 (Structural Arrow-like impossibility). For any JA structure SJ with a strongly connected agenda, there exists no aggregation function which satisfies U , I and NoDict . It is so that agendas on implicative issues in a many-valued setting are strongly connected. Proposition 18 (Structural properties of JA→ [0,1] agendas). Implicative agendas are strongly connected. As a consequence, Proposition 17 applies directly to JA→ [0,1] . 18 19 20

The property of minimal inconsistency was introduced in [20]. This part of the condition is known to be equivalent to the requirement, introduced in [8], that the set Iss be a non-affine subset of {0, 1}Iss [6]. We borrowed the term path-connectedness from [6]. A better-known term for it is total-blockedness (see [16]).

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49

→ Corollary 19 (Arrow-like impossibility in JA→ [0,1] ). For any JA[0,1] J   structure S , there exists no aggregation function which satisfies U , I and NoDict .

Proof. Follows directly from Proposition 17 and Proposition 18.

 

Obviously, via J−1 , Corollary 19 can obtain a formulation in PA terms. However, how is such formulation related to Arrow’s theorem itself or, equivalently, how are Corollary 12 and Corollary 19 related? Are they two formulations of the same impossibility? Next section is devoted to answering these questions. Comparing Impossibilities. Both Corollary 12 and Corollary 19 are JA impossibilities, the former is a notational variant of Arrow’s theorem, and the latter looks quite similar, but are they equivalent? The answer to the question rests on the type of logical relationships holding between the translation via J of the PA aggregation conditions, and the JA aggregation conditions. We might think that J(U), J(I) and J(NoDict) are nothing but instances of the general conditions U , I and NoDict . However, this is not the case. Proposition 20 (Relative strength of aggregation conditions in JA→ [0,1] ). The following logical relations between aggregation conditions hold for JA structures on many-valued implicative issues: U implies J(U), I implies J(I), and J(NoDict) implies NoDict . All implications are strict. We can therefore conclude that Corollaries 12 and 19 concern, in fact, two differTp ent kinds of impossibility for JA→ [0,1] (and therefore for PA ). Corollary 12 shows that a strong version of non-dictatorship, i.e., J(NoDict) is incompatible with the conjunction of a weak version of unanimity, i.e., J(U), and a weak version of independence, i.e., J(I). Corollary 19 proves, instead, that a weak version of non-dictatorship, i.e., NoDict is incompatible with the conjunction of a strong version of unanimity, i.e., U , and a strong version of independence, i.e., I . The considerations above highlight—in our opinion—the fact that the essential difference between PA and JA rests on the difference of the aggregation conditions considered by the two frameworks, and not on the aggregation structures themselves as it might look at first. In other words, it does not matter much whether you aggregate judgment or preferences. What matters are rather the underlying logical structure, and the conditions you impose on the aggregation. In the next section we shall substantiate this very same conclusion, although by proceeding to a correspondence between standard JA and a specific class of PA problems.

4

Judgments as Preferences

The key insight developed in the previous section consisted in identifying ranking functions with propositional many-valued interpretation functions, in order to use propositional implications to encode preferences (Formula 8). In this section

50

D. Grossi

we proceed the other way around, identifying propositional binary valuation functions with special kind of ranking functions. Our argument here will follow the very same structure of Section 3. A few preliminary words are in order. We shall work with the version of the JA problem based on propositional logic. In this case, agendas are simply sets ag(Iss) = {φ |φ ∈ Iss} ∪ {¬φ |φ ∈ Iss}, and judgment sets are consistent (w.r.t. the standard binary semantics) and complete (w.r.t. the set Iss) subsets of the agenda. The JA problem for propositional logic was the first one to be investigated in the literature [15]. To avoid confusion, we shall denote such setting JAfull 0,1 where the subscript 0, 1 stands for the two admitted values, and the superscript full denotes that we are not limited to implications but we allow all Boolean connectives. 4.1

Boolean Preferences

What strikes at first as an essential difference between PA and JA is the fact that while PA concerns the aggregation of “atomic” issues, JA concerns the aggregation of compounded issues displaying logical form. Viewing the binary valuation functions of propositional logics as ranking functions with specific properties allows us to bridge this gap. Let 2 = {0, 1}, , , −, 0, 1 be the Boolean algebra on the support {1, 0}, where  and  are the min, respectively, max operations, − is the involution defined as −x = 1 − x, and 0 and 1 are the designated elements. The total preorders generated by a propositional interpretation function are called Boolean preferences. Definition 21. (Boolean preferences) A Boolean preference (BP) is a total preorder on a set of formulae Φ which can be mapped to the total preorder {0, 1}, ≤ by a function f ∗ : Φ −→ {1, 0} such that: i) f ∗ is a homomorphism from Φ to 2;21 ii) ∀x, y ∈ Φ, x  y iff f ∗ (x) ≤ f ∗ (y). The set of all Boolean preferences on a set Φ is denoted Bp(Φ). Intuitively, BPs are total preorders on sets Φ of formulae for which there exists an order-embedding map to the order in 2 consisting of the top and bottom elements, i.e., {0, 1}, ≤ , which preserves the standard meaning of propositional connectives according to 2. Observe that function f ∗ is, in fact, a binary ranking function. It follows that BPs are a special class of dichotomous preferences22 which exhibit a logical behavior. A few considerations are in order. Notice first of all that, within a BP, ≺-paths have maximum length 1. In fact, stating that y ≺ x is equivalent to assign value 1 to x and value 0 to y. Notice also that a total preorder containing x ≺ y, y ≺ x∧y cannot be a Boolean preference since there exists no function assigning 1 and 0 21 22

Technically, f ∗ is the restriction to Φ of the homomorphism from the smallest term algebra including Φ and 2. Dichotomous preferences are such that ∀x, y, z ∈ IssP there exists a bipartition (G, B) of IssP such that x ≺ y if and only if x ∈ B and y ∈ G [14].

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51

to x and y, which preserves  on ≤ and, at the same time, ∧ on min. In addition, notice that all BPs trivially contain the pair (⊥, ) and trivially lack the pair (, ⊥), exactly like all judgment sets trivially accept  and reject ⊥. In other words,  and ⊥ denote elements which, in all BPs, are contained in the set of all -maximal and, respectively, all -minimal elements. 4.2

Boolean Preferences and Propositional Valuations

We proceed here exactly like in Section 3.1, that is, we shall first notice, unsurprisingly, that each BP is representable by a propositional valuation function, and then that it is equivalent to a set of binary valuation functions which all agree on the interpretation of a set Φ of formulae. The set Φ0 denote the set of atoms occurring in Φ. To get started, notice that the following fact is a direct consequence of Definition 21. Fact 22 (Representation of BPs). For any ∈ Bp(Φ) with Φ countable, there exists a propositional valuation function f of the atoms occurring in Φ s.t. ∀x ∈ Φ:   x iff 1 ≤ f ∗ (x)

(11)

where f ∗ is the inductive extension of f . We now provide a definition of the semantic content of a Boolean preference mirroring Definition 6. Definition 23 (Semantic content of BPs). Let ∈ Bp(Φ) with Φ countable. The semantic content of  is defined as follows: u() := {f | ∀x ∈ Φ, 1 ≤ f ∗ (x) iff   x}

(12)

where f is a propositional valuation function of the atoms occurring in Φ and f ∗ its inductive extension. In other words, u() consists of all the valuation functions of the atoms in Φ0 which all represent the Boolean preference . As a consequence of Fact 22 and Definition 23, another characterization of u() can be provided. Fact 24. Let ∈ Bp(Φ) with Φ countable, and let f be a binary valuation of Φ0 representing : u() = |f |Φ .

(13)

We have thus established a connection between the semantic content of a Boolean preference over Φ, and the set of propositional valuations which are indistinguishable with respect to Φ. Like in Section 3, this simple fact enables the bridge upon which the results in the next two sections build, where judgment sets are shown to be equivalent to BPs.

52

D. Grossi Table 5. Discursive dilemma as a PA paradox p |= |= |=  |=

4.3

p→q |= |=  |= |=

q |= |=  |=  |= 

{p, }

p

p p≺

p

{q, p} p q q≺p p q p q

{q, }

q q≺ q≺ q≺

The Discursive Dilemma as a PA Paradox

In the discursive dilemma [22] propositionwise majority voting leads to the specification of an impossible valuation of the propositions at issue, as showed in the left-hand side of Table 5. To rephrase the dilemma in PA terms, Formula 8 comes in handy again. From propositional logic we know that each valuation f is such that, for any proposition x: f |= x iff f |=  → x

(14)

that is, a formula is true iff it is implied by truth. Keeping in mind that f |= x iff 1 ≤ f ∗ (x) we can pull Formulae 11 and 14 together and obtain an instance of Formula 8 for BPs: x

iff 1 ≤ f ∗ (x) iff

f |=  → x.

(15)

We thus obtain a direct bridge between propositional judgments and a specific type of preferences, by elaborating again upon the ranking function intuition suggested by Formula 4. By exploiting Formula 15 we obtain a neat reformulation of the Discursive dilemma as a weak form of Condorcet’s paradox, as depicted in the right-hand side of Table 5. Recall that, also in propositional logic, f |= x → y iff f ∗ (x) ≤ f ∗ (y). The aggregated preference violates the transitivity of . Like in the cases of Examples 8 and 14, the translation of a paradox from the one framework to the other opens up interesting insights. Example 25 (Classifying Platypus: choosing the expert). After noticing that it was impossible to reach an agreement between the two experts Prof. A and Prof. B (Example 14) the three zoologists decide to vote in order to decide which of the two experts to believe. They decide to proceed like this: each of them will cast an “approval” or “rejection” vote for both Prof. A and Prof. B but, at the same time, they will cast an “approval” or “rejection” vote also about whether to approve at least one of the two, since they want to make sure that if the pole ends up with the rejection of both professors then they should have collectively approved such outcome. The majority on each issue wins. However, they soon discover that they could find themselves in the situation depicted in Table 6, which is just a variant of the dilemma in Table 5. It might be instructive to notice that the faulty procedure devised by the three zoologist in the example is an instance of approval voting [3] on the domain of BPs.

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Table 6. Example 25 depicted A |= |= |=  |= 

4.4

B |= |=  |= |= 

A∨B |=  |= |= |=

{A, } A≺

A A≺ A≺

{B, } B≺ B≺

B B≺

{A ∨ B, } A∨B ≺

A∨B

A∨B

A∨B

Bp JAfull 0,1 = PA

In this section we shall show that any judgment set J for standard propositional logic on IssJ can be translated to a total preorder in Bp(IssJ ), in such a way that all and only the formulae accepted by J are ranked as  in its translation. J

Definition 26 (Translating Js). The function P : J(IssJ ) −→ 2Iss defined as follows:

×IssJ

is

P(J) := {(x, y) | ¬x ∈ J} ∪ {(x, y) | y ∈ J} Informally, P sends judgment sets in propositional logic to sets of pairs of issues drawn from IssJ . Such sets of pairs can be proven to be exactly all the Bp(IssJ ). We observe the following fact, recalling Definition 23. Fact 27. Let J be a judgment set for IssJ in JAfull 0,1 and f a valuation satisfying J: u(P(J)) = |f |IssJ

(16)

In other words, the values of function P correspond—semantically—to sets of propositional valuations which are all equivalent with respect to the judgment set inputted in the translation. As an immediate consequence we obtain that  and hence that P is a bijection between J(IssJ ) and Bp(IssJ ). u(P(J)) = J, We can thus prove the correspondence we are after. Theorem 28 (Correspondence between J(IssJ ) and P(J(IssJ ))). Let J be a judgment set for IssJ in JAfull 0,1 and := P(J): x  y iff J |= x → y.

(17)

Theorem 28 guarantees the analogue of Theorem 11 to hold between JAfull 0,1 and PABp : any judgment set in JAfull can be translated to a semantically equivalent 0,1 BP. It is a matter of trivial application of propositional logic to produce the analogue of Table 4. J As a consequence, every JAfull 0,1 structure S can be translated to an equivalent PA structure SP := P(SJ ) by just stating: AgnJ := AgnP ; IssP := IssJ ; PrfP := P(PrfJ ) (where P over sets of profiles is defined in the natural way23 ); Bp AggP (P(j)) := P(AggJ (j)). We thus find a surjective map from JAfull . 0,1 to PA 23

P

More precisely, P(PrfJ ) = Bp(IssJ )Agn .

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4.5

D. Grossi

Importing and Comparing Impossibilities

In the previous section we have proven a correspondence between JA in standard propositional logic, and PA restricted to the domain of Boolean preferences. We now proceed to import/export results between the two frameworks and to compare them along the line we followed in Section 3.5. Unlike in Section 3, we shall notice how the transfer of results highlights a match between the two aggregation problems also at the level of the aggregation conditions. Bp From JAfull . Function P naturally yields PA translations of the typ0,1 to PA ical JA aggregation conditions. Also in this case we denote the translation of a condition by prefixing it with P. For example, P(U ) denotes the following condition: (∀x ∈ IssP )(∀p ∈ P(PrfJ )) [[[(∀i ∈ AgnP ) i x] ⇒   x]] & [[(∀i ∈ AgnP )x ≺i ] ⇒ x ≺ ]]]. The following result, which corresponds to one of the first impossibility theorems proven for JAfull 0,1 [15], can now be directly imported to PABp .

Corollary 29. (Impossibility for Boolean Preferences under JA conditions) Let SP contain a set of issues IssP s.t. {p, q, p ∧ q} ⊆ IssP (where ∧ can be substituted by ∨ or →) and PrfP is the set of Boolean preference profiles on IssP . There exists no aggregation function which satisfies P(U ), P(Sys ) and P(NoDict ). Proof. It follows from List & Pettit’s theorem [15], Definition 26 and Theorem 28.   Bp From PABp to JAfull impossibility results to 0,1 . Can we also import PA full JA0,1 under the PA conditions of U, Sys and NoDict? The answer is affirmative, and we obtain the following result.

Proposition 30. (Impossibility for Boolean Preferences under Arrovian conditions) Let SP contain a set of issues IssP s.t. {p, q, p ∧ q} ⊆ IssP (where ∧ can be substituted by ∨ or →) and PrfP is the set of Boolean preference profiles on IssP . There exists no aggregation function which satisfies U, Sys and NoDict. Proposition 30 can be obviously reformulated in JA terms via P−1 . So, would such formulation perfectly overlap with the List & Pettit’s theorem [15] which has been used in the previous section to import a JA impossibility in PA? The next section answers this question. Comparing Impossibilities. Unlike in Section 3, the question can actually be answered affirmatively. In fact, it can be proven that Corollary 29 and Proposition 30 are really the same result, thus showing a match between PABp and JAfull 0,1 also at the level of their aggregation conditions. Proposition 31. (Relative strength of aggregation conditions in PABp ) The following relations hold under BPs: P(U ) is equivalent to U; P(Sys ) is equivalent to Sys; NoDict is equivalent to P(NoDict ).

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Moving away from technicalities, Proposition 31 guarantees that, when we restrict the PA framework to BPs, and the JA framework to plain propositional logic, then we are not only dealing with the same type of aggregation structures, but also with the very same type of aggregation conditions. This is particularly interesting, especially in the light of the discrepancies observed at the end of Section 3. The PA and JA problems fit perfectly when the underlying semantic framework consists of binary ranking functions. Instead, they can diverge when based on many-valued ranking functions. It is worth stressing, nevertheless, that such match essentially relies on the strong properties of BPs (in particular on the fact that they are dichotomous), and on the presence of the logical constants  and ⊥. Without constants it would be impossible, from a PA point of view, to be able to distinguish the preferences which rank all the elements as maximal, from the preferences which rank all elements as minimal.

5

Conclusions

By borrowing ideas from logical semantics, the paper has investigated how to embed classes of PA problems into classes of JA problems and vice versa. While an embedding of PA in JA had already been studied in the literature [6], the embedding in the opposite direction is a novelty. What is also new is the way followed to pursue such embeddings. While [6] does it axiomatically by imposing further constraints on a first-order logic agenda (i.e., the axioms of strict total orders), the present paper has done it semantically, by ranking the issues in IssP on the [0, 1] interval and then interpreting preferences as implications. Concretely, the paper has proven two correspondence results: the first (Theorem 11) between the standard framework of PA, i.e., PA on total preorders (PATp ) and the framework of JA on many-valued logics with implicative issues

PATp

1st correspondence

JA→ [0,1] JA

PA PABp

2nd correspondence

JAfull 0,1

Fig. 1. Correspondences between classes of PA and JA problems

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(JA→ [0,1] ); the second (Theorem 28) between the JA framework based on standard propositional logic (JAfull 0,1 ), and the PA framework restricted to the domain of Boolean preferences (PABp ). Such correspondences are depicted in Figure 1. The big arrow from PATp to JA→ [0,1] represents the translation J described in Section Bp 3, while the big arrow from JAfull represents the translation P intro0,1 to PA duced in Section 4. As made explicit in the picture the second correspondence concerns a small subset of total preorders on the PA side, and of sets of valuation functions—and hence judgment sets—on the JA side. The correspondence results have allowed the transfer of several impossibilities between the frameworks, and showed how each impossibility result can be given a PA as well as a JA formulation, thereby also allowing for a systematic comparison of the strength of some of the impossibilities considered by the literature in the two fields.

References 1. Arrow, K.: A Difficulty in the Concept of Social Welfare. J. Political Econ. 58(4), 328–346 (1950) 2. Arrow, K.: Social Choice and Individual Values, 2nd edn. John Wiley, New York (1963) 3. Brams, S.J., Fishburn, P.C.: Approval Voting. Am. Political Sci. Rev. 72(3), 831– 847 (1978) 4. Debreu, G.: Represetation of a Preference ordering by a Numerical Function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. (eds.) Decision Processes. John Wiley, Chichester (1954) 5. Dietrich, F.: A Generalised Model of Judgment Aggregation. Soc. Choice Welf. 28(4), 529–565 (2007) 6. Dietrich, F., List, C.: Arrow’s Theorem in Judgment Aggregation. Soc. Choice Welf. 29, 19–33 (2007) 7. Dietrich, F., List, C.: A Liberal Paradox for Judgment Aggregation. Soc. Choice Welf. 31, 59–78 (2008) 8. Dokow, E., Holzman, R.: Aggregation of Binary Evaluations, Working Paper, Technion Israel Institute of Technology (2005) 9. Gaertner, W.: A Primer in Social Choice Theory. Oxford University Press, Oxford (2006) 10. Geanakoplos, J.: Three Brief Proofs of Arrow’s Impossibility Theorem. Econ. Theory 26, 211–215 (2005) 11. Gottwald, S.: Many-valued Logics. In: Jacquette, D. (ed.) Philosophy of Logic. Handbook of the Philosophy of Sciences, vol. 5, pp. 675–722. North-Holland, Amsterdam (2007) 12. H¨ ahnle, R.: Advanced Many-valued Logics. In: Gabbay, D.M., Guenther, F. (eds.) Handbook of Philosophical Logic, 2nd edn., vol. 2, pp. 297–395. Kluwer, Dordrecht (2001) 13. H´ ajek, P.: Making Fuzzy Description Logics More General. Fuzzy Sets and Systems 154(1), 1–15 (2005) 14. Inada, K.: A Note on the Simple Majority Decision Rule. Econometrica 32(4), 525–531 (1964)

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15. List, C., Pettit, P.: Aggregating Sets of Judgments: Two Impossibility Results Compared. Synthese 140, 207–235 (2004) 16. List, C., Puppe, C.: Judgment Aggregation: A Survey. In: Anand, P., Pattanaik, P., Puppe, C. (eds.) Handbook of Rational and Social Choice. Oxford University Press, Oxford (2009) 17. Makinson, D.: Combinatorial Versus Decision-theoretic Components of Impossibility Theorems. Theory and Decision 40, 181–190 (1996) 18. Musil, R.: The Man Without Qualities. Picador (1997); original title: Der Mann ohne Eigenschaften (1943) 19. Nehring, K.: Arrow’s Theorem as a Corollary. Econ. Letters 80, 379–382 (2003) 20. Nehring, K., Puppe, C.: Strategy-proof Social Choice on Single-peaked Domains: Possibility, Impossibility and the Space Between. University of Califorina at Davis (working paper) 21. Pauly, M., van Hees, M.: Logical Lonstraints on Judgment Aggregation. J. Philos. Log. 35, 569–585 (2006) 22. Pettit, P.: Deliberative Democracy and the Discoursive Dilemma. Philos. Issues 11(1), 268–299 (2001) 23. Post, E.: Introduction to a General Theory of Propositions. Am. J. Math. 43, 163–185 (1921) 24. Sen, A.K.: The Impossibility of a Paretian Liberal. J. Political Econ. 78, 152–157 (1970) 25. Taylor, A.D.: Social Choice and the Mathematics of Manipulation. Cambridge University Press, Cambridge (2005) 26. van Hees, M.: The Limits of Epistemic Democracy. Soc. Choice Welf. 28, 649–666 (2007)

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Appendix Proof of Theorem 11. From Definition 6 we have that: for all x, y in X, it holds that ∀u ∈ u(), x  y iff u(x) ≤ u(y) and hence that if x  y then ∀u ∈ u(), u(x) ≤ u(y). From Fact 5 we also know that u()  = ∅, that is, ∃u ∈ u(), x  y iff u(x) ≤ u(y). It therefore follows that if ∀u ∈ u(), u(x) ≤ u(y) then x  y. We thus obtain that, for all x, y ∈ X, ∀u ∈ u(), u(x) ≤ u(y) iff x  y. Now,  = u(), from which the desired result by Fact 7 and Fact 10 we have that: J() follows.   Proof of Proposition 18 (Sketch). It is easy to prove that agendas on implicative issues are minimally connected. The argument for proving path-connectedness is more complicated. Clearly, all the irreflexive implications of an agenda on implicative issues are contingent. We reason by cases, and take α = x → y or x → y and β = x → y  or x → y  . Take [x = x , y = y  ]. We distinguish four ∅

→ x → y. ii) [α = x → y, β = x → y]. For sub-cases: i) [α = β = x → y]: x → y − X1

X2

X

3 z ∈ Iss − {x, y}, x → y −−→ x → z −−→ y → z −−→ x → y with X1 = {z → y}, X2 = {x → y}, X3 = {x → z}. iii) [α = x → y, β = x → y]. For z ∈ Iss− {x, y},

X

X

X

1 2 3 x → y −−→ x → z −−→ y → z −−→ x → y with X1 = {y → z}, X2 = {y → x}, X3 = {z → x}. iv) [α = β = x → y]: Similar to case 1.i. The application of this argument to the remaining cases completes the proof.  

Proof of Proposition 20. It suffices to spell out the definitions. [Unanimity] In many-valued implicative issues, U requires that if all agents accept x → y then the aggregated judgment accepts it too, and if they reject it then the aggregated judgment rejects it too. On the other hand J(U) only requires that if all agents reject x → y, i.e., if x is strictly preferred to y by all agents (see Table 4), then so does the aggregated judgment. [Independence] Assume I and suppose J(I) is false, that is: (∃x → y ∈ im(IssP ))(∃j, j ∈ PrfP ) [(∀i ∈ AgnP )[Ji  |= x → y ⇔ P       Ji  |= x → y] & [J  |= x → y & J |= x → y]]. Since (∀i ∈ Agn )[Ji  |= x → y ⇔ J i  |= x → y] strictly implies (∀i ∈ AgnP )[Ji |= x → y ⇔ J i |= x → y] we can conclude by I that J |= x → y ⇔ J |= x → y, thus obtaining a contradiction. [Non-dictatorship] While NoDict requires the aggregation not to be the projection function for some agent i, J(NoDict) requires something weaker, namely that the aggregation function is not such that if i rejects x → y, the aggregated judgment also always does.   Proof of Theorem 28. Since := P(J), by Definition 23 we obtain that ∀x, yIssJ if x  y, then ∀f ∈ u() f |= ¬x or f |= y. By Fact 22, u()  = ∅, hence if ∀f ∈ u() f |= ¬x or f |= y, then x  y. Finally, by Fact 27, we know  and therefore by propositional logic that ∀x, yIssJ , x  y iff that u() = J, J |= x → y.   Proposition 30 (Sketch). A proof can be obtained along the same lines of Arrow’s original proof [2]. Let us first introduce some terminology and adjust the notions of decisive and almost decisive voters to Boolean preferences. A set V ⊆ AgnP

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is almost decisive for issue x over issue y (in symbols, ADV (x, y)) iff (∀p = (i )i∈AgnP )[[∀i ∈ V, y ≺i x] & [∀i  ∈ V, x i y]] ⇒ y ≺ x], that is to say, if all the agents outside a group V weakly prefer y to x but the agents in V strictly prefer x over y, then the aggregated function agrees with V . A set V ⊆ AgnP is decisive for issue x over issue y (in symbols, DV (x, y)) iff (∀p = (i )i∈AgnP )[[∀i ∈ V, y ≺i x] ⇒ [y ≺ x]. Obviously, for any x, y ∈ IssP : DV (x, y) implies ADV (x, y). If V is a singleton, e.g. {i}, then we use the notation ADi (x, y) and Di (x, y). To prove the desired result we need the two following lemmas. Lemma 32 (Contagion property). Let SP contain a set of issues IssP s.t. {p, q, p ∧ q} ⊆ IssP (where ∧ can be substituted by ∨ or →) and PrfP is the set of Boolean preference profiles on IssP . If there exists an individual i ∈ AgnP such that ADi (x, y) for some pair (x, y) then, under the conditions U and Sys, Di (x, y) for any pair of issues, that is, i is a dictator. Proof. We shall prove the case of conjunction. The cases for the other connectives can be proven with similar arguments. For each of these pairs (x, y) we show that if ADi (x, y) then i is decisive for at least one of the remaining pairs. Let I denote AgnP − {i}. The following cases follow directly from the properties of Boolean preferences: ADi (, ⊥) ⇒ Di (, ⊥); ADi (, q) ⇒ Di (, p ∧ q) and similarly for p; ADi (p ∧ q, ⊥) ⇒ Di (q, ⊥) and similarly for p; ADi (p ∧ q, q) ⇒ Di (p, q) & D(, q) and similarly for p; ADi (p, q) ⇒ Di (p, p ∧ q) and vice versa. We sketch the proof for the remaining more interesting cases. Claim: ADi (p, ⊥) ⇒ Di (q, ⊥) & Di (, q). Assume ADi (p, ⊥). There are two possible profiles: (i) ⊥ ≺i p . . . ⊥ ≺i p ∧ q . . . p ≺I  . . . p ∧ q ≺I  . . . and

(ii) ⊥ ≺i p . . . p ∧ q ≺i  . . . p ≺I  . . . p ∧ q ≺I  . . .

We prove that i is decisive for both (q, ⊥) and (, q): ⊥ ≺i q iff ⊥ ≺ q. i) If ⊥ ≺i q, then by Sys we should conclude that ⊥ ≺ p∧q (since p is above ⊥ iff p∧q is above ⊥ in this profile) and, therefore, that ⊥ ≺ q; ii) If p ∧ q ≺i  then, by by U we conclude that p ∧ q ≺I  and hence that q ≺i , which proves our claim. With a similar argument we can prove that ADi (q, ⊥) ⇒ Di (p, ⊥) & Di (, p) and that ADi (, p ∧ q) ⇒ Di (, p) & Di (, q) & Di (p, q) & Di (q, p) & Di (p, p ∧ q) & Di (q, p ∧ q). It is now easy to see that by assuming the almost-decisiveness on a pair we can infer the decisiveness on all other pairs.   Lemma 33 (Existence of an almost decisive voter). Let SP contain a set of issues IssP s.t. {, ⊥, p, q, p ∧ q} ⊆ IssP (where ∧ can be substituted by ∨ or →) and PrfP is the set of Boolean preference profiles on IssP . If the aggregation function satisfies U, then there exists an agent i ∈ AgnP such that ADi (x, y). Proof. For condition U, there always exists for each pair of issues a set which is decisive for that pair, that is, AgnP . Let us proceed per absurdum assuming

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that there is no almost decisive agent. That means that for any pair of issues (x, y) there exists a set V such that, for any profile, ADV (x, y) and 1 < |V |. Let V be the smallest (possibly not unique) of such sets, and let J := V − {i} and K := AgnP − V . Let us suppose then ADV (p, ⊥). Consider now the following profile: ⊥ ≺i p ⊥ ≺i q ⊥ ≺i p ∧ q . . . ⊥ ≺J p q ≺J  p ∧ q ≺J  . . . p ≺K  ⊥ ≺K q p ∧ q ≺I  . . . By ADV (p, ⊥), it follows that ⊥ ≺ p. There are two options for q: i) if q ≺ , then J would decide q ≺ , which contradicts our hypothesis; ii) if ⊥ ≺ q, then i would decide ⊥ ≺i p ∧ q, again against the hypothesis. It is easy to see that the same argument holds also under the assumption of almost-decisiveness w.r.t. different pair of issues.   Now Proposition 30 follows directly from Lemmas 32 and 33.

 

Proof of Proposition 31. The proof exploits the availability of the ranking constants ⊥ and . As an example, we show the equivalence between P(U) and U. From left to right. Take any pair of issues (x, y). If (∀i ∈ AgnP )y ≺i x then, for the properties of BPs, y ≺i  and  i x. By P(U) it follows that y ≺  and   x and hence that y ≺ x. From right to left. Given any pair of issues (x, y) if (∀i ∈ AgnP ) i x and (∀i ∈ AgnP )y ≺i  then, by U, y ≺ x and, by the properties of BPs, y ≺  and   x. The same type of argument applies to all other conditions.  

Logic and Data Exchange: Which Solutions Are “Good” Solutions? André Hernich and Nicole Schweikardt Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt am Main, Postfach 11 19 32, 60054 Frankfurt am Main, Germany {hernich,schweika}@informatik.uni-frankfurt.de

Abstract. This paper gives an introduction into the area of data exchange, with a special emphasis on the question of which solutions can be considered “good” solutions. We shall concentrate on notions of “good” solutions for query answering, in particular, universal solutions, the core of the universal solutions, and CWA-solutions.

1

Introduction

Data exchange deals with the problem of translating data that is structured in an “old” format into data structured in a “new” format. The “old” and the “new” format are called source schema and target schema, respectively. The task in data exchange can be described as follows: given (i) a database instance over the source schema and (ii) a specification of the relationship between the source and the target, construct a solution, i.e., a database instance over the target schema that satisfies the given relationship. Preferably, in case that solutions exist at all, one would like to find particular solutions that reflect the given source data as accurately as possible. Such data exchange problems occur in many real-world applications of database systems; and systems for solving this task have been developed and implemented in the last few years [31,22]. From a logician’s point of view, the setting in data exchange can be stated as follows: The source schema and the target schema are finite relational vocabularies, the database instances over the source schema and the target schema are finite structures over these vocabularies, and the specification of the relationship between the source and the target is given by a set of formulas of a suitable logic. The present paper’s goal is to give an introduction into the area of data exchange, addressed to readers who have a background in logic, but who are not necessarily experts in database theory. A special emphasis is put on the question of which solutions can be considered “good” solutions. The paper is structured as follows: In Section 2 we give an introduction to the basic problem of data exchange, including the notions of schema mappings and solutions. In Section 3 we concentrate on the first two notions of “good” solutions: the universal solutions and the core. Section 4 gives a summary on how to compute solutions. Section 5 deals with the question of how to answer G. Bonanno, B. Löwe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 61–85, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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queries that are formulated with respect to the target schema, and points out certain deficiencies of some notions of semantics of queries. Section 6 presents the CWA-solutions as a notion of “good” solutions for which these deficiencies do not occur. Finally, Section 7 concludes the paper by pointing out further topics of research in the area of data exchange. We are aware that data exchange does not belong to the core topics of the LOFT conference. However, we believe that data exchange still is of reasonable interest for the LOFT community: it deals with fundamental questions in mathematical logic, it is concerned with deciding on the “right” semantics of certain logics, and in connection with peer data exchange [9,19,20] it deals with the interaction of multiple peers. Furthermore, game theoretic concepts help to gain a better understanding of a particular notion of “good” solutions for data exchange: in Section 6 we present a new, game theoretic characterization of the CWA-solutions. Of course, the present paper can only give a very brief introduction to some basic concepts and results in data exchange. As a starting point for gaining an in-depth knowledge of the area of data exchange we refer the interested reader to Kolaitis’ excellent survey [26], to Barceló’s very recent survey [7], and to the articles on data exchange published in the proceedings of the ACM Symposium on Principles of Database Systems (PODS) and the International Conference on Database Theory (ICDT).

2

Basic Notions

A schema is a finite relational vocabulary, i.e., a finite set of relation symbols where each relation symbol is associated with an arity. An instance I over a schema σ assigns to each relation symbol R ∈ σ a finite relation RI of the same arity as R. Thus, an instance over σ is a finite σ-structure. Given two schemas σ and τ (called source schema and target schema, respectively), and a specification of the relationship between the source and the target, the goal in data exchange is to transform an instance S (the source instance) over σ into an instance T (a target instance) over τ that satisfies the given specification. Let us illustrate this with an example. Example 1. Consider the scenario of the fusion of two airlines, say KLA and Air Flight (AF , for short), into a single airline. Suppose we have a source instance S over the source schema σ = {KLA, AF }, where KLAS and AF S , respectively, contain the flights of KLA and Air Flight, represented by triples (departure_city, arrival_city, flight_number). We want to create a target instance T over the target schema τ = {New } such that New T contains the flights of the new airline, represented by triples (departure_city, arrival_city, aircraft_type). Moreover, we require that the new airline offers (at least) the same direct connections as KLA, and that for each route of Air Flight, the cities remain reachable with at most one change of planes. 

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Throughout this article we shall assume that σ and τ are disjoint. We write S ∪T to denote the instance of schema σ ∪ τ where each R ∈ σ is assigned the relation RS , and each R ∈ τ is assigned the relation RT . Furthermore, we assume that all elements that occur in a tuple that belongs to a relation RI of some (source or target) instance I belong to a fixed infinite set Dom of potential data values. 2.1

Schema Mappings

Requirements such as those described in Example 1 are formalized by schema mappings: Definition 2. Let σ and τ be two disjoint schemas. (a) A schema mapping M from σ to τ is a non-empty subset of { (S, T ) : S is an instance over σ, and T is an instance over τ }. We call σ the source schema of M , and an instance over σ is called source instance for M . Analogously, we call τ the target schema of M , and an instance over τ is called target instance for M . (b) Given a source instance S for a schema mapping M , a solution for S under M is a target instance T for M such that (S, T ) ∈ M .  Note that schema mappings are not required to be mappings (i.e., functions) in the mathematical sense. According to Definition 2, a schema mapping can be an arbitrary relation between the class of instances over the source schema and the class of instances over the target schema. Thus, for a given source instance S there might exist either no solution, or exactly one solution, or more than one solution. Example 3. The requirements from Example 1 can be formalized by a schema mapping M from the source schema {KLA, AF } to the target schema {New } that consists of precisely all tuples (S, T ), where S is a source instance and T is a target instance for M such that (1) for every tuple (x1 , x2 , y) ∈ KLAS , there is an aircraft type z such that (x1 , x2 , z) ∈ New T , and (2) for every tuple (x1 , x2 , y) ∈ AF S , there either is an aircraft type z such that (x1 , x2 , z) belongs to New T , or there is an intermediate stop z1 and aircraft  types z  and z  such that (x1 , z1 , z  ) and (z1 , x2 , z  ) belong to New T . One of the goals in data exchange is: Given a schema mapping M and a source instance S, decide whether a solution for S under M exists — and if so, compute such a solution T . In particular, this leads to the following decision problem: Existence-of-Solutions(M ) Input: A source instance S for M . Question: Is there a solution for S under M ? Preferably, in case that solutions exist at all, one would like to find a solution that reflects the given data as accurately as possible.

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2.2

How to Specify Schema Mappings

For specifying schema mappings in a high-level, declarative way, it seems natural to consider formulas of a certain logic, e.g., first-order logic. Several formalisms for schema specification are conceivable, among them, for example: (1) Logical interpretations, e.g., first-order interpretations: Here, a tuple f := (ϕR )R∈τ of formulas of vocabulary σ associates with every source instance S the target instance f (S) where, for each relation symbol R ∈ τ , Rf (S) consists of all tuples a with S |= ϕR (a). The interpretation f thus specifies the schema mapping Mf := { (S, f (S)) : S is a source instance }. Note that such logical interpretations f allow for specifying only schema mappings where each source instance S has exactly one solution. For realworld applications, this is far too restrictive. Usually, one does not intend to fully specify the target, but only to describe certain properties that the target should have as, e.g., in the data exchange scenario described in Example 1. One therefore usually considers the following, more general way of specifying schema mappings. (2) A set Σ of formulas of a certain logic, e.g., first-order logic over vocabulary σ ∪ τ , specifies the schema mapping   S is a source instance, T a target instance, MΣ = (S, T ) : . and S ∪ T |= Σ Thus, the formulas in Σ can be viewed as a formal specification of the relationship between the source and the target. However, if the class of formulas allowed for choosing Σ is too powerful, the problem Existence-of-Solutions(MΣ ) may be undecidable. The following example exposes a particular first-order sentence ϕ such that Existence-of-Solutions(M{ϕ}) is undecidable.1 Note that the difficulty in proving the undecidability is that the schema mapping is assumed to be fixed. If the schema mapping is regarded as part of the input, undecidability is a simple corollary to Trakhtenbrot’s theorem. Example 4. Consider Post’s correspondence problem, which is well-known to be undecidable. The problem is defined as follows: Post’s correspondence problem Input: A number k ≥ 1 and a list L = (u1 , v1 ), . . . , (uk , vk ) of k pairs of non-empty words ui and vi over an alphabet {a1 , . . . , a }. Question: Are there a number m ≥ 1 and indices i1 , . . . , im ∈ {1, . . . , k} such that ui1 ui2 · · · uim = vi1 vi2 · · · vim ? 1

A different example (based on the halting problem for Turing machines rather than Post’s correspondence problem) is sketched in [26].

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The input to Post’s correspondence problem can be represented by an instance S over the source schema σ = {Ord, U, V }, where Ord is a binary relation symbol, U and V are ternary relation symbols, • Ord S consists of all pairs (p, q) of integers such that 1 ≤ p ≤ q ≤ max{k, , |u1|, . . . , |uk |, |v1 |, . . . , |vk |}, and • U S (resp. V S ) consists of all triples (i, p, λ) of integers where 1 ≤ i ≤ k, 1 ≤ p ≤ |ui | (resp., |vi |), and 1 ≤ λ ≤ , such that the p-th letter in ui (resp., in vi ) is aλ . We represent a solution m ≥ 1,

i1 , . . . , im ∈ {1, . . . , k},

w = ui1 ui2 · · · uim = vi1 vi2 · · · vim

to Post’s correspondence problem by an instance T over the target schema τ = {Ord  , P, WU , WV }, where Ord  and P are binary relation symbols, and WU and WV are ternary relation symbols, as follows: For n := |w|, T

• Ord  consists of all pairs (p, q) of integers such that 1 ≤ p ≤ q ≤ max{n, m, k, , |u1|, . . . , |uk |, |v1 |, . . . , |vk |}, • P T consists of all pairs (j, ij ) with 1 ≤ j ≤ m, • WU T consists of all triples (r, j, p) of integers such that 1 ≤ r ≤ n, 1 ≤ j ≤ m, 1 ≤ p ≤ |uij |, and r = |ui1 | + · · · + |uij−1 | + p (i.e., triple (r, j, p) corresponds to the information that the r-th position in w is the p-th position in the j-th segment of the decomposition of w into ui1 · · · uim ) • WV T consists of all triples (r, j, p) of integers such that 1 ≤ r ≤ n, 1 ≤ j ≤ m, 1 ≤ p ≤ |vij |, and r = |vi1 | + · · · + |vij−1 | + p (i.e., triple (r, j, p) corresponds to the information that the r-th position in w is the p-th position in the j-th segment of the decomposition of w into vi1 · · · vim ). Now it is an easy (but somewhat tedious) exercise to construct a first-order sentence ϕ of vocabulary σ ∪τ , such that for any source instance S which represents an input to Post’s correspondence problem, and for any target instance T , the following is true:2 S ∪ T |= ϕ if, and only if, T is isomorphic to an instance which represents a solution to Post’s correspondence problem. The undecidability of Post’s correspondence problem thus implies that the  problem Existence-of-Solutions(M{ϕ} ) is undecidable. As a language for specifying schema mappings one therefore usually does not allow full first-order logic, but restricts attention to certain fragments of first-order logic. Ideally, such a fragment should be expressive enough to specify schema mappings used in practical applications, and simple enough to allow for efficient algorithms for the Existence-of-Solutions problem. A systematic study of suitable schema specification languages has started just recently (cf., [10]). Up to date, most publications in the area of data exchange 2

Recall from the beginning of Section 2 that, by definition, all instances are finite.

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deal with schema mappings MΣ that are specified by a set Σ that consists of so-called dependencies of the following kind. Definition 5 (s-t-tgds, t-tgds, egds) (a)

(i) A source-to-target tuple generating dependency ( s-t-tgd, for short) is a formula δ of the form   ∀¯ x ∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) , where • x ¯, y¯, z¯ are finite sequences of pairwise distinct first-order variables, • ϕ(¯ x, y¯) is a conjunction of relational atoms of the form R(u1 , . . . , ur ) where R ∈ σ, r is the arity of R, and each ui is either a variable in x ¯, y¯ or an element in Dom, and • ψ(¯ x, z¯) is a conjunction of relational atoms of the form R(u1 , . . . , ur ) where R ∈ τ , r is the arity of R, and each ui is either a variable in x ¯, z¯ or an element in Dom. (ii) A target tuple generating dependency ( t-tgd, for short) is a formula δ of the same form as in (i), where both, ϕ(¯ x, y¯) and ψ(¯ x, z¯) are conjunctions of atoms over τ . (iii) An equality generating dependency ( egd, for short) is a formula of the form   ∀¯ x ϕ(¯ x) → xi = xj ,

where ϕ(¯ x) is a conjunction of atoms over τ , x¯ = x1 , . . . , xk is a finite sequence of first-order variables, and i, j ∈ {1, . . . , k}. (b) If δ is a s-t-tgd or a t-tgd, S is a source instance, and T a target instance, we say that S ∪T satisfies δ (in symbols: S ∪T |= δ), if for all interpretations of x¯, y¯ with elements a ¯, ¯b in Dom for which each of the conjuncts in ϕ(¯ a, ¯b) is true in S ∪ T , there exists an interpretation of z¯ with elements c¯ in Dom such that each of the conjuncts in ψ(¯ a, c¯) is true in S ∪ T . Similarly, if δ is an egd, we say that S ∪ T satisfies δ, if for all interpretations of x ¯ = x1 , . . . , xk with elements a1 , . . . , ak in Dom for which each of the conjuncts in ϕ(¯ a) is true in S ∪ T , it holds that ai = aj .  Definition 6. We write M = (σ, τ, Σst , Σt ) to denote that M is a schema mapping from σ to τ defined via M := MΣ , where Σ = Σst ∪ Σt and Σst (the so-called set of source-to-target dependencies) is a finite set of s-t-tgds, and Σt (the so-called set of target dependencies) is a finite set of t-tgds and egds.  Example 7. Let us consider the schema mapping M described in Example 3. Condition (1) can be formalized by the s-t-tgd   δ1 := ∀x1 ∀x2 ∀y KLA(x1 , x2 , y) → ∃z New (x1 , x2 , z) . Condition (2) can be formalized by the formula  ∀x1 ∀x2 ∀y AF (x1 , x2 , y) →   ∃z New (x1 , x2 , z) ∨ ∃z∃z  ∃z  (New (x1 , z, z  ) ∧ New (z, x2 , z  )) .

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Note, however, that this is not an s-t-tgd since, according to Definition 5, disjunctions are not allowed in s-t-tgds. In order to formalize (a variant3 of) condition (2) we therefore use the s-t-tgd δ2 :=    ∀x1 ∀x2 ∀y AF (x1 , x2 , y) → ∃z∃z  ∃z  New (x1 , z, z  ) ∧ New (z, x2 , z  ) and the t-tgd   δ3 := ∀x1 ∀x2 ∀y New (x1 , x2 , y) → ∃z New (x2 , x2 , z) . With Mairline we shall henceforth denote the schema mapping (σ, τ, Σst , Σt ) with σ = {KLA, AF }, τ = {New }, Σst = {δ1 , δ2 }, and Σt = {δ3 }.  2.3

Solutions

Recall that for a schema mapping M = (σ, τ, Σst , Σt ) and a source instance S, a solution for S under M is a target instance T such that S ∪ T |= Σst ∪ Σt . Example 8. Let us consider the schema mapping Mairline from Example 7 and the source instance Sairline with KLASairline := 

( AMS, TXL, KL1812 ), ( FRA, AMS, KL1772 ), ( FRA, AMS, KL1431 )

AF Sairline := 

and

( CDG, TXL, AF123 ), ( CDG, AMS, AF837 )

Then, each of the following target instances T1 and T2 is a solution for Sairline under Mairline : New T1 := 

( AMS, TXL, F100 ), ( FRA, AMS, A310 ), ( FRA, AMS, F100 ), ( CDG, TXL, F100 ), ( TXL, AMS, B707 ), ( TXL, TXL, stay-here ), ( AMS, AMS, stay-here )

New T2 := 

( AMS, TXL, F100 ), ( FRA, AMS, A310 ), ( CDG, AMS, B707 ), ( TXL, TXL, stay-here ), ( AMS, AMS, stay-here )

 In light of the fact that, for a given schema mapping M and a source instance S, either no solution or more than just one solution may exist, the question “Which solutions are particularly “good” solutions?” occurs in a natural way. One kind of particularly good solutions are the universal solutions introduced in the next section. 3

It should be mentioned that we use a little “hack” here: δ3 makes sure that a “selfloop” is added to every destination airport.

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Universal Solutions and the Core

Universal solutions were introduced by Fagin, Kolaitis, Miller, and Popa in [14] as a formalization of an intuitive notion of “most general solutions”. Before giving the formal definition of universal solutions, let us first note that target instances may contain two different kinds of values: first of all, there may be values that are present already in the given source instance. E.g., in Example 8, the values AMS, TXL, FRA, CDG are present in Sairline and in every solution T for Sairline under Mairline . Such values are called constants. But the solutions T1 and T2 in Example 8 also contain values that are not present in Sairline , e.g., the values for aircraft_type (in T1 and T2 , these are the values F100, A310, B707, and stay-here). The dependencies in Σst and Σt enforce that such values are present in every solution, but they do not specify the precise choice of these values. E.g., when replacing in T1 the tuple (AMS, TXL, F100) with the tuple (AMS, TXL, sunshine), one obtains another solution for Sairline under Mairline . It would therefore be nice to have “placeholders” at hand, i.e., particular values which tell us that we know that (a) some value has to be inserted here, and that (b) no constraint on the precise choice of this value is given. Such placeholders are called nulls. From now on, we therefore assume that the set Dom of potential values is the union of two infinite, disjoint sets Const (the set of constants) and Null (the set of nulls). We shall often use the symbols ⊥, ⊥ , ⊥1 , ⊥2 , . . . to denote nulls. Source instances will usually contain only constants, whereas target instances may contain both, constants and nulls. The dependencies fixed in Definition 5 are allowed to contain constants, but no nulls (i.e., in Definition 5 (a), Dom must be replaced by Const). Two target instances T and T  are called isomorphic if they are identical up to renaming of nulls. Let us now come back to the question “Which solutions are particularly “good” solutions?”. Intuitively, we are particularly interested in those solutions that carry no more and no less information than is required by S and M . In [14], the universal solutions were introduced as a kind of “most general solutions” in this sense. The following notion of homomorphism is crucial for the formal definition of universal solutions. Definition 9 (homomorphism). Let T and T  be two instances over τ . A homomorphism from T to T  is a mapping h : Dom → Dom which has the following two properties: (1) h is the identity on Const, i.e., h(c) = c for all c ∈ Const, and (2) for all R ∈ τ and all tuples (a1 , . . . , ar ) ∈ RT , we have  h(a1 ), . . . , h(ar ) ∈ RT .



Note that, by definition, a homomorphism has to be the identity on Const, but it may map a null onto either a null or a constant. Definition 10 (universal solution [14]). Let M be a schema mapping and let S be a source instance for M . A solution T for S under M is a universal

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solution if for every solution T  for S under M there exists a homomorphism h from T to T  .  Thus, a universal solution is a “most general solution” in the sense that it can be homomorphically mapped into every other solution. Example 11. Consider the schema mapping Mairline from Example 7 and the source instance Sairline from Example 8. Then, none of the solutions T1 and T2 from Example 8 is a universal solution. However, the solutions T3 and T4 given in Figure 1 are universal solutions. 

New T3 := 

( AMS, TXL, ⊥1 ), ( FRA, AMS, ⊥2 ), ( FRA, AMS, ⊥3 ), ( CDG, ⊥4 , ⊥5 ), ( ⊥4 , TXL, ⊥6 ), ( CDG, ⊥7 , ⊥8 ), ( ⊥7 , AMS, ⊥9 ), ( TXL, TXL, ⊥10 ), ( AMS, AMS, ⊥11 ), ( ⊥4 , ⊥4 , ⊥12 ), ( ⊥7 , ⊥7 , ⊥13 )

New T4 := 

( AMS, TXL, ⊥1 ), ( FRA, AMS, ⊥2 ), ( CDG, ⊥4 , ⊥5 ), ( ⊥4 , TXL, ⊥6 ), ( CDG, ⊥7 , ⊥8 ), ( ⊥7 , AMS, ⊥9 ), ( TXL, TXL, ⊥10 ), ( AMS, AMS, ⊥11 ), ( ⊥4 , ⊥4 , ⊥12 ), ( ⊥7 , ⊥7 , ⊥13 )

Fig. 1. Two universal solutions T3 and T4 for Sairline under Mairline

Note that there is no guarantee that universal solutions exist. There are examples of schema mappings and source instances for which solutions do exist, but universal solutions do not: Example 12. Let M = (σ, τ, Σst , Σt ) be the schema mapping  where σ = {F }, τ =  {E}, Σst = { ∀x1 ∀x2 F (x1 , x2 ) → E(x1 , x2 ) } and Σt = { ∀x∀y E(y, x) → ∃z E(x, z) }. Let S be the source instance with F S = {(a, b)}. Then, the target instance T with E T = {(a, b), (b, b)} is a solution for S under M . However, it is not difficult to see that there is no universal solution for S under M : Assume, for contradiction, that T  is a universal solution for S under  M . Viewing E T as the set of edges of a directed graph, let n be the size of the smallest cycle in this graph (a cycle must exist since T  is finite and M enforces that every node has out-degree at least one). Now let T  be a solution for S under M that consists of a cycle on n+1 nodes. Since T  is a universal solution, there must be a homomorphism h from T  to T  . Letting v1 , . . . , vn be the nodes on the cycle of length n in T  , the homomorphic images h(v1 ), . . . , h(vn ) of these nodes must form a cycle in T  . However, by definition, the shortest cycle in T  has length n+1. 

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It therefore makes sense to consider, apart from the problem Existence-of-Solutions(M ), also the following problem: Existence-of-Universal-Solutions(M ) Input: A source instance S for M . Question: Is there a universal solution for S under M ? In [16], Fagin, Kolaitis, and Popa introduced the notion of a unique “minimal” universal solution, the so-called core of the universal solutions. To give the formal definition of the core of the universal solutions, we need to fix the notion of a sub-instance. An instance C over τ is a sub-instance of an instance T (in symbols: C ⊆ T ) if RC ⊆ RT , for every R ∈ τ . Definition 13 ([23]). Let T be an instance over τ . An instance C ⊆ T is a core of T , if there is a homomorphism h from T to C, and there is no homomorphism = C.  from T to any C  with C  ⊆ C and C  It can be easily seen (cf., [23]) that every instance over τ has a core, and that the core is unique up to isomorphism. Furthermore, it has been shown in [16] that all universal solutions for S under M have the same core (up to isomorphism), and this core is a universal solution itself. Definition 14. If universal solutions for S under M exist, then CoreM (S) denotes the core of the universal solutions for S under M .  Note that universal solutions for S under M exist if, and only if, CoreM (S) exists.

4

How to Compute Solutions

An important question in the area of data exchange is: Given a schema mapping M and a source instance S for M , how can we compute a solution T for S under M , how can we compute a universal solution, and how can we compute CoreM (S)? Let us first note that the class of schema mappings fixed in Definition 6 still contains schema mappings for which the problems Existence-of-Solutions, respectively, Existence-of-Universal-Solutions are undecidable: Theorem 15 (Kolaitis, Panttaja, Tan [27]). There is a schema mapping  , Σt ) (consisting of one s-t-tgd, two t-tgds, and one egd) such M  = (σ  , τ  , Σst that Existence-of-Solutions(M  ) is undecidable.  Theorem 16 (Hernich, Schweikardt [24]). There is a schema mapping M   = (σ  , τ  , Σst , Σt ) (containing s-t-tgds and t-tgds, but no egds) such that Existence-of-Universal-Solutions(M  ) is undecidable. 

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The proof of Theorem 15 is by a reduction from the embedding problem for finite semigroups, which is known to be undecidable. The proof of Theorem 16 is by a reduction from the halting problem for Turing machines. Interestingly, M  and M  can be chosen in such a way that the problems Existence-of-Solutions(M  ) and Existence-of-Universal-Solutions(M  ) are trivial. It is not difficult to see that, if M has only source-to-target dependencies (i.e., the set Σt of target dependencies is empty), then every source instance has at least one universal solution, i.e., the problems Existence-of-Solutions(M ) and Existence-of-Universal-Solutions(M ) are trivial. Furthermore, a subclass of the schema mappings fixed in Definition 6 has been identified, for which the Existence-of-Solutions problem is decidable, and solutions can be computed efficiently: the so-called weakly acyclic schema mappings. Definition 17 (weakly acyclic schema mappings [14,12]) Let M = (σ, τ, Σst , Σt ) be a schema mapping. • A position over τ is a pair (R, i) such that R ∈ τ and i ∈ {1, . . . , r}, where r is the arity of R. Given a conjunction ϕ of relational atomic formulas, and a variable x, we say that x appears at position (R, i) in ϕ if ϕ contains a conjunct R(u1 , . . . , ur ) with ui = x. • The dependency graph of M is a directed graph, whose vertices are the positions  over τ , and whose  edges are given as follows: For every t-tgd ¯, and every position ∀¯ x ∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) in Σt , every variable x in x (R, i) at which x appears in ϕ, there is – a copying edge from (R, i) to every position at which x appears in ψ, and – an existential edge from (R, i) to every position at which some variable from z¯ appears in ψ. • M is called weakly acylic if no cycle in the dependency graph of M contains an existential edge.  Σt conExample 18. Consider a schema mapping M  1 = (σ, τ, Σst , Σt ), where  tains exactly one t-tgd δ1 of the form ∀x∀y E(x, y) → ∃z E(x, z) . Then, M1 is weakly acyclic (cf., the dependency graph of M1 , illustrated in Figure 2). On the other hand, if M2 = (σ, τ, Σst , Σt ) is such  that Σt contains exactly one t-tgd δ2 of the form ∀x∀y E(y, x) → ∃z E(x, z) , then M2 is not weakly acyclic (see Figure 2). 

(E, 1)

(E, 2)

(E, 1)

(E, 2)

Fig. 2. The dependency graphs of M1 (left) and M2 (right). Copying edges are grey, existential edges are black.

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The following theorem shows that for weakly acyclic schema mappings, the problems Existence-of-Solutions(M ) and Existence-of-Universal-Solutions(M ) coincide, and solutions can be computed in polynomial time. Theorem 19 (Fagin, Kolaitis, Miller, Popa [14]) Let M = (σ, τ, Σst , Σt ) be a weakly acyclic schema mapping. (a) For any source instance S, the following is true: There is a solution for S under M if and only if there is a universal solution for S under M . (b) There is a polynomial-time4 algorithm which, given a source instance S, tests whether a solution for S under M exists and, if so, produces a universal solution T for S.  The algorithm from Theorem 19(b) is based on the chase procedure. For describing the chase procedure, it is convenient to identify an instance I with the set atoms(I) of all atoms in I, i.e., atoms(I) consists of all statements R(¯ u) where R is a relation symbol in the underlying schema and u ¯ is a tuple of elements in Dom such that u¯ ∈ RI . Similarly, if A is a finite set of atoms, we write inst(A) to denote the instance I with atoms(I) = A. Now let M = (σ, τ, Σst , Σt ) be a weakly acyclic schema mapping. For a given source instance S, the chase procedure starts with the set A1 := atoms(S) and proceeds with steps i = 1, 2, 3, . . . as follows: If inst(Ai ) satisfies all dependencies in Σst ∪ Σt , then stop and output the target instance T with atoms(T ) = Ai \ atoms(S) (obviously, T then is a solution for S under M ). Otherwise, there must exist a s-t-tgd in Σst or a t-tgd in Σt or an egd in Σt that is not satisfied by inst(Ai ). We distinguish between two cases.   Case 1: Σt contains an egd of the form ∀¯ x ϕ(¯ x) → xi = xj for which there is a tuple a ¯ of elements in Dom such that each of the conjuncts in ϕ(¯ a) is satisfied, = aj . If ai and aj are both constants, then the chase procedure stops but ai and outputs a failure notice. Otherwise, if one of ai is a null, say ⊥, and the other one is either a constant or a null, say c, then Ai+1 is obtained from Ai by replacing every occurrence of ⊥ by c. This finishes step i, and the algorithm proceeds with step i + 1. Case 2: Σst ∪ Σt contains an s-t-tgd or a t-tgd of the form   ∀¯ x∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) , for which there are tuples a ¯ and ¯b of elements in Dom such that each of the ¯ conjuncts in ϕ(¯ a, b) belongs to Ai , but there is no tuple c¯ over Dom such that each of the conjuncts in ψ(¯ a, c¯) belongs to Ai . Then let c¯ be a tuple of pairwise distinct nulls that do not occur in Ai , and let Ai+1 be the union of Ai with all 4

Note that here the schema mapping is assumed to be fixed and the complexity of the algorithm is measured only in terms of the size of the source instance S. The complexity of the variant of the problem where the schema mapping M is part of the input has been studied in [27].

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conjuncts in ψ(¯ a, c¯). This finishes step i, and the algorithm proceeds with step i + 1. It is shown in [14] that weak acyclicity of M enforces that the chase procedure stops after a number of steps that is polynomial in the size of the source instance S. If the output of the procedure is a failure notice, then there exists no solution for S under M ; otherwise the output of the procedure is a universal solution for S under M . Solutions produced by the chase procedure are sometimes referred to as canonical universal solutions. For example, the universal solution T4 from Example 11 is a canonical universal solution. It is known that for weakly acyclic mappings also the core (i.e., the “smallest” universal solution) can be computed in polynomial time — the algorithm, however, is much more involved than the chase procedure. Theorem 20 (Gottlob, Nash [21]). For every weakly acyclic schema mapping M = (σ, τ, Σst , Σt ), there is a polynomial-time algorithm which, given a source instance S, tests whether a solution for S under M exists and, if so, it  produces CoreM (S). Recently, Deutsch, Nash, and Remmel [11] introduced a generalization of the notion of weakly acyclic schema mappings, the so-called stratified schema mappings, and showed that the chase procedure can be used to generalize Theorem 19 to the class of all stratified schema mappings. Note, however, that there also exist schema mappings which are not stratified, but for which solutions still can be computed efficiently: Example 21. Let M = (σ, τ, Σst , Σt ) be the schema mapping from Example 12. From Example 18 we know that M is not weakly acyclic. In fact, M is not stratified in the sense of [11]. However, it is not difficult to see that (a) solutions always exist (and can be computed easily), and (b) universal solutions exist for a source instance S if, and only if, the target instance T obtained from S by renaming F into E (i.e., E T := F S ) satisfies the t-tgd in Σt .  It remains an important future task to identify further classes of schema mappings for which solutions can be computed efficiently.

5

Query Answering in Data Exchange

Another important question in the area of data exchange is: What are the “right” semantics for answering queries that are formulated with respect to the target schema, if only information about the source instance and the schema mapping is available? More precisely: Given a schema mapping M , a source instance S for M , and a query q that is formulated with respect to the target schema, what does “answering q with respect to S and M ” mean? Let us illustrate this with an example.

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Example 22. Let Mairline and Sairline be the schema mapping and the source instance from Example 7 and 8, respectively. Let q be the query that asks for all tuples (c1 , c2 ) of cities that are connected by a direct flight of the new airline. This query can be formalized in first-order logic via q(x1 , x2 ) := ∃z New (x1 , x2 , z) . For a target instance T we write q(T ) to denote the set of tuples (c1 , c2 ) of cities for which the formula q(x1 , x2 ) is satisfied in T when interpreting the variables x1 , x2 with the constants c1 , c2 . We have already seen that several solutions for Sairline under Mairline exist — and evaluating the query q on different solutions may lead to different query answers. Considering, e.g., the solutions T1 and T2 from Example 8, the tuple (CDG,TXL) belongs to q(T1 ), but not to q(T2 ).  In general, if M , S, and q are given, and q is evaluated on two different solutions T1 and T2 for S under M , it is possible that two different sets of answers, q(T1 ) and q(T2 ), are produced. But what should then be the precise semantics of query answering in data exchange? Fagin, Kolaitis, Miller, and Popa [14] proposed to adopt the concept of certain answers that had already been successfully used in the area of incomplete databases (see, e.g., [1,25]). Precisely, the set of certain answers of q on S with respect to M is the set5

certainM (q, S) := { q(T ) : T is a solution for S under M }. I.e., the set of certain answers contains only those tuples that belong to q(T ), for every solution T for S under M . Example 23. It is not difficult to see that for the query q from Example 22, the set certainMairline (q, Sairline ) consists of the following tuples: (AMS,TXL), (FRA,AMS), (TXL,TXL), (AMS,AMS).  In [14], the authors identified a large class of queries, for which the set certainM (q, S) is determined by an arbitrary universal solution: the unions of conjunctive queries. These queries are disjunctions of formulas ∃¯ y ϕ(¯ x, y¯), where x ¯ and y¯ are (possibly empty) finite tuples of first-order variables, and ϕ(¯ x, y¯) is a conjunction of atoms of the form R(u1 , . . . , ur ), where R ∈ τ , r is the arity of R, and each ui is either one of the variables in x¯, y¯ or an element in Const. Theorem 24 (Fagin, Kolaitis, Miller, Popa [14]). If M = (σ, τ, Σst , Σt ) is a schema mapping, S is a source instance for M , q is a union of conjunctive queries with respect to the target schema τ , and T is an arbitrary universal solution for M under S, then certainM (q, S) = q(T )↓ , where q(T )↓ is the set of all “null-free” tuples in q(T ) (i.e., tuples c¯, where each component in c¯ is an element in Const).  5

If M is a set whose elements are sets, we write



M to denote the set



N∈M

N.

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Together with Theorem 19 (b) this, in particular, tells us that for every weakly acyclic schema mapping M and every query q that is a union of conjunctive queries, there is a polynomial-time algorithm which, given a source instance S, computes the certain answers certainM (q, S). In [16], the authors introduced a variant of the certain answers semantics, the certain universal answers, which are defined as follows:

certainM { q(T ) : T is a universal solution for S under M }. univ (q, S) := It is obvious that certainM (q, S) is a subset of certainM univ (q, S). From Theorem 24 it immediately follows that certainM (q, S) = certainM univ (q, S) if q is a union of conjunctive queries. Furthermore, it has been shown in [16] that CoreM (S) can be used to compute certainM univ (q, S) if q is an existential query (that may include negation). However, it has been noted (cf., e.g., [3,28]) that the certain answers semantics and the certain universal answers semantics give rise to some anomalies where the semantics behave in a counterintuitive way, i.e., queries produce results that intuitively do not seem to be accurate. Let us illustrate this with the following example. Example 25. Consider a variant of the airline setting Mairline from Example 7,  namely the setting Mairline = (σ, τ, Σst , Σt ) where σ and τ are chosen in the same way as in Mairline , Σst = {δ1 , δ2 }, and Σt = ∅, where   δ1 := ∀x1 ∀x2 ∀y KLA(x1 , x2 , y) → ∃z New (x1 , x2 , z) ,   δ2 := ∀x1 ∀x2 ∀y AF (x1 , x2 , y) → ∃z New (x1 , x2 , z) .  Intuitively, the schema mapping Mairline tells us that the new airline simply takes the union of all the connections of the two airlines KLA and Air Flight and replaces the information on flight numbers with information on aircraft types. Now consider the query    q(x) := ∃y1 ∃z1 New (x, y1 , z1 ) ∧ ∀y2 ∀z2 New (x, y2 , z2 ) → y2 = y1 ,

which asks for all cities x from which exactly one city can be reached by a direct flight of the new airline. Intuitively, one would expect that for the source instance Sairline from Example 8, the query q(x) should yield the results AMS and FRA.  The certain answers certainMairline (q, Sairline ) and the certain universal answers  Mairline certainuniv (q, Sairline ), however, yield as result the empty set. To see why, note that the two target instances T1 and T2 with 

New T1 := 

( AMS, TXL, ⊥1 ), ( FRA, AMS, ⊥2 ), ( CDG, TXL, ⊥3 ), ( CDG, AMS, ⊥4 )





New T2 := New T1 ∪ and



( AMS, ⊥5 , ⊥6 ), ( FRA, ⊥7 , ⊥8 )

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 (and ⊥i = ⊥j whenever i = j) are universal solutions for Sairline under Mairline , and that q(T2 ) = ∅. 

The same counterintuitive behaviour of the certain (universal) answers semantics occurs in so-called copying scenarios, i.e., schema mappings of the form M = (σ, τ, Σst , Σt ) where σ = {R1 , . . . , Rn }, τ = {R1 , . . . , Rn }, Σt = ∅, and Σst contains, x Ri (¯ x) →  for each relation symbol Ri in σ, a s-t-tgd of the form ∀¯  x) . Intuitively, such a schema mapping just changes the names of the relation Ri (¯ symbols. One would therefore expect that a query q that is formulated with respect to the target schema, can be rewritten into an equivalent query q˜ over the source schema by just replacing each occurrence of a relation symbol Ri by its old name Ri . Thus, one would expect that evaluating q on “good” solutions for S under M yields the same result as evaluating q˜ on S. However, for queries similar ˜(S) to query q from Example 25, one obtains that certainM univ (q, S) = ∅ although q is non-empty. In [28], Libkin observed that the reason for this counterintuitive behaviour is the fact that the presence of “incomplete” information (i.e., nulls) in target instances is ignored, and the certain answers semantics and the certain universal answers semantics are defined with respect to sets of solutions treating nulls in the same way as constants.

6

CWA-Solutions

At the end of Section 5 we pointed out that the certain answers semantics and the certain universal answers semantics give rise to some anomalies where the semantics behave in a counterintuitive way, i.e., queries produce results that intuitively do not seem to be accurate. To resolve these anomalies, let us again consider the question “Which solutions are particularly “good” solutions?”. Intuitively, “good” solutions are those solutions that carry no more and no less information than required by S and M . — And for defining the semantics of queries, only such solutions should be taken into account. In [28], Libkin formulated the following three requirements that such solutions should meet, and he used the name CWA-solutions to address such solutions:6 (1) The presence of each atom in a CWA-solution is justified by the source instance and the dependencies in the schema mapping. (2) Justifications should not be overused. I.e., each justification for introducing atoms does not generate more atoms than necessary. (3) Each fact that is true in a CWA-solution follows from the source instance and the dependencies in the schema mapping.7 6 7

Similarly as in Section 4, we identify an instance I with the set atoms(I) of atoms in I. I.e., CWA-solutions are not allowed to contain “invented” facts that cannot be inferred by the schema mapping and the source instance.

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The name “CWA-solution” is chosen due to the fact that CWA-solutions are based on the closed world assumption (cf., [32]) in the sense that all facts which are true in a CWA-solution must be justified by the source instance and the dependencies in the schema mapping. For schema mappings M = (σ, τ, Σst , Σt ) where Σt = ∅, the requirements (1)–(3) were formalized by Libkin in [28]. A formalization for the general case (where Σt is not necessarily empty) was given by Hernich and Schweikardt in [24]. This formalization is explained in the next two subsections; for simplicity in presentation, however, we shall restrict attention to the special case where Σt contains no egds (details on the general case where Σt may contain egds as well as t-tgds can be found in [24]). 6.1

Formalization of Requirements (1) and (2)

The first two requirements can be formalized in terms of a game. Let M = (σ, τ, Σst , Σt ) be a schema mapping, let S be a source instance for M , and let T be a target instance for M . For the description of the game we also need the set  ¯, ¯b) : δ is a s-t-tgd in Σst , or a t-tgd in Σt , of the form JM := (δ, a   ∀¯ x∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) , and a ¯ and ¯b are interpretations of x¯ and y¯ with elements in Dom }.   An element (δ, a ¯, ¯b) ∈ JM , where δ = ∀¯ x∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) , can be used to justify the presence (in T ) of each of the conjuncts in ψ(¯ a, c¯) (for some interpretation c¯ for the variables in z¯) with (i) the presence (in S ∪ T ) of each of the conjuncts in ϕ(¯ a, ¯b), and (ii) the fact that the presence of the conjuncts in ψ(¯ a, c¯) is a possible consequence of the presence of the conjuncts in ϕ(¯ a, ¯b) with respect to δ. The game, denoted by G(M, S, T ), is played by two players, the verifier and the falsifier. The verifier’s goal is to show that T satisfies the requirements (1) and (2) with respect to S and M , whereas the falsifier’s goal is to show the converse. The game has at most |atoms(T )| + 1 rounds. In round 0, the verifier fixes the two sets A1 := atoms(T ) and J1 := JM and picks a linear order ≺ on A1 . Intuitively, A1 is the set of atoms that need to be justified, and J1 is the set of potential justifications that may be used for this purpose. The linear order ≺ determines that for justifying an atom R(¯ u), only atoms that either belong to S or are smaller than R(¯ u) (w.r.t. ≺) can be used. For each i ≥ 1, round i then proceeds as follows: First, the falsifier picks an atom R(¯ u) in Ai (or loses if Ai is empty). Then, the verifier has to “justify” R(¯ u) by picking a tuple (δ, ¯ a, ¯b) ∈ Ji , where δ has the form ∀¯ x∀¯ y ϕ(¯ x, y¯) →  ∃¯ z ψ(¯ x, z¯) , and an interpretation c¯ of z¯ with elements in Dom such that: • S ∪ T |= ϕ(¯ a, ¯b), T |= ψ(¯ a, c¯), and R(¯ u) is a conjunct in ψ(¯ a, c¯), and u ) in ϕ(¯ a, ¯b), we either have R (¯ u ) ∈ atoms(S), or • for each conjunct R (¯ R (¯ u ) ≺ R(¯ u).

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If this is not possible, the verifier loses. Otherwise, the game proceeds with round i + 1, where Ai+1 consists of all atoms in Ai that do not occur as a conjunct in ψ(¯ a, c¯), and Ji+1 := Ji \ {(δ, a ¯, ¯b)}. Note that by removing all conjuncts in ψ(¯ a, c¯) from Ai , we ensure that these conjuncts do not need to be justified again (they are already justified by (δ, a ¯, ¯b)). ¯ By removing (δ, a ¯, b) from the set Ji , we ensure that every justification is used at most once (this corresponds to requirement (2)). A target instance T for M is called CWA-presolution for S under M if T is a solution for S under M and the verifier has a winning strategy in the game G(M, S, T ). Intuitively, the CWA-presolutions are precisely the solutions that meet the requirements (1) and (2). Example 26. Consider the schema mapping Mairline from Example 7 and the source instance Sairline from Example 8. Then, the universal solution T4 from Example 11 is a CWA-presolution for Sairline under Mairline . Indeed, it is not difficult to see that the verifier can win the game G(Mairline , Sairline , T4 ) by picking, in round 0 a linear order ≺ on atoms(T4 ) such that the atoms New (⊥4 , ⊥4 , ⊥12 ) and New (⊥7 , ⊥7 , ⊥13 ) are the two largest atoms with respect to ≺. On the other hand, the solutions T5 and T6 with New T5 := New T4 ∪ {(LHR, GVA, ⊥14 ), (GVA, GVA, ⊥15 )}, New T6 := New T4 ∪ {(AMS, TXL, ⊥1 )} are no CWA-presolutions for Sairline under Mairline . For example, to win the game G(Mairline , Sairline , T5 ), the falsifier simply picks, in round 1, the atom New (LHR, GVA, ⊥14 ), which the verifier cannot justify. In order to win the game G(Mairline , Sairline , T6 ), the falsifier can choose New (AMS, TXL, ⊥1 ) in round 1,  and New (AMS, TXL, ⊥1 ) in round 2. A different, but equivalent, definition of CWA-presolutions relies on the αchase procedure [24]. For simplicity, we here only describe the special case of schema mappings without egds (see [24] for details on how to handle egds). Let α be a mapping that maps ¯, ¯b) ∈ JM , where δ has the form  every tuple (δ, a ∀¯ x∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) , to an interpretation c¯ of z¯ with elements in Dom. The α-chase procedure is defined in the same way as the chase procedure described in Section 4, except for the following: Suppose that in step i, there is a  x∀¯ y ϕ(¯ x, y¯) → ∃¯ z ψ(¯ x, z¯) for which s-t-tgd or a t-tgd δ in Σst ∪ Σt of the form ∀¯ there are tuples a ¯ and ¯b of elements in Dom such that each of the conjuncts a, c¯), where in ϕ(¯ a, ¯b) belongs to Ai , and at least one of the conjuncts in ψ(¯ c¯ := α(δ, a ¯, ¯b), does not belong to Ai . Then, Ai+1 is the union of Ai with all conjuncts in ψ(¯ a, c¯), and the procedure continues with step i + 1. The procedure stops only if in some step there is no such s-t-tgd or t-tgd. It is not difficult to verify that for every solution T for S under M , the following is true: There is a mapping α such that the α-chase procedure outputs T if, and only if, the verifier has a winning strategy in the game G(M, S, T ).

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79

CWA-Solutions

To formalize requirement (3) and thus complete the definition of CWA-solutions, we need the following notion: A fact (over a schema τ ) is a Boolean conjunctive query over τ , i.e., a first-order sentence of the form ∃¯ x ϕ(¯ x), where x ¯ is a finite sequence of first-order variables and ϕ(¯ x) is a conjunction of atoms of the form R(u1 , . . . , ur ) where R ∈ τ , r is the arity of R, and each ui is either one of the variables in x¯ or a constant (i.e., an element in Const). An example of a fact Mairline in Example 7 is  over the target schema of the schema mapping  ∃x New (FRA, AMS, x) ∧ New (AMS, TXL, x) . It expresses that the same aircraft type can be used to fly from FRA to AMS and from AMS to TXL. Definition 27 (CWA-solution [28,24]). Let M = (σ, τ, Σst , Σt ) be a schema mapping, and let S be a source instance for M . A CWA-solution for S under M is a CWA-presolution T for M such that each fact that is true in T , is true in every solution for S under M .  Example 28. Consider, once again, the schema mapping Mairline from Example 7 and the source instance Sairline from Example 8. Then, both, T3 and T4 in Example 11 are CWA-solutions for Sairline under Mairline . On the other hand, the solutions T1 and T2 for Sairline from Example 8, and the solutions T5 and T6 from Example 26 are no CWA-solutions, since they are not even CWA-presolutions for Sairline under Mairline . Finally, let T7 be the solution for Sairline that is obtained from T4 by replacing the value ⊥2 with ⊥1 . Then T7 is a CWA-presolution. However, it is no CWAsolution for Sairline under Mairline  , since, for example, the fact ∃x New (FRA,  AMS, x) ∧ New (AMS, TXL, x) is true in T7 , but it is not true in T4 . The next theorem draws the connection between CWA-solutions and the universal solutions from Section 3. Theorem 29 (Libkin [28], Hernich and Schweikardt [24]) Let M = (σ, τ, Σst , Σt ) be a schema mapping, and let S be a source instance for M . Then the following is true: (a) For every target instance T for M , the following are equivalent: • T is a CWA-solution for S under M . • T is a universal solution for S under M , and T is a CWA-presolution for S under M . (b) If CoreM (S) exists, then CoreM (S) is a CWA-solution for S under M .  In particular, Theorem 29 implies that if CoreM (S) exists, then it is the unique “minimal” CWA-solution for S under M , in the sense that every CWA-solution T for S under M has a sub-instance that is isomorphic to CoreM (S). If M contains no target dependencies, then there also is a unique CWA-solution Tmax , which is “maximal” in the sense that every CWA-solution T for S under M is a homomorphic image of Tmax [28]. However, [24] gives an example of a schema mapping M with Σt = ∅, where there is, for each n ∈ N, a source instance Sn of

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size O(n) for which (at least) 2n “maximal” CWA-solutions T1 , T2 , . . . , T2n exist, each of which is (up to the renaming of nulls) no homomorphic image of another CWA-solution. See Figure 3 for an illustration of the space of CWA-solutions. Tmax

T2 T1

T2n

T

T ⊆ Core(S)

⊆

Core(S)

Fig. 3. A representation of the space of CWA-solutions for the case that M has no target dependencies (left), and for the case that M has target dependencies (right). The arcs indicate homomorphisms.

6.3

How to Compute CWA-Solutions

Given a schema mapping M and a source instance S for M , how can we decide whether a CWA-solution for S under M exists — and if so, how can we compute a CWA-solution? First of all, note that from Theorem 29 we know that for every schema mapping M = (σ, τ, Σst , Σt ) and every source instance S for M , the following are equivalent: (i) there is a CWA-solution for S under M , (ii) there is a universal solution for S under M , (iii) CoreM (S) exists. Theorem 16 thus implies that there is a schema mapping M  = (σ, τ, Σst , Σt ) such that the problem Existence-of-CWA-Solutions(M  ) Input: A source instance S for M  . Question: Is there a CWA-solution for S under M  ? is undecidable. On the other hand, if M is a weakly acyclic schema mapping, then, by Theorem 19, there is a polynomial-time algorithm which, given a source instance S for M , tests whether a universal solution for S under M exists, and if so, computes a universal solution for S. Since a CWA-solution for S exists if and only if a universal solution for S exists, the same algorithm can be applied to test whether there is a CWA-solution for S. However, it is easy to construct weakly acyclic schema mappings for which the algorithm’s result is in general no CWA-solution. Still, if CWA-solutions for S under M exist, we can compute a CWA-solution for S under M , namely CoreM (S), using the polynomial-time algorithm from Theorem 20. Similarly as with Section 4, it is an important future task to identify broader classes of schema mappings for which CWA-solutions can be computed efficiently.

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81

Query Answering Using CWA-Solutions

Let us now turn back to the task of query answering. In the following, we shall introduce the CWA-solution-based query answering semantics that have been proposed by Libkin [28] in order to get rid of the anomalies pointed out at the end of Section 5. Let M = (σ, τ, Σst , Σt ) be a schema mapping, let S be a source instance for M , and let q be a query over τ . To obtain the result of q under any of the semantics from [28], one takes the intersection or the union of the answers to q on CWA-solutions for S. To give the precise definition, we first explain how q is answered on a single CWA-solution for S. Notice that a CWA-solution T for S under M may contain nulls, which are placeholders for existent, but unknown (constant) values (see Section 3). That is, T represents a number of solutions Tˆ for S without nulls, and these solutions can be obtained from T by replacing every null with a constant in Dom. Let RepM (T ) denote the set of all solutions Tˆ for S under M that are possibly represented by T , i.e., all solutions Tˆ for S under M which can be obtained from T by replacing every null with an element in Const. The certain answers to q on T are then defined as follows (cf., e.g., [1,25]):

 q(Tˆ ) : Tˆ ∈ RepM (T ) . M q(T ) := This results in the certain CWA answers semantics and the potential certain CWA answers semantics, respectively, which are defined as

 certainM M q(T ) : T is a CWA-solution for S under M ,  (q, S) :=  certainM M q(T ) : T is a CWA-solution for S under M . ♦ (q, S) := I.e., the set of certain CWA answers certainM  (q, S) contains only those tuples that appear as certain answers for all CWA-solutions, whereas the set of potential certain CWA answers certainM ♦ (q, S) contains those tuples that appear as certain answers for at least one CWA-solution. Example 30. Recall the source instance Sairline from Example 8 and the schema  mapping Mairline and the solution T1 from Example 25. It is not difficult to see  that T1 is a CWA-solution. In fact, there are only two different CWA-solutions,   namely T1 and the solution T3 with New T3 := New T1 ∪ {(FRA, AMS, ⊥5 )}. Therefore, for the query q from Example 25, the certain CWA answers of the M form certain airline (q, Sairline ) and the potential certain CWA answers of the form M

certain♦ airline (q, Sairline ) yield the results AMS and FRA, as intuitively expected.   Furthermore, note that for any copying scenario M (cf., the end of Section 5), each source instance S has a unique CWA-solution, namely its “copy” TS which is obtained from S by renaming each relation Ri into Ri (i.e., Ri TS := RiS ). Therefore, for each query q that is formulated with respect to the target schema,

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M one obtains that certainM ˜(S), as one intuitively ex (q, S) = certain♦ (q, S) = q pects (here, q˜ is the query obtained from q by replacing each occurrence of Ri with Ri ). The following theorem summarizes some of the basic properties of the new CWA-solution-based query semantics.

Theorem 31 (Libkin [28], Hernich and Schweikardt [24]) Let M = (σ, τ, Σst , Σt ) be a schema mapping, and let S be a source instance for M such that S has at least one CWA-solution under M . Then the following is true for each query q over τ : (a) certainM ♦ (q, S) = M q(CoreM (S)). (b) If M has no target dependencies, then certainM  (q, S) = M q(Tmax ), where Tmax is the unique “maximal” CWA-solution for S under M mentioned at the end of Section 6.2. (c) If q is preserved under homomorphisms (e.g., a union of conjunctive queM M ries), then certainM   (q, S) = certain♦ (q, S) = certain (q, S). In particular, for a weakly acyclic schema mapping M and a query q that is a union of conjunctive queries, Theorem 31 (c), Theorem 24, and Theorem 19 (b) tell us that there is a polynomial-time algorithm which, given a source instance M S for M , computes certainM  (q, S) (resp., certain♦ (q, S)). However, moving from unions of conjunctive queries to more expressive query languages, the data complexity8 quickly increases to co-NP-hard. Table 1 summarizes what is known about the data complexity (with respect to certain and certain♦ ) of first-order queries (FO), unions of conjunctive queries (UCQ), and unions of conjunctive queries with at most one inequality per disjunct. Here, richly acyclic schema mappings [24] are particular weakly acyclic mappings; full tgds are s-t-tgds or t-tgds in which no existential quantifier occurs, i.e., they are of the form ∀¯ x∀¯ y (ϕ(¯ x, y¯) → ψ(¯ x)).

7

Further Topics

In this paper we gave a brief introduction into the area of data exchange, with a special emphasis on the question “Which solutions are “good” solutions?”. Of course, there is a long list of topics that have not been discussed in this paper. First of all, we should note that it is still not clear which solutions can be considered to be “the best” solutions. Even for the CWA-solution-based query semantics it is possible to construct examples where the semantics behave in a counterintuitive way. For this reason, Libkin and Sirangelo [29] recently proposed a notion of solutions that is a mixture between the solutions from Section 2 and the CWA-solutions from Section 6. Moreover, Afrati and Kolaitis [2] argued that, for answering so-called aggregate queries, one needs a rather strong restriction of CWA-solutions. 8

Here, the term data complexity refers to the fact that M and q are assumed to be fixed, and the complexity is measured only in terms of the size of the source instance.

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M Table 1. Data complexity of computing certainM  (q, S) and certain♦ (q, S) for certain restrictions of M and q [24]

query language

restriction of schema mapping

UCQ

UCQ with at most one inequality per disjunct

weakly acyclic

co-NP-hard

richly acyclic only s-t-tgds, egds

FO

co-NP-complete PTIME

only full tgds, egds

A lot of work in data exchange has been done concerning Bernstein’s metadata management framework [8], in particular, on the composition of schema mappings [17,29] and the inverse of a schema mapping [13,18,6]. Moreover, [15] studied the problem of schema mapping optimization, which is closely related to metadata management. The complexity of query answering for various query languages has been studied, e.g., in [14,16,30,27,4]. Closely related to the query answering problem is the query rewriting problem [3], dealing with the following goal: Given a query q, find a query q  , called rewriting of q, such that the result of q  over a materialized solution T is equivalent to the result of q with respect to some particular query answering semantics, e.g., the certain answers semantics or the certain CWA answers semantics. In the present paper, we restricted attention on the problem of exchanging relational data from a single source to a single target. Of course, several variations are conceivable. Arenas and Libkin [5], for example, considered the problem of XML data exchange. And the exchange of data with multiple sources and targets is the subject of peer data exchange [9,19,20]. Let us conclude by pointing out two directions for future research. Certainly, it remains an important future task to identify broader classes of schema mappings, for which “good” solutions can be computed efficiently. Furthermore, as described at the beginning of Section 7, the question “Which solutions are “good” solutions?” has still not been fully answered. In the end, it seems that, for different types of schema mappings and different types of queries that one wants to support, different notions of “good” solutions need to be developed.

References 1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading (1995) 2. Afrati, F.N., Kolaitis, P.G.: Answering Aggregate Queries in Data Exchange. In: Lenzerini, M., Lembo, D. (eds.) PODS 2008, pp. 129–138. ACM, New York (2008)

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3. Arenas, M., Barceló, P., Fagin, R., Libkin, L.: Locally Consistent Transformations and Query Answering in Data Exchange. In: Deutsch, A. (ed.) PODS 2004, pp. 229–240. ACM, New York (2004) 4. Arenas, M., Barceló, P., Reutter, J.: Query Languages for Data Exchange: Beyond Unions of Conjunctive Queries. In: Fagin, R. (ed.) ICDT 2009, pp. 73–83. ACM, New York (2009) 5. Arenas, M., Libkin, L.: XML Data Exchange: Consistency and Query Answering. In: Li, C. (ed.) PODS 2005, pp. 13–24. ACM, New York (2005) 6. Arenas, M., Pérez, J., Riveros, R.: The Recovery of a Schema Mapping: Bringing Exchanged Data Back. In: Lenzerini, M., Lembo, D. (eds.) PODS 2008, pp. 13–22. ACM, New York (2008) 7. Barceló, P.: Logical Foundations of Relational Data Exchange. SIGMOD Record 38(1), 49–58 (2009) 8. Bernstein, P.A.: Applying Model Management to classical meta-data problems. In: CIDR 2003, pp. 209–220 (2003) 9. Bertossi, L.E., Bravo, L.: Query answering in peer-to-peer data exchange systems. In: Lindner, W., Mesiti, M., Türker, C., Tzitzikas, Y., Vakali, A.I. (eds.) EDBT 2004. LNCS, vol. 3268, pp. 476–485. Springer, Heidelberg (2004) 10. ten Cate, B., Kolaitis, P.G.: Structural Characterizations of Schema-mapping Languages. In: Fagin, R. (ed.) ICDT 2009, pp. 63–72. ACM, New York (2009) 11. Deutsch, A., Nash, A., Remmel, J.B.: The Chase Revisited. In: Lenzerini, M., Lembo, D. (eds.) PODS 2008, pp. 149–158. ACM, New York (2008) 12. Deutsch, A., Tannen, V.: Reformulation of XML queries and constraints. In: Calvanese, D., Lenzerini, M., Motwani, R. (eds.) ICDT 2003. LNCS, vol. 2572, pp. 225–241. Springer, Heidelberg (2002) 13. Fagin, R.: Inverting Schema Mappings. In: Vansummeren, S. (ed.) PODS 2006, pp. 50–59. ACM, New York (2006) 14. Fagin, R., Kolaitis, P.G., Miller, R.J., Popa, L.: Data Exchange: Semantics and Query Answering. Theor. Comput. Sci. 336(1), 89–124 (2005) 15. Fagin, R., Kolaitis, P.G., Nash, A., Popa, L.: Towards a Theory of Schema-mapping Optimization. In: Lenzerini, M., Lembo, D. (eds.) PODS 2008, pp. 33–42. ACM, New York (2008) 16. Fagin, R., Kolaitis, P.G., Popa, L.: Data exchange: Getting to the core. ACM Trans. on Database Syst. 30(1), 174–210 (2005) 17. Fagin, R., Kolaitis, P.G., Popa, L., Tan, W.C.: Composing schema mappings: Second-order dependencies to the rescue. In: Deutsch, A. (ed.) PODS 2004, pp. 83–94. ACM, New York (2004) 18. Fagin, R., Kolaitis, P.G., Popa, L., Tan, W.C.: Quasi-inverses of schema mappings. In: Libkin, L. (ed.) PODS 2007, pp. 123–132. ACM, New York (2007) 19. Fuxman, A., Kolaitis, P.G., Miller, R.J., Tan, W.C.: Peer Data Exchange. ACM Trans. Database Syst. 31(4), 1454–1498 (2006) 20. Giacomo, G.D., Lembo, D., Lenzerini, M., Rosati, R.: On Reconciling Data Exchange, Data Integration, and Peer Data Management. In: Libkin, L. (ed.) PODS 2007, pp. 133–142. ACM, New York (2007) 21. Gottlob, G., Nash, A.: Efficient Core Computation in Data Exchange. J. ACM 55(2), 1–49 (2008) 22. Haas, L.M., Hernández, M., Ho, H., Popa, L., Roth, M.: Clio Grows up: From Research Prototype to Industrial Tool. In: Özcan, F. (ed.) SIGMOD 2005, pp. 805–810. ACM, New York (2005) 23. Hell, P., Nesetril, J.: The Core of a Graph. Discrete Math. 109(1-3), 117–126 (1992)

Logic and Data Exchange: Which Solutions Are “Good” Solutions?

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24. Hernich, A., Schweikardt, N.: CWA-solutions for Data Exchange Settings with Target Dependencies. In: Libkin, L. (ed.) PODS 2007, pp. 113–122. ACM, New York (2007) 25. Imielinski, T., Lipski, W.: Incomplete Information in Relational Databases. J. ACM 31(4), 761–791 (1984) 26. Kolaitis, P.G.: Schema Mappings, Data Exchange, and Metadata Management. In: Li, C. (ed.) PODS 2005, pp. 61–75. ACM, New York (2005) 27. Kolaitis, P.G., Panttaja, J., Tan, W.C.: The Complexity of Data Exchange. In: Vansummeren, S. (ed.) PODS 2006, pp. 30–39. ACM, New York (2006) 28. Libkin, L.: Data Exchange and Incomplete Information. In: Vansummeren, S. (ed.) PODS 2006, pp. 60–69. ACM, New York (2006) 29. Libkin, L., Sirangelo, C.: Data Exchange and Schema Mappings in Open and Closed Worlds. In: Lenzerini, M., Lembo, D. (eds.) PODS 2008, pp. 139–148. ACM, New York (2008) 30. Mądry, A.: Data Exchange: On the Complexity of Answering Queries with Inequalities. Inform. Process. Lett. 94, 253–257 (2005) 31. Miller, R.J., Haas, L.M., Hernández, M.: Schema Mapping as Query Discovery. In: El Abbadi, A., Brodie, M.L., Chakravarthy, S., Dayal, U., Kamel, N., Schlageter, G., Whang, K.-Y. (eds.) VLDB 2000, pp. 77–89. Morgan Kaufmann, San Francisco (2000) 32. Reiter, R.: On Closed World Data Bases. In: Gallaire, H., Minker, J. (eds.) Logic and Data Bases, pp. 119–140. Plenum Publ. Co., New York (1978)

Preference Change Triggered by Belief Change: A Principled Approach J´erˆome Lang1 and Leendert van der Torre2, 1

Laboratoire d’Analyse et Mod´elisation des Syst`emes pour l’Aide ` a la D´ecision (LAMSADE), Universit´e Paris-Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775 Paris, France [email protected] 2 Computer Science and Communications, Universit´e de Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg, Luxembourg [email protected]

Abstract. Various tasks need to consider preferences in a dynamic way. To evaluate and classify methods for preference change, we introduce eight properties for preferences evolving after some new fact has been learned. Four properties are concerned with persistence of preferences when something being preferred is (partly) satisfied or dissatisfied, and formalize that preference change indicates that the ideal state has not been reached or has become unreachable. Four other properties are concerned with persistence of preferences when, roughly, the agent learns something she already expected to hold, and formalizes that preference change is due to surprise. We define a family of preference change operators, parameterized by a revision function on epistemic states and a semantics for interpreting preferences over formulas, and we give conditions on the revision function and the semantics of preference for each of the eight conditions to hold.

1

Introduction

The behaviour of a rational agent — namely, the actions she decides to perform — is a function of her beliefs about the current state of the world and the predicted state of the world after performing such or such course of action, and of her preferences about those states of world she wants to bring about (or avoid to) and the actions she wants to perform (or avoid to). This classical distinction between beliefs and preferences comes from practical reasoning and decision theory, where the most common approach consists in modelling beliefs by probability distributions, and preferences by utility functions. Also in cognitive approaches one can find concepts corresponding to beliefs and preferences as they are used in decision theory, typically referring to concepts like knowledge or belief on the one hand, and preference, desire or goals on the other hand, together with other 

We wish to thank the anonymous reviewers of a previous version of this paper for helpful comments. J´erˆ ome Lang is supported by the ANR Project ANR-05-BLAN0384. A much shorter and preliminary version of this paper appeared as [21].

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 86–111, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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cognitive and social concepts like intentions, commitments, obligations, and so on. In this introduction we use this terminology “beliefs” and “preferences” with the meaning conveyed by practical reasoning and decision theory: beliefs refer to the uncertainty the agent has about the current and future states of the worlds, and preferences refer to her satisfaction when performing an action sequence and obtaining a sequence of states. For the formal framework and results in this paper, we discuss our assumptions in more detail in Section 1.3. Thus, in contrast to some papers, we do not use “preferences” as a mere synonym of “ranking”, independently of whether this ranking expresses relative plausibility of relative satisfaction. For example, Freund [13,14] investigates preference revision in the following meaning: how should an initial ranking (a “chain”) over a set of worlds be revised by the addition, retraction of modification of the links of the chain? See also Chomicki [9] for a similar approach to preference revision, in a database querying context. In these papers, “preference” has to be understood as “ranking over a set of worlds” rather than its decision-theoretic sense, and the results apply indifferently whether the ranking is interpreted in terms of decisiontheoretic preferences or in terms of comparative plausibility. Our work is based on the fundamental distinction between belief and preference, since changes of preferences are often the repercussion of changes of beliefs. Beliefs are dynamic, because they change every time the agent learns something about the state of the world (notably via observations) or performs an action. The effects of learning some information or performing some action on the agent’s beliefs has been extensively studied, not only in the classical Bayesian setting (where learning an information amounts to Bayesian conditioning) but also in logical and ordinal settings, where beliefs most often consist of rankings over worlds rather than probability distributions. Started by Alchourr` on, G¨ ardenfors and Makinson [1], families of theory change operators have been axiomatically characterized, by means of representation theorems. There is now a huge literature on belief change (see, e.g., Rott’s recent survey [25]), and, to some extent, a general agreement about the meaning of the various classes of belief change operators such as revision or update. Now, the question of whether preferences evolve, and how, is just as relevant, and yet the literature on preference change is much sparser than the one on belief change. A few recent articles have focused on this issue (see [4] for a recent overview) but much remains to be done. This particular paper provides an AGM-style approach to preference and preference change. A lot of the work in this paper is conceptual and programmatic, in the sense that we are searching for the right formulation of AGM-style axioms for preference change. Before we discuss the need for our approach, and how our approach can deal with problems that are not accessible to other approaches, we start with an overview of the work in preference change. 1.1

Kinds of Preference Change

While what “belief change” conveys is fairly agreed upon, the recent literature describes a variety of very different processes that can be called “preference

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change”, which roughly can be clustered in three groups, depending mainly on the nature of the mathematical object that changes and the nature of the input that leads this object to change. “Direct” preference change, or revision of preferences by preferences The first group consist in approaches viewing preference change as parallel to belief change: just as belief revision aims at incorporating newly acquired beliefs into an old belief state, this kind of preference change (that we may call intrinsic preference revision) consists in incorporating new preferences into an old preference state: preferences are revised by preferences so as to lead to new preferences. This kind of preference change can be modelled in a way that mirrors belief change, in the sense that preferences are revised by preferences, and lead to new preferences, without beliefs to intervene in the process. This kind of preference change has been given an in depth analysis by Hansson [17,18]. He addresses not only preference revision and contraction, but also preference addition (respectively subtraction), where preference evolves after an alternative is added to (respectively removed from) the set of alternatives. Preference change triggered by “commands” or “suggestions”, as considered in van Benthem and Liu [5], can be seen as a variant of the former class of approaches, the difference being that the “input” is exogeneous: a command is an imperative from an authority (“see to it that ϕ!”) whose effect is that the agent now prefers ϕ-worlds over ¬ϕ-worlds, and a suggestion is a milder kind of preference upgrade. Example 1. [5] Let’s take a trip! Example 1 is a command or a suggestion, depending of whether the preference for a trip must hold in the resulting preference state or not. Van Benthem and Liu [5] build an extension of dynamic epistemic logic for reasoning both with beliefs and preferences, in which these two kinds of preference change are defined and studied. See also Girard [15] for a more extensive study of preference change in modal logics and [27] for preference-based deontic logic. Preference change trigerred by belief change The second group consist in approaches where preferences change in response to changes in belief. Example 2. Initially, I desire to eat sushi from this plate (e). Then I learn that this sushi has been made with old fish (¬f ). Now I desire not to eat this sushi. The event that trigerred the preference change does not primarily concern preference, but beliefs. Learning that the sushi was made from old fish made me believe that I could be sick, and as a consequence my initial preference for sushi has been replaced for the opposite preference. A different kind of preference can be put in the same group: preferences that change when the world changes:

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Example 3. It is a nice afternoon and I would like to have a walk. Then it starts to rain. I do not want to have a walk anymore. Here the change in preferences is triggered by a change of the world, because it was not raining and now it does. Things are quite similar to change trigerred by belief revision as discussed just above, with the difference that the belief change process is not a revision, but an update [20]. Preference change as a result of belief change has been considered only recently. Bradley [8] argues that changes in preference can have two sorts of possible causes: “what might be called change in tastes” (cf. Example 5) and change in beliefs, where “preference change is induced by a redistribution of belief across some particular partition of the possibility space.” Then he develops a Bayesian formalization of this principle. De Jongh and Liu [19] (see also Sections 3.4 and 3.5 of [24]) consider also preference change due to belief change. Preferences are induced from priorities (over formulas) and beliefs, using various possible strategies, as illustrated on their following example. Example 4. [19,24] Alice is looking for a flat. She considers price more important than neighbourhood. She believes that flat d1 has a low cost and is in a bad neighbourhood. She has no information about the price of flat d2 , and believes it is in a good neighbourhood. For instance, the so-called “decision stategy” compares two alternatives by focusing on the most important criterion that one alternative is believed to satisfy and the other one is not, and here would lead to preferring d1 over d2 . When beliefs change, preference may change as well: for instance, if Alice learns that d2 has a low cost, she will now prefer d2 over d1 . This preference change trigerred by belief change contrasts with preference change due to changes in her priorities (see also Example 6). Preferences that change when the agent evolves Example 5. [18] I grow tired of my favourite brand of mustard, A, and start to like brand B better. Here, a change in preference reflects a modification of the agent’s tastes, possibly due to an event the agent is subject to. It could be discussed whether it is relevant to distinguish preference change due to the evolution of the rational agent to preference change due to the evolution of the world. This is primarily a choice to be made when we model the process, and it thus comes down to deciding whether the rational agent should be part of the world of not. Consider the following example from Liu [24], a variation of Example 4: Example 6. [24] Alice is looking for a flat. She considers price more important than quality. After she wins a lottery prize of ten million dollars, she considered quality most important.

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Depending on whether the agent is part of the world, this is an instance of a preference change triggered by a change in the world or by an evolution of the tastes of the agent. 1.2

Evaluating and Classifying Preference Change Methods

Our survey of the three kinds of preference change illustrates a wide variety in the kinds of preference change studied in the literature, even when we restrict ourselves to the notions of preference and belief studied in practical reasoning and decision theory. Our central research question is therefore: How should we evaluate and classify preference change methods? This breaks down in the following research questions: 1. Which language do we need to represent postulates of preference change? 2. Which postulates should we use to evaluate and classify methods for preference change? 3. How to use the postulates to evaluate or classify existing or new approaches to preference change, or to develop new approaches? The success criterium of our postulates is that they are able to distinguish a variety of approaches. To illustrate our postulates and their use to evaluate or classify preference change methods, we propose a general family of operators for preferences evolving after a new fact has been learned, parameterized by a revision function on epistemic states and a semantics for interpreting preferences over formulas, and we give sufficient conditions on the revision function and the semantics of preference for each of these axioms to hold. Our overall methodology is inspired by the so-called AGM framework of theory change [1], a formal framework to evaluate and classify change methods, originally developed as a framework to describe and classify both normative systems and belief change (though only normative system change is mentioned explicitly in [1] as an example of theory change). The AGM framework studies how a set of propositions should change in view of possibly new conflicting information, by providing a set of postulates that the new theory should satisfy. Typically there are several operators that satisfy the conditions and no solution about which one to chose is provided. Summarizing, we are searching for the right formulation of AGM-style postulates for preference change. AGM theory respects a number of postulates which may be useful in the setting of preference change, like minimality. However, it has been used for belief change only, not for preference or norm change. We might wonder whether the AGM postulates can still be of any use for preference change. Unfortunately, they do do not seem very helpful to define properties for preference change. For example, the most often discussed postulates, like success, do not make sense in a preference change setting when the trigger of the preference change is a belief change. We are therefore looking for other postulates.

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Our ultimate goal is to characterize classes of preference revision operators by AGM-like axioms. Even if the revision of preferences by beliefs has been considered in several places, there exists so far no principled study. Such a study would allow to shed light on what these operators mean. Obtaining a full characterization is an ambitious goal, that we do not aim at reaching in this paper. Another long term goal is to use our theory to develop new preference change methods. 1.3

Three Assumptions of Our Language for Preference Change Postulates

The evaluation and classification of preference change methods are not accessible to other approaches discussed thus far, because most papers aim to define a precise notion of preference and preference change by fixing the meaning of the concepts. The AGM approach to theory change can be used to evaluate and classify belief change methods, because it is based on a minimal number of assumptions: a belief base is represented by a theory, the belief change is triggered by new incoming information, and the result is again a theory. The AGM theory is abstract, there is not even a reference to belief in the AGM approach to theory change. However, it is not the case that the most general framework is the best one to evaluate and classify methods, but there is a tradeoff in generality: on the one hand we want to be general to cover a large class of approaches, on the other hand specialized enough to be able to represent useful properties. For example, the AGM postulate of success implies that unsuccessful revisions cannot be represented. In the case of preference change, we have to make some additional assumptions on the language we use to represent the postulates, though we should not limit applicability by fixing the meaning of the concepts. The first assumption we make is that we accept a distinction between beliefs and preferences. This is a very weak assumption, which holds in most models that use the notion of preference. More precisely, we assume a language in which we can talk about beliefs and preferences. These two classes can be found in decision theory by probabilities and (expected) utilities, but also in cognitive science or in agent theories in computer science, which makes our postulates generally applicable. For example, whereas in decision theory there are additional assumptions on how utility and probability can be combined to calculate expected utilities, which are not made in cognitive science, we do not make any assumptions on the way belief and preference can be combined. Moreover, whereas in cognitive approaches it is assumed that belief and preference interact with other social-cognitive concepts like intentions or obligations, which is not assumed in decision theory, we do not make such assumptions either. Summarizing, in our approach it is not important what belief or preference precisely mean, since we give a general framework to compare existing methods, and to guide the development of future preference change methods. The second assumption of our framework is that preference change is due to belief change. In other words, we focus on the second group of approaches,

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namely, preference change trigerred by belief revision. The first reason is that we find it more natural and more widely applicable than other types of preference change. What triggers changes in the mental state of an agent (hence changing her present or future behaviour) generally consists of inputs that come from the world or from other agents (via observations, communication etc.) and primarily affects the agent’s beliefs. We do not mean that these inputs do not affect in any way the agent’s preferences, but that they often do so because they change her beliefs in the first place. The second reason that we consider only change of belief, is that we think that in most cases, and in particular in the class of situations that we consider here, preferences can be assumed to be static. This is analogous to approaches in decision theory, that assume that the utility function is fixed while probabilities change. The third assumption we make is that belief change can be appropriately represented by the AGM approach to theory change, together with some more recent extensions to deal with iterated theory change (we do not need iteration stricto sensu, but we need to refer to revision operators acting on belief states, i.e., plausibility or normality rankings on worlds). The reason of this assumption is that we need a framework of belief change, and the AGM framework is the most generally accepted one. 1.4

Postulates

In the AGM approach to theory change, an important guideline for finding postulates is to formalize the idea of minimal change, as, for example, formulated in the Ramsey test. This is also our first guideline, in the sense that all properties we consider in this paper are of the form: if there is a preference for α, and some other conditions hold, then after learning β, the preference for α still holds. They are therefore a kind of persistence properties, explaining a kind of minimal change properties for preference change. Moreover, our second guideline to find the properties discussed in this paper is the interaction between belief and preference change. In belief change, an important distinction between so-called revision and expansion is that the former not only adds some new facts to the belief base, but when the new information is inconsistent with the previously held beliefs, it also has to drop some beliefs. Thus, we may say that revision also handles the surprise of learning something which the agent expected to be false. In the case of preference change, our properties represent whether preference change is due to surprise, where we distinguish between weak surprise (something new is learned as handled by both expansion and revision) and strong surprise (something is learned which was believed to be false, as handled by revision only). The first four properties (P1 to P4) consider the case in which we learn that our preferences are (partly) satisfied or dissatisfied: (P1)

P α → [α]P α

(P1) intuitively means that learning something that is preferred leaves this preference unchanged. For instance, if I already prefer to have my paper accepted

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then I still prefer this state of fact to the opposite after learning that it has actually been accepted. Alternatively, as another example, if I desire to be rich, then after becoming rich, this state does not become undesirable. This persistence of preference seems natural in most contexts (except perhaps for some pathological agents who are always unhappy with what they have). This principle should not be confused with the dropping of so-called achievement goals, once they are fulfilled. Consider an agent who has the goal to run the marathon once in her life. After she achieves this goal, we may expect her to drop the goal from her goal base (this kind of reasoning is common in planning and BDI theories of agent action, see, e.g., [10]). This, however, does not conflict with our preference persistence postulate. In our setting, preferences bear on propositions, not on actions. “Run the marathon” is an action, whereas “having already run the marathon in one’s life” is a proposition (denote it by m). If the agent initially has a preference for m, and m becomes true, then she keeps preferring m over ¬m (which has now become unacessible). In other words, once she has run the marathon, she’s happy with this state of fact and does not wish she had never done it. By symmetry, things are similar when revising by a dispreferred formula: (P2)

P α → [¬α]P α

Suppose now that we learn that what we want to hold, in fact partially holds. In that case, it would be intuitive that the preference persists. (P3) (P3’)

P α ∧ ¬N (¬β|¬α) → [(α ∨ β)]P α

P α ∧ ¬N (¬β|¬α) ∧ ¬N (α|α ∨ β) → [(α ∨ β)]P α

The following example illustrates the normality condition in (P3). Example 7. Consider the following version of property (P3) without normality condition: (P3*) P α → [(α ∨ β)]P α. Assume there is a lottery in which you can win a car, and let α be “you win a BMW in the lottery” and let β be “you win a Mercedes in the lottery.” Moreover, assume that it is unlikely that you win a car, but you desire to win one without a preference of either brand. Now if you receive the information that you win a car in the lottery, then you still do not care whether it is a BMW or a Mercedes. Before, you preferred to win a BMW since you compared the alternatives of winning a BMW and winning nothing, but with this new information that you won a car, you compare the alternatives of winning a BMW and winning a Mercedes. It satisfies the property (P3), since of the normal worlds in which you do not win the BMW, there is no world in which you win the Mercedes. In other words, normally you do not win the Mercedes but you win nothing, which explains why the preference does not persist. Therefore (P3*) is too strong. (P4) and (P4’) are similar to (P3) and (P3’): (P4)

P α ∧ ¬N (¬β|α) → [∗(¬α ∨ β)]P α

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(P4’)

P α ∧ ¬N (¬β|α) ∧ ¬N (¬α|¬α ∨ β) → [∗(¬α ∨ β)]P α

The next four properties are concerned with the case in which we learn something we expected to hold. (P5)

P α ∧ N β → [β]P α

Property P5 is logically equivalent to the principle that preference change implies surprise: P α∧¬[β]P α → ¬N β. Note that this is only a weak notion of surprise, since it does not imply that β is exceptional, only that β is not normal (a stronger notion of surprise is considered in (P8) below). Property (P5) expresses that if we learn something we already expected to hold, then none of our preferences should change. While this property seems intuitively satisfactory, it is sometimes too strong. Consider the following example. Example 8. Take α to be “my paper is accepted” and β to be “my paper is bad”, because, for instance, the proof of the main result is flawed (and I am not good enough to detect it myself). I initially prefer α, and since I hold myself in a bad self-esteem I believe β. Now, suppose there is a strong correlation between ¬α and β: if my paper is bad, then it is very likely that it will be rejected. I prefer my paper to be accepted (and in case I learn this I will revise my belief that the paper is bad and start believing the opposite), but I would surely not want my paper to be accepted if the main result is false; this is consistent with β being normal, provided that states where β ∧ α hold are at least as exceptional as states where ¬β hold. A weaker form of (P5) is that preference for α should remain unchanged if we learn something that is normal both given α and given ¬α. (P6) P α ∧ N (β|α) ∧ N (β|¬α) → [β]P α While Example 8 conflicts with (P5), it does not conflict with (P6), because N (β|α) does not hold (if my paper is accepted then it is likely that it is good). Having β normal both given α and given ¬α ensures that when comparing α and ¬α, the most normal worlds, that is, the ones I focus one, remain the same, which is a strong reason for α to remain preferred. Another weaker form of (P5), which is when one learns something which is believed (normal), and the preference bears on a proposition α such that neither α nor ¬α is exceptional. (P7)

P α ∧ N β ∧ ¬N α ∧ ¬N ¬α → [∗β]P α

Again, the reason why we need α and ¬α to be non-exceptional is that together with β being believed, it guarantees some stability of the most normal α-worlds and most normal ¬α-worlds before and after revision by β. Consider Example 8 again; it does not contradict (P7), because N α does not hold (N β and N (¬α|β) imply N ¬α).

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This condition that both β and ¬β are non-exceptional is intuitively desirable in many contexts, especially when β (and ¬β) refers to something that is controllable by the agent. For instance, on Example 2: M |= P e ∧ ¬N ¬e ∧ ¬N ¬e ∧ N f : the agent initially believes that the fish is fresh and, of course, does not considers eating, nor not eating, as exceptional. As a result, after learning that the fish is fresh, she still prefers eating the sushi. Now, when revising by something that is not exceptional (not disbelieved), we would expect some form of preservation of preference as well. (P8) P α ∧ ¬N (¬β|α) ∧ ¬N (¬β|¬α) → [β]P α (P8) means that if α is initially preferred and is no longer preferred after learning β, then not only β should not be normal, but it should be exceptional, either given α or given ¬α. Again we need these two conditions ¬N (¬β|α) and ¬N (¬β|¬α). Suppose for instance that ¬N (¬β|¬α) holds (as in Example 8). Then revising by β may change radically the meaning of ¬α. There are dependencies between the postulates. However, stating and proving them formally needs us to be more precise anout the semantics of normality, hence we leave this to Section 3. We just state here informally that under the usual semantics for normality, the following relationships hold: (P5) implies (P6), (P5) implies (P7) and (P8) implies (P6). (For any other pair of postulates (Pi), (Pj) than these three, (Pi) and (Pj) are independent.) 1.5

Using Our Postulates

To illustrate our postulates and their use to evaluate or classify preference change methods, we propose a general family of operators for preferences evolving after some new fact has been learned, parameterized by a revision function on epistemic states and a semantics for interpreting preferences over formulas, and we give conditions on the revision function and the semantics of preference for each of our postulates to hold. We give here an informal presentation of the operators (formal details will be given in Section 2). In order to express preference revision trigerred by belief revision we need to consider both relative plausibility between worlds (or normality) and preference between worlds in our model. While a classical decision-theoretic approach would model plausibility and preference by probability distributions and utility functions respectively, we stick here to a purely ordinal modelling, following a long-standing tradition in the belief change community. Our models consist of two orderings on a set of worlds, one for normality and one for preference, as illustrated on Figure 1. On the left hand side, the mental state of the agent is represented by two complete weak orders expressing respectively normality and preference, and new incoming information f results in the shift of f worlds towards normality, leaving the preference order unchanged. On the right hand side, the two complete weak orders are visualized more compactly by a two-dimensional structure. The striking out parts of the right hand side show the normality shift of the f¯-worlds.

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normality 6 preference 6     ef    ef e¯f    ef¯   e¯f6 e¯f¯  ∗f  e¯f¯6   f¯   e6

normality 6 preference 6    ¯ ¯  ef    e¯f ef    e¯f   e¯f6 e¯f¯   ef6   f¯   e6

preference ef¯ e¯f¯ ef¯

e¯f e¯f¯

ef

normality 6

Fig. 1. Learning f : f becomes most normal, preference remains unchanged

The layout of the rest of the paper is as follows. In Section 2 we define the general model for the evolution of an agent’s preferences after revision by a new fact. In Section 3 we investigate the range of validity of our postulates within this family of preference change operators. We conclude and evoke further developments in Section 4.

2 2.1

Preference Change Triggered by Belief Change: A General Model Notations

Throughout the paper we consider a propositional language L formed from a fixed, finite set of propositional symbols and the usual connectives (this language is enriched with modalities in the following subsection). Propositional symbols are denoted by p, q, etc. Propositional formulas of L are denoted by α, β, ϕ etc. The set of all truth assignments (or valuations) satisfying a formula ϕ is denoted by Mod(ϕ). Valuations are denoted by w, w etc. We use the following notation for valuations: a¯bc denotes the valuation where a and c are assigned to true and b to false. The set of all valuations is denoted by W . A complete weak order is a reflexive, transitive and complete relation  on the set of valuations. L(W ) denotes the set of all complete weak orders on W . The relations ∼ and  are defined from  in the usual way: for any valuations s, s , s , we have s ∼ s iff s  s and s  s and s  s iff s  s and not (s  s). If X ⊆ W , Max (X) is the set of maximal elements in X: Max (X) = {w ∈ X | there is no w such that w  w}. Below we shall make use of two complete weak orders on the set of worlds: a normality relation N and a preference relation P (where ‘preference’ is here employed in its decision-theoretic meaning, cf. the first paragraph of Section 1). 2.2

Beliefs and Preferences

We now consider in more detail the scenario illustrated informally on Example 2. Whether preferences have really changed is a complicated question. This primarily depends on what we mean by “preference”. The preference relation on complete states of the worlds remains static – only the relative plausibility of these states of the world change, and thus the agent’s beliefs. Let S = ¯ e¯f, e¯f¯} be the set of possible states of the world. Some may argue that {ef, ef,

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e is an action rather than a static proposition. To resolve this ambiguity, just consider that e precisely refers to “being in the process of eating”. At first, it is reasonable to assume that I believe the sushi to be made out of fresh fish, or, at least, to assume that I do not believe that the fish is not fresh, even if this is not said explicitly. The reason is that if I already believed that the fish is not fresh, then the new information would have had no impact on my beliefs, and likewise, no impact on my future behaviour. After I am told that the fish is not fresh, it is reasonable to expect that my belief that the fish is fresh gets much lower. As for my preferences, they may initially be ef P e¯f ∼P e¯f¯ P ef¯ i.e., I prefer eating fresh sushi over not eating sushi, and I prefer not eating sushi over eating sushi made out of old fish; if I do not eat the sushi, then I do not care whether the fish is old or not. Now, my preferences after learning that ¬f is true or likely to be true are exactly the same. even if I now consider ef hardly plausible, I still prefer this world to e¯f and e¯f¯, and these two to ef¯. Thus, beliefs change, but preferences remain static. Still, it is no less true that I used to prefer e over ¬e and I no longer do. However, e and ¬e are not single states, but formulas or, equivalently, sets of ¯ and ¬e to {¯ states (e corresponds to the set of states {ef, ef}) ef, e¯f¯}). When expressing an initial preference for e I mean that when I focus on those states where e is true, I see ef as the most plausible state, and similarly when I focus on those states where ¬e is true, I see e¯f as the most plausible state. Because I prefer ef to e¯f , I naturally prefer e to ¬e: in other terms, I prefer e to ¬e because I prefer the most plausible state satisfying e to the most plausible state satisfying ¬e. Of course, after learning the information about the fish, these typical states are now ef¯ and e¯f¯, and after focusing, I now prefer ¬e to e. One may argue also that whether preferences over states change or not is a question of language granularity. If both e and f are in the language, then preference over states do not change, but if the language contains only the propositional symbol e, then they do change, and in this case, it is not possible to express that we learn ¬f , therefore the only way of modeling the input is a “direct preference change”: the world sends a “command” to the user, asking her to now prefer e¯ to e. This informal discussion on Example 2 allows us state the general principle of preference change triggered by belief change: – the agent has some initial beliefs and preferences over possible states of the world; these preferences over states can be lifted to preferences over formulas; – then she learns a new piece of information α about the world; – therefore she revises her prior beliefs by α and keeps the same preference on states; however, preferences over formulas may change in reaction to the change of beliefs. We see that a formalization needs at least two semantical structures: one for beliefs and one for preferences. For the sake of simplicity, and because we have

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to start with some specific assumptions, we stick to the ordinal way of modeling beliefs and preferences, which is common in the belief change literature. Thus, as in Boutilier [7] and subsequently in Lang, van der Torre and Weydert [23], we use a normality ordering together with a preference ordering. Definition 9. A model M is a triple W, N , P , where W is a set of valuations of a set of propositions, and N and P are total pre-orders on W . We do not distinguish worlds from valuations, so each valuation occurs precisely once, and W is finite. s N s means that s is at least as plausible (or normal) as s , whereas s P s means that s is at least as preferred as s . The model for Example 2 is visualized in the figure below. The normality ordering is visualized vertically, where higher worlds are more normal. The most normal worlds are worlds in which the fish is fresh, and exceptional worlds are worlds in which the fish is not fresh: f e ∼N f e¯ N f¯e ∼N f¯e¯. Preferences are visualized horizontally, where the more to the right are the more preferred worlds. Eating fresh sushi is preferred to not eating sushi, which itself is preferred to eating not fresh sushi: ef P e¯f ∼P e¯f¯ P ef¯. preference e¯f ef ef¯ e¯f¯

normality 6

As in Boutilier [7] and Lang, van der Torre and Weydert [23], we extend the propositional language with two dyadic modalities: N for normality and P for preference. As usual, N (ψ|ϕ) is true if the most normal ϕ-worlds are ψ-worlds. N (ϕ| ) is abbreviated as N (ϕ). Definition 10 (normality) M |= N (ψ|ϕ) iff MaxN (Mod(ϕ)) ⊆ Mod(ψ) Things are less easy with preference, for two reasons. First, there are several ways of lifting preferences from the level of worlds to the level of sets of worlds. A canonical family of lifting operators is obtained by comparing two sets of worlds A and B by comparing the best (or the worst) elements of A with the best (or the worst) elements of B – this gives four ways of alternating quantifiers, henceforth, four lifting operators. This principle is classical, as it has been used in many places, including [16,2,22,5]. There are other families of lifting operators, notably ceteris paribus preferences [28,6] and other kinds of similarity-based preferences [18]. While these would also be worth considering, in this paper we restrict our study to the “canonical” lifting operators defined below.

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strong lifting W1 str W2 if W1  = ∅, W2  = ∅, and ∀w ∈ W1 ∀w ∈ W2 : w P w : the worst worlds in W1 are preferred to the best worlds in W2 , or equivalently, every world in W1 is preferred to every world in W2 . optimistic lifting W1 opt W2 if W1  = ∅, W2  = ∅, and ∃w ∈ W1 such that ∀w ∈ W2 , w P w : the best worlds in W1 are preferred to the best worlds in W2 (or equivalently, the best ϕ ∨ ψ worlds are ¬ψ worlds). pessimistic lifting = ∅, W2  = ∅, and ∀w ∈ W1 ∃w ∈ W2 such that w P w : W1 pess W2 if W1  the worst worlds in W1 are preferred to the worst worlds in W2 . We deliberately omit to define the fourth case, corresponding to two existential quantifiers, because the resulting lifting operator is much too weak, as it makes P ϕ ∧ P ¬ϕ consistent. This weak lifting operator is left out of our study. Recall that the set of truth assignments is finite; therefore, there cannot be any infinite ascending or descending chains of worlds, and the last two definitions always make sense. An equivalent definition of opt , which does not need the finiteness assumption, is: ∀w ∈ W2 ∃w ∈ W1 such that w ≺P w . Second, as argued in [7,23], in the presence of uncertainty or normality expressed by N , preferences cannot be interpreted from P alone, but from P and N . There are at least two ways of interpreting a preference for ϕ over ¬ϕ in this context, that we name B and LTW after their authors.1 Let  be one of str , opt , or pess . B “among the most normal φ-worlds, ψ is preferred to ¬ψ”[7]: M |= P (ψ|ϕ) iff MaxN (Mod(ϕ)) ∩ Mod(ψ)  MaxN (Mod(ϕ)) ∩ Mod(¬ψ)) LTW “the most normal ψ ∧ φ-worlds are preferred to the most normal ¬ψ ∧ φworlds” [23]: M |= P (ψ|ϕ) iff MaxN (Mod(ϕ ∧ ψ))  MaxN (Mod(ϕ ∧ ¬ψ)) P (ϕ| ) is abbreviated in P (ϕ). Note that B and LTW are not equivalent, because either the most normal ψ∧φ worlds or the most normal ¬ψ∧φ worlds may be exceptional among the φ worlds. The two approaches are based on distinct intuitions. In LTW, the intuition is that an agent is comparing two alternatives, and for each alternative he is considering the most normal situations. Then he compares the two alternatives and expresses a preference of the former over the latter. The difference between 1

Another way, for example, is to compare all worlds in the preference ranking up to minimal rank of MaxN (ψ ∧ φ) and MaxN (¬ψ ∧ φ).

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both approaches, already discussed in [23], is a matter of choosing the worlds to focus on. The two approaches coincide if there exist both most normal ψ ∧ φworlds and most normal ¬ψ ∧ φ-worlds, that is, if ¬N (ψ|φ) ∧ ¬N (¬ψ|φ) holds. We have thus defined six semantics for interpreting P (.|.), since we have three ways of lifting preference from worlds to formulas, and two ways of focusing on normal worlds. We denote the corresponding six modalities using the superscript LTW B or LTW, and one of the three subscripts str, opt or pess. For instance, Popt refers to the semantics in [23] and the optimistic way of lifting preferences. However we shall try to avoid using these subscripts and superscripts when it is clear from the context. From the P modality we may also define a dyadic > modality, where ϕ > ψ means “I prefer ϕ to ψ”, defined by (ϕ > ψ) ≡ P (ϕ|(ϕ ∧ ¬ψ) ∨ (ψ ∧ ¬ϕ)) P (.|.) and . > . are interdefinable (see [18]). 2.3

Revision Functions

Given a model M = W, N , P , its revision by belief α is a new model M = M  α consisting of the same W , the same P (since preferences over worlds do not change), and the revision of the initial plausibility ordering N by α. This requires the prior definition of a revision function  acting on plausibility orderings. Such functions have been extensively considered in the literature of iterated belief revision (e.g., [11,26]). Definition 11. A revision function  is a mapping from L(W ) × L to L(W ), i.e., it maps a complete weak order over W and a formula α into a complete weak order over W . For the sake of notation we note α N instead of N α. Revision functions on plausibility orderings are usually required to obey some properties. For example,  satisfies success iff for every N and every satisfiable α, Max(α N , W ) ⊆ [α]. Hence, the most normal worlds after revising by α satisfy α. In the rest of the paper we need the following properties. A revision function  satisfies – positive uniformity iff for any two worlds w, w such that w |= α and w |= α,   w α N w iff w N w ; – negative uniformity iff for any two worlds w, w such that w |= ¬α and   w |= ¬α, w α N w iff w N w . – responsiveness iff for any two worlds w, w such that w |= α and w |= ¬α,  w N w implies w α N w . – stability iff the following holds: if all most normal worlds in N satisfy α then α N =N ;

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– top-stability iff the following holds: if all most normal worlds in N satisfy α then Max(α N , W ) = Max(N , W ); Positive and negative uniformity are named respectively (CR1) and (CR2) by Darwiche and Pearl [11]. Note that success implies that M  α |= N α. Topstability is weaker than stability, and top-stability is implied by positive uniformity together with responsiveness. Definition 12. Given a model M = W, N , P , a revision function , and a formula α, the revision of M by α, is the model M  α defined by M  α = W, α N , P

3

Back to the Postulates

As explained in Section 1.4, perhaps the easiest way to describe the behavior of preference change, is to aim for an AGM style representation with postulates. To do so, we use dynamic modalities to refer to revisions, as by van Ditmarsch, van der Hoek and Kooi [12] and van Benthem [3]. M, w |= [α]ϕ iff M  α, w |= ϕ In Section 1 we introduced the following eight postulates that preference change may fulfill. All properties are concerned with conditions in which a preference for α persists when new information is learned. The first four properties P1-P4 consider the case in which we learn that our preferences are (partly) satisfied or dissatisfied, and the following four properties P5-P8 are concerned with the case in which we learn something which we expected. (P1) P α → [α]P α (P2) P α → [¬α]P α (P3) P α ∧ ¬N (¬β|¬α) → [(α ∨ β)]P α (P3’) P α ∧ ¬N (¬β|¬α) ∧ ¬N (α|α ∨ β) → [(α ∨ β)]P α (P4) P α ∧ ¬N (¬β|α) → [∗(¬α ∨ β)]P α (P4’) P α ∧ ¬N (¬β|α) ∧ ¬N (¬α|¬α ∨ β) → [∗(¬α ∨ β)]P α (P5) P α ∧ N β → [β]P α (P6) P α ∧ N (β|α) ∧ N (β|¬α) → [β]P α (P7) P α ∧ N β ∧ ¬N α ∧ ¬N ¬α → [∗β]P α (P8) P α ∧ ¬N (¬β|α) ∧ ¬N (¬β|¬α) → [β]P α The relationships between the postulates are the following. When saying that (Pi) implies (Pj) we mean that (Pi) implies (Pj) whatever the chosen semantics for preference, provided that the semantics for normality is fixed to the classical semantics for normality (as defined above). We state these relationships without proof (they are straightforward). – (P5) implies (P6) and (P7); – (P8) implies (P6).

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Any two properties (Pi) and (Pj) other than the ones above are independent. We are now going to look for sufficient conditions, on the belief revision operator  used and the choice of the semantics for interpreting preference, for each of these postulates to be satisfied. 3.1

Preference Satisfaction (or Dissatisfaction)

We first consider (P1). (P1)

P α → [α]P α

or, equivalently: if M |= P α then M  α |= P α. Proposition 13. (P1) is satisfied: – if  satisfies positive and negative uniformity, and – for any lifting operator ∈ {str , opt , pess }, with the LTW semantics. Proof. Positive uniformity implies that Max(α N , [α]) = Max(N , [α]), and negative uniformity that Max(α , [¬α]) = Max( N , [¬α]): the most normal αN worlds are the same before and after revision by α, and similarly for the most normal ¬α-worlds. Now, for any lifting operator, whether P α holds in the LTW semantics depends only on the preference between the most normal α-worlds and the most normal ¬α-worlds, from which the result follows. Let us give the details for opt (things are similar for the proof for str and pess , and for any lifting operator). We have (1) Max(P , Max(α N [α])) = Max(P , Max(N , [α])) and (2) Max(P , Max(α N [¬α])) = Max(P , Max(N , [¬α])). Suppose M |= P α, i.e., (3) Max(P , Max(N , α)) opt Max(P , Max(N , ¬α)). From (1), (2) α and (3) we get Max(P , Max(α N , α)) opt Max(P , Max(N , ¬α)), that is, M  α |= P α follows.   Positive and negative uniformity are necessary. Consider for instance the drastic revision operator that preserves the relative ranking of α-worlds and then pushes all ¬α-worlds towards the bottom, irrespectively of their relative initial ranking:     w α N w iff (a) w |= α, w |= α and w N w ; or (b) w |= α and w |= ¬α. ∗ satisfies positive uniformity, but not negative uniformity. In Figure 2 we initially have pq N p¯q¯  p¯ q  p¯q and p¯q P pq P p¯q¯  p¯ q. After revision by p we have p¯ q  p ¯ q ∼ p ¯ q ¯ , therefore, with the optimistic lifting we have M |= P p and pq p N M |= [p]P ¬p. (P1) is meaningless or arbitrary for Boutilier’s semantics, because we have the property ¬[α]P α for satisfiable α: If the most normal worlds become α worlds, then the intersection of most normal worlds and ¬α worlds is empty. Moreover, this property also suggests that there is no other property in the spirit of (P1) we can define for B semantics. By symmetry, things are similar when revising by a dispreferred formula: (P2)

P α → [¬α]P α

Preference Change Triggered by Belief Change: A Principled Approach

normality 6 preference 6    pq p¯q        6  p¯6 q¯   pq     ∗p 6 p¯ q   p¯6 q¯      6 q   p¯ q   p¯6

normality 6 preference 6    pq p¯q        6 6  p¯ q   pq     p¯q¯   p¯6 q¯  p¯q 6   6 q   p¯

preference pq p¯q¯ p¯ q p¯q¯ p¯q

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normality 6

Fig. 2. Learning p: p becomes least normal, preference remains unchanged

Proposition 14. (P2) is satisfied: – if  satisfies positive and negative uniformity, and – for any lifting operator with the LTW semantics. Proof. Positive uniformity implies that the most normal ¬α-worlds are the same before and after revision by ¬α, and negative uniformity that the most normal α-worlds are the same before and after revision by ¬α. The rest of the proof is exactly as in the proof of Proposition 13.   (P2) does not hold for B semantics, for similar reasons as (P1) does not hold: we have the property ¬[¬α]α for satisfiable ¬α. We now consider (P3), (P3’), (P4) and (P4’). (P3) (P3’)

P α ∧ ¬N (¬β|¬α) → [(α ∨ β)]P α

P α ∧ ¬N (¬β|¬α) ∧ ¬N (α|α ∨ β) → [(α ∨ β)]P α

(P3) is equivalent to P α ∧ ¬[(α ∨ β)]P α → N (¬β|¬α), which expresses that preference change in case of partial preference satisfaction is due to abnormality of β in case of ¬α. Proposition 15. (P3) is satisfied: – if  satisfies positive and negative uniformity, and responsiveness, and – for strong or optimistic lifting with the LTW semantics. (P3’) is satisfied: – if  satisfies positive and negative uniformity, and responsiveness, and – for strong or optimistic lifting with the B semantics. Proof. Consider first the proof of (P3). By positive uniformity, α ∨ β-worlds are shifted uniformly when revising by α ∨ β. This applies in particular to α-worlds, therefore (1) the most normal α-worlds remain the same. Assume M |= ¬N (¬β|¬α): then at least one most normal ¬α-world satisfies β. Let w be such a world. After revision by α∨β, w is still a most normal ¬α-world. (α∨β) w. To see this, assume there exists a world w such that w |= ¬α and w N

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If w |= ¬α ∧ β then by positive uniformity, w N w, which contradicts w being a most normal ¬α-world. If w |= ¬α ∧ ¬β then by responsiveness, w N w, which again contradicts w being a most normal ¬α-world. Analogously, for any (α∨β) other most normal ¬α-world w , i.e. w N w, if w |= ¬α ∧ β then w N w by positive uniformity, and if w |= ¬α ∧ ¬β then w N w by responsiveness. (α∨β) is contained in the Therefore, (2) the set of most normal ¬α-worlds in N set of most normal ¬α-worlds in N . (α∨β) Therefore, if w1 is a most normal α-world in N and w2 is a most normal (α∨β) ¬α-world in N , then (1) implies that w1 is a most normal α-world in N , and (2) implies that w2 be a most normal ¬α-world in N . =str (α∨β) (α∨β) Assume M |= P α. Let w1 ∈ Max(N , [α]) and w2 ∈ Max(N , [¬α]), which implies w1 ∈ Max(N , [α]) and w2 ∈ Max(N , [¬α]). From M |= P α, we now have w1 P w2 . Therefore, every most normal α-world (α∨β) (α∨β) in N is preferred to every most normal ¬α-world in N , that is, M |= [(α ∨ β)]P α. =opt (α∨β) Assume (3) M |= P α. Let (4) w1 ∈ Max(P , Max(N , [α])) and (5) (α∨β) , [¬α])). Again, from (4) and (5) we get (6) w1 ∈ Max(N w2 ∈ Max(N , [α])) and (7) w2 ∈ Max(N , [¬α]). Suppose now that w1 is not a most preferred world in w1 ∈ Max(N , [α])), that is, that there exists w3 ∈ Max(N , [α])) such that (8) w3 P w1 . Because w1 and w3 are both most normal in N , we have w1 ∼N w3 , which by positive uniformity (and because w1 (α∨β) and w3 both satisfy α ∨ β) implies w1 ∼N w3 , which, together with (8), contradicts (4). Therefore we have (9) w1 ∈ Max(P , Max(N , [α])). Now, from (3), the most preferred worlds in Max(N , [α]) are preferred to the most preferred worlds in Max(N , [¬α]), therefore they are preferred to all worlds in Max(N , [¬α]), which implies that w1 P w2 , from which the result follows. Consider now the proof of (P3’). Assume in addition that M |= ¬N (α|α ∨ β), i.e. there is a ¬α ∧ β world among the most normal α ∨ β worlds - let us call it w . The proof is analogous, with the extra condition that from M |= P α it follows that the most normal α worlds of M and the most normal ¬α worlds of M are among the most normal worlds of M, and we have to prove that a similar condition holds for M  (α ∨ β). Due to positive uniformity and strong responsiveness, it follows that the most normal α worlds of M  (α ∨ β) as well as w are among the most normal worlds of M  (α ∨ β). From the inclusion of w it follows that the most normal ¬α worlds of M  (α ∨ β) are among the most normal worlds of M  (α ∨ β).   Note that (P3) does not hold for the pessimistic semantics, since if the worst world used to be an ¬α-world, then after the revision the worst world may be an α-world. Nor does it hold for the B-semantics, because after revision by α ∨ β the ¬α-worlds may disappear from the top cluster.

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The case for (P4) and (P4’) is similar. (P4) (P4’)

P α ∧ ¬N (¬β|α) → [∗(¬α ∨ β)]P α

P α ∧ ¬N (¬β|α) ∧ ¬N (¬α|¬α ∨ β) → [∗(¬α ∨ β)]P α

Proposition 16. (P4) is satisfied if: –  satisfies positive and negative uniformity, and responsiveness, and –  = strong or pessimistic lifting with the LTW semantics. (P4’) is satisfied if: –  satisfies positive and negative uniformity, and responsiveness, and –  = strong or pessimistic lifting with the B semantics. The proof is similar to the proof of Proposition 15. 3.2

Preference Change Implies Surprise

We start by (P5). (P5)

P α ∧ N β → [β]P α

Proposition 17. (P5) is satisfied: – if  satisfies stability, and – for any lifting operator with the LTW semantics. or – if  satisfies top-stability, and – for any lifting operator with the B semantics. Proof 1. take any lifting operator with the LTW semantics. and assume that  satisfies stability. Assume M |= N β. Then stability implies that N does not change after revision by β, that is, β N =N . Therefore, most normal α-worlds are the same before and after revision by β, and similarly for ¬α-worlds, from which we get that M |= P α implies M |= [β]P α. 2. take any lifting operator with the B semantics. and assume that  satisfies top-stability. If M |= N β then all most normal worlds in N satisfy β, therefore revising by β leaves these most normal worlds (that is, Max(N , W )) unchanged; since the truth of P (.|.) depends only on MaxN (W ), preferences remain unchanged after revision by β, therefore M |= P α implies M |= [β]P α.  

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Figure 3 illustrates that item 1. of the proof of Proposition 17 no longer holds if  does not satisfy stability, because revising by β may change the most normal α-worlds or the most normal ¬α-worlds. We have N : pq  p¯ q  p¯q¯  p¯q; P : p¯q  pq  p¯q¯  p¯ q; and  such that that in β N , all β-worlds are ranked above all ¬β-worlds. That is: q ¯q  p¯ q  p¯q¯. Before learning q, the most N : pq  p normal p-world is pq and the most normal ¬p-world is p¯q¯, therefore M |= P p for any kind of lifting. After learning q, the most normal p-world is still pq and the most normal ¬p-world is p¯q, therefore M |= P ¬p, again for any kind of lifting. normality 6 preference 6    pq   p¯q      6 6  p¯ q   pq   ∗q   p¯6 q¯   p¯6 q¯      6 q   p¯ q   p¯6

normality 6 preference 6    pq   p¯q      6  p¯6 q   pq     6 p¯ q   p¯6 q¯      6 q¯   p¯ q   p¯6

preference pq p¯q p¯ q p¯q¯ p¯q

normality 6

Fig. 3. Learning q: q becomes least normal, preference remains unchanged

We now consider (P6). (P 6) P α ∧ N (β|α) ∧ N (β|¬α) → [β]P α Proposition 18. (P6) is satisfied: – if  satisfies positive uniformity and responsiveness, and – for any lifting operator with the LTW semantics. or – if  satisfies top-stability, and – for any lifting operator with the B semantics. Proof 1. take any lifting operator with the LTW semantics, and assume that  satisfies positive uniformity and responsiveness. Moreover, assume M |= N (β|α)∧N (β|¬α). Because M |= N (β | α), the most normal α-worlds in N are β ∧ α-worlds. This, together with positive uniformity, implies that (1) the most normal β ∧ α-worlds are the same before and after learning β. Indeed, let w ∈ Max(N , [β ∧ α]) and w |= β ∧ α. We have w N w ,  therefore, by positive uniformity, w β N w . Similarly, again using positive uniformity, (2) the most normal β ∧ ¬α-worlds are the same before and after learning β. Now, the most normal α-worlds are also most normal β ∧ α-worlds (because M |= N (β | α), therefore, these worlds remain among the most normal α-worlds after revising by β. We now have to prove that no other world can

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become a most normal β ∧ α-world after learning β. Let w ∈ Max(N , [α])  and assume there is a w ∈ Max(β ∈ Max(N , [α]). EiN , [α]) such that w     ther w |= β ∧ α or w |= ¬β ∧ α. If w |= β then by (1), w ∈ Max(N , [α]), a contradiction. If w |= ¬β ∧ α then w N w , because M |= N (β | α). Then, we have w |= β, w |= ¬β and w N w , therefore by responsiveness β   we get w β N w , which contradicts w ∈ Max(N , [α]). Therefore, the most normal α-worlds before and after revision by β are the same. Similarly, we show that the most normal ¬α-worlds before and after revision by β are the same. The result then follows. 2. consequence of the second part of Proposition 17, using the fact that (P5) implies (P6).   We now consider (P7). (P7)

P α ∧ N β ∧ ¬N α ∧ ¬N ¬α → [∗β]P α

Proposition 19. (P7) is satisfied: – if  satisfies top-stability, and – for any lifting operator with the LTW semantics. or – if  satisfies top-stability, and – for any lifting operator with the B semantics. Proof. The second part (with the B semantics) is a direct consequence of Proposition 17 together with the fact that (P5) implies (P7). As for the first part, take any lifting operator with the LTW semantics and let  satisfying top-stability. Assume M |= N β ∧ ¬N α ∧ ¬N ¬α ∧ P α. Top-stability and M |= N β imply that (1) the most normal worlds are the same in N and in β N . Now, all most normal worlds satisfy β; moreover, because M |= ¬N α ∧ ¬N ¬α, at least one of these satisfy α and one of these satisfies ¬α. Therefore, Max(N , [α]) = Max(N , [β ∩α]) ⊆ Max(N , [β]) and similarly Max(N , [¬α]) ⊆ Max(N , [β]). This, together with (1), implies that the most normal α-worlds are the same before and after revision by β, and similarly for ¬α-worlds, from which the result follows.   This condition that both β and ¬β are non-exceptional is intuitively desirable in many contexts, especially when β (and ¬β) refers to something that is controllable by the agent. For instance, on Example 2: M |= P e ∧ ¬N ¬e ∧ ¬N ¬e ∧ N f : the agent initially believes that the fish is fresh and, of course, does not considers eating, nor not eating, as exceptional. As a result, after learning that the fish is fresh, he still prefers eating the sushi. Lastly, we consider (P8). (P8) P α ∧ ¬N (¬β|α) ∧ ¬N (¬β|¬α) → [β]P α

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Proposition 20. (P8) is satisfied: – if  satisfies positive uniformity and responsiveness, and – for the strong lifting operator with either the LTW or the B semantics. Proof 1. Take first the strong lifting operator with the LTW semantics, and assume (1) M |= ¬N (¬β | α), (2) M |= ¬N (¬β | ¬α) and (3) M |= P α. (1) implies that there exists a world w1 in Max(≥N , [α]) ∩ [β]. (2) implies that there exists a world w2 in Max(≥N , [¬α]) ∩ [β]. Let w3 ∈ Max(≥β N , [α]), which β implies w3 N w1 . Two cases: – w3 |= β. In this case, w3 β N w1 , together with w1 |= β and positive uniformity, implies w3 N w1 . – w3 |= ¬β. In this case, w3 β N w1 , together with w1 |= β and responsiveness, implies w3 N w1 . Therefore, w3 N w1 . Together with w1 ∈ Max(≥N , [α]), this implies (4) w3 ∈ Max(≥N , [α]). Similarly, let w4 ∈ Max(≥β N , [¬α]), then we show in the very same way (using (2) instead of (1)) that (5) w4 ∈ Max(≥N , [¬α]). Lastly, from (3), (4) and (5) we get w3 P w4 . This being true for β any w3 ∈ Max(≥β N , [α]) and any w4 ∈ Max(≥N , [¬α]), we conclude that M |= [β]P α. 2. Take now the strong lifting operator with the B semantics, and assume (1), (2) = ∅ and and (3) hold. Again, (1) and (2) imply Max(≥N , [α ∧ β])  = ∅. Moreover, let w ∈ Max(β ) ∩ [α] and Max(≥N , [¬α ∧ β])  N β  w ∈ Max(N ) ∩ [¬α]. From positive uniformity, responsiveness, and the nonemptiness of Max(≥N , [α ∧ β]) and of Max(≥N , [¬α ∧ β]) (which follows from (1) and (2) respectively), we have that w ∈ Max(≥N , [α ∧ β]) and w ∈   Max(≥N , [¬α ∧ β]); from (3) we have w P w , and the result follows. However this no longer holds with the other kinds of lifting, as can be seen on the following example: N : pq ∼ p¯ q  p¯q ∼ p¯q¯ and P : p¯ q  p¯q  pq  p¯q¯. We have M |= P p for any of =opt or =pess . After learning q, for any q “reasonable” revision operator , including drastic revision, we have pq q N p¯ and p¯q  p¯q¯. Therefore, the most normal p-world is pq and the most normal ¬p-world is p¯q, which implies that we have M |= [q](P ¬p ∧ ¬P p).

4

Conclusion

There is a wide variety in the kinds of preference change studied in the literature, even when we restrict ourselves to the notions of preference and belief studied in practical reasoning and decision theory. Since the AGM approach to theory change can be used to evaluate and classify belief change methods, because it is based on a minimal number of assumptions, we propose an analogous approach to evaluate and classify preference change methods. We assume a distinction

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between beliefs and preferences, without assuming that they can be combined (like probabilities and utilities can be combined in expected utility) or extended with other concepts (like beliefs and desires can be extended with intentions in cognitive theories). Moreover, we assume that preference change is due to belief change, because we find it more natural and more widely applicable than other approaches we discussed in the introduction of this paper, and because we think that in most cases, preferences can be assumed to be static (like the utility function is fixed while probabilities change). Finally, we assume that belief change can be appropriately represented by the AGM approach to theory change, together with some more recent extensions to deal with iterated theory change, because the AGM framework is the most generally accepted one for belief change. We introduce a standard language to represent postulates for preference change triggered by belief change, based on a dyadic modal operator for normality or belief operator, represented by N (α | β) for “α is normal or believed given β,” and P (α | β) for “α is preferred given β.” Moreover, to represent the updates, we extend this modal language with an update operator, represented by [∗α]β for “after learning the new information α, β holds.” We introduce the following eight postulates to evaluate and classify preference change methods. All properties are concerned with conditions in which a preference for α persists when new information is learned. The first four properties P1-P4 consider the case in which we learn that our preferences are (partly) satisfied or dissatisfied, and the following four properties P5-P8 are concerned with the case in which we learn something which we expected or which did not surprise us. (P1) P α → [α]P α (P2) P α → [¬α]P α (P3) P α ∧ ¬N (¬β|¬α) → [(α ∨ β)]P α (P3’) P α ∧ ¬N (¬β|¬α) ∧ ¬N (α|α ∨ β) → [(α ∨ β)]P α (P4) P α ∧ ¬N (¬β|α) → [∗(¬α ∨ β)]P α (P4’) P α ∧ ¬N (¬β|α) ∧ ¬N (¬α|¬α ∨ β) → [∗(¬α ∨ β)]P α (P5) P α ∧ N β → [β]P α (P6) P α ∧ N (β|α) ∧ N (β|¬α) → [β]P α (P7) P α ∧ N β ∧ ¬N α ∧ ¬N ¬α → [∗β]P α (P8) P α ∧ ¬N (¬β|α) ∧ ¬N (¬β|¬α) → [β]P α Moreover, we show how to use our postulates to evaluate and classify preference change methods. We define a family of operators for preferences evolving after some new fact has been learned, parameterized by a revision function on epistemic states and a semantics for interpreting preferences over formulas. Moreover, we give conditions on the revision function and the semantics of preference for each of these conditions to hold, as listed in Table 1. Roughly, all of them hold for LTW semantics under some conditions, whereas (P1) and (P2) are not meaningful for B semantics, whereas some of the others need stronger or other conditions. Summarizing, in this paper we have given an investigation of the properties of preference change in reponse to belief change, depending on the choice of

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Table 1. Results for some operators: PU = positive uniformity, NU = negative uniformity, R = responsiveness, S = stability, TS = top-stability

(P1) PU, NU (P2) PU, NU (P3) PU, NU, R (P3’) (P4) PU, NU, R (P4’) (P5) S (P6) PU, R (P7) TS (P8) PU, R

LTW str , opt , pess str , opt , pess str , opt

B not applicable not applicable PU, NU, R

str , pess str , opt , pess str , opt , pess str , opt , pess str

str , opt

PU, NU, R str , pess TS str , opt , pess TS str , opt , pess TS str , opt , pess PU, R str

a revision operator and the choice of a semantics of semantics for preference. Even if we have obtained sufficient conditions for several significant properties of preference change, what is still missing is a series of representation theorems of the form: this list of properties is satisfied if and only if  satisfies this set of properties and  this other set of properties. Obtaining such a result is a longterm goal due to the high number of parameters that can vary.

References 1. Alchourr` on, C., G¨ ardenfors, P., Makinson, D.: On the Logic of Theory Change: Partial Meet Functions for Contraction and Revision. J. Symb. Log. 50 (1985) 2. Barber` a, S., Bossert, W., Pattanaik, P.: Ranking Sets of Objects. In: Barber` a, S., Hammond, P., Seidl, C. (eds.) Handbook of Utility Theory, pp. 895–978. Kluwer Academic Publishers, Dordrecht (2004) 3. van Benthem, J.: Dynamic Logic for Belief Revision. J. Appl. Non-Class. Log. 17(2), 129–156 (2007) 4. van Benthem, J.: For Better or for Worse: Dynamic Logics of Preference. In: Gr¨ uneYanoff, T., Hansson, S.O. (eds.) Preference Change. Approaches from Philosophy, Economics and Psychology. Theory and Decision Library A, vol. 42, pp. 57–84. Springer, Netherlands (2009) 5. van Benthem, J., Liu, F.: Dynamic Logic of Preference Upgrade. J. Appl. NonClass. Log. 17(2), 157–182 (2007) 6. van Benthem, J., Roy, O., Girard, P.: Everything Else Being Equal: A Modal Logic Approach to ceteris paribus Preferences. J. Philos. Log. 38(1), 83–125 (2009) 7. Boutilier, C.: Toward a Logic for Qualitative Decision Theory. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) KR 1994, pp. 75–86. Morgan Kaufmann, San Francisco (1994) 8. Bradley, R.: The Kinematics of Belief and Desire. Synthese 156(3), 513–535 (2007) 9. Chomicki, J.: Database Querying under Changing Preferences. Ann. Math. Artif. Intell. 50(1-2), 79–107 (2007) 10. Cohen, P., Levesque, H.: Intention is Choice + Commitment. Artif. Intell. 42(2-3), 213–261 (1990)

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11. Darwiche, A., Pearl, J.: On the Logic of Iterated Belief Revision. Artif. Intell. 89, 1–29 (1997) 12. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library, vol. 337. Springer, Heidelberg (2007) 13. Freund, M.: On the Revision of Preferences and Rational Inference Processes. Artif. Intell. 152(1), 105–137 (2004) 14. Freund, M.: Revising Preferences and Choices. J. Math. Econ. 41, 229–251 (2005) 15. Girard, P.: Modal Logic for Belief and Preference Change. Ph.D. thesis, Stanford University, ILLC Publications DS-2008-04 (2008) 16. Halpern, J.: Defining Relative Likelihood Inpartially Ordered Preferential Structures. J. Artif. Intell. Res. 7, 1–24 (1997) 17. Hansson, S.O.: Changes in Preferences. Theory and Decision 38, 1–28 (1995) 18. Hansson, S.O.: The structure of Values and Norms. Cambridge University Press, Cambridge (2001) 19. de Jongh, D., Liu, F.: Optimality, Belief and Preference. In: Artemov, S., Parikh, R. (eds.) Proceedings of the Workshop on Rationality and Knowledge. ESSLLI, Malaga (2006) 20. Katsuno, H., Mendelzon, A.: On the Difference Between Updating a Knowledge Base and Revising it. In: Allen, J.F., Fikes, R., Sandewall, E. (eds.) KR 1991, pp. 387–394. Morgan Kaufmann, San Francisco (1991) 21. Lang, J., van der Torre, L.: From Belief Change to Preference Change. In: Ghallab, M., Spyropoulos, C.D., Fakotakis, N., Avouris, N.M. (eds.) ECAI 2008, pp. 351– 355. IOS Press, Amsterdam (2008) 22. Lang, J., van der Torre, L., Weydert, E.: Utilitarian Desires. J. Auton. Agents Multi-Agent Syst. 5, 329–363 (2002) 23. Lang, J., van der Torre, L., Weydert, E.: Hidden Uncertainty in the Logical Representation of Desires. In: Gottlob, G., Walsh, T. (eds.) IJCAI 2003, pp. 685–690. Morgan Kaufmann, San Francisco (2003) 24. Liu, F.: Changing for the Better. Preference Dynamics and Agent Diversity. Ph.D. thesis, Universiteit van Amsterdam, ILLC Publications DS-2008-02 (2008) 25. Rott, H.: Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning. Oxford University Press, Oxford (2001) 26. Rott, H.: Shifting Priorities: Simple Representations for Twenty-seven Iterated Theory Change Operators. In: Makinson, D., Malinowski, J., Wansing, H. (eds.) Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg. Uppsala Philosophical Studies, vol. 53, pp. 359–384. Uppsala Universitet (2006) 27. van der Torre, L.: Reasoning About Obligations: Defeasibility in Preference-based Deontic Logic. Ph.D. thesis, Erasmus University Rotterdam (1997) 28. von Wright, G.: The Logic of Preference. Edinburgh University Press (1963)

A Runs-and-Systems Semantics for Logics of Announcements Riccardo Pucella1 and Mehrnoosh Sadrzadeh2, 1

2

College of Computer and Information Science, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, United States of America [email protected] Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom [email protected]

Abstract. Logics of announcements are logics of knowledge to reason about agents that communicate by broadcasting interpreted messages. These logics are typically given a semantics in terms of updatable Kripke structures, which tend to be abstract. We revisit the semantics of logics of announcements and develop a concrete semantics using runs and systems. The advantage is that we can devise models that capture scenarios without having to express properties of those scenarios within the logic itself. In this concrete setting, we study honesty as well as belief in the presence of announcements that are not broadcast to all agents in a system.

1

Introduction

Logics of knowledge provide the ability to reason about knowledge of agents in situations where they interact. Roughly speaking, logics of knowledge, or epistemic logics, are logics with an operator typically written Ki ϕ used to capture the notion that agent i knows fact ϕ. Such logics have found applications in philosophy, artificial intelligence, economics, game theory, and distributed computing. How knowledge is interpreted varies widely, but a common interpretation based on possible worlds goes back to Hintikka [10], where an agent knows a fact if that fact is true at all the worlds that the agent consider possible alternatives to the actual world. Kripke structures [12] are used to formalize this semantics for knowledge. Most early uses of logics of knowledge involved reasoning about agents that communicate by exchanging uninterpreted messages. Here, by uninterpreted messages, we mean messages that are not intended to be interpreted as formulas, and therefore do not require a priori knowledge of the semantics of the 

The original ideas behind this paper were formed while attending the “Mathematical Methods for Security Protocols” workshop organized by Prakash Panangaden at the Bellairs Research Institute of McGill University in March 2007. The authors would like to express their gratitude to Prakash and Bellairs. M.S. would also like to thank Lena Kurzen for fruitful discussions on Moore sentences at LOFT’08.

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 112–134, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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logic to make sense. For instance, an agent in a system sending a message containing the agent’s name is sending an uninterpreted message. In contrast, there are situations where agents interact by exchanging interpreted messages, that is, messages intended as logical formulas whose meaning depends on the semantics of the logic such as statements about their own knowledge. To reason about such scenarios, logics of announcements were developed [3,8,2,1,11]. Logics of announcements can be seen as a combination of epistemic logic and dynamic logic [9]. Dynamic logic takes as a basic logical operator the operator [α]ϕ, interpreted as ϕ holding after actions α have been performed. Logics of announcements specialize the actions to dynamic logic by permitted agents to announce statements, including statements about their own knowledge. Logics of announcements are traditionally given a semantics in terms of updating Kripke structures. Roughly speaking, a formula such as “ϕ is true after announcing that agent i knows ψ” is true at a state of some Kripke structure M if ϕ is true in the new Kripke structure obtained by updating M with the result of announcing Ki ψ. There are several difficulties with this approach to reasoning about announcements. How to update Kripke structures after a particular announcement is not immediately clear, for instance. But from a practical perspective, one difficulty is that traditional models for logics of announcements are quite abstract. It is usually nontrivial to take a given scenario and model it for analysis. In this paper, we revisit the semantics of logics of announcements, and focus on giving a concrete independently motivated semantics to such logics. The idea is simple: our models, based on the runs-and-systems framework of Fagin et al. [7,16], describe a concrete scenario to be analyzed in terms of the possible evolutions of a situtation, where the possible evolutions include announcements made by the various agents in the scenario. The models are purely descriptive. The logic is used to state properties of the scenario via an intuitive notion of truth. This approach is in contrast to that used in DEL [2], for instance, where modeling a scenario is done completely within the logic, by stating properties of the scenario within the logic itself as axioms, and reasoning about what can be derived from those axioms. As we argue, this sometimes raises the question of how to ensure that the scenario has been accurately captured in the logic. Such a question is less problematic to answer with a concrete semantics, because the model of a scenario is a description that is very close to the way in which the scenario is informally described. Our models are quite simple, and they already highlight several subtle issues with reasoning about announcements. For instance, synchrony, that is, whether agents know the time, impacts what knowledge agents gain after hearing an announcement, which is especially relevant when agents only announce statements that they know to be true. Another subtlety that our framework highlights is the role of strategies to understand how to model belief. Belief arises naturally when considering announcements that are not broadcast to all agents, but instead are heard by only a subset of agents. Belief, according to some approaches, can be taken to be what an agent knows if she assumes that not having heard an announcement means that that announcement was never made. Making this

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precise, we argue, can only be done if agents already have an idea about what strategies other agents in the systems are using to choose their actions. This paper is structured as follows. In §2, we introduce the syntax and semantics of the logic of announcement we use. In §3, we try to make precise the notion of honesty, where agents only announce true statements. In §4, we compare our approach with traditional logics of announcements. In §5, we turn to the question of announcements that are not broadcasted to all agents, and try to develop reasonable notions of belief in this context. We conclude in §6.

2

A Logic of Announcements

We start by defining a class of structures that correspond somewhat closely to execution of protocols or programs. We use the runs and systems framework of Fagin et al [7], which takes as models sets of execution traces. An interpreted system is a pair (R, π), where R is a set of runs, each run representing a possible execution of the system, and π is an interpretation for the primitive positions (or atoms). A run is a map from time to global states, where r(m) is the global state in run r at time m. (For simplicity, we take time to range over the natural numbers.) We call a pair (r, m) a point of r. Thus, each point corresponds to a global state of the system. A global state is of the form (se , s1 , . . . , sn ), where se is the local state of the environment, and s1 , . . . , sn are local states for each agent. If r(m) = (se , s1 , . . . , sn ), we write re (m) for se and ri (m) for si . Intuitively, the local state for an agent records the observations that the agent has made. Here, we consider observations to be announcements that the agent has received. We also record in the local state possible initial observations made by the agent. For instance, in the muddy children puzzle, initial observations for an agent include which other children are dirty. The local state of the environment records information which is not available to the agents. For example, in a coin-tossing scenario, the result of the coin toss and also the sequence of all announcements that have been made can be stored in the local state of environment. The announcements that an agent has received will be a subsequence of this sequence of announcements. An announcement consists of a message θ (in some language that we describe below, but is irrelevant for now), with an associated announcer a and a group β of recipients. We require that the announcer a be in β. We represent an announcement by the event θ!aβ appearing in the local state of agents receiving the announcement, that is, agents in β. As we shall see below, for bookkeeping purposes, we record all announcements in the local state of the environment as well. An announcement is public if its associated group of recipients is {1, . . . , n}, otherwise, it is private. Building announcements out of a language raises subtle issues, including whether we want to distinguish announcements that are logically equivalent. For instance, announcement (p ∧ q)!β is often taken to be the same announcement as (q ∧ p)!β . However, there are situations where “how” the announcement

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is presented is relevant. After all, an announcement is a message, and concretely, the message has a representation in the system. For instance, an announcement may be represented as a bit-string sent over a network link, and an agent may be able to distinguish two different bit-strings corresponding to two different presentations of the same announcement. If all one cares about is the interpretation of the announcement, as is the case in several announcement logics where announcements are taken to be sets of states, then there is no distinction between announcing p ∧ q and announcing q ∧ p. But this view is somewhat limiting. It turns out that the results of this paper do not depend on how we resolve this choice. For simplicity, we take two announcements to be the same when they are structurally equal, with the understanding that we could impose an arbitrary equivalence relation over announcements to capture announcements we want to consider equivalent (such as logical equivalence). We want to interpret knowledge in such systems. We take a standard Hintikka semantics for knowledge, where an agent knows a fact in some global state s if that fact is true at all the global states she considers possible alternatives to state s. To do so, we define, for each agent i, a relation over points of the system capturing which points agent i cannot distinguish. Intuitively, two points are indistinguishable to agent i if i has the same local state at both points. We define (r, m) ∼i (r , m ) if ri (m) = ri (m). Note that this makes ∼i an equivalence relation. We also define the reflexive transitive closure of ∼i relative to a subset β of agents, by taking (r, m) ∼∗β (r , m ) if there exists a sequence i1 , . . . , ik of agents in β with (r, m) ∼i1 (r1 , m1 ) ∼i2 · · · ∼ik−1 (rk−1 , mk−1 ) ∼ik (r , m ). The above description of systems is extremely general, suitable for defining a broad class of multiagent systems, and in fact more general than we want for defining announcement systems. In particular, we assume a specific structure to the local state of agents and the environment in announcement systems. We assume that the local state of an agent (including the environment) contains an initial local state ι and a list of sets of observations Sj , that is si = ι, S1 , · · · , Sm  where ι is an initial local state and each Sj is a set of observations, for i ∈ {e, 1, . . . , n}. We write θ!aβ ∈ ri (m) if θ!aβ appears in one of the sets of observations in the local state of agent i, including the environment, that is, if θ!aβ ∈ Sj for 1 ≤ j ≤ m and ri (m) = ι, S1 , . . . , Sm  . The following properties of a run r capture these properties and characterize systems that interest us: R1. Facts do not change during the run. Thus, for all primitive propositions p and times m, m ≥ 0, we have π(r, m)(p) = true if and only if π(r, m )(p) = true. R2. Announcements are accumulated in the local state of the environment. Thus, for all times m ≥ 0, if re (m) = ι, S1 , . . . , Sm  where ι is an initial state and S1 , . . . , Sm are sets of announcements, then re (m + 1) = ι, S1 , . . . , Sm , Sm+1 , where Sm+1 is a (possibly empty) set of announcements. R3. An announcement θ!aβ appears in the local state of the environment if and only if it appears in the local state of every agent in β. Thus, for all times m ≥ 0, announcements θ!aβ and agents i ∈ β, we have θ!aβ ∈ re (m) (as defined above) if and only if θ!aβ ∈ ri (m).

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R4. The local state for agent i at every point is obtained from the local state of the environment at that point, filtered to keep only announcements pertaining to i. Thus, for all times m ≥ 0 and agents i ∈ β, if re (m) = ι, S1 , . . . , Sm  where ι is an initial state and S1 , . . . , Sm are sets of an nouncements, then ri (m) = ι , S1 , . . . , Sm  where Sj = {θ!aβ ∈ Sj | i ∈ β} for all j ∈ {1, . . . , m}. Properties R2 and R3 together ensure that agents have perfect recall, that is, they do not forget announcements that they have heard in the past. R3 imposes no constraint on how the local state of agents records announcements, beyond the fact that all pertinent announcements that have been made appear in the local state. In particular, from a round to the next, if no new announcement is heard by the agent, then the local state of the agent need not change. In other words, an agent need not be aware of time passing if she has not heard any new announcements. Thus, R3 allows for asynchronous behavior [7]. Property R4 forces the representation of agent local states to record the number of rounds and the announcements made at each round. Thus, in a precise sense, if every run has property R4 in a system, then agents know the current time in a system—at a point (r, m), they will not consider a point with a different time m possible. Definition 1. An announcement system is a system in which every run satisfies R1–3. An announcement system is synchronous if every run additionally satisfies R4. All our work until now has remained completely independent of the language of announcements. We write announcements in an epistemic propositional language. Start with a set Φ0 of primitive propositions, representing the basic facts that we care about, and form the language LK n of announcements over n agents by closing Φ0 under ∧, ¬, and the modal operator Ki . As usual, we take ϕ ∨ ψ to be an abbreviation for ¬(¬ϕ ∧ ¬ψ), and ϕ ⇒ ψ an abbreviation for ¬ϕ ∨ ψ. We use θ to range over formulas in LK n. for reasoning about announcements, which inWe define a simple logic LK,A n cludes LK as a sublanguage. The logic has essentially only epistemic operators. n Again, we start with the set Φ0 of primitive propositions. We consider two additional kinds of propositions specifically for dealing with announcements: propositions of the form annj (θ, β), read “agent j has just announced θ to agents in β”, and announced(j, θ, β), read “agent j announced θ to agents in β at some point in the past.” There is some flexibility in the choice of propositions for the logic, as usual, and making a distinction between current announcements and past announcements turns out to be convenient for some classes of specifications. include ϕ1 ∧ ϕ2 , ¬ϕ, Ki ϕ (read “agent i knows ϕ”), and Cβ ϕ Formulas of LK,A n (read “it is common knowledge among the agents in β that ϕ”). ϕ, ψ ::= p | anna (θ, β) | announced(a, θ, β) | ϕ ∧ ψ | ¬ϕ | Ki ϕ | Cβ ϕ Again, we take ϕ∨ψ to be an abbreviation for ¬(¬ϕ∧¬ψ), and ϕ ⇒ ψ an abbreviation for ¬ϕ∨ψ. We also write EG ϕ as an abbreviation for ∧i∈G Ki ϕ. Since LK n

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is a sublanguage of LK,A , we consider θ as a formula of LK,A . Working with a n n two-layered syntax yields a simple logic with a straightforward semantics, avoiding problems caused by announcements such as anna (annb (θ, γ), β)—expressing, for instance, that the statement of an announcement has just been announced— which are difficult to interpret semantically. These nested announcements tend to be uncommon in applications. We interpret LK,A with respect to announcement systems and say formula ϕ n is true at a point (r, m) of system A, written (A, r, m) |= ϕ, defined inductively as (A, r, m) |= p iff π(r, m)(p) = true ∈ re (m − 1)) (A, r, m) |= anna (θ, β) iff θ!aβ ∈ re (m) and (m = 0 or θ!aβ  (A, r, m) |= announced(a, θ, β) iff θ!aβ ∈ re (m) (A, r, m) |= ϕ1 ∧ ϕ2 iff (A, r, m) |= ϕ1 and (A, r, m) |= ϕ2 (A, r, m) |= ¬ϕ iff (A, r, m)  |= ϕ (A, r, m) |= Ki ϕ iff for all (r , m ) ∼i (r, m), (A, r , m ) |= ϕ (A, r, m) |= Cβ ϕ iff for all (r , m ) ∼∗β (r, m), (A, r , m ) |= ϕ Some remarks about this semantics are in order. First, note that anni (θ, β) is true the first time θ!iβ is announced; subsequent announcements of θ!i] β will make the proposition false. This is not a problem for the examples we shall consider in this paper, and indeed, for most scenarios of interest. If repeated announcements are needed, one possibility is to add primitive propositions to the announcements along the lines of “this is the second such announcement”.1 Second, there is no specific proposition expressing that an agent i has actually heard an announcement. It is already expressible in our logic: Ki announced(j, θ, β)captures agent i hearing annoucement θ!jβ . We sometimes write ann(θ, β) for i∈β anni (θ, β), which can be interpreted as “some agent (necessarily in β) just announced θ to agents in β.” We say ϕ is valid in A if (A, r, m) |= ϕ for all r and m. A formula ϕ is valid with respect to a class of announcement systems C if A |= ϕ for all A in C. Example 2 (The muddy children puzzle). Consider a scenario with n+1 agents, n of which are children (numbered 1, . . . , n), one of which is their father (numbered 0). The primitive propositions are dirty[1], . . . , dirty[n], interpreted as “child i is dirty”. We can construct a system Amuddy for this scenario, with a run rD for every subset of {1, . . . , n} that we interpret as the children that are initially dirty. The initial local state of each child is a single bit recording whether that child is dirty or not. The interpretation π of primitive propositions is chosen so that dirty[i] is true in a run exactly when the bit of agent i in the initial local state is set. In the first round, if at least one child is dirty, the father announces to everyone that there is a dirty child: n   dirty[i] !0{0,1,...,n} θ1  i=1

1

Another possibility is to augment announcements in local state of agents with tags to differentiate the same textual announcement made at different times. A cursory look at our results reveals they would not be affected by this change.

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For the following k − 1 rounds, where k is the number of dirty children in the run, the father announces at round j: θj 

n  i=1

 ¬Ki dirty[i] ∧ ¬Ki ¬dirty[i] !0{0,1,...,n}

Note that the run r∅ corresponding to the case where there are no dirty children has no announcement. Let C ann be the class of all announcement systems. Let C syn be the class of all synchronous announcement systems. We can axiomatize reasoning in announcement systems as follows. The first set of axioms concerns propositional reasoning: Prop. Any instance of a valid formula of propositional logic MP. From ϕ and ϕ ⇒ ψ infer ψ The second set of axioms concerns reasoning about knowledge and common knowledge, and is the usual S5 semantics: K1. K2. K3. K4. K5.

Ki ϕ ∧ Ki (ϕ ⇒ ψ) ⇒ Ki (ψ) Ki ϕ ⇒ ϕ Ki ϕ ⇒ Ki Ki ϕ ¬Ki ϕ ⇒ Ki ¬Ki ϕ From ϕ infer Ki ϕ

The following axioms concern common knowledge: C1. CG ϕ ⇒ EG (ϕ ∧ CG ϕ) C2. From ϕ ⇒ EG (ψ ∧ ϕ) infer ϕ ⇒ CG ψ Finally, the following axioms capture properties of announcements: A1. annj (θ, β) ⇒ announced(j, θ, β) A2. announced(j, θ, β) ⇒ Ki announced(j, θ, β), where i ∈ β Axiom A2 extends to common knowledge immediately. Proposition 3. announced(j, θ, β) ⇒ Cβ announced(j, θ, β) is provable from the previous axioms. Proof. A straightforward derivation using C2 and A2. Let AXann be the axiomatization {Prop, MP, K1–5, C1–2, A1–2}. Recall that an axiomatization is sound relative to a class of models M if whenever a formula L is provable from the axioms, then that formula is valid in M. Theorem 4. AXann is a sound axiomatization for LK,A relative to C ann . n

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Proof. To prove soundness, it suffices to prove that every axiom in AXann is valid, and that every inference rule in AXann preserves validity. Axioms and inference rules Prop, MP, K1-5, and C1-2 are straightforward. Axiom A1 is clearly valid: if (A, r, m) |= annj (θ, β), then θ!jβ ∈ re (m) and either m = 0 or θ!aβ  ∈ re (m − 1); because θ!jβ ∈ re (m), then (A, r, m) |= announced(j, θ, β). Axiom A2 is also clearly valid: if (A, r, m) |= announced(j, θ, β), then θ!jβ ∈ re (m); because θ!jβ ∈ re (m) and i ∈ β, by property R3 of announcement systems, we have θ!jβ ∈ ri (m). Let (r , m ) ∼i (r, m). By definition of ∼i , θ!jβ ∈ ri (m ), and again by R3, θ!jβ ∈ re (m ), so (A, r , m ) |= announced(j, θ, β). Therefore, we have (A, r, m) |= Ki announced(j, θ, β). Consider the following seemingly reasonable alternative to A2: A2’. annj (θ, β) ⇒ Ki annj (θ, β) Axiom A2’ is not in fact sound for announcement systems. Roughly, this is because agents cannot know exactly when an announcement is made, since they may not be aware of time passing. If we restrict ourselves to synchronous announcement systems, however, A2’ is now sound. Let AXsyn be the axiomatization obtained by adding A2’ to AXann . Theorem 5. AXsyn is a sound axiomatization for LK,A relative to C syn . n Proof. We show the validity of A2’. If (A, r, m) |= annj (θ, β), then θ!jβ ∈ re (m) ∈ re (m − 1). Take any (r , m ) ∼i (r, m). By definition of and either m = 0 or θ!jβ  ∼i , θ!jβ ∈ re (m ). Because A is synchronous, we have m = m. Consider two cases. If m = 0, then m = 0, so (A, r , m ) |= annj (θ, β). Otherwise, θ!jβ  ∈ re (m − 1). By property R4, θ!jβ  ∈ ri (m − 1). By definition of ∼i , θ!jβ  ∈ ri (m − 1). By property R3 and m = m, θ!jβ  ∈ re (m − 1), and thus (A, r , m ) |= annj (θ, β).   Since (r , m ) was chosen arbitrarily, we have (A, r, m) |= Ki annj (θ, β). Note that A2 is in fact immediately derivable from A1 and A2’ via K1 and K5. We leave this easy verification to the reader. 2.1

Public Announcement Systems

A special class of announcement systems studied in the literature are public announcement systems. For instance, the announcement system in Example 2 is a public announcement system. Definition 6. An announcement system A is public if for every announcement θ!aβ in A, we have β = {1, . . . , n}. Let C pub be the class of all public announcement systems.

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The following axiom distinguishes public announcement systems: A3. ¬announced(j, θ, β), for β a proper subset of {1, . . . , n} As usual, axiom A3 is really an axiom schema; sample instances of A3 in LK,A 3 include ¬announced(1, p, {1, 2}) and ¬announced(2, p ∨ K2 q, {2, 3}). Let AXpub be the axiomatization AXann ∪ {A3}. relative to C pub . Theorem 7. AXpub is a sound axiomatization for LK,A n Proof. Public announcement systems are announcement systems, so by Theorem 4, all axioms and inference rules in AXann are sound. Axiom A3 is trivially valid, because there is no announcement θ!aβ with β  = {1, . . . , n} in any public announcement system. As before, we can obtain a sound axiomatization for synchronous public announcement systems by adding A2’ to AXpub .

3

Honest Announcements

Up until now, we have been unable to reason about the content of announcements. Intuitively, this inability comes from there being no restrictions on what can be announced. If we do restrict what can be announced, then we can reason to some extent about the content of announcements. A common restriction is to have agents be honest. Intuitively, an announcement is honest if its content is true. One interpretation of honesty, then, is to simply require that whenever an announcement θ!aβ is made, θ holds. But this seems too weak. In particular, it permits agents to be honest “by accident”, so to speak. We tend to think of honesty as a mental attitude of an agent saying true things. Not only is the agent saying true things, but the agent knows that the things she is saying are true. This interpretation of honesty is used in the security literature, where it is often weakened to require agents to simply believe that what they say is true [5]. We shall have more to say about belief in §5. If we take this last definition of honesty, then we can define an honest announcement system to be an announcement system in which agents only announce what they know is true. For instance, the announcement system in Example 2 is honest. We can characterize honest announcement systems semantically. Doing so is somewhat subtle because we have to take time into account. In particular, an honest announcement of θ!aβ requires that agent a knows θ. But when is that knowledge supposed to hold? Intuitively, knowledge that the announcement is true should hold just before the announcement is made. Because we take announcements to be instantaneous relative to the granularity of time in our models—an announcement is heard at the same time step it is made—it is easy to be inconsistent were we to require an announcement to be known to be true exactly when it is made, which is the same as when it is heard.

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To see this, consider the announcement (p∧¬K1 p)!2{1,2} of the Moore sentence [15] p ∧ ¬K1 p. Suppose for simplicity that this announcement is made in a synchronous system. (This argument does not depend on synchrony, but synchrony enables a simpler example.) If this is an honest announcement, then agent 2 knows p ∧ ¬K1 p when the announcement is made, and in particular K2 ¬K1 p holds at the time the announcement is made. When the announcement is heard by agent 1, then by synchrony agent 1 knows that agent 2 knows p∧¬K1 p, and in fact agent 2 knows that agent 1 knows p ∧ ¬K1 p, and in particular, because K2 p implies p, we have K2 K1 p holding when the announcement is heard by agent 1. If knowledge of the content of an announcement for honesty is determined at the same time as the announcement is heard, then we have both K2 ¬K1 p and K2 K1 p holding at the same time, an impossibility given our S5 interpretation of knowledge. This argument justifies a temporal characterization of honest announcements: R5. For all r, θ, a, β: (A, r, 0) |= ¬anna (θ, β) and for all m ≥ 1, if (A, r, m) |= anna (θ, β), then (A, r, m − 1) |= Ka θ. Note that Unlike R1–4, property R5 is not a property of individual runs, but rather a property of systems as a whole, because of the presence of the knowledge operator. Thus, honesty is a global property of a system. Definition 8. An announcement system is honest if it satisfies property R5. Let C hon be the class of all honest announcement systems. To axiomatize honest announcement systems, we need a more expressive language than the one we have been considering. In particular, we need to have formulas that relate two different time steps, to capture knowledge update. There are a few ways of doing this, and it is not clear a priori which is the more rea- ϕ, read “ϕ is sonable. For now, we add a past time temporal operator [13],  true in the previous time step, if one exists”, with the following semantics. -ϕ (A, r, 0) |=  - ϕ iff (A, r, m) |= ϕ (A, r, m + 1) |=  This is a so-called weak previous-time operator, meaning that it does not guarK,A, antee that a previous time actually exists. Let Ln be the logic obtained by K,A -. augmenting Ln with operator  To axiomatize honest announcement systems, we need axioms to capture  - ϕ, on top of an axiom to capture property R5. Axioms for  - ϕ are straightforward. P1. P2. P3. P4. P5. P6.

 - ϕ∧ - (ϕ ⇒ ψ) ⇒  -ψ  - ϕ ⇔ ( - false ∨ ¬ - (¬ϕ)) - false ∧  -p⇒p ¬ -p p⇒ ¬ - false ∧  - announced(j, θ, β) ⇒ announced(j, θ, β) -ϕ from ϕ infer 

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Note that P2 essentially says that ϕ holds in the previous time step either when there is no previous time step (which can be conveniently expressed as  - false) or if ¬ϕ does not hold in the previous time step. Axioms P3–4 capture the fact that primitive propositions do not change truth value during a run. Axiom P5 captures the fact that once an announcement has been made it remains in the local state of the receiving agents. Axiom P6 is the necessitation rule for the previous time modality. The honesty property R5 of announcement systems is now easy to capture: A4.  - false ⇒ ¬anna (θ, β) A5. anna (θ, β) ⇒  - Ka θ Let AXhon be the axiomatization AXann ∪ {P1–5, A4–5}. K,A, -

Theorem 9. AXhon is a sound axiomatization for Ln

relative to C hon .

Proof. It is trivial to check that P1–5 are valid in all announcement systems, not just honest ones. Clearly, A4 is valid in honest announcement systems, which do not allow announcements at time 0 by R5. We check that A5 is valid in honest announcement systems. Let (A, r, m) |= anna (θ, β). We consider two cases. If - Ka θ trivially by the semantics of  - . If m > 0, then m = 0, then (A, r, m) |=  by property R5, (A, r, m − 1) |= Ka θ, so that (A, r, m) |=  - Ka θ. What about what other agents can infer when an honest message is announced? The best we can do is to prove the following property. Proposition 10. For all agents i, Ki anna (θ, β) ⇒ Ki  - θ is provable in AXhon . Proof. This is a straightforward derivation first using A5 and K2 together - θ, and then using K1 and K5 to derive with P6 to derive anna (θ, β) ⇒  Ki anna (θ, β) ⇒ Ki  - θ, along with the usual propositional reasoning involved in such derivations. In other words, if an agent i knows that θ has been announced, then that agent knows that θ was true in the previous time step. If all the agents know that θ has been announced, we can derive a similar result for common knowledge: - θ is provable Proposition 11. For all groups G of agents, CG anna (θ, β) ⇒ CG  in AXhon . Proof. Similar to the derivation in the proof of Proposition 10, except for using the S5 properties of Cβ instead of Ki , and using C2 instead of the necessitation rule K5. Note that this result is completely independent of whether the group G is the same as the group β of recipients of the announcement. We see now that announcing a Moore sentence such as p ∧ ¬K1 p, of the kind we saw at the beginning of this section, is not a problem with our interpretation of honesty. Again, consider making such an announcement in the

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context of a synchronous system. By Proposition 10 and P1-P3, we have that when agent 1 hears agent 2’s announcement (given that he knows when exactly the announcement has been made), he gets to know that p, but also that before the announcement p was true and he did not know it then, that - ¬K1 p. For agent 2 by axiom A5 we have is ann2 (p ∧ ¬K1 p, {1, 2}) ⇒ K1 p ∧ K1  that just before he made his announcement agent 2 knew that p was true and agent 1 did not know p, that is, ann2 (p ∧ ¬K1 p, {1, 2}) ⇒  - K2 (p ∧ ¬K1 p). For another example, consider the honest announcement system Amuddy for the muddy children puzzle in Example 2. By proposition 10 we have that when the children hear their father’s announcement and k − 1 rounds of children’s announcements (given that they know exactly when each announcement has been made—note that Amuddy is a synchronous system), each of the dirty children i ∈ {1, · · · , n} knows that he must have previously been dirty, that is, if i ∈ D in run rD , then (Amuddy , rD , k) |= Ki  - dirty[i]. But they learn this after exactly k − 1 rounds, while just after they heard k − 2 announcements, they did not know that they are dirty, i.e. (Amuddy , rD , k − 1) |= ¬Ki  - dirty[i]. Clearly, a sound axiomatization for honest public announcement systems, in which all announcements are honest and public, can be obtained by taking AXhon ∪ {A3}. When the system under consideration is synchronous, then we can do a bit better than Propositions 10 and 11. First, synchronous systems induce a new axiom relating time and knowledge. - Ki ϕ ⇒ Ki  - ϕ. KP.  Intuitively, KP says that agents have total recall, and do not forget that they knew facts in the past—if an agent knew ϕ, then that agent knows now that ϕ was true then. The converse is not valid, as the following system A illustrates: suppose a system with two runs, r1 and r2 , and a primitive proposition p. Suppose that p is true on run r1 and false on run r2 . Suppose further that agent 1 consider (r1 , 0) and (r2 , 0) equivalent (and no other points equivalent to those), but does not consider any point equivalent to (r1 , 1). It is easy to check that (A, r1 , 1) |= K1  - p, but that (A, r1 , 1)  |=  - K1 p. Let AXhs = AXhon ∪ {A2’, KP}. K,A, -

Theorem 12. AXhs is a sound axiomatization for Ln nous announcement systems in C hon .

relative to synchro-

Proof. We show KP is valid in every synchronous announcement systems, not - Kϕ. We show (A, r, m) |= K  - ϕ. Conjust honest ones. Suppose (A, r, m) |=  sider two cases. First, the case where m = 0. Let (r , m ) ∼i (r, m). By synchrony, m = 0. By definition of  - , (A, r , m ) |=  - ϕ. Because (r , m ) was arbitrary, (A, r, m) |= K  - ϕ. Second, the case where m > 0. Let (r , m ) ∼i (r, m). We claim that (A, r , m ) |=  - ϕ. By synchrony, m = m > 0. So,  consider (r , m − 1). By property R4 of synchronous announcement systems, because (r , m) ∼i (r, m), we have (r , m − 1) ∼i (r, m − 1). By assumption

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(A, r, m − 1) |= Kϕ, meaning that (A, r , m − 1) |= ϕ. Thus, (A, r , m) |=  - ϕ. Since (r , m ) was arbitrary, we have (A, r, m) |= K  - ϕ, as required. In honest synchronous announcement systems, when an announcement is made, then every recipient of the announcement knows that θ was true before the announcement was made, and in fact, it is common knowledge amongst every subset of the recipients. Proposition 13 (a) anna (θ, β) ⇒ Ki  - θ is provable in AXhs , for i ∈ β; (b) anna (θ, β) ⇒ Cβ   - θ is provable in AXhs , for β  ⊆ β. Proof. Part (a) follows directly from A2’ and Proposition 10. Part (b) follows from part (a) and C2. A sound axiomatization for synchronous honest public announcement systems is obtained by taking AXhps = AXhon ∪ {A2’, A3, KP}. A variant of honest announcement systems are dishonest announcement systems in which agents can only lie, that is announce the negation of something they know is true. These are defined by replacing Ka θ by ¬Ka θ in the definition of honest announcement systems. A sound axiomatization can be obtained by replacing Ka θ by ¬Ka θ in A5.

4

Relationship with Public Announcement Logic

We compare our logic LK,A with Public Announcement Logic (PAL) [6] The n syntax of PAL is as follows F, G ::= p | F ∧ G | ¬F | Ki F | [N !j ]F where Ki F stands for “agent i knows that F holds” and [N !j ]F stands for “after agent j announces N , proposition F holds”,2 where N is given by N, M ::= p | N ∧ M | ¬N | Ki N In order to be compatible with the syntax of our logic, we work here with the original syntax of PAL where announcement formulas cannot themselves be announced. As we pointed out in Section 2, it is not immediately clear how to concretely interpreted nested announcements, and indeed even the abstract semantics of PAL is not entirely intuitive on this subject. We therefore focus on the simpler logic here, and save the investigation of nested announcements for future work. 2

Strictly speaking, the syntax of PAL does not explicitly mention announcers. Announcements are simply written N !. Announcers can usually be inferred from the context; here we make them explicit to more easily compare PAL to our logic.

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We show that our models validate the axioms of PAL. To make this statement precise, we translate PAL formulas to formulas in our logic in a way that translates PAL-provable formulas to provable formulas with a similar interpretation in our logic. We denote the translation of a PAL formula F by [[F ]] and define it inductively as follows [[p]]  p [[F ∧ G]]  [[F ]] ∧ [[G]] [[¬F ]]  ¬[[F ]] [[Ki F ]]  Ki [[F ]] [[[N !j ]F ]]  [[N !j ]]([[F ]]) where [[N !j ]]([[F ]])  annj ([[N ]], {1, · · · , n}) ⇒ [[F ]] Other than the usual propositional logic and epistemic axioms of S5, PAL uses the following specialized axioms to deal with announcements [N !j ]p ⇔ (prec(N !j ) ⇒ p) j

j

j

[N ! ]¬F ⇔ (prec(N ! ) ⇒ ¬[N ! ]F ) j

j

j

[N ! ]Ki F ⇔ (prec(N ! ) ⇒ Ki [N ! ]F )

(D1) (D2) (D3)

D1 is referred to as preservation of facts and says that epistemic actions do not affect the truth value of primitive propositions. D2 is referred to as partial functionality, and D3 is the action-knowledge axiom and says that agents know the consequences of announcements: they know a proposition after an announcement exactly when they know that making the announcement yields that proposition. The prec(N !j ) stands for the precondition of an announcement, that is, the formulas that should be true in order for the announcement to be made. Since the runs of our systems correspond to the actual evolution of a scenario independently of the logic, our runs encode when each announcement is being made. So we can simply translate the precondition of an announcement to the proposition that says the announcement has just been made, as follows: [[prec(N !j )]]  annj ([[N ]], {1, · · · , n}) For instance, in honest public announcement systems, this will imply  - Kj [[N ]] and thus  - [[N ]]; in dishonest public announcements it implies  - ¬[[N ]]. It is this flexibility, this possibility we have of interpreting preconditions in different ways, that lets our logic be more flexible than PAL, for instance by not being restricted to scenarios where every agent is honest.3 3

In most versions of PAL, the precondition is the formula that is being announced prec(N !j ) = N , which implies that in a scenario every true formula could be announced. This is of course not the case for most scenarios. In more general versions of PAL, such as Dynamic Epistemic Logic (DEL) [2], the precondition is kept as a variable whose instantiation depends on the scenario under examination.

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Consider the axiomatization AXhps for synchronous honest public announcement systems given in §3. The following theorem shows that our translation of PAL formulas preserves provability. Theorem 14. If F is a formula provable in PAL, then [[F ]] is provable in AXhps . Proof. It suffices to show that all the axioms of PAL are provable in AXhps . The propositional axioms of PAL translate to the propositional axioms Prop and MP, and the epistemic axioms translate to axioms K1–5, so it is enough to show that PAL’s announcement axioms are provable in AXhps . For simplicity, in this proof, we shorthand the translation of a formula as the formula itself, that is we write N instead of [[N ]] and F instead of [[F ]]. Since all the announcements in the proof are to the same receivers, we also shorthand annj (N, {1, · · · , n}) to annj (N ). – D1 translates to the following tautology.     annj (N ) ⇒ p ⇔ annj (N ) ⇒ p – D2 translates to the following, which is easily derivable using Prop.     annj (N ) ⇒ ¬F ⇔ annj (N ) ⇒ ¬(annj (N ) ⇒ F ) – D3 translates to      annj (N ) ⇒ Ki F ⇔ annj (N ) ⇒ Ki annj (N ) ⇒ F   We first show (annj (N ) ⇒ Ki F ) ⇒ annj (N ) ⇒ Ki (annj (N ) ⇒ F ) . Using Prop, this is equivalent to (annj (N ) ∧ annj (N ) ⇒ Ki F ) ⇒ Ki (annj (N ) ⇒ F ). To prove this implication, we first assume annj (N ) and annj (N ) ⇒ Ki F . By MP, we get Ki F . By synchrony (A2’) and MP we also get Ki annj (N ). Therefore, we can conclude Ki (annj (N )) ∧ Ki F , which is equivalent to Ki (annj (N ) ∧ F ) by K1–5, which is equivalent to Ki (annj (N ) ∧ (annj (N ) ⇒ F )) by K1–5 and Prop, which is equivalent to Ki annj (N ) ∧ Ki (annj (N ) ⇒ F ) by K1–5, which in turns implies Ki (annj (N ) ⇒ F ). Thus, (annj (N ) ∧ annj (N ) ⇒ Ki F ) ⇒ Ki (annj (N ) ⇒ F ), as required.  The reverse implication is similar, and simpler. First, to show annj (N ) ⇒  Ki (annj (N ) ⇒ F) ⇒ (annj (N ) ⇒ Ki F ), it is sufficient toprove the equivalent (by Prop) annj (N ) ∧ annj (N ) ⇒ Ki (annj (N ) ⇒ F ) ⇒ Ki F . Therefore, assume annj (N ) and annj (N ) ⇒ (Ki (annj (N ) ⇒ F )). By MP, we get Ki (annj (N ) ⇒ F ). By K1 and Prop, this last is equivalent to Ki annj (N ) ⇒ Ki F . By synchrony (A2’) and by the assumption that annj (N ) holds, we and therefore by MP that Ki F holds, know that Ki annj (N ) holds,   which is what we wanted. Thus annj (N ) ∧ annj (N ) ⇒ Ki (annj (N ) ⇒ F ) ⇒ Ki F , as required.

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Informally speaking, for the ⇒ direction, assume that if an announcement has just happened, then agent i gets to know that F is true. By synchrony, this also implies that he knows that the announcement has just happened, so he knows the conjunction of both, which implies the right hand side. For the ⇐ direction, assume that if an announcement has just happened, then if agent i knows that the announcement has just happened then he knows that F is true. But by synchrony, i already knows that an announcement has just happened, so he knows that F is true, which is what the left-hand side is expressing. Thus every derivable formula of PAL is so (after translation) in AXhps . Although we do not yet have a similar result for truth (that our translation preserves the truth of PAL formulas), a weaker corollary about preservation of validity (truth in all models) immediately follows. Proving that our translation preserves truth of PAL formulas within a specific model requires developing a faithful translation of the PAL models to the models of our logic. We save this for future work, and return to this topic in the conclusion. Corollary 15. If F is a valid formula of PAL, then [[F ]] is valid in synchronous honest public announcement systems. Proof. Assume that F is a valid formula of PAL, since PAL is complete, F is derivable from the axioms of PAL. As a result, by theorem 14 its translation [[F ]] becomes derivable in AXhps and since AXhps is sound, we obtain that [[F ]] is valid in synchronous honest public announcement systems. Example 16. In the muddy children puzzle in Example 2, n  

n   dirty[i] !0 Ki dirty[i]

i=1

i=1

is derivable in PAL. Its translation to our logic ann0 (

n 

dirty[i], {0, 1, . . . , n}) ⇒ Ki

i=1

n 

dirty[i]

i=1

is derivable: Proposition 13 derives ann0 (

n 

dirty[i], {0, 1, . . . , n}) ⇒ Ki  -

i=1

n 

dirty[i]

i=1

  - dirty[i], since  - preThis implies ann0 ( ni=1 dirty[i], {0, 1, . . . , n}) ⇒ Ki ni=1  serves disjunction in one direction. The latter implies ann0 (

n 

i=1

dirty[i], {0, 1, . . . , n}) ⇒ Ki

n 

dirty[i]

i=1

by P3 and the fact that there is a previous time step before the father’s announcement.

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So PAL is interpretable in our logic. What about the other way around? Is our logic interpretable in PAL, or is our logic more expressive? One way to answer this question is to provide a converse to Theorem 14. Part of the problem here is the previous-time modality of our logic.4 As we saw, the previous-time modality seems necessary to handle Moore sentences. This suggests that our logic is more expressive, in some sense, than PAL. Recently, PAL has been extended with temporal modalities, including a previous-time operator [18,19]. The study of how these logics relate to ours is left to future work. Theorem 14 and corollary 15 extend to Public Announcement Logic with Common Knowledge. The syntax of this logic is an extension of PAL’s with a common knowledge operator Cβ F , which we translate as [[Cβ F ]]  Cβ [[F ]]. Its axiomatization is the same as PAL’s except that axiom D3 changes to [N !]Cβ F ⇔ (prec(N !) ⇒ Cβ [N !]F ).

(D3’)

Theorem 17. If F is a formula provable in PAL with common knowledge, then [[F ]] is provable in AXhps . Proof. D3’ follows from Cβ annj (N, {1, · · · , n}) ⇔ annj (N, {1, · · · , n}), which is derivable from axioms of common knowledge and synchrony. Corollary 18. If F is a valid formula of PAL with common knowledge, then [[F ]] is valid in synchronous honest public announcement systems. Proof. Assume that F is a valid formula of PAL with common knowledge, since PAL with common knowledge is complete, F is derivable from the axioms of PAL with common knowledge. As a result, by theorem 14 its translation [[F ]] becomes derivable in AXhps and since AXhps is sound, we obtain that [[F ]] is valid in synchronous honest public announcement systems.

5

Private Announcements and Belief

Private announcements, that is, announcements not broadcasted to every agent in the system, have held no special status in this work until now. Several researchers have argued, however, that we can say something more in the presence 4

A natural translation that comes to mind is the map that simply forgets the previousto PAL, such that [[ - ϕ]]−1  [[ϕ]]−1 , time modality, that is, the map [[ ]]−1 from LK,A n sends the implication by an announcement (for both ann and announced formulas) to the dynamic modality, i.e. [[annj (ϕ, β) ⇒ ψ]]−1  [[[ϕ]]−1 !j ][[ψ]]−1 , and acts as identity on all other formulas. It is not hard to show that this translation does not preserve provability; the announcement of a Moore sentence such as the one in §3 provides a counterexample.

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of private announcements. Roughly speaking, they propose that when an agent does not hear an announcement, she should assume that no announcement has been made, and reason accordingly. Consider the following sample scenario with three agents, Alice, Bob, and Zoe. Alice tosses a coin, the outcome of which is not seen by any agent other than Alice. Alice then announces to herself and to Bob the result of the toss. Suppose we have two propositions, H and T , true when the result of the coin toss is heads and tails, respectively. It is simple to model this scenario in our framework using a system consisting of two runs rH and rT , where the initial state of Alice and the environment in each run records whether the coin lands heads (run rH ) or tails (run rT ). At time 1, Alice announces the result of the coin toss, her announcement is recorded in the local states of Alice and Bob. Thus, the local states of the environment are reT (1) = T, T !A {A,B} , and the local states T T A T of agents are rA (1) = T, T !A , r (1) = −, T ! B {A,B} {A,B} , but rZ (1) = −, and H similarly for rZ (1). It is easy to check that (rH , 0) |= ¬KB (H) ∧ ¬KB (T ), while H (r , 1) |= KA (H) ∧ KB (H) ∧ ¬KZ (H) ∧ ¬KZ (T ), and (rH , 1) |= ¬KZ KB H. If we do take for granted an interpretation of private announcements with the view that an agent having not heard an announcement assumes that that announcement was never made, then Zoe thinks that ¬KB (H) at the second state of rH . We say “thinks” here instead of “knows” because knowledge is generally taken to satisfy the truth axiom Ki ϕ ⇒ ϕ, and so clearly Zoe cannot know ¬KB (H). In some sense, this interpretation of private announcements, and most importantly, of what information agents can derive from private announcements, amounts to a form of belief. Intuitively, Zoe believes that Bob does not know H because, having not seen the private announcement, she assumes that there was no announcement. DEL [2], for instance, defines i to be a belief operator with that kind of interpretation. In DEL, for the above scenario, it is possible to derive that Z ¬B (H). DEL’s definition of belief is somewhat abstract, however, and given a purely proof-theoretic interpretation. Our concrete semantics, based on actual system evolution, let us explore how to rationally capture such an interpretation for belief. Intuitively, to capture such an interpretation, we need to define belief in such a way that the outsider agent i who did not hear any announcement considers possible other states where other agents have in fact not received the purported private announcement. But of course, those states where agents have not received the announcement are not “real” states, that is, they are not states in the scenario being modeled, and therefore need to be added to the model. The question becomes, then, what states, and therefore what runs, should be added to the system to capture this kind of belief. It turns out that it is in general not wholly possible to extract such an interpretation for a description of the scenario being modeled, but rather we have to make assumptions on the initial knowledge of the various agents in the scenario about the strategy followed by all the agents in the scenario. In order to make all of this precise, we need to carefully examine how systems are, in fact, generated. We follow an approached inspired by [7, Chapter 5].

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Intuitively, a system is generated by each agent following a strategy (a protocol, a program), which is simply a description of what actions the agent may take as a function of her local state. Actions, for our purposes, will consist of making announcements (although we could incorporate other actions, such as tossing coins to randomize strategies). A joint strategy P consists of a strategy for every agent in the system, and describes how the system evolves. We can formalize the general idea above as follows. For illustration purposes, here, we define only how to generate synchronous announcement systems. Fix a set Li of local states for agent i (the local states that arise in some system) and a set Ai of possible actions that agent i can perform. An action is any (finite) number of announcements. This can be formalized by taking Ai to be the class of all finite subsets of {ann(θ, β) | θ ∈ LK n , β ⊆ {1, . . . , n}}; we view ∅ as the action of making no announcement. A strategy Pi for agent i is a function that associates with every local state in Li a nonempty subset of actions in Ai . Intuitively, Pi (s) is the set of actions that agent i may perform in local state s. Notice that agent i’s actions can depend only on her local state. Also, and in contrast to some definitions of strategy in other fields, strategies for us are nondeterministic—they do not prescribe which actual action to take in a specific state, but give the set of possible actions that can be performed. To consider the effect of all the agents’ strategies on the global state of the system, we define a joint strategy (P1 , . . . , Pn ), consisting of a strategy for each of the agents, that associates with each global state a subset of possible joint actions, a subset of A ⊆ A1 × · · · × An . Joint actions transform global states. To capture their effect, associate with every joint action (a1 , . . . , an ) a function from global states to global states. Formally, we let τ (a1 , . . . , an ) be a function taking a global state (se , s1 , . . . , sn ) to (se , s1 , . . . , sn ), where si = si · {θ!jβ | ann(θ, β) ∈ aj and i ∈ β} se

= se ·

{θ!jβ

for i ∈ {1, . . . , n}

| ann(θ, β) ∈ aj }

and · denotes sequence concatenation. Given a joint strategy (with the above associated interpretation τ for joint actions) and a set of initial global states, we can generate a system in a straightforward way. Intuitively, the system consists of all the runs that are obtained by running the joint strategy from one of the initial global states. More formally, say that run r is consistent with joint strategy P if it could have been generated by P , that is, for all m, r(m + 1) is the result of applying a joint action a that could have been performing according to joint strategy P to r(m). (More precisely, there exists a joint action (a1 , . . . , an ) such that ai ∈ Pi (ri (m)) and r(m + 1) = τ (a1 , . . . , an )(r(m)).) Given a set I of global states, a joint strategy P , and an interpretation π, the announcement system AP,I = (RP,I , π) consists of all the runs RP,I consistent with joint strategy in P that starts in some initial global state in I.

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In our example at the beginning of the section, initial states for Alice include the result of the coin toss, either T or H, the initial state for Bob is empty, the initial state for Zoe is empty. Let IABZ be the corresponding set of initial states. The strategy PA for Alice is to announce H or T to {A, B} in the first state, and do nothing thereafter. The strategy PB for Bob is to do nothing. The strategy PZ for Zoe is to do nothing. The strategies are simple, and it is easy to check that (with the appropriate interpretation for the primitive propositions) A(PA ,PB ,PZ ),IABZ is the system we described at the beginning of the section. To understand the notion of belief where an agent behaves as though no announcement has been made at all when she does not hear an announcement, we need to understand what states the agent believes exist, and we propose to use the notion of strategy, and most importantly, those strategies that an agent believes other agents are using, as opposed to the real ones they are using. Presumably, in the actual scenario, each agent is using a strategy.5 What we need to add to the scenario is the strategy that each agent believes the other agents are using. Thus, for instance, Zoe, in the above example, may believe that Alice tosses a coin but does not send an announcement to anyone. (This is of course not Alice’s actual strategy, who does in fact make an announcement to Bob.) We take an announcement system for belief to be an announcement system generated by a joint strategy P = (P1 , . . . , Pn ) and a set of initial states I, along with a map Pi for every agent i indicating the strategy Pi (j) that agents i believes agent j is using. Because agents are generally not mistaken about their own strategy, we take Pi (i) to be the actual strategy Pi that agent i is using, If A is the system for belief as defined above, define Ai = A(Pi (1),...,Pi (n)),I for each agent i, that is, the system generated by the strategies that agent i believes the other agents are using. In other words, Ai represents agent i’s view of the scenario. Returning to our coin-tossing example, we can suppose that Zoe does not believe Alice does any announcement, meaning that PZ (A) is the strategy where Alice does nothing. We can take PZ (B) to be the strategy where Bob does nothing as well. (PZ (Z) is Zoe’s actual strategy, again to do nothing.) Thus, AZ here is the announcement system where there are no announcement made on any run. We can define a new modality Bi that captures what agents believe in an announcement system for belief. At a point (r, m) of A, take (A, r, m) |= Bi ϕ iff for all (r , m ) in Ai with (r , m ) ∼i (r, m), (Ai , r , m ) |= ϕ We extend ∼i to all points of A and Ai by again taking two points to be ∼i equivalent if agent i has the same local states at both points. An agent believes ϕ at some point (r, m) of A if ϕ is true at every point of the system that the 5

For simplicity, here, we consider scenarios where agents use a single strategy. It is not difficult to generalize to agents using one of a set of strategies, but the result does not offer more insight, although it may prove more applicable in practice.

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agent believes is the system describing the actual scenario in which she has the same local state. We can now check that if A is the announcement system for belief generated by joint strategy (PA , PB , PA Z) and initial states IABZ corresponding to the cointossing example, and letting PZ be as above, we get (A, rH , 1) |= BZ ¬KB (H), that is, Zoe believes that Bob does not know that H was the result of the coin toss, clearly a false belief, but consistent with the interpretation of belief we set out to capture. That Bi ϕ represents belief is justified by the fact that it satisfies the KD45 axioms for belief—it satisfies all the axioms for knowledge except for Bi ϕ ⇒ ϕ, at least on the runs corresponding to the real system A. It is then easy to show that the translation of axiom D3 where Ki is replaced with Bi is still derivable in our logic. Note that if an agent i believes that Pi (j) = Pj for all j, that is, that the strategies that the other agents are using are the actual strategies they are using, then Bi ϕ ⇒ Ki ϕ holds for all runs of A, so that Bi ϕ ⇒ ϕ holds for all runs of A.

6

Conclusion

Logics of announcements have proved popular as formal languages for reasoning about agents that explicitly exchange messages involving statements about their and other agents’ knowledge. The main question that these logics must address is how to update the state of knowledge of an agent based on hearing an announcement of some other agent’s knowledge. Assessing whether this knowledge update is suitable for a specific application of a logic of announcements requires a clear semantics that makes the interpretation of the update apparent. Unfortunately, the semantics of most logics of announcements has remained fairly abstract. There has been comparatively little work on developing natural semantics for logics of announcement grounded in the actual execution of a system, or evolution of a scenario. We have initiated such a development in this paper, describing a concrete semantics for a simple logic of announcements, where the semantics is given in terms of sets of runs of a system, each run recording exactly what is happening in the system during each execution. This means, in particular, that our models are independent of the choice of the logic used to reason about them. In fact, announcements have no special status in our models. An announcement is just a message like any other. Announcements impact the knowledge of other agents like any other message in systems of communicating agents. And to model situations where announcements have a specific meaning that can affect other agents in specific ways, it suffices to impose constraints on the possible executions of a system. We examine honesty as a particularly simple instance of constraints on executions allowing for a connection to be made between a message being announced and information about the content of the message being disseminated. Our work here is a first step in the development of natural grounded semantics for logics of announcements. There are several questions to be resolved, several

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choices to explore. A question that we have tried to address, but to which we do not yet have a satisfactory answer, is that of the exact relationship between PAL (for instance), and our logic. As it stands, the relationship we exhibit in this paper is a preservation of provability (and therefore of validity): if a formula F is provable in PAL, then its translation is provable in our logic. While this is the right starting point, it does not completely capture the intuition that our translation preserves the meaning of PAL formulas. A stronger theorem would establish that a PAL formula that is true in a model M remains true after translation in the systems AM corresponding to model M . This aspect of the relationship between PAL and our logic remains to be studied, possibly along the lines of recent results of [4,18] on relating models of ETL [17] to models of DEL [2]. To more firmly ground our concrete semantics, we need to extend our logic to be able to interpret more expressive logics of announcements [6], dynamic versions of PAL such as DEL. The issues that arise here amounts to understanding the relationship between the branching time semantics of DEL with the more linear time semantics of the runs-and-systems framework. One possibility is to move to a branching-time form of runs and systems, such as found in [14]. Aside from looking at more dynamic forms of PAL, it would also be interesting to look at temporal extensions of PAL, such as recent work by Sack [18] and Yap [19]. Finally, we believe some of the sound axiomatizations we presented in this paper are in fact complete for their respective classes of announcement systems, or close to be. It would be interesting to establish such completeness results.

References 1. Baltag, A., Coecke, B., Sadrzadeh, M.: Epistemic a Actions as Resources. J. Log. Comput. 17(3), 555–585 (2007) 2. Baltag, A., Moss, L.S.: Logics for Epistemic Programs. Synthese 139, 165–224 (2004) 3. van Benthem, J.: One is a Lonely Number. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) LC 2002. Lecture Notes in Logic, vol. 27, pp. 96–129. AK Peters, Wellesley (2006) 4. van Benthem, J., Gerbrandy, J., Pacuit, E.: Merging Frameworks for Interaction: DEL and ETL. In: Samet, D. (ed.) TARK 2007, pp. 72–81. ACM Press, New York (2007) 5. Burrows, M., Abadi, M., Needham, R.: A Logic of Authentication. ACM Trans. Comput. Syst. 8(1), 18–36 (1990) 6. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library, vol. 337. Springer, Heidelberg (2007) 7. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995) 8. Gerbrandy, J.: Dynamic Epistemic Logic. In: Moss, L.S., Ginzburg, J., de Rijke, M. (eds.) Logic, Language, and Information, vol. 2, pp. 67–84. CSLI Publication, Stanford (1999) 9. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000) 10. Hintikka, J.: Knowledge and Belief. Cornell University Press (1962)

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11. van der Hoek, W., Wooldridge, M.: Time, knowledge, and cooperation: Alternatingtime temporal epistemic logic and its applications. In: Arbab, F., Talcott, C. (eds.) COORDINATION 2002. LNCS, vol. 2315, p. 4. Springer, Heidelberg (2002) 12. Kripke, S.: A Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi. Z. Math. Log. Grundl. Math. 9, 67–96 (1963) 13. Lichtenstein, O., Pnueli, A., Zuck, L.: The Glory of the Past. In: Parikh, R. (ed.) Logic of Programs 1985. LNCS, vol. 193, pp. 196–218. Springer, Heidelberg (1985) 14. van der Meyden, R., Wong, K.: Complete Axiomatizations for Reasoning about Knowledge and Branching Time. Stud. Log. 75(1), 93–123 (2003) 15. Moore, G.E.: A Reply to my Critics. In: Schilpp, P.A. (ed.) The Philosophy of G. E. Moore. The Library of Living Philosophers, vol. 4, pp. 535–677. Northwestern University (1942) 16. Panangaden, P., Taylor, K.: Concurrent Common Knowledge. Distributed Comp. 6, 73–93 (1992) 17. Parikh, R., Ramanujan, R.: A Knowledge Based Semantics of Messages. J. Log. Lang. Inf. 12(4), 453–467 (2003) 18. Sack, J.: Temporal Languages for Epistemic Programs. J. Log. Lang. Inf. 17(2), 183–216 (2008) 19. Yap, A.: Product Update and Looking Backwards. ILLC Publications PP-2006-39, Universiteit van Amsterdam (2006)

From Nash to Dependency Equilibria Wolfgang Spohn Fachbereich Philosophie, Universit¨ at Konstanz, Universit¨ atsstraße 10, 78464 Konstanz, Germany [email protected]

Abstract. As is well known, Nash equilibria assume the causal independence of the decisions and the actions of the players. While the independence of the actions is constitutive of normal form games, the independence of the decisions may and should be given up. This leads to the wider and quite different notion of a dependency equilibrium; e.g., cooperation in the single-shot prisoners’ dilemma is a dependency equilibrium. The paper argues this notion to be meaningful and significant and sketches some of its consequences.

1

Introduction

Game theory is now 65 years old, and it has had a breath-taking career. It has become the foundational theory of all economics; countless economic phenomena have found a game theoretic explanation; countless economic problems have found a game-theoretic solution. No doubt, these achievements are at least comparable to the Bourbaki program in mathematics. Indeed, its claim reaches far into all of social sciences; even here it exhibits great success, although the extension of its claim is contested. The picture of a rationally deciding social individual that game theory explicates dominates large parts of our cultural and political ideology. In all that the notion of a Nash equilibrium is the foundation of game theory. Almost all theoretical efforts deal with it or build on it. Many equilibrium concepts have been invented in the meantime, but almost all lie between the narrowest, the notion of a strict Nash equilibrium, and the widest, the notion of a Nash equilibrium (see the survey diagrams in [14, pp. 335f]). The notion was and remains extremely compelling, also for me. I have become doubtful, though. The notion rests on an assumption of which all are aware and which appears entirely obvious, namely the assumption of the causal independence of the decisions and actions of the players. This assumption is unjustified, as I will show not with the help of exotic scenarios, but through a straightforward way of reasoning. If one drops the assumption, one is automatically led to the wider notion of a dependency equilibrium, as I call it. Dependency 

I am most grateful for having received very valuable comments from Christoph Fehige, Christian List, Ita Sher, the editors of this volume, and two anonymous referees.

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 135–150, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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equilibria behave in a significantly different way; for instance, mutual cooperation is a dependency equilibrium in the single shot prisoners’ dilemma. This indicates that the foundations of game theory might undergo dramatic changes, if that notion were taken seriously—changes that would not deny, but fully turn over the picture of a rationally deciding individual and that would thus have significant theoretical and even ideological consequences. I shall attempt to make these claims credible in the next four sections. In Section 2 I shall briefly explain the notion of a Nash equilibrium and how the said assumption is built into it. In Section 3 I shall explain which equilibrium concept ensues when this assumption is given up. In Section 4 I shall justify why the denial of this assumption is not only not absurd, but natural and often mandatory. Section 5 concludes with a few more general and comparative remarks. I should emphasize at the outset that this is not a formal paper. I have made some rudimentary glimpses at the formal theory of dependency equilibria in [35,37]. There is not much sense, though, in developing it further, unless its meaningfulness is clearly established, something rather obscured by formal activism. Therefore, Section 4 will be the core section of this paper where I shall try to explain its meaningfulness in as plain terms as possible. Once this is accepted, we may, and indeed should, continue elaborating the formal theory.

2

Nash Equilibria

Let us focus on two-person games in normal form. The conceptual generalization of our considerations should be obvious; the transfer to other forms of game representation, in particular to the extensive form, should be thought through. Let us call the two players Ann and Bob. Ann has a set A = {a1 , . . . , am } of options; these may be few simple actions as in scissors, paper, stone or many complex strategies as in chess that prescribe a response to each possible course of the game. Correspondingly, Bob has the set B = {b1 , . . . , bn } of options. The actual complete course of the game, the outcome, depends not only on their decisions, but possibly on further contingencies not fully controlled by the players, the so-called moves of nature. The outcomes or complete courses of the game are evaluated by the players in similar or different ways. In the normal form, however, Ann’s and Bob’s evaluations get reduced to evaluations of their possible strategy combinations that already contain their expectations with respect to the more or less favorable outcomes ensuing from the strategy combinations. So, let u be the evaluation or utility function of Ann and v that of Bob; both are functions from A × B, the set of strategy combinations, into R, the set of reals. According to the standard theory, Ann and Bob do not only have pure, but also mixed strategies. Let S be the set of Ann’s mixed strategies, i.e., the set of all probability distribution s over A; similarly, T is to be the set of all distributions t over Bob’s set B of pure strategies. So, if the mixed strategy combination s, t is played, Ann’s expected utility is

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s(ai ) · t(bj ) · u(ai , bj )

i=1 j=1

and likewise for Bob. Now, as is well known, a strategy combination s, t is a Nash equilibrium if and only if no player can improve by individually deviating from the equilibrium, i.e., if for all of Ann’s mixed strategies s ∈ S,   s(ai ) · t(bj ) · u(ai , bj ) ≥ s (ai ) · t(bj ) · u(ai , bj ) i,j

i,j

or, equivalently, for all ak ∈ A   s(ai ) · t(bj ) · u(ai , bj ) ≥ t(bj ) · u(ak , bj ) i,j

j

and if the corresponding inequalities hold for Bob. Each game has at least one Nash equilibrium in mixed strategies. s, t is a strict Nash equilibrium iff each player can only lose by individually deviating, i.e., if “≥” is replaced by “>” in the above inequalities. Strict Nash equilibria need not exist; but if they do, they do so only in pure strategies (where s(ai ) = 1 and t(bj ) = 1 for some i and j). Traditionally, game theorists assumed that players actually choose mixed strategies by employing some random device determining the pure strategy to be played. In epistemic game theory, grounding in many papers of John Harsanyi, more explicitly conceived, e.g., in [1] and [33], and finally established in the theory of rationalizability of [7] and [27], a different interpretation emerged that is more appropriate for our context: A Nash equilibrium s, t may also be conceived as an equilibrium of opinions. For, why should Ann choose mixed strategy s1 ? This can only be reasonable when she does not care which of the pure strategies ai with s(ai ) > 0 results from playing s. But how can she be indifferent? Only when all a i with s(ai ) > 0 are equally good for her, i.e., have the same expected utility j t(bj ) · u(ai , bj )—where t now represents Ann’s opinion about Bob’s pure strategies. This indifference is guaranteed in the equilibrium s, t. The same holds for Bob’s pure strategies bj with t(bj ) > 0, when s represents his opinion about Ann’s possible actions. Only in such an equilibrium of opinions can the opinions of the players be mutual or common knowledge among the players (as it has been assumed in game theory all along with respect to the structure of the game and the utility functions of the players)2. Ann cannot stick to her opinion t about Bob and at the same time guess that Bob may not 1 2

Cf. also the old critical considerations of [13]. Common knowledge usually denotes the full infinite hierarchy of mutual (and reflexive) beliefs. As observed in [33, sect. 4] and emphasized in [4] first-order and mutual second-order beliefs suffice, at least in the two-person case; Aumann & Brandenburger ([4]) call this mutual knowledge. For this paper it is not important to carefully distinguish between mutual and common knowledge (and, for that matter, between knowledge and belief).

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have the opinion s about her (as long as she is convinced that he is an expected utility maximizer). Now one may further ponder about the justification of such equilibria. One may derive the rationality of the opinion equilibrium from the rationality of the mixed strategy equilibrium; if it is rational for Ann to play the mixed equilibrium strategy s and if Bob takes Ann to be rational, then Bob must rationally have the opinion s about Ann. This is how one can view the matter provided one has already justified the rationality of the mixed strategy equilibrium in some other way. If one prefers to do without that presupposition—as I do—, one may try to directly show the rationality of the opinion equilibrium or to derive if from common knowledge or common prior assumptions. I have extensively discussed all this in [33], with the somewhat skeptical conclusion that justification carries only up to the rationalizable strategies, as they were called and more deeply investigated by Bernheim ([7]) and Pearce ([27]). The issue is further elaborated by Aumann & Brandenburger ([4]), in particular for the more complicated morethan-two-person case. Be this as it may, let us accept the familiar assumption that what is rational must be sustainable in public, i.e., may be mutual or common knowledge. Then, obviously, only Nash equilibria can be maintained as rational. However, this is the gist of the matter, the crucial assumption indicated in the introduction is already contained in this abstract representation of the social situation of Ann and Bob. Let me explain: In a mixed strategy equilibrium s, t Ann and Bob play their strategies each by their own; there is no playing of a joint distribution p over A × B, as it is the case with Aumann’s [2,3] objective correlated equilibria. The tossing of the one has no causal influence on the tossing of the other; s and t are assumed to be causally independent. Of course, this holds as well for the actions or pure strategies selected; what Ann does has no influence on what Bob does, and vice versa.3 Well, how could Ann’s action have an influence on Bob’s? The usual way would be that Bob sees, or is in some other way informed about, what Ann does and behaves accordingly. However, this is explicitly excluded; it would have to be modeled in a different way. Of course, subconscious influences or even more exotic scenarios are excluded all the more. The same holds for the preferred weaker interpretation of s, t as an opinion equilibrium. Ann’s opinion consists in an unconditional distribution t over Bob’s possible actions which she thereby accepts as something she cannot influence, just as, say, tomorrow’s weather. If she thought to have at least probabilistic influence on Bob’s actions, then her probabilities for Bob’s actions should depend on her own actions, i.e., unlike t 3

At this point it is advisable to conceive pure strategies as single actions. The latter clearly stand in causal relations. Whether one can say so of complex contingency plans is at least questionable, since they are temporarily and modally extended, i.e., they plan for many possible situations most of which do not realize. In fact, “plan” already contains the ambiguity I shall emphasize later on: the adoption of a plan may be a local event capable of causal relations, its execution is not.

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they should vary with her own actions. Vice versa for Bob. That is, if we conceive the opinion equilibrium as a Nash equilibrium, we have built into it the belief of the players in the mutual causal independence of their actions.4 To be sure, the last conclusion is not entirely cogent. It presupposes a relation between conditional subjective probabilities and causal opinions that is in need of justification; after all, deterministic and probabilistic causation are most contested notions. However, the relation as I have just presupposed it appears most plausible and has strong backings.5 I shall return to it in Section 4. This is the snag. I shall certainly not doubt the causal independence of Ann’s and Bob’s actions; that would be absurd. The causal independence of Ann’s and Bob’s decisions or intentions, however, is something subtly different. It is the point I shall question; and this will then have specific consequences for the form of Ann’s and Bob’s subjective probabilities. In Section 4 I shall explain what this may mean. For the moment, I only want to dismiss the conclusion that Ann’s and Bob’s opinions about each other consist in unconditional subjective probabilities. If we give this up, the concept of a Nash equilibrium is no longer applicable. What could stand in its place? This is the topic of the next section.

3

Dependency Equilibria

What we have to do now is to allow that Ann’s conditional probabilities for Bob’s actions or pure strategies vary. Thus, her opinions now take the form q(bj | ai ) where for each ai ∈ A, q(· | ai ) is a probability distribution over B. Reversely, Bob’s opinions now take the form r(ai | bj ), where for each bj ∈ B, r(· | bj ) is a distribution over A. What could it mean under these assumptions for Ann to be rational? It means to maximize conditional expected utility, i.e., to choose an ai for which  q(b j | ai ) · u(ai , bj ) is maximal. This was the important progress of Fishburn i ([17]) over Savage ([30]). Savage assumed the subject to have only unconditional probabilities for action-independent circumstances or states of the world. Fishburn found Savage’s distinction of states of the world and consequences and the accompanying conception of acts as functions from the former to the latter to be problematic and to be made superfluous by his assumption that the subject has probabilities conditional on each of her possible acts for the rest of world (where the acts are a primitive ingredient of the decision model); of course, this allows that some propositions, e.g., Savage’s states of the world, are probabilistically independent of the actions. In this conception, the subject then maximizes conditional expected utility. This is, I believe, generally accepted. So far, Ann’s and Bob’s subjective probabilities were not constrained by any conditions. However, we shall now take over the leading idea entailing Nash 4

5

Aumann & and Brandenburger ([4]) outright assume the opinions or conjectures about the other players to be of that unconditional form entailing causal independence. Hence, their results do not constrain my further considerations. Cf. [32, sect. 3.3] [31, Sect. 3.7], [25], and [28, Ch. 3 and 4].

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equilibria and assume that these probabilities are no secrecy, but must be sustainable in public, i.e., can be or even are common knowledge. This entails two substantial constraints. The first constraint did not show up in Nash equilibria, since they satisfied it, anyway. It is that Ann’s and Bob’s conditional probabilities must combine into a single joint distribution p over A × B, i.e., there must be such a p such that for all i and j, p(bj | ai ) = q(bj | ai ) and p(ai | bj ) = r(ai | bj ). This condition may fail to be satisfied, since q and r together have more degrees of freedom than p. The combinability of Ann’s q and Bob’s r into a joint distribution p follows, however, from the possibility of common knowledge. If such a joint distribution p does not exist, then Ann cannot know Bob’s probabilities r, moreover know that Bob knows her probabilities q and still stick to her q. This is so far just a consistency constraint that, as mentioned, was automatically satisfied in the case of Nash equilibria and that has the consequence that from now on we may restrict attention to that joint distribution p. The second constraint is induced by the common knowledge of rationality. According to the p in question we have p(ai ) > 0 for some of Ann’s actions ai ∈ A; hence, each such ai must have positive probability in Bob’s view conditional on at least some of his actions. How can this be? Bob knows that Ann is rational, i.e., that Ann maximizes conditional expected utility. If Ann achieves this only with ai and no other action, we should have p(ai ) = 1 and Bob should be certain that Ann does ai . If, however, several of Ann’s actions are optimal in Ann’s view and Bob rightly assigns them positive probability, then all of them must have equal and maximal conditional expected utility for Ann. Mutatis mutandis, the same holds for Bob. (Certainly, the derivation of the two constraints should be carried out in formal detail.) Thus, the mutual or common knowledge or knowability assumption leads us to the following equilibrium concept. The probability distribution p over A × B is a dependency equilibrium iff for all i with p(ai ) > 0 and all k = 1, . . . , m   p(bj | ai ) · u(ai , bj ) ≥ p(bj | ak ) · u(ak , bj ) j

j

and reversely, for all j with p(bj ) > 0 and all l = 1, . . . , n   p(ai | bj ) · v(ai , bj ) ≥ p(ai | bl ) · v(ai , bl ) i

i

i.e., if all of Ann’s and Bob’s actions that are not excluded and have positive probability according to p have, respectively, the same maximal expected utility for Ann and Bob. There is no space and no need to formally develop this concept here. I restrict myself to a few remarks. First, for some ak ∈ A or bl ∈ B we may have p(ak ) = 0 or p(bl ) = 0 so that no conditional probabilities are defined for them and the definition just given makes no sense. This defect may, however, be removed in a precise and adequate way (cf. [37, Sect. 2]). Second, I should remark that dependency equilibria are not to be confused with the correlated equilibria of [2,3]. An objective correlated equilibrium is also a joint

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distribution p over A×B. However, the issue here is roughly whether or not it pays for a player to break the dependency given by p and do something instead that is optimal under the marginal distribution over the actions of the other player given by p. If there is no such advantage, then no player will break the dependency and p is a correlated equilibrium. Even this very coarse description shows that correlated and dependency equilibria are driven by different ideas. Dependency equilibria form a wider class than Nash equilibria. Those distributions p over A×B that factorize into independent s over A and t over B—Nash equilibria apply only to such p—are obviously Nash equilibria if and only if they are (degenerated) dependency equilibria according to our definition. The way in which dependency equilibria go beyond Nash equilibria (and also diverge from correlated equilibria) is illustrated in [37, Sect. 3] with some significant examples. The most important of them is the single-shot prisoners’ dilemma. Its normal form is given, for instance, by the utility matrix of Table 1, where Ann is the row chooser and Bob the column chooser and where c stands for “cooperate” and d for “defect”. It is obvious that d strictly dominates c and is thus preferred to c, given the independence of the other’s choice. Hence, d, d is the only Nash equilibrium and even the only correlated equilibrium in the prisoners’ dilemma. Table 1. The prisoners’ dilemma. For a pair (u, v), u represents the utility of Ann, and v that of Bob. c d c (2,2) (0,3) d (3,0) (1,1)

However, there surprisingly are two whole families of dependency equilibria, one in which the players are asymmetrically or negatively correlated and one in which they are symmetrically or positively correlated. The latter are given by the matrix of Table 2 for all x with 0 ≤ x ≤ 1. Table 2. Two families of dependency equilibria for the prisoner’s dilemma p

c

d

1 c 12 x(1 + x) x(1 − x) 2 1 1 d 2 x(1 − x) 2 (1 − x)(2 − x)

The fact that both, cooperation and defection, have the same conditional expected utility according to such a p must be checked by calculation. It is, however, immediately obvious that p(c, c) converges to 1 if x does and that p(c, c) = 1 for x = 1. That is, p(c, c) = 1 or certain mutual cooperation is a dependency equilibrium that consists in each player’s belief that the other will cooperate if and only if he or she cooperates. It is even the weakly Paretodominant dependency equilibrium; in no other dependency equilibrium do the

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players have a higher conditional expected utility.6 For details see again [37, Sect. 3]. I shall return to the example. I admit, however, that so far the theory of dependency equilibria is developed only in a most rudimentary way. This is due to their novelty and their perhaps doubtful significance, but also due to mathematical intricacies (in nperson games one has to solve systems of quadratic equations for many variables instead of linear equations). At least, though, we have in the two-person case that each pure strategy combination ai , bj  or each p with p(ai , bj ) = 1 that weakly Pareto dominates a Nash equilibrium is a dependency equilibrium (for a proof see [35, pp. 208f.]). Ita Sher pointed out to me that the same sort of proof should apply for showing that exactly those pure strategy combinations are dependency equilibria that weakly Pareto dominate the maximin strategies of the players. That suggests at least that exactly those combinations of pure strategies that are at least as good as the maximin strategies and that are not Pareto dominated by other strategy combinations are the most interesting from the point of view of the theory of dependency equilibria. Since Nash equilibria are also dependency equilibria, the existence of the latter is guaranteed. However, the selection problem is graver for the latter. With respect to Nash equilibria this problem was that even in view of many Nash equilibria one may hope to be able to justify a rational selection from them and thus to come to more specific recommendations. Whether this is feasible is contested within the standard theory.7 As for dependency equilibria, I have presently nothing to say on this issue. Still, already these few remarks suggest that game theory might considerably change when based on the notion of dependency instead of Nash equilibria. This brings us back to the question: Why should we take this notion seriously? So far, it must be understood as imputing to both players the belief that their actions have a causal influence on the other’s action. Since such causal loops are impossible—it would be bizarre to deny this—, at least one of the players and presumably both must be massively in error. However, a notion that builds on such error is uninteresting; it could not be common knowledge. At the end of Section 2 I had claimed that this is not our problem. This claim is still very mysterious. The next section attempts to solve the mystery.

4

The Causation of and Correlation between Actions

The argument I shall give now may sound involved; its core, however, is quite simple. In [35, sect. 3-5] I have elaborated a formal version. Here, I venture an 6

7

This consideration seems to resemble the justification of mutual cooperation by the so-called mirror principle (cf. [16]). However, this justification always seemed to me to be inconclusive ([35, p. 250]) and to receive sufficient support only through dependency equilibria. Moreover, the mirror principle works only when the situation of the players is perfectly symmetric, whereas the theory of dependency equilibria is not restricted in this way. [20] certainly was the most heroic attempt at a general solution of the selection problem.

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informal short version that should be much more perspicuous, even though it does not render the formal version superfluous. As stated, our problem is to find an explanation of the action-dependent probabilities for the other player’s actions assumed by dependency equilibria, an explanation that does not impute mad causal beliefs to the players. The first thought may be that there is no problem at all; this is simply the old story of cause and correlation. Of course, any two variables and thus also two variables like the actions of the two players may be correlated, i.e., probabilistically dependent without being causally dependent, i.e., without the earlier exerting causal influence on the later. The most we can assert is Reichenbach’s principle of the common cause according to which two correlated variables must have a common cause, indeed a complete common cause rendering them independent, if neither causally depends on the other.8 In general, this is correct, no doubt, and I shall return to it. However, the second thought must be that this general point does not apply at all from the point of view of an agent, as far as his own actions are concerned. From his point of view, i.e., within his model of the decision situation, his actions are exogenous variables that have only effects, but no causes within the model. The rational agent wants to optimize the probable consequences of his actions; causes of his actions, whether they consist in his practical deliberation itself or lie elsewhere, simply play no role in the optimization of consequences.9 Hence, the agent cannot view a correlation between his actions and other variables as indicative of a common cause. In this special case, the correlation always represents a causal dependency. This insight is the whole point of the long-standing discussion about Newcomb’s problem (cf. e.g., [12]), in which the great majority has adopted the stance of causal decision theory that is characterized by this insight. If, counterfactually, my desire to smoke as well as my increased risk of lung cancer would exclusively be genetically caused and thus correlated, it would be silly to let the correlation ruin my desire; by refraining to smoke I cannot reduce my risk of lung cancer. The exogeneity of the action variables has been theoretically developed in the theory of truncated graphs ([28, sect. 3.2]) and of manipulated graphs ([31, sect. 3.7]); but see already [32, sect. 3.3 and 5.2]). Intuitively, though, the point was clear long before all those theoretical discussions. As a consequence, the only possibility to conceive equilibria was to conceive them as Nash equilibria. In general, also this second thought is correct. However, it has a gap in turn: When the deciding of the agent, i.e., his decision situation (that is nothing but his view of it), causally influences, in his view, not only his action, but also 8

9

The principle is widely, though not universally accepted; it is, for instance, a consequence of the Causal Markov Condition, the basic causal axiom of [31, sect. 3.4.1]. In my view it is even provable on the basis of a suitable explication of probabilistic causation; cf. [34]. I think that the uncausedness of one’s own actions from the perspective of one’s own practical deliberation is a fundamental sense of freedom of action and of the will; cf. [36].

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some other variable X, then and only then it is the case that his action, though not causally influencing the variable X in his view, is nevertheless correlated with the variable X, in a way relevant to his practical deliberation, i.e., to the conditional expected utility of his actions. That’s the crux I want to explain. Apparently, it is now somehow important that the actions are correlated with other variables due to common cause relations. Hence, we must first ask how actions are caused at all. Obviously in most complex ways; the most multifarious circumstances have an influence on our doings. However, since we consider only rational action by rational agents, all the influences must channel through the beliefs and the desires of the agent, through his representation of the decision situation including his subjective probabilities and utilities that result in an intention or decision for a certain action. This decision situation is the direct cause of that action according to the causal theory of agency of Hempel ([22]) and Davidson ([15]) that is still the dominating one today. (Of course, there must also exist some opaque mediation from mental state to bodily movement, something we need not consider here.) Each decision situation causes at least one action, indeed an action optimal according to it. It might also cause several actions; possibly, a whole course of action comes to be decided. Reversely, in the view of the agent each action can be caused only by exactly one decision situation: at least one, since otherwise it would not have been an intentional action, and at most one, because one cannot think to decide twice about one and the same action; if the agent envisages to take a second decision on the same action later on, he cannot think to make a decision at the first time. Moreover, a decision situation is a complete cause of the actions decided in it; there are no other causes of them besides the ingredients of the situation. Finally, the decision situation is only causally, but not necessarily temporally directly before the action; the decision is not always taken in the last moment. No doubt, this is a somewhat crude, idealized picture of the relation between decision situations, i.e., complex sets of graded beliefs and desires, and actions. I think, though, that what we can say about this idealized picture holds as well for more realistic, softer pictures. In particular I should emphasize that what I have called a decision situation in which a decision is made about the possible actions or options is nothing very well or sharply defined. The conscious introspection of one’s desires and beliefs and the explicit practical deliberation that results in a determinately localizable formation of an intention or decision is rather seldom; to carry out this exercise all the time would be much too demanding. Otherwise, however, it is often not so clear what the relevant beliefs and desires are and when exactly an intention has been formed. This is not to say, however, that the intention is never really formed or only in the last moment, when the action is about to be executed and can no longer be stopped. It is important to realize that in principle decision and action are temporally decoupled. Often I go to bed with a plan in mind that I simply execute the next morning. In our strange times, where to make the next vacation

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is something to be decided many months before. And I lay down my last will in the hope it will take 30 years or more till it comes into force.10 The crucial point now is this: If we thus explicitly consider actions as caused by decision situations, then we must as well allow that such complexes of beliefs, desires, and intentions have other effects besides the relevant actions and hence are common causes of actions and other things. We sometimes declare our intentions (even though this should perhaps be modeled as a further action). Often, our intentions, being closely connected to our emotions, can be glanced from our mimics, gestures, and other emotional responses. This is most important for human intercourse.11 Extremely controlled persons who allow a glimpse into their inner life only through their actions are somehow eerie. The much discussed toxin puzzle (see [23]) focuses exactly on this point by its fictitious story where only the forming of an intention (as measured by a cerebroscope) and not the intentional action itself is rewarded. How should the agent deal with such a possibility? Should he take into account such side effects of his decision? Yes, by all means. If in the toxin case the forming of an intention is rewarded, then I form the intention, provided the reward outweighs the negative consequences of carrying out the intention.12 If in the smoker’s case neither the relevant genes nor the smoking itself, but, in some mysterious ways, only the desire to smoke disposes to lung cancer, then I better do not have the desire (and consequently do not smoke). Yet, how can this point be accounted for in the representation of a decision situation? So far, this representation only contains the possible actions, all the other things or propositions the beliefs and desires are about, and, of course, the beliefs and desires themselves, but it does not, as it were, reflexively contain the possible decision situation itself as an additional variable. How could it? As we said, the causes of one’s own actions are not part of the practical deliberation. At this point, things become a bit involved.13 One can also introduce reflexive decision models and study their relation to the non-reflexive models hitherto considered. This relation results from the fact that the reflexive and the nonreflexive model represent in a sense the same decision situation (cf. [35, sect. 4.3]). The consequence of such a refined modeling should be clear, though. In the nonreflexive model the side effects of the decision situation show up precisely in a correlation between these side effects and the actions. Such a correlation finds no explanation at all in the non-reflexive model and could then, as we saw, never be regarded as admissible. In the reflexive expansion, however, we see a common cause of this correlation, namely the (non-reflexive) decision situation itself. It is in this case and only in this case that in the perspective of the agent his own 10 11 12 13

The temporal decoupling of decision and action is also an essential ingredient of Bratman’s account of intention, planning, and agency; see [11] and [10, chs. 1-4]. Frank ([18]) profoundly elaborates on this aspect of rational action. I have argued this in detail in [35, sect. 5.1]. One may even resort to Barwise ([5]) who devised model theoretic means based on set theory without the foundation axiom in order to accommodate such reflexive phenomena. I think, though, that such a move is not required.

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actions can be correlated with other things or variables without (probabilistically) causally influencing them. In this way, my line of argument returns to the dependency equilibria. For, we have seen now that the correlation between the actions of the players stated in these equilibria need not be interpreted as the false belief of causally influencing the other’s actions by one’s own actions. It can instead point to a common cause in the way indicated. What is the common cause in this case? If each player’s action is caused exclusively by her or his decision situation, and if this decision situation is moreover to exert influence on the other player’s action, then the common cause can only consist in the joint formation of the player’s decision situations. Is this plausible? Yes, I think so. I had mentioned already that such a decision situation is not sharply defined. It is temporally extended, unfolding in this extension; this leaves enough room for interactions and mutual dependencies that may arise from any form of communication between the players. In particular, it seems entirely plausible to me in the original prisoners’ dilemma story that the two gangsters do not take their decisions in the separate prison cells, as maliciously suggested by the police, but are decided all along, entangled in the cooperative dependency equilibrium that has formed in their sworn community during their raids. Let me summarize once more the picture that has thus emerged with respect to the prisoners’ dilemma. The players’ decision situations may develop over a certain stretch of time, and they may causally interact during this time. The effect is that the one player’s decision situation is causally responsible not only for her or his own action, but also for the other player’s decision situation and thus indirectly for his or her resulting action. If this process evolves under conditions of mutual or common knowledge, it must result in a dependency equilibrium. That may be any, but rationally it should be the Pareto-optimal cooperative dependency equilibrium in which each player believes the other to cooperate if and only if she or he cooperates and in which cooperation maximizes conditional expected utility. Of course, the players are free to break the mutual dependency; as stated, Nash equilibria are also dependency equilibria, though degenerated ones. The time of decision can always be chosen so late that causal interaction is excluded not only between the actions, but also between the decision situations. Sometimes, however, it is more reasonable to maintain the dependency than to break it. This is so at least in the prisoners’ dilemma.

5

Afterthoughts

I have thus reached the primary aim of my paper: namely to identify the apparently indispensable assumption that committed us to Nash equilibria, to explain its dispensability and thus to provide the wider concept of dependency equilibria with sense and significance. However, one will ask, did more than 50 years of proliferous game-theoretic thinking not produce similar ideas? Yes and no.

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First, I should mention that my proposal closely resembles the old theory of conjectural variation. It seems, though, that this theory was abandoned precisely for the reason that “at the level of simultaneous decisions in a single-period model, conjectural variation is not meaningful” ([19, p. 107]). So, one tried to make sense of it rather in the context of repeated games (cf. [19, sect. 5.1 and 9.3] and [29]). If I am right, this reason need not apply. Hence, it might be worthwhile elaborating on this resemblance. Conversely, of course, the theory of dependency equilibria needs to be extended to the context of repeated games, which might provide a theoretical explanation of how such a dependency may arise. There is a growing literature on correlation within the context of evolutionary game theory (cf., e.g., [6]). In particular, there are evolutionary explanations of the emergence of cooperation in the prisoners’ dilemma (cf. [21]. However, evolutionary game theory is governed by a different interpretation, an issue I cannot pursue here.14 Cooperative game theory does not look at how groups of players function internally, but rather presupposes correlation within possible coalitions, turning then to its own difficult problems. However, my proposal is located within non-cooperative game theory, and it might help understanding that inner functioning. For instance, it seems that any value of the characteristic function of a cooperative n-person game is the outcome of some dependency equilibrium of that game. In non-cooperative game theory as well correlation plays an increasing role. The main strand certainly originates from Aumann’s [2,3] invention of correlated equilibria. However, I had indicated how dependency equilibria diverge. The divergence continues with more recent inquiries. For instance, it is important to Brandenburger & Dekel ([8]) and Brandenburger & Friedenberg ([9]) that in the eyes of each player the acts or strategies of the other players may be correlated (in the more-than-two person case where there are at least two other players); the latter even say (p. 32) that this “is really just an adaptation to game theory of the usual idea of common-cause correlation”. Even one’s own acts may be correlated with those of the other players due to uncertain external circumstances on which all players’ acts depend according to their strategies. Still, the optimality of strategies is assessed according to their unconditional expected utility, as is characteristic of correlated equilibria, and not according to their conditional expected utility.15 There also are attempts to model mutual dependency as described in the previous section within what one might call standard non-cooperative game theory. In particular, Harsanyi & Selten ([20, pp. 4-7 and 18-23]) have enlarged the prisoners’ dilemma by a game of previous self-commitment moves and then showed that the Nash equilibrium of the enlarged game includes cooperation in the 14 15

In [35, pp. 251ff.] I have made more extensive comparative remarks. This is true also of the a posteriori equilibria of [8, p. 1395]), which are based on a comparison of conditional expected utilities, but in a different sense; each act is compared with other acts conditional on the same information.

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prisoners’ dilemma. And Myerson ([26, pp. 249–257]) has generally explained how a theory of so-called preplay communication is able to reduce Aumann’s correlated equilibria and the dependence encoded in them to Nash equilibria. These considerations are most instructive, and one should find out whether they might be applied to dependency equilibria as well. However, such a move should not be interpreted as possibly reducing away dependency equilibria (or, for that matter, correlated equilibria). Of course, one may try to describe such dependencies as the result of special games within the theory of Nash equilibria. Or one may reversely consider the existence of such dependencies as given and then, as I started to do, develop a theory about which behavior is rational when standing in such dependencies and thus also about how such dependencies are rationally to be shaped. The point then is not which theory is more general; that depends on the perspective. Dependency equilibria are obviously more general than Nash equilibria; and if the ideas just mentioned were successful, one might reversely be able to represent dependency equilibria as Nash equilibria in special games. The point is rather the radical change of our conception of rationality in game situations that comes with directly considering dependency equilibria. For instance, one usually thinks that defection would be completely rational in the one-shot prisoners’ dilemma and that cooperation in the finitely iterated prisoners’ dilemma can only be understood as some form of bounded rationality. It is, however, just the other way around. Cooperation is perfectly rational in the single-shot as well as in the iterated prisoners’ dilemma, and defection can only be explained by insufficient trust in the rationality of the other player or by insufficient common knowledge of rationality. One must free oneself from the idea that standard decision and game theory define what is rational; they only propose an explication of rationality that has deficiencies which show up in the iterated prisoners’ dilemma and elsewhere in a particularly drastic way and that must hence be improved. (And, as I would like to add, changes in one’s explication of rationality also change one’s conception of bounded rationality.) Broadening the perspective, one may even say that the ideological picture propagated in particular by economic theorizing is thereby falsified, the liberalistic picture of the freely and independently deciding individual, against whose self-interest the rationality of cooperation is hard to explain. This picture turns out wrong on the basis of the present considerations. We always stand in interpersonal dependencies; and individual rationality may tell us to acknowledge these dependencies to our own benefit (and that of others) and thus to lean towards a communitarian perspective. Genuine dependency equilibria always involve an element of commitment. I mentioned above that the time of decision can always be chosen so late as to exclude causal interaction between the decision situations. Not deferring the decision thus means remaining committed. Surely, this is another large topic for economics as well as for philosophy16 that I should not enter. However, it seems within the reach of my considerations to provide rationality criteria for the 16

E.g., see again Bratman ([11,10]).

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choice of decision times, for early commitment or late decision, for dependence or independence and thus to integrate socalled sophisticated and resolute choice introduced and discussed by McClennen ([24]) as two competing decision rules into a unified theory. Turning even more to philosophy, a last big topic involved in dependency equilibria is that of shared or joint or we-intentions and attitudes17 as pursued by Raimo Tuomela for decades (see, e.g., [38]) and others (see, e.g., [10, ch 5-8]). Perhaps, dependency equilibria help accounting for the internal structure of such joint intentions, a point related to my brief remark on cooperative game theory. Again, though, this is a suggestion I cannot pursue here. These associations open a rich agenda. Simply mentioning them is certainly bad scientific style. However, I wanted to suggest at least that the notion argued here to make good sense is indeed at the intersection of many pressing issues. Good reason to pursue it further.

References 1. Armbruster, W., B¨ oge, W.: Bayesian Game Theory. In: Moeschlin, O., Pallaschke, D. (eds.) Game Theory and Related Topics, pp. 17–28. North-Holland, Amsterdam (1979) 2. Aumann, R.J.: Subjectivity and Correlation in Randomized Strategies. J. Math. Econ. 1, 67–96 (1974) 3. Aumann, R.J.: Correlated Equilibrium as an Expression of Bayesian Rationality. Econometrica 55, 1–18 (1987) 4. Aumann, R.J., Brandenburger, A.: Epistemic Conditions for Nash Equilibrium. Econometrica 63, 1161–1180 (1995) 5. Barwise, J.: On the Model Theory of Common Knowledge. In: Barwise, J. (ed.) The Situation in Logic. CSLI Lecture Notes, vol. 17, CSLI, Cambridge (1990) 6. Bergstrom, T.C.: The Algebra of Asortative Encounters and the Evolution of Cooperation. Int. Game Theory Rev. 5, 211–228 (2003) 7. Bernheim, B.D.: Rationalizable Strategic Behavior. Econometrica 52, 1007–1028 (1984) 8. Brandenburger, A., Dekel, E.: Rationalizability and Correlated Equilibria. Econometrica 55, 1391–1402 (1987) 9. Brandenburger, A., Friedenberg, A.: Intrinsic Coerraltion in Games. J. Econ. Theory 141, 28–67 (2008) 10. Bratman, M.E.: Faces of Intention. Selected Essays on Intention and Agency. Cambridge University Press, Cambridge (1999) 11. Bratman, M.E.: Intentions, Plans, and Practical Reasons. Harvard University Press, Cambridge (1987) 12. Campbell, R., Sowden, L. (eds.): Paradoxes of Rationality and Cooperation. University of British Columbia Press, Vancouver (1985) 13. Chernoff, H.: Rational Selection of Decision Functions. Econometrica 22, 422–443 (1954) 14. van Damme, E.: Stability and Perfection of Nash Equilibria, 2nd edn. Springer, Berlin (1991) 17

Thanks to Christian List for pointing this out to me.

150 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

30. 31. 32. 33. 34.

35. 36.

37. 38.

W. Spohn Davidson, D.: Actions, reasons, and causes. J. Philos. 60, 685–700 (1963) Davis, L.: Prisoners, Paradox, and Rationality. Am. Philos. Q. 114, 319–327 (1977) Fishburn, P.C.: Decision and Value Theory. Wiley, New York (1964) Frank, R.H.: Passions Within Reason. The Strategic Role of the Emotions. W. W. Norton & Company, New York (1988) Friedman, J.: Oligopoly Theory. Cambridge University Press, Cambridge (1983) Harsanyi, J.C., Selten, R.: A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge (1988) Heiner, R., Albert, M., Schmidtchen, D.: Rational Contingent Cooperation in the One-shot Prisoner’s Dilemma (2000) (unpublished manuscript) Hempel, C.G.: Rational action. Proc. Addresses APA 35, 5–23 (1961) Kavka, G.S.: The Toxin Puzzle. Analysis 43, 33–36 (1983) McClennen, E.F.: Rationality and Dynamic Choice. Cambridge University Press, Cambridge (1990) Meek, C., Glymour, C.: Conditioning and Intervening. Br. J. Philos. Sci. 45, 1001– 1021 (1994) Myerson, R.B.: Game Theory. Analysis of Conflict. Harvard University Press, Cambridge (1991) Pearce, D.G.: Rationalizable Strategic Behavior and the Problem of Perfection. Econometrica 52, 1029–1050 (1984) Pearl, J.: Causality. Models, Reasoning, and Inference. Cambridge University Press, Cambridge (2000) Sabourian, H.: Rational Conjectural Equilibrium and Repeated Games. In: Dasgupta, P., Gale, D., Hart, O., Maskin, E. (eds.) Economic Analysis of Markets and Games, pp. 228–257. MIT Press, Cambridge (1992) Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954); 2nd edn., New York, Dover (1972) Spirtes, P., Glymour, C., Scheines, R.: Causation, Prediction, and Search. Springer, Berlin (1993); 2nd edn. (2000) Spohn, W.: Grundlagen der Entscheidungstheorie. Ph.D. thesis, Universit¨ at M¨ unchen (1976) Spohn, W.: How to make sense of game theory. In: Stegm¨ uller, W., Balzer, W., Spohn, W. (eds.) Philosophy of Economics, pp. 239–270. Springer, Berlin (1982) Spohn, W.: On reichenbach’s principle of the common cause. In: Salmon, W.C., Wolters, G. (eds.) Logic, Language, and the Structure of Scientific Theories, pp. 215–239. Pittsburgh University Press, Pittsburgh (1994) Spohn, W.: Dependency equilibria and the causal structure of decision and game situations. Homo Oeconomicus 20, 195–255 (2003) Spohn, W.: The core of free will. In: Machamer, P., Wolters, G. (eds.) Thinking About Causes. From Greek Philosophy to Modern Physics, pp. 297–309. Pittsburgh University Press, Pittsburgh (2007) Spohn, W.: Dependency equilibria. Philos. Sci. 74, 775–789 (2007) Tuomela, R.: Cooperation. Kluwer, Dordrecht (2000)

Bridges between Dynamic Doxastic and Doxastic Temporal Logics Johan van Benthem1,2 and C´edric D´egremont1, 1

Institute of Logic, Language and Computation, Universiteit van Amsterdam, Postbus 94242, 1090 GE Amsterdam, The Netherlands [email protected], [email protected] 2 Department of Philosophy, Stanford University, Stanford CA 94305, United States of America

Abstract. This paper builds bridges between the two main families of modal logics of belief change, both based on plausibility pre-orders: dynamic doxastic logics computing stepwise updates, and temporal doxastic logics describing global system evolutions. Following earlier results linking dynamic-epistemic and epistemic-temporal logics, we prove representation theorems showing under which conditions a doxastic temporal model can be represented as the stepwise evolution of a doxastic model under successive ‘priority updates’. This allows for merges, where, in particular, the notion of a ’temporal protocol’ defining a global information process (for instance of communication or learning) can be introduced into the more local dynamic perspective.

1

Introduction

Analyzing the behavior of agents in a dynamic environment requires describing the evolution of their knowledge as they receive new information. But equally crucial are agents’ beliefs over time, since most of our decisions and actions involve uncertainty, from going to work to selling shares. Beliefs, too, are information-based, and when refuted, they have to be revised in systematic ways. These phenomena have been studied in many different formal frameworks, including, game theory [14,5], belief revision theory [1], and formal learning theory [24,22]. In this paper, however, we are concerned with two logic-based approaches. One are dynamic logics for changing beliefs that have been developed recently (van Benthem [7], Baltag and Smets [3]) using plausibility relations between worlds to represent agents’ beliefs and conditional beliefs. An act of 

The first author acknowledges a Weilun Visiting Professorship of Humanities, Philosophical Institute, Tsinghua University, Beijing, 2008–2009. The second author was supported by a GLoRiClass fellowship of the European Commission (Research Training Fellowship MEST-CT-2005-020841). We thank Tomohiro Hoshi and Eric Pacuit for useful discussions. Also, Olivier Roy, Hans van Ditmarsch, Nina Gierasimczuk, Valentin Goranko, R. Ramanujam, Yu Junhua and Jonathan Zvesper gave helpful comments. Finally, the extensive comments of two referees helped us produce a more focused and hopefully clearer paper.

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 151–173, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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revision is then a single step of change in such a relation, triggered by some new incoming, hard or soft, information. Of course, such single steps can be iterated, leading to longer sequences. The other approach that we consider are doxastic temporal logics (cf. Halpern and Friedman [21], Bonanno [15]), representing time as a Grand Stage of possible histories where informational processes unfold. Dynamic and temporal logic seem the two major logical paradigms for agency, and this paper is a contribution to clarifying their connections. In doing so, we do not operate in a void. Similar questions have been solved for knowledge in van Benthem and Pacuit [12], van Benthem, Gerbrandy, Hoshi and Pacuit [11], in the form of representation theorems showing how sequences of models produced by ‘product update’ in dynamic-epistemic logic form a special subclass of epistemic temporal models in the sense of Fagin, Halpern, Moses and Vardi [20] and Parikh and Ramanujam [25]. In particular, these are the temporal models for agents endowed with Perfect Recall and ‘No Miracles’, learning by new observations only, possibly constrained by epistemic protocols. Our aim is to do the same for the dynamic doxastic logic of plausibility change by ‘priority update’, relating them to models of doxastic temporal logic. We shall identify the crucial agent features behind dynamic doxastic belief revision, and position them inside the broader temporal setting. This is not just a simple generalization of the epistemic case, but the benefits are similar: comparability of frameworks, and interesting new research questions once they are merged. In this paper, we concentrate on the representation aspect. Further development of the merged theory of dynamic agents in a doxastic temporal language and logic is found in the follow-up paper van Benthem and D´egremont [9]. We start in the next section with basic terminology and background on earlier results for the epistemic setting. In Section 3 we introduce plausibility models that model static multi-agent doxastic situations. We then present the dynamic step by step approach to belief change (Section 4), in particular, defining priority update. Next, the global temporal approach to beliefs over time is presented in Section 5. In Section 6 we show how step by step priority updates of a doxastic model, perhaps constrained by a protocol, generate a doxastic temporal model. The key temporal doxastic properties that characterize priority updaters are then identified and motivated in Section 7. In section 8 we prove our main result linking the temporal and dynamic frameworks, for the special case of total pre-orders, and then in general in Section 9. We discuss some variations and extensions in Section 10. Finally, in a last section, we state our conclusions, mention follow-up questions involving formal languages and complete logics, and discuss possible applications to belief revision theory and learning theory.

2

Introduction: Background Results

Epistemic temporal trees and dynamic logics with product update are complementary ways of looking at multi-agent information flow. Representation theorems linking both approaches were proposed for the first time in [6]. A nice presentation of these early results can be found in [23, ch5]. We briefly state a

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recent version from [11], referring the reader to that paper for a proof, as well as generalizations and variations. Definition 1 (Epistemic and Event Models, Product Update) = ∅, for – An epistemic model M is of the form W, (∼i )i∈N , V  where W  each i ∈ N , ∼i is a binary relation1 on W , and V : Prop → ℘(W ). = ∅, and for each i ∈ N , ∼i – An event model  = E, (∼i )i∈N , pre has E  is a relation on E. Finally, there is a precondition map pre : E → LEL , where LEL is the usual language of epistemic logic. We shall consider some generalizations of this precondition language later. – The product update of an epistemic model M = W, (∼i )i∈N , V  with an event model  = E, (∼i )i∈N , pre is the model M ⊗  whose states are the pairs (w, e) such that w satisfies the precondition of the event e and whose epistemic relations are defined as: (w, e) ∼i (w , e ) iff e ∼i e , w ∼i w and whose valuation is defined by (w, e) ∈ V (p) iff w ∈ V (p) An epistemic model describes what agents currently know, while product update creates the new epistemic situation after some informational event has taken place. Telling illustrations of the strength of this simple mechanism can be found in [2]. Next we turn to epistemic temporal models, introduced by [25] as a Grand Stage of unfolding informational events. In what follows, Σ ∗ is the set of finite sequences on any set Σ, which naturally forms a branching ‘tree’. Definition 2 (Epistemic Temporal Models). An epistemic temporal model (‘ET L model’) H is a tuple Σ, H, (∼i )i∈N , V  with Σ a finite set of events, and H ⊆ Σ ∗ closed under non-empty prefixes. For each i ∈ N , ∼i is a binary relation on H, and there is a valuation V : Prop → ℘(H). Here the set of histories H functions as a protocol defining all admissible trajectories of an informational multi-agent process. While such ET L models are very general, many special constraints are possible. Some are the usual assumptions in epistemic logic, like having accessibility be an equivalence relation for S5-agents. But more important here are properties connecting epistemic accessibility with flow of time, defining general properties of an informational process and the agents participating in it. Such agents can have more idealized or more bounded powers of observation, memory, and other cognitive features. In particular, the following epistemic temporal properties drive the main representation theorem in [11]: 1

The ∼i are often taken to be equivalence relations, if only for convenience, but such options are orthogonal to our main results.

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Definition 3 (Basic Agent Properties). Let H = Σ, H, (∼i )i∈N , V  be an ET L model. H satisfies: – Propositional stability : Whenever h is a finite prefix of h , then h and h satisfies the same proposition letters. – Synchronicity : Whenever h ∼i h , we have len(h) = len(h ). – Bisimulation Invariance : Whenever h and h are epistemically bisimilar, we have h e ∈ H iff he ∈ H, for all events e. – Perfect Recall : Whenever ha ∼i h b, we also have h ∼i h . – Uniform No Miracles : Whenever ga ∼i g  b then, for every h a, hb ∈ H, we also have h a ∼i hb.2 Dynamic-epistemic logic has borrowed once crucial idea from epistemic temporal logics. An epistemic protocol P maps states in an epistemic model to sets of finite sequences of pointed event models closed under taking prefixes. In general, this allows branching choices in a tree-like structure. This again defines the admissible runs of some informational process: not every observation may be available, or appropriate. More formally, let E be the class of all pointed event models, having one ‘actual event’ marked. Then the set of protocols is P rot(E) = {P ⊆ E∗ | P is closed under finite prefixes }. Next comes the more general notion used in the recent literature: Definition 4 (Local Protocols). Given an epistemic model M, a local protocol for M is a function P : |M| → P rot(E). In the particular case where the P is a constant function (mapping each world to the same set of sequences), we call the protocol uniform. Finally when the local protocol maps worlds to just a unique linear sequence of event models, we say that it is a line protocol. To avoid technicalities, in this paper we state results with uniform line protocols. But our results generalize: see [11] for the epistemic case. Indeed, under suitable renaming of events, making different event models disjoint, line protocols even have the same expressive power as general branching protocols. Now, given an epistemic model M as our initial situation, plus a uniform protocol P , we can define the resulting temporal evolution as an epistemic temporal model Forest(M, P ) = ∈P M ⊗ , the ‘epistemic forest generated by’ M through sequential application of the pointed event models in P using product update ⊗. Finally, we can state what iterated dynamic-epistemic update means in the broader setting of epistemic-temporal logic: Theorem 5 (van Benthem et al. [11]). Let H be an arbitrary epistemictemporal ET L model. The following two assertions are equivalent: – H is isomorphic to the temporal evolution Forest(M, P ) of some epistemic model M and uniform line protocol P , 2

This says essentially that agents only get new information by acts of observation.

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– H satisfies Propositional Stability, Synchronicity, Bisimulation Invariance, Perfect Recall, and Uniform No Miracles. Thus, epistemic temporal conditions describing idealized epistemic agents characterize just those trees that arise from performing iterated product update governed by some protocol. [11] and [23, ch5] have details. As stated in the Introduction, our paper extends this analysis to the richer setting of belief revision, where plausibility orders of agents evolve as they observe possibly surprising events. But to do so, we first need appropriate belief models, plus an appealing systematic revision mechanism. Important remark about languages. Before moving on, it is important to stress one feature of the preceding representation theorem and results in its family. The precondition languages for event models should exactly match the notion of bisimulation. This means that the language should be invariant under such bisimulations, and also, that it should be strong enough to characterize a pointed model up to such bisimulations. Two technical observations follow from this: 1. To get the right definability, we should either restrict attention to finitely branching ET L models (as in [11]), or alternatively, let the precondition function of product models take values in an infinitary epistemic logic. 2. These theorems can be parametrized, in the epistemic case, and even more so, the doxastic setting. We stay at a semantic level in this paper, and state our results up to language choice.3

3

Plausibility Models: Static Doxastic Situations

As with knowledge, we first introduce static models that encode current prior (conditional) beliefs of agents. These carry a pre-order ≤ between worlds standing for a plausibility relation. Often this relation is taken to be total, but when we think of beliefs in terms of multi-criteria decisions, a pre-order allowing for incomparable situations may be all we get [19]. We shall thus state our results for both total and arbitrary pre-orders. We write a b (‘indifference’) if a ≤ b and b ≤ a, and a < b if a ≤ b and b  ≤ a. In what follows, N = {1, . . . , n} is a fixed finite set of agents. The following definition is like the models in [14,18,3]: Definition 6 (Doxastic Plausibility Model). A doxastic plausibility model M = W, ( i )i∈N , V  has W  = ∅, while, for each i ∈ N , i is a pre-order on W , and V : Prop → ℘(W ). 3

The issue of language choice returns briefly in Section 10. The reader may also consult our companion paper [9] for an extensive discussion of syntactic issues, including other desiderata on the language, such as its expressive power for specifying the relevant properties of informational processes and the agents involved in them.

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w ≤i w means that w is considered at least plausible as w by agent i. Intuitively, the plausibility pre-orders encode current beliefs of agents. Here, we have taken them to be binary for convenience, but such relations can depend on states. An appealing intermediate case arises when we combine plausibility with an epistemic relation encoding ‘hard information’: Definition 7 (Doxastic Epistemic Model). A doxastic plausibility model = ∅, for each i ∈ N , i is a pre-order on M = W, ( i )i∈N , (∼i )i∈N , V  has W  W and ∼i is binary equivalence relation on W , and V : Prop → ℘(W ). We write: Ki [w] = {v ∈ W | w ∼i v}. A belief operator for i is really necessity with respect to the most plausible states (i.e. the ≤-minimal elements) of an information partition. Though this paper does not discuss syntactic issues, it may help to state how models like these support a natural epistemic-doxastic language: M, w  Ki φ iff ∀v with v ∈ Ki [w] we have M, v  φ M, w  Bi φ iff ∀v with v ∈ M in≤i (Ki [w]) we have M, v  φ The setting also supports new modalities. In fact, the necessity operator for ≥ ∩ ∼ is a weakly defeasible (S4)-knowledge operator of ‘safe belief’ ([3]): M, w  i φ iff ∀v with v ≤i w and w ∼i v we have M, v  φ Remark: Alternatives. Some authors use models with just primitive plausibility relations. One can then define epistemic accessibility for a single agent as the union of that relation with its converse, accessing also less plausible worlds. We return to this perspective briefly in Subsection 10.3. In what follows, we concentrate on pure plausibility models of our simplest sort, though our analysis will also work for more complex structures. We must now consider how such models evolve as agents observe events.

4

Dynamic Logics of Stepwise Belief Change (DDL)

Just like epistemic models, doxastic plausibility models change when appropriate triggering events are observed. It has become clear recently that a general mechanism for doing so works like the earlier product update ([3]). We start with the structures that describe complex doxastic events, crucially including the ways in which they appear to agents: Definition 8 (Plausibility Event Model). A plausibility event model (‘event = ∅, each i is a premodel’, for short)  is a tuple E, ( i )i∈N , pre with E  order on E, and pre : E → L, where L is the basic doxastic language. As in the epistemic case, our analysis will work for various precondition languages for doxastic events. One specific choice is found at the end of Section 7.

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Combining perspectives, an ’epistemic plausibility event model’ is a plausibility event model together with a collection of equivalence relations (∼i )i∈N on E. In the following update rule, a new event itself comes with instructions as to how prior beliefs may be overridden. The principle is like that of ‘Jeffrey Update’ for probabilities: we follow the preferences of the plausibility event model, but if it leaves things open, we stick with prior preferences: Definition 9 (Priority Update; [3]). Priority update of a plausibility model M = W, ( i )i∈N , V  and an event model  = E, ( i )i∈N , pre produces the plausibility model M ⊗  = W  , ( i )i∈N , V   defined as follows: – W  = {(w, e) ∈ W × E | M, w  pre(e)} – (w, e) i (w , e ) iff either e≺i e , or e i e and w i w – V  ((s, e)) = V (s) In the doxastic epistemic setting, Priority Update by epistemic plausibility event model combines the preceding mechanism with Product Update, i.e. it has one more clause: – (w, e) ∼i (w , e ) iff w ∼i w and e ∼i e More motivation for this rule can be found in [3], and at the end of this section. First here is a concrete example. As mentioned doxastic plausibility models are naturally combined with information partitions to describe scenarios involving both knowledge and beliefs. In this case Priority Update is applied to the plausibility ordering while product update is applied to information partition. We shall discuss this issue in connection with the temporal models in Section 10. Let us for now present a concrete scenario that involves both knowledge and beliefs. Reading the figures. In the following figures, the actual state (resp. event taking place) is the shaded one. Epistemic equivalence classes are represented by rectangles or ellipses. We use < to display the strict plausibility ordering within such classes. Our example assumes that all agents have the same plausibility ordering. i believes φ at w is interpreted as φ holds in the i-most plausible states within i-information partition Ki [w]. An agent’s beliefs at the actual state are thus displayed by an arrow from the actual state to the ones she considers most plausible, often just one. Thus, an arrow from x to y labelled by the agent Enzo means that y is the ≤e -minimal state within Ke [x]. A similar convention applies to the event-model. Finally, we omit reflexive arrows throughout. Example 10. Failed invitation. C´eline and Enzo would like to invite to Denis to their Wii party. The party has been decided but none of them has informed Denis yet. Denis considers it a priori more plausible that no Wii party is taking place unless informed otherwise. This initial situation is common knowledge between C´eline and Enzo. In the following figures, plain rectangles (or ellipses) will represent Denis’ epistemic partition, dashed ones Enzo’s and dotted ones C´eline’s. w and w are state names.

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<

w:¬p

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Fig. 1. No Wii Party unless stated otherwise. Initial model.

The key event model. The telephone rings and C´eline pick up the phone. Enzo hears part of the conversation and concludes that C´eline is inviting Denis. In fact C´eline is not on the phone with Denis. C´eline think it was clear from the conversation that she was not talking to Denis. Enzo

e: <

f : >

g:p

C´eline Fig. 2. Event model of a misleading phone call

We are now able to compute the new doxastic epistemic situation. The misunderstanding is now complete. In fact one can check that Enzo wrongly believes that it is now common knowledge between C´eline and Denis that there is a Wii party while C´eline wrongly believes that it is common belief between her and Enzo that Denis still does know about the Wii party and even that Denis still believes that there is no Wii party. Remark (Priority Update). In AGM style belief revision theory [1], new information is simply a new formula ‘to be believed’ by the agent. This allows for many different ‘revision policies’, from radical to conservative – a line also followed in a DDL setting by van Benthem [7]. It is important to appreciate that Priority Update is not just one such policy among many, but a general mechanism that can mimic many different policies depending on the richer structure of its triggers, viz. the plausibility event models [4]. If the event model has ‘strong views’, the update is radical, otherwise, the update remains conservative. Interestingly, this mechanisms also shifts the variety in belief revision away from fixed agent types, to case-by-case decisions: I can be radical with one input, and conservative with another.

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Denis Enzo

we

wf

we

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C´eline Fig. 3. Product model of a misunderstanding

We feel that a logic should describe a ‘universal’ mechanism, instead of a jungle of styles. This is why we have chosen Priority Update in this paper, leading to one representation that covers all special cases.

5

Doxastic Temporal Models: The Global View

We now turn to the temporal perspective on multi-agent belief revision, as an informational process over time with global long-term features. The following models are a natural doxastic enrichment of the temporal ET L models of [25]. They are also close to the temporal doxastic models of [15,21]. First the doxastic temporal models: Definition 11 (Doxastic Temporal Models). A doxastic temporal model (‘DoTL model’ for short) H is of the form Σ, H, (≤i )i∈N , V , where Σ is a finite set of events, H ⊆ Σ ∗ is closed under non-empty prefixes, for each i ∈ N , ≤i is a pre-order on H, and V : Prop → ℘(H). Doxastic Epistemic Temporal models (DET L models for short) are Doxastic Temporal models extended by a collection of epistemic equivalence relations (∼i )i∈N on H. Given some history h ∈ H and event e ∈ Σ, we let he stand for the concatenation of h with e. Given that plausibility links are not themselves events, the model H may again be viewed as a ‘forest’, a disjoint union of event trees. We sometimes refer to DoTL models as doxastic temporal forests. Figure 4 gives a concrete illustration of a practical setting with this abstract format. It display the evolution of a doctor’s knowledge (dashed rectangles) and belief (diagnosis) about what is wrong with her patient - as she performs medical tests and observe their positive or negative results (labelled edges). An arrow towards a state labelled Environ means that at this stage of the diagnostic process, the doctor think the patient’s symptoms have an environmental cause. We omit reflexive and symmetric arrows. Our models also gain concretenes by considering doxastic temporal languages interpreted on them. While these are the subject of our follow-up paper [9], we display a few truth conditions:

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H, h  eφ iff ∃h ∈ H with h = he and H, h  φ H, h  i φ iff ∀h  with h ≤i h and h ∼i h we have H, h  φ H, h  Ki φ iff ∀h  with h ∼i h we have H, h  φ H, h  Bi φ iff ∀h  with h ∈ M in≤i Ki [h] we have H, h  φ D´egremont [16] has comparisons of this framework with others, such as ‘belief functions’, or the models in [15].

Fig. 4. A medical investigation over time

6

From DDL Models to Doxastic Temporal Models

Now we come to the main question of this paper. Like AGM-style belief revision theory, Dynamic Doxastic Logic analyses one-step update scenarios. But unlike, AGM heory, it has no problem with iterating these updates to form longer sequences. Indeed let us put Example 10 together: Figure 5 looks like a doxastic epistemic forest model already. We shall make this precise now, but as in the epistemic case, we need one more ingredient. In many informational processes, such as learning, or belief revision in games, the information that agents receive may be highly constrained. Thus, there is crucial information in the set of admissible histories of the process, its ‘protocol’. This notion can be defined formally just as before in Definition 4. Let E be the class of all pointed plausibility event models. The set of protocols P rot(E) = {P ⊆ (E∗ | P is closed under finite prefixes}. What we need is again a slightly more flexible version:

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¬ed

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o nz

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ni s De C ´e li ne

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w-dd

Fig. 5. The Wii-party misunderstanding in temporal perspective

If we take a look at the figure describing Example 10 we see that it really looks like an doxastic (epistemic) forest already. Actually we could continue the story, and the further updates would generate a larger forest. More generally, priority update of a plausibility model according to a protocol generates a doxastic temporal forest. Definition 12 (Doxastic Protocols). Given a doxastic plausibility model M, a local protocol for M is a function P : |M| → P rot(E). If P is a constant function, the protocol is called uniform. When P maps states to a linear nested sequence of event models, we call it a line protocol. In line with Section 2, we state our main theorems in terms of uniform line protocols, leaving variations and extensions to [16]. Iterated Priority Update of a doxastic plausibility model according to a uniform line protocol P generates a doxastic temporal forest model. We construct the forest by induction, starting with the doxastic plausibility model and and then checking which events can be executed according to the precondition and to the protocol. Finally the new plausibility order is updated at each stage according to Priority Update. Since Priority Update describes purely doxastic, non-ontic change, the valuation stays the same as in the initial model. (For ways of adding real factual change, cf. [10].) For simplicity, we write P (w) =  where  is a finite sequence of event models. Definition 13 (DoTL model generated by a sequence of updates). Each initial plausibility model M = W, ( i )i∈N , V  and each sequence of plausibility sevent models  = (j )j∈ω where j = Ej , ( ji )i∈N , prej  yields a generated DoTL plausibility model Σ, H, (≤i )i∈N , V as follows:

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 – Let Σ := m i=1 Ei . – Let H1 := W , and for each 1 < n ≤ m, let Hn+1 := {(we1 . . . en ) | (we1 . . . en−1 ) ∈ Hn and M ⊗ 1 ⊗ . . . ⊗ n−1  pren (en )}.  Finally let H = 1≤k≤m Hk .  – If h, h ∈ H1 , then h ≤i h iff h M i h.     – For 1 < k ≤ m, he ≤i h e iff 1. he, h e ∈ Hk , and 2. either e≺ki e , or e ki e and h ≤i h . – Finally, set wh ∈ V(p) iff w ∈ V (p).

Our task is to identify just when a doxastic temporal model is isomorphic to the ‘forest’ thus generated by a sequence of priority updates. In particular, this will uncover the key doxastic properties of agents assumed in this belief revision mechanism.

7

Crucial Frame Properties for Priority Update

We first get a few more general properties of our information process out of the way. The first of these merely says that in that process, the facts of the world do not change, only agents’ beliefs about it: Definition 14. Let H = Σ, H, (≤i )i∈N , V  be a DoTL model. H satisfies propositional stability if, whenever h is a finite prefix of h , h and h satisfy the same proposition letters. Note that this can be generalized to include real world change. Next comes a basic property of the events that we allowed as revision triggers: 7.1

Bisimulation Invariance

Since the aim of this notion is to guarantee the existence of pre-conditions behind events in some modal language. Depending on the language parameter we choose, one has to choose the corresponding Bisimulation notion. As mentioned in Section 2 we shall state our results up to language choice, therefore we give an abstract definition of bisimulation below. We shall however give a concrete example of language instantation when stating a corollary of our result for doxastic epistemic models. Let τ be a finite collection of binary relations R1 , . . . , Rn  on H × H. Definition 15 (τ -Bisimulation) Let H and H be two DoTL models based on the same alphabet Σ. A relation Z ⊆ H × H  is a τ -Bisimulation if, for all h ∈ H, h ∈ H  and all Ri ∈ τ (prop) h and h satisfy the same proposition letters, (back) If hZh and hRi j, then there is a j  ∈ H  with jZj  and h Ri j  , (forth) If hZh and h Ri j  , then there is a j ∈ H with jZj  and hRi j. If Z is a τ -bisimulation and hZh , we say h and h are τ -bisimilar.

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Definition 16 (τ -Bisimulation Invariance). A DoTL model H satisfies τ bisimulation invariance if, for all τ -bisimilar histories h, h ∈ H, and all events e, h e ∈ H iff he ∈ H. Note that these definitions apply also to DET L models. Here is an example. (∼i ∩ ≤i )i∈N -Bisimulation Invariance will leave all formulas of the basic doxastic language with safe belief invariant, and hence our earlier preconditions for events. If we want these preconditions to be richer, then we need more clauses in the bisimulation – and the same is true if we want the bisimulation to preserve explicit temporal formulas involving events. 7.2

Agent-Oriented Properties

Now we come to the relevant agent properties. These depend on single agents i only, and hence we shall drop agent labels and prefixes “for each i ∈ N ” for the sake of clarity. Also, in what follows, when we write ha for events a, we assume that ha ∈ H. Definition 17. Let H = Σ, H, (≤i )i∈N , V  be a DoTL model. H satisfies: – Synchronicity Whenever h ≤ h , we have len(h) = len(h ). This says intuitively that agents have a correct belief about the exact stage the process is in. The following two properties trace the belief revising behavior of priority-updating agents more precisely: – Preference Propagation if ja ≤ j  b, then h ≤ h implies ha ≤ h b. – Preference Revelation If jb ≤ j  a, then ha ≤ h b implies h ≤ h . What do the latter properties say? In the earlier epistemic representation theorems, the corresponding properties of Perfect Recall and No Miracles described observational agents with ideal memory, the two basic features behind the Product Update rule. Likewise, our new properties express the two basic features ‘hard-wired into’ the Priority Update rule, its ‘radicalism’ and its ‘conservatism’. Preference Propagation says that, if the last-observed events ever allowed a plausibility preference, then they always do – or stated contrapositively, if they ever ‘over-rule’ an existing plausibility, then they always do. This reflects the first radical clause in the definition of Priority Update. Next, Preference Revelation says that when an agent has no strict plausibility preference induced by two observed events, then she will go with her prior plausibility. This reflects the second, conservative clause in Priority Update. As we have said before, this is a qualitative description of a ‘Jeffrey-style’ updating agent in a probabilistic setting.

8

The Main Representation Theorem

Now we prove our main result relating DDL and DTL models, both with total orders.

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Theorem 18. Let H be any doxastic-temporal model with a total plausibility order. Then the following two assertions are equivalent: 1. There exists a total plausibility model M and a sequence of total plausibility event models  such that H is isomorphic to the forest generated by the Priority Update of M by the sequence . 2. H satisfies Propositional Stability, Synchronicity, Bisimulation Invariance, Preference Propagation, and Preference Revelation. Proof. Necessity (1 =⇒ 2). We show that the given conditions are satisfied by any DoTL model generated through successive priority updates along some given protocol sequence. Here, Propositional Stability and Synchronicity are straightforward from the definition of generated forests. Preference Propagation. Assume that ja ≤ j  b (1). It follows from either clause in the definition of priority update that a ≤ b (2). Now assume that h ≤ h (3). It follows from (2), (3) and again priority update that ha ≤ h b. Preference Revelation. Assume that jb ≤ j  a (1). It follows from the definition of priority update that b ≤ a (2). Now assume ha ≤ h b (3). By the definition of priority update, (3) can happen in two ways. Case 1: a < b (4). It follows from (4) by the definition of < that b  ≤ a (5). But (5) contradicts (2). We are therefore in Case 2: a b (6), and so h ≤ h (7). Note that we did not make use of totality in this direction of the proof. Sufficiency (2 =⇒ 1). Given a DoTL model M satisfying the stated conditions, we show how to construct a matching doxastic plausibility model and a sequence of event models. Construction. Here is the initial plausibility model M0 = W, ( i )i∈N , Vˆ : – W := {h ∈ H | len(h) = 1}. – Set h i h iff h ≤i h . – For every p ∈ Prop, Vˆ (p) = V (p) ∩ W . Now we construct the j-th event model j = Ej , ( ji )i∈N , prej : – Ej := {e ∈ Σ | there is a history he ∈ H with len(h) = j} – Set a ji b iff there are ha, h b ∈ H such that len(h) = len(h) = j and ha ≤i h b. – For each e ∈ Ej , let prej (e) be the formula that characterizes the set {h | he ∈ H and len(h) = j}. By general modal logic, our condition of Bisimulation Invariance guarantees that there is such a formula. Again as mentioned at the end of Section 2 this sentence may be an infinitary one in general (if we don’t assume the doxastic temporal models to be finitely branching). We give a concrete instantiation when we discuss the epistemic doxastic corollary of our result.

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Now we show that the construction is correct in the following sense: Claim (Correctness). Let ≤ be the plausibility relation in the given doxastic temporal model. Let F DDL be the plausibility relation in the forest model induced by priority update over the just constructed plausibility model F and the constructed sequence of event models. We have:  h ≤ h iff h F DDL h .

Proof of the claim. The proof is by induction on the length of histories. The base case is obvious from the construction of our initial model M0 . Now comes the induction step: From DoTL to Forest(DDL). Assume that h1 a ≤ h2 b (1). It follows that in the constructed event model a ≤ b (2). Case 1: a < b. By priority update we have h1 a F DDL h2 b, whatever relationship held between h1 and H2 in F . Case 2: b ≤ a (3). This means that there are h3 b, h4 a such that h3 b ≤ h4 a. But then by Preference Revelation and (1) we have h1 ≤ h2 in the original doxastic temporal model M. It follows by the inductive hypothesis that h1 F DDL h2 . But then, since a and b are indifferent by (2) and (3), priority update gives us h1 a  F DDL h2 b. From Forest(DDL) to DoTL. Now let h1 a F DDL h2 b. Again we follow the two clauses in the definition of priority update: Case 1: a < b. By definition, this implies that b  ≤ a. But then by the above ≤ h4 a. In particular we construction, for all histories h3 , h4 ∈ H we have h3 b  have h2 b  ≤ h1 a. But then by totality(this is the only place where we use this property), h1 a ≤ h2 b. Case 2: a b (4) and h1 F DDL h2 . For a start, by the inductive hypothesis, h1 ≤ h2 (5). By (4) and our construction, there are h3 a, h4 b with h3 a ≤ h4 b (6). But then by Preference Propagation, (5) and (6) imply that we have h1 a ≤ h2 b.   Remark. Corollary for the Doxastic Epistemic case. We get a representation result for the doxastic epistemic case as an immediate corollary from Theorem 18 and Theorem 5. Moreover we give a concrete instantiation of this corollary by choosing the language of Safe Belief. In the result below we refer to Priority Update as the results of applying Product update to the epistemic relations and Priority Update to the plausibility orderings. Corollary 19. Let H be any doxastic epistemic temporal model with a total plausibility order. Then the following two assertions are equivalent:

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1. There exists a total an epistemic plausibility model M and a sequence of total epistemic plausibility event models  taking preconditions in the modal language of Safe Belief such that H is isomorphic to the forest generated by the Priority Update of M by the sequence . 2. H satisfies Propositional Stability, Synchronicity, Perfect Recall, Uniform No Miracles, (∼i ∩ ≤i )i∈N - Bisimulation Invariance, Preference Propagation, and Preference Revelation. This result shows how to find, inside the much broader class of all doxastic temporal models, those whose plausibility pattern was produced by a systematic priority update process.

9

Extension to Arbitrary Pre-orders

The preceding result generalizes to the general case of pre-orders, allowing incomparability. Here we need a new notion that was hidden so far: Definition 20 (Accommodating Events). Two events a, b ∈ Σ are pairwise accommodating if, for all ga, g  b: (g ≤ g  ↔ ga ≤ g  b), i.e. a, b preserve and anti-preserve plausibility. We can now define our new condition on doxastic-temporal models: – Accommodation Events a and b are accommodating in the sense of Def. 20 if both ja ≤ j  b and ha  ≤ h b for some j, j  , h, h . Accommodation is a uniformity property saying that, if two events allow both plausibility orders for histories, then they are always ‘neutral’ for determining plausibility order.This property only comes into its own with pre-orders allowing incomparable situations: Fact 21. If ≤ is a total pre-order and H satisfies Preference Propagation and Preference Revelation, then H satisfies Accommodation. ≤ h b. By totality, the latter implies Proof. Assume that ja ≤ j  b (i) and ha    hb ≤ h a (ii). Now let g ≤ g . By Preference Propagation and (i), ga ≤ g  b. Conversely, assume that ga ≤ g  b. By Preference Revelation, (i) and (ii), we   have g  ≤ g. We can also prove a partial converse without assuming totality: Fact 22. If H satisfies Accommodation, it satisfies Preference Propagation. Proof. Let ja ≤ j  b (1) and h ≤ h (2). Assume that ha  ≤ h b. Then by Accom   modation, for every ga, g b, g ≤ g ↔ ga ≤ g b. So, in particular, h ≤ h ↔ ha ≤ h b. But since h ≤ h , we get ha ≤ h b: a contradiction.   Finally, an easy counter-example shows that, even with ≤ total:

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Fact 23. Accommodation does not imply Preference Revelation. Proof. Take a simplest model where the following holds: h b ha j  a jb and h < h j  j.   With arbitrary pre-orders we need to impose Accommodation: Theorem 24. Let H be any doxastic-temporal model with a plausibility preorder. Then the following two assertions are equivalent: 1. There exists a plausibility model M, and a sequence of plausibility event models  such that H is isomorphic to the forest generated by the Priority Update of M by the sequence . 2. H satisfies Bisimulation Invariance, Propositional Stability, Synchronicity, Preference Revelation, and Accommodation. By Fact 22, Accommodation also gives us Preference Propagation. Proof. Necessity of the conditions. (1 =⇒ 2) Checking the conditions in the Section 7 did not use totality. So we focus on the new condition: Accommodation. Assume that ja ≤ j  b (1). It follows by the definition of priority update that a ≤ b (2). Now let ha  ≤ h b (3). This implies by priority update that a  < b (4). By definition, (2) with (4) imply that a b (5). Now assume that g ≤ g  (6). It follows from (5), (6) and priority update that ga ≤ g  b. The other direction is similar. Sufficiency of the conditions. (2 =⇒ 1) Given a DoTL model, we again construct a DDL plausibility model plus a sequence of event models: Construction. The plausibility model M0 = W, ( i )i∈N , Vˆ  is as follows: – W := {h ∈ H | len(h) = 1}, – Set h i h whenever h ≤i h , – For every p ∈ Prop, Vˆ (p) = V (p) ∩ W . We construct the j-th event model j = Ej , ( ji )i∈N , prej  as follows: – Ej := {e ∈ Σ | there is a history of the form he ∈ H with len(h) = j} – For each i ∈ N , define a ji b iff either (a) there are ha, h b ∈ H such that len(h) = len(h) = j and ha ≤i h b, or (b) [a new case] a and b are accommodating, and we put a b (i.e., both a ≤ b and b ≤ a). – For each e ∈ Ej , let prej (e) be the basic doxastic formula characterizing the set {h | he ∈ H and len(h) = j}. Bisimulation Invariance guarantees that there is such a formula (maybe infinitary). Again we show that the construction is correct in the following sense:

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Claim (Correctness). Let ≤ be the plausibility relation in the doxastic temporal model M. Let F DDL be the plausibility relation in the forest F induced by successive priority updates of the plausibility model by the sequence of event models we just constructed. We have:  h ≤ h iff h F DDL h .

Proof of the claim. We proceed by induction on the length of histories. The base case is clear from our construction of the initial model M0 . Now for the induction step, with the same simplified notation as earlier: From DoTL to Forest(DDL). We distinguish two cases.  Case 1. ha ≤ h b, h ≤ h . By the inductive hypothesis, h ≤ h implies h F DDL h (1). Since ha ≤ h b, it follows by the construction that a ≤ b (2). Then, by (1),  (2) and priority update, we get ha F DDL h b.

≤ h . Clearly, then, a and b are not accommodating and Case 2. ha ≤ h b, h  thus the special clause has not been used to build the event model, though we do have a ≤ b (1). By the contrapositive of Preference Revelation, we also conclude ≤ ja (2). Therefore, our construction gives that for all ja, j  b ∈ H, we have j  b  b ≤ a (3), and we conclude that a < b (4). But then by priority update, we get  ha F DDL h b. From Forest(DDL) to DoTL. We again distinguish two cases.  F  F Case 1. ha F DDL h b, h DDL h . By the definition of priority update, ha DDL  h b implies that a ≤ b (1). There are two possibilities.

Case 1.1: The special clause of the construction has been used, and a, b are   accommodating (2). By the inductive hypothesis, h F DDL h implies h ≤ h (3).  But (2) and (3) imply that ha ≤ h b. Case 1.2: Clause (1) holds because for some ja, j  b ∈ H in the DoTL model,   ja ≤ j  b (4). By the inductive hypothesis, h F DDL h implies h ≤ h (5). Now it  follows from (4), (5) and Preference Propagation that ha ≤ h b.   Case 2. ha F F DDL h b, h  DDL h . Here is where we put our new accommodation  F  clause to work. Let us label our assertions: h  F DDL h (1) and ha DDL h b (2). It follows from (1) and (2) by the definition of priority update that a < b (3), and hence by definition, b  ≤ a (4). Clearly, a and b are not accommodating (5): for otherwise, we would have had a b, and hence b ≤ a, contradicting (4). Therefore, (3) implies that there are ja, j  b ∈ H with ja ≤ j  b (6). Now assume for contradictio that (in the DoTL model) ha  ≤ h b (7). It follows from (6) and (7) by Accommodation that a and b are accommodating, contradicting (5). Thus we must have ha ≤ h b.  

Given a doxastic temporal model describing the evolution of the beliefs of a group of agents, we have determined whether it could have been generated by successive ‘local’ priority updates of an initial plausibility model.

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More Extensions and Variations of the Theorem

Several further scenarios can be treated in the same manner. In particular, it is easy to combine the epistemic analysis in Section 1 with ours to include agents having both knowledge and belief. Here are three more directions: 10.1

From Uniform to Local Protocols

So far we have considered uniform line protocols. We have already suggested that line protocols are powerful enough to mimic branching protocols through renaming of events, and then taking a disjoint union of all branching alternatives. But uniformity is a real restriction, and it can be lifted. Local protocols allow the set of executable sequences of pointed events models forming our current informational process to vary from state to state. Indeed, agents need not even know which protocol is running. As was done in [11] for the epistemic case, we can still get our representation theorems, by merely dropping the condition of Bisimulation Invariance. While this seems a simple move, local protocols drastically change the complete dynamic-doxastic logic of the system (cf. [9] and [16] for details). 10.2

Languages and Bisimulations

As we have noted in Section 5, our doxastic-temporal models support various languages and logics. These will be pursued in [9], but we do make a few points here. One is that complete doxastic-temporal logics for the above special model classes will have valid principles reflecting the reduction axioms of dynamicdoxastic logic. In fact, these doxastic-temporal correspond to Preference Propagation and Preference Revelation in the sense of modal correspondence theory. Thus, our structural analysis of priority-updating agents extends to the level of valid reasoning. Proposition 25. The following law is sound for plausibility change:   , e≤i φ ↔ (pre(e) ∧ (≤i  {f φ : e i f } ∨ E {gφ : e
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In fact, a doxastic-temporal language has two main purposes in our setting: (a) stating ‘local’ preconditions for events, (b) specifying ‘global’ properties of the temporal evolution of the current process. As is well-known [13] a choice of language here corresponds with a choice of a semantic invariance relation, usually some weaker or stronger variant of bisimulation. In stating our results, we kept to the simplest version, of a modal bisimulation adequate for the static language of safe belief. But this can be varied, and one can also have stronger notions of bisimulation, respecting more structure, that work for more expressive doxastic languages, or for combined doxastic-epistemic languages. If we add epistemic structure, the bisimulation also needs back-and-forth clauses for the intersection of the epistemic accessibility and doxastic plausibility relations. For instance, if the precondition language contains a belief operator scanning the intersection of a plausibility ≤i relation and an epistemic indistinguishability relation ∼, then the back and forth clauses should not only apply to ≤i and ∼i separately, but also to ≤i ∩ ∼i . (Indeed ∩ is not safe for bisimulation.) And things get even more complicated if we allow temporal operators in our languages (cf. [11]). We do not want to commit to any specific choice here, since the choice of a language seems orthogonal to our main concerns. Even so, we shall discuss formal languages in the next section, taking definability of our major structural constraints as a guide. 10.3

Alternative Model Classes

Finally, we mentioned in Section 5 that one can also work with a primitive plausibility relation that merges epistemic indistinguishability and doxastic plausibility. This is done for Priority Update in [3], and we indicate briefly the notions involved in the corresponding results: Definition 26. The priority update of a unified plausibility model M of type W, (i )i∈N , V  and a -event model  = E, (i )i∈N , pre is the unified plausibility model M ⊗  = W  , (i )i∈N , V   constructed as follows: – W  = {(w, e) ∈ W × E | M, w  pre(e)}, – (w, a) i (w , b) iff either 1. a i b, b a and w  w ∨ w  w or 2. a i b, b  a and w  w , – V  ((s, e)) = V (s). Here are our basic temporal doxastic agent properties in this setting: – – – –

-Perfect Recall If ha  h b we have h  h ∨ h  h. -Preference Propagation If h  h and ja  j  b then also ha  h b. -Preference Revelation If ha  h b ∧ jb  j  a, also h  h . -Accommodation If (ja  j  b, h  h and ha  h b), for all ga, g  b ∈ H     (g  g ↔ ga  g b), and for all g a, gb ∈ H (g  g ↔ gb  g  a).

[9,16] show how these conditions drive a general representation theorem similar to the one in Section 8 and 8.

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171

Conclusion

Agents that update their knowledge and revise their beliefs leave an epistemic and doxastic ‘trace’ over time of epistemic and doxastic relations. We have determined the special constraints that capture agents operating with the ‘local updates’ of dynamic doxastic logic. This took the form of representation theorems that state just when a general doxastic temporal model is equivalent to the forest model generated by successive priority updates of an initial doxastic model by a protocol sequence of event models.

DDL DT L

Fig. 6. DDL inside DTL

Thus, we have determined an area where the idealized belief changers of dynamic-doxastic logic live. This identification becomes even stronger through its axiomatization in suitable doxastic-temporal languages, as explored in our follow-up paper [9]. As for open problems, this paper has already indicated several technical issues along the way. Here we list a few more: – Languages and logics. There are many issues of expressive power of different languages over our models and their complexity effects (cf. [12] for the epistemic case). In particular, belief revision for individual agents has a natural companion of belief merge for groups of agents, and various update processes that might create common beliefs. These create complications in the epistemic case [10] and are even more challenging here. For all such languages, there are also natural questions of complete axiomatization. Indeed, we have proved axiomatic completeness for the ‘protocol logic’ of revision processes in [9,16]. This is analogous to the epistemic theory of observation and conversation protocols initiated in [11]. – Agent diversity. Beyond the realm of priority-updating agents studied in this paper, there are also other zones of interest inside DTL models, including processes involving agents with memory bounds, that may revise beliefs in different ways, and may even see their knowledge fade into belief, and finally, oblivion. – Related Approaches. A comparison of our ‘constructive’ DDL-inspired approach to DTL universes with the more abstract AGM-style postulational approach of [15] remains to be made.

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– Concrete scenarios: game theory and learning theory. We intend to take our analysis to knowledge and belief dynamics in the concrete setting of extensive games, following up on the initial studies [8,5]. It would also be of interest to link our doxastic-temporal models with more concrete learning scenarios. In the temporal universes of formal learning theory [22], learning rules are belief revision or plausibility update rules producing the new hypothesis after the next observed event. D´egremont and Gierasimczuk [17] is a first systematic attempt at connecting this with DDL and DTL models.

References 1. Alchourr´ on, C., G¨ ardenfors, P., Makinson, D.: On the Logic of Theory Change: Partial Meet Contraction and Revision Functions. J. Symb. Log. 50(2), 510–530 (1985) 2. Baltag, A., Moss, L.S.: Logics for Epistemic Programs. Synthese 139(2), 165–224 (2004) 3. Baltag, A., Smets, S.: Dynamic Belief Revision over Multi-agent Plausibility Models. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) LOFT 2006, University of Liverpool, pp. 11–24 (2006) 4. Baltag, A., Smets, S.: A Qualitative Theory of Dynamic Interactive Belief Revision. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) LOFT 2006. Texts in Logic and Games, vol. 3, pp. 9–58. Amsterdam University Press (2008) 5. Baltag, A., Smets, S., Zvesper, J.A.: When All is Done but Not (Yet) Said: Dynamic Rationality in Extensive Games. In: van Benthem, J., Pacuit, E. (eds.) Proceedings of Workshop on Logic and Intelligent Interaction, ESSLLI (2008) 6. van Benthem, J.: Games in Dynamic Epistemic Logic. Bull. Econ. Res. 53(4), 219– 248 (2001) 7. van Benthem, J.: Dynamic Logics for Belief Change. J. Appl. Non-Class. Log. 17(2), 129–155 (2007) 8. van Benthem, J.: Logical Dynamics of Information Flow and Interaction. Cambridge University Press, Cambridge (to appear) 9. van Benthem, J., D´egremont, C.: Building bridges between dynamic and temporal doxastic logics. ILLC Publications PP-2008-17. Universiteit van Amsterdam (2008) 10. van Benthem, J., van Eijck, J., Kooi, B.P.: Logics of communication and change. Inform. Comput. 204(11), 1620–1662 (2006) 11. van Benthem, J., Gerbrandy, J., Hoshi, T., Pacuit, E.: Merging Frameworks for Interaction: DEL and ETL. J. Philos. Log. (2009) 12. van Benthem, J., Pacuit, E.: The Tree of Knowledge in Action: Towards a Common Perspective. In: Governatori, G., Hodkinson, I., Venema, Y. (eds.) AiML 6, pp. 87– 106. College Publications (2006) 13. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001) 14. Board, O.: Dynamic interactive epistemology. Games Econ. Behav. 49, 49–80 (2004) 15. Bonanno, G.: Axiomatic Characterization of the AGM Theory of Belief Revision in a Temporal Logic. Artif. Intell. 171(2-3), 144–160 (2007) 16. D´egremont, C.: Logics of Information and Interaction. Ph.D. thesis, Universiteit van Amsterdam (2010); ILLC Publications DS-2010-03

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17. D´egremont, C., Gierasimczuk, N.: Can Doxastic Agents Learn? On the Temporal Structure of Learning. In: He, X., Horty, J., Pacuit, E. (eds.) LORI 2009. LNCS, vol. 5834, pp. 90–104. Springer, Heidelberg (2009) 18. van Ditmarsch, H.: Prolegomena to Dynamic Logic for Belief Revision. Synthese 147, 229–275 (2005) 19. Eliaz, K., Ok, E.A.: Indifference or Indecisiveness? Choice-theoretic Foundations of Incomplete Preferences. Games Econ. Behav. 56(1), 61–86 (2006) 20. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995) 21. Friedman, N., Halpern, J.Y.: Modeling Belief in Dynamic Systems, part I: Foundations. Artif. Intell. 95(2), 257–316 (1997) 22. Kelly, K.T.: The Learning Power of Belief Revision. In: Gilboa, I. (ed.) TARK 1998, pp. 111–124. Morgan Kaufmann, San Francisco (1998) 23. Liu, F.: Changing for the Better: Preference Dynamics and Agent Diversity. Ph.D. thesis, Universiteit van Amsterdam (2008); ILLC Publications DS-2008-02 24. Martin, E., Osherson, D.: Elements of Scientific Inquiry. MIT Press, Cambridge (1998) 25. Parikh, R., Ramanujam, R.: A Knowledge Based Semantics of Messages. J. Log. Lang. Inf. 12(4), 453–467 (2003)

Multi-agent Belief Revision with Linked Preferences Jan van Eijck and Floor Sietsma Centrum voor Wiskunde en Informatica (CWI), Science Park 123, 1098 XG Amsterdam, The Netherlands [email protected], [email protected]

Abstract. In this paper we forge a connection between dynamic epistemic logics of belief revision on one hand and studies of collective judgement and multi-agent preference change on the other. Belief revision in the spirit of dynamic epistemic logic uses updating with relational substitutions to change the beliefs of individual agents. Collective judgement in social choice theory studies the collective outcomes of individual belief changes. We start out from the logic of communication and change (LCC) without constraints, and then study the effects of imposing a single constraint, namely the constraint that the agent’s preference relations are linked. Finally, we show that the resulting framework can be used to model consensus seeking procedures. We focus on the case of plenary Dutch meetings. In Dutch meetings, a belief change (or rather: preference change) is performed for all agents in the meeting if a majority believes (or: is in favour of) the proposition that is under discussion. A special case of these meetings is judgement aggregation, and we apply our framework to the discursive dilemma in this field. Our framework has obvious connections to coalition logic and social choice theory.

1

Introduction

In [17] the following problem in judgement aggregation is considered. There are three judges a, b, c with a, b agreeing that p, and b, c agreeing that q, so that both p and q command a majority, but p ∧ q does not. The example shows that majority judgement is not closed under logical consequence. It seems that there is a collective belief in p and in q, but not in p ∧ q. To see the relevance of the example for the practice of law, assume that p expresses that the defendant has done action X, and q expresses that the defendant is under a legal obligation not to do X. Then p ∧ q expresses that the defendant has broken his contract not to do X. So two of the three judges agree that the defendant has done X, two of the three judges agree that the defendant is under a legal obligation not to do X, but there is no majority 

We wish to thank two LOFT’08 reviewers and three reviewers for the postproceedings volume for their useful comments on earlier drafts, and are grateful to the editors of this volume for their patience and encouragement.

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 174–189, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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for the verdict of ‘guilty’. This is a standard paradox in judgement aggregation called the discursive dilemma or doctrinal paradox. The discursive dilemma is an example of a situation where multiple agents have different beliefs, and where a naive way of combining these beliefs — majority voting on each individual belief — leads to an implausible outcome. What seems to be needed is a more sophisticated way, or more sophisticated ways, to combine individual beliefs into collective beliefs. We will present an epistemic/doxastic framework that can be used to model such situations, and present ways to update these frameworks with new beliefs. In the above example, this gives various protocols for judgement aggregation that are different from the above simple majority rule. The framework of (our version of) doxastic/epistemic logic is quite abstract. It starts out from a basic preference relation for each agent, without any constraints on the preference relations. In particular, we do not insist that the preference relations are reflexive or transitive. The relations in our framework can be seen as plausibility relations, and used for modeling the beliefs of agents, as is the case in the above example. But we can also view them as modelling preference, and arrive at an interpretation that connects the framework to the area of social choice theory. Henceforth, we will stick to ‘preference relation’, but we do not intend to exclude other interpretations.

2

Preference Relations without Constraints

A preference model M for set of agents Ag and set of basic propositions Prop is a tuple (W, P, V ) where W is a non-empty set of worlds, P is a function that maps each agent a to a relation Pa (the preference relation for a), and V is a map from W to P(Prop) (a map that assigns to each world a Prop-valuation). The tuple (W, P ) is called the frame of the model. There are no conditions at all on the Pa . Appropriate conditions will be imposed by constructing the operators for belief and knowledge by means of PDL operations. We fix a PDL style language for talking about preference. Assume p ranges over a set of basic propositions Prop and a over a set of agents Ag. Then the language of epistemic/doxastic PDL is given by: φ ::=  | p | ¬φ | φ1 ∧ φ2 | [π]φ π ::= a | aˇ |?φ | π1 ; π2 | π1 ∪ π2 | π ∗ We use PROG for the set of program expressions (expressions of the form π) of this language. The intuitive meaning of aˇ is ‘converse of a’, i.e., reversing the directions of the arrows. The construct π1 ; π2 expresses sequential composition of relations. In the standard interpretation of PDL, π ∗ expresses program iteration. In our epistemic/doxastic interpretation, it expresses transitive closure of a (possibly complex) preference relation. The language is to be interpreted in the usual PDL manner, with [[[π]]]M giving the relation that interprets relational expression π in M = (W, V, P ). [π]φ is true

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in world w of M if for all v with (w, v) ∈ [[[π]]]M it holds that φ is true in v. We adopt the usual abbreviations of ⊥, φ1 ∨ φ2 , φ1 → φ2 , φ1 ↔ φ2 and πφ. The following additional abbreviations allow us to express knowledge, strong belief, conditional belief and plain belief: knowledge [∼a ]φ abbreviates [(a ∪ aˇ)∗ ]φ. The formula [∼a ]φ expresses that agent a knows that φ. strong belief [≥a ]φ abbreviates [a∗ ]φ. The formula [≥a ]φ expresses that agent a strongly believes that φ. The only difference between knowledge and strong belief is that knowledge has the property of negative introspection (if a does not know φ, then a knows that she does not know φ), while strong belief lacks this property. conditional belief [→φa ]ψ abbreviates ∼a φ → ∼a (φ ∧ [≥a ](φ → ψ)). The formula [→φa ]ψ expresses that in all of the agent’s preferred φ worlds, ψ holds. plain belief [→a ]ψ abbreviates [→ a ]ψ. The formula [→a ]ψ expresses that in all of the agent’s preferred worlds, ψ holds. Note that this can be expressed equivelently as ∼a [≥a ]φ. The definition of →φa (conditional belief for a, with condition φ) is from Boutillier [10]. This definition, also used in [4], states that conditional to φ, a believes in ψ if either there are no accessible φ worlds, or there is an accessible φ world in which the belief in φ → ψ is safe. The definition of →φa matches the well-known accessibility relations →P a for each subset P of the domain, given by: →P a := {(x, y) | x∼a y ∧ y ∈ MIN≤a P }, where MIN≤a P , the set of minimal elements of P under ≤a , is defined as {s ∈ P : ∀s ∈ P (s ≤a s ⇒ s ≤a s )}. This logic is completely axiomatized by the standard PDL rules and axioms ([21,16]) plus the following axioms that describe the relation between the basic programs a and aˇ:  φ → [a] aˇφ

 φ → [aˇ] aφ

Any preference relation Pa can be turned into a pre-order by taking its reflexive transitive closure Pa ∗ . The abbreviation for strong belief introduces the ≥a as names for these pre-orders. The knowledge abbreviation introduces the ∼a as names for the equivalences given by (Pa ∪ Paˇ)∗ . If the Pa are well-founded, MIN≤a P will be non-empty for non-empty P . The canonical model construction for PDL yields finite models. Therefore, the Pa relations in the canonical model will be well-founded, for each relation on a finite model is well-founded. Thus, there is no need to impose well-foundedness as a relational condition. We get it for free by the fact that PDL has the finite model property.

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This yields a very expressive complete and decidable PDL logic for preference, to which we can add mechanisms for belief update and for belief revision. Our logic is able to express knowledge and strong belief as single modalities (as (a ∪ aˇ)∗ and a∗ , respectively). We were unable to find single modalities for conditional belief and plain belief, however. Note that expressing conditional belief in terms of single unary modalities is different from expressing conditional belief as a single unary modality. So we have the following conjecture, which we were unable to prove: Conjecture. It is impossible to describe conditional belief and plain belief as single unary modalities in converse PDL. Intermezzo: Conditional Belief, Complement and Intersection Instead of using PDL with converse as a basis for constructing logics of belief revision, one might wish to use PDL with converse and complement, i.e., the result of adding operations aˇ and π to the language, with π to be interpreted as relational complementation. Unfortunately, the resulting logic is undecidable [15]. Still, the advantage of having complement available is that it allows us to express all our knowledge and belief constructs as single unary modalities, as follows: unary unary unary unary

modality modality modality modality

for for for for

knowledge (a ∪ aˇ)∗ . strong belief a∗ . conditional belief (a ∪ aˇ)∗ ; ?(φ ∧ [a∗ − a∗ˇ]¬φ). plain belief (a ∪ aˇ)∗ ; ?[a∗ − a∗ˇ]⊥.

This uses a∗ − a∗ˇ, which can be expressed with complement as a∗ ∪ a∗ˇ. Note that a∗ − a∗ˇ represents the moves to a strictly a-better world: w is related to w by a∗ − a∗ˇ iff there is an a-path from w to w but not vice versa. It is easy to see that ?(φ ∧ [a∗ − a∗ˇ]¬φ) is true in a world w iff w is an a-best φ world. But perhaps this shift to an undecidable logic is unnecessary. Maybe adding intersection and converse to PDL is enough. Intersection PDL with converse is like PDL, but with a construct for atomic converse and an operation π1 ∩ π2 for relation intersection. Like PDL with converse, intersection PDL with converse is decidable [18]. Here is an open question: Question. Is it possible to describe conditional belief and plain belief as single modalities in intersection PDL with converse?

3

Belief Revision with Linked Preference Relations

The preference relations that serve as the basis for construction of a preference pre-order in Section 2 leave something to be desired. Compare an optometrist who collects answers for a number of lenses she tries out on you:

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“Better or worse?”, (change of lens), “Better or worse?” (change of lens), “Better or worse?”. . . . If you reply “worse” after a change of x to y, and “worse” after a change from y to z, she will most probably not bother to collect your reaction to a change from x to z. But what if you answer “better” after the second swap? Then, if she is reasonable, she will try to find out how x compares to z. It does make sense to impose this as a requirement on preference relations. There are several ways to do this. Recall that we did not impose a requirement of transitivity on the basic preference relations. Here is a definition that does not imply transitivity, but yields that the transitive closures of the basic preferences are well-behaved. See Figure 1. The intended meaning is that if the black arrows are present, either the grey arrow on the left or the grey arrow on the right has to be present as well.

R

R∗

R

R∗

R∗

R∗

Fig. 1. Forward linkedness of a binary relation R

Definition 1. A binary relation R is forward linked if the following holds: ∀x, y, z((xRy ∧ xR∗ z) → (yR∗ z ∨ zR∗ y)). R is linked if both R and Rˇ are forward linked. Note that this is different from the notion of weak connectedness (terminology of [13]): a relation R is weakly connected if ∀x, y, z((xRy ∧ xRz) → (yRz ∨ y = z ∨ zRy)). Theorem 2. R is forward linked iff R∗ is weakly connected. Proof. The right to left direction is immediate. For the left to right direction, assume R is forward linked. Let wR∗ w1 and wR∗ w2 . Then there is an n ∈ N with wRn w1 . We prove the claim by induction on n. If n = 0 then w = w1 and w1 R∗ w2 , and we are done. Otherwise, assume the claim holds for n. We have to show it holds for n + 1. Suppose wRn+1 w1 . Then for some w , wRw Rn w1 . By forward linking of R, either w R∗ w2 or w2 R∗ w . In the first case, use the induction hypothesis to get w1 R∗ w2 or w2 R∗ w1 . In the second case, it follows from w2 R∗ w and w Rn w1 that w2 R∗ w1 . Starting from relations that are linked, one can use the method of constructing knowledge and belief relations by means of the regular operations that was

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presented in the previous section to construct ‘preference models’ in the style of Grove [14], Board [9], and Baltag and Smets [3,4] (who call them ‘multi-agent plausibility frames’). It is well-known that the following principle characterizes weak connectedness of Pa (cf. [13]): [a]((φ ∧ [a]φ) → ψ) ∨ [a]((ψ ∧ [a]ψ) → φ). The notion of forward linking is characterized by: [a]((φ ∧ [a∗ ]φ) → ψ) ∨ [a∗ ]((ψ ∧ [a∗ ]ψ) → φ).

(*)

Theorem 3. Principle (*) holds in a belief revision frame iff Pa is forward linked. Proof. Let (W, P ) be a frame where Pa is forward linked, and let M = (W, P, V ) be some model based on the frame. We show that (*) holds. Let w be a world in M. Assume M  |=w [a]((φ ∧ [a∗ ]φ) → ψ). We have to show: M |=w [a∗ ]((ψ ∧ ∗ |=w [a]((φ ∧ [a∗ ]φ) → ψ), we get that there [a ]ψ) → φ). From the fact that M  is a world w1 with wPa w1 and M |=w1 φ ∧ [a∗ ]φ ∧ ¬ψ. Let w2 be an arbitrary world with wPa∗ w2 . Then by forward linking of Pa , either w1 Pa∗ w2 or w2 Pa∗ w1 . In the first case, it follows from M |=w1 [a∗ ]φ that M |=w2 φ, and therefore M |=w2 (ψ ∧ [a∗ ]ψ) → φ. In the second case, it follows from M |=w1 ¬ψ that M |=w2 ¬[a∗ ]ψ, and therefore M |=w2 (ψ ∧ [a∗ ]ψ) → φ. So in both cases, M |=w2 (ψ ∧ [a∗ ]ψ) → φ, and since w2 was an arbitrary world with wPa∗ w2 , it follows that M |=w [a∗ ]((ψ ∧ [a∗ ]ψ) → φ). Next, assume a frame (W, P ) where Pa is not forward linked. We will construct a model M = (W, P, V ) and an instance of (*) that does not hold. If Pa is not forward linked, there are w, w1 , w2 with wPa w1 , wPa∗ w2 , and neither w1 Pa∗ w2 nor w2 Pa∗ w1 . We proceed to make an instance of (*) fail in w. Construct the valuation of M by setting p true in w1 and in all worlds w with w1 Pa∗ w , and false everywhere else, and setting q true in w2 and in all worlds w with w2 Pa∗ w , and false everywhere else. Note that since not w1 Pa∗ w2 , p will be false in w2 , and that since not w2 Pa∗ w1 , q will be false in w1 . So we get M |=w1 p ∧ [a∗ ]p ∧ ¬q and M |=w2 q ∧ [a∗ ]q ∧ ¬p. It follows that M |=w a(p ∧ [a∗ ]p ∧ ¬q) ∧ a∗ (q ∧ [a∗ ]q ∧ ¬p), i.e.,

M |=w [a]((p ∧ [a∗ ]p) → q) ∨ [a∗ ]((q ∧ [a∗ ]q) → p),

showing that this instance of (*) does not hold in M. Let us make a remark: As one of the referees pointed out, (*) is a so-called Sahlqvist formula. Such formulas are canonical for the properties they define. Canonicity of (*) for the property of forward linkedness means that the canonical frame for a normal modal logic containing (*) is forward linked, and that (*) is valid on any class of forward linked frames. See [8] for further details.

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In the multi-agent case there is a further natural constraint. Consider a situation where Alice and Bob have to decide on the chairperson of a program committee. Carol is mediator. Alice says she prefers y to x. Bob counters by saying that he prefers z to x. What should Carol do? Clearly, she should urge both of them to compare y and z. Translating this example to our logic of belief, we want to require that if x ≥a y and x ≥b z, then either y ≥a z or z ≥a y and either y ≥b z or z ≥b y. Definition 4 (Connectedness). A set of binary relations R on a domain W is forward connected if for all R, S in R, if xRy and xSz, then either ySz or zSy. R is backward connected if the set {Rˇ | R ∈ R} is forward connected. R is connected if R is both forward and backward connected. Clearly, our motivating requirement that if x ≥a y and x ≥b z then either y ≥a z or z ≥a y expresses that ≤a and ≤b are forward connected. Another natural requirement is that if Alice prefers x to y and Bob prefers x to z, then Alice and Bob should both compare y and z. If x ≤a y and x ≤b z then either y ≤a z or z ≤a y (and similarly for ≤b ). This expresses that ≤a and ≤b are backward connected. The following definition extends the notion of linkedness to the multi-agent case (see Figure 2).

R

S∗

R

S∗

S∗

S∗

S

S

R∗

R∗

R∗

R∗ Fig. 2. Forward linkedness of a pair of relations {R, S}

Definition 5 (Linkedness). A set of binary relations R on a domain W is forward linked if for all R, S in R, if xRy and xS ∗ z, then either yS ∗ z or zS ∗ y. R is backward linked if the set {Rˇ | R ∈ R} is forward linked. R is linked if R is both forward and backward linked.

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It follows from Definition 5 that the set {R} is forward linked iff R is forward linked according to Definition 1. So Definition 5 gives a natural extension of linking (and of local connectedness) to the multi-agent case. The following theorem shows that Definition 5 can indeed replace Definition 4. Theorem 6. R and S are linked iff R∗ and S ∗ are connected. Proof. The right to left direction is immediate. For the left to right direction, assume R and S are linked. Suppose xR∗ y and xS ∗ z . We will prove that for any w on the R path from x to y, either wS ∗ z or zS ∗ w. This clearly holds for w = x. Suppose w is the successor of w on the path, and the result holds for w . Suppose w S ∗ z. Since w Rw the result holds by forward linking of R and S . Suppose yS ∗ w . From w Rw and w S ∗ w we get that either w S ∗ w or wS ∗ w . In the first case trivially yS ∗ w. Assume the second case. We will show that for any v on the S path from w to w , either vS ∗ y or yS ∗ v. This clearly holds for v = w . Suppose v is the predecessor of v  on the path, and the result holds for v  . Suppose v  S ∗ y. Then since vSv  , trivially vS ∗ y. Suppose yS ∗ v  . Then the result holds by backward linking of S. This shows that R∗ and S ∗ are forward connected. The proof that R∗ and S ∗ are backward connected is similar. If R and S are preference relations, (R ∪ Rˇ)∗ and (S ∪ Sˇ)∗ are knowledge relations. The common knowledge relation (for the agents corresponding to R and S) is the reflexive transitive closure of the two knowledge relations, so common knowledge is expressed by: ((R ∪ Rˇ)∗ ∪ (S ∪ Sˇ)∗ )∗ This is easily seen to be equivalent to: (R ∪ Rˇ ∪ S ∪ Sˇ)∗ As we will now show, if R and S are linked relations, then this can be simplified even further. If R and S are linked then common knowledge equals the union of strong common belief and strong reverse common belief: Theorem 7. If R and S are linked, then (R ∪ Rˇ ∪ S ∪ Sˇ)∗ = (R ∪ S)∗ ∪ (Rˇ ∪ Sˇ)∗ . Proof. The inclusion from right to left is obvious. For the inclusion from left to right, assume x(R ∪ Rˇ ∪ S ∪ Sˇ)∗ y. Letting X and Y range over R and S, observe that each X ◦ Y ˇ∗ link can be replaced by either a Y ∗ or a Y ˇ∗ link, and similarly for Xˇ ◦ Y ∗ links, by linking of R and S. Continuing this process until all one-step links are of the form R ∪ S or of the form Rˇ ∪ Sˇ, this yields x(R ∪ S)∗ y or x(Rˇ ∪ Sˇ)∗ y. The theorem shows that linking of relations simplifies the notion of common knowledge. The modal characterization of relation linking is given by: [a]((φ ∧ [b∗ ]φ) → ψ) ∨ [b∗ ]((ψ ∧ [b∗ ]ψ) → φ)

(LINK)

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Theorem 8. The set of LINK principles (with a, b ranging over the set of all agents) holds in a preference model iff the basic preference relations in the model are forward linked. Proof. Analogous to the proof of Theorem 3.

4

Belief Update and Belief Change

For the definition of action models A and of the update product operation ⊗ that combines an epistemic model and an action model, to produce an updated epistemic model, we refer to [2]. In [7] it is shown how extending the PDL language with an extra modality [A, e]φ, where A is an action model and e is a designated action of A, does not change its expressive power. Interpretation of the new modality: [A, e]φ is true in w in M if success of the update of M with action model A to M ⊗ A implies that φ is true in (w, e) in M ⊗ A. The logic of PDL with action modalities added was called LCC or logic of communication and change. a 0:h

1:h

0:h

(0, 0) : h

Fig. 3. Belief about h, belief update, and its result

LCC, as proposed in [7], has a design flaw: it starts out from individual agent accessibity relations with the appropriate properties for knowledge or for belief. But updates with non S5 action models may destroy these properties: the update product of an S5 model with a non-S5 action model is in general not an S5 model. This means that the resulting relations cannot be interpreted as epistemic accessibilities any longer. Similarly, if one interprets the agent accessibilities as belief. A KD45 model can easily change into a non KD45 model by an appropriate action update. This issue is remedied in [12], where it was first proposed to construct the relational properties for knowledge from more basic relations, by means of the PDL operations. The present paper proposes the same for (conditional) belief. Action update by means of the product construction can now be viewed as belief update. In Figure 3 belief update about a coin toss is modelled. Agent a initially believes ¬h, while h (heads) is actually the case,. for h is true in the actual world, shown in grey. Next, h is announced. The result of the belief update is that a now believes h. Belief change is something different from belief update. Update can never reverse the directions of relational arrows, or introduce new arrows, for the arrows in the update result are the arrows that are present in the situation before the update and in the action model. Belief change is something more radical than this: replacing existing preference relations by new ones.

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Here, we will focus on belief change rather than belief update. Belief change can be compared to factual change. A coin shows head, you toss it again, and now it shows tails. This can be modeled as a propositional substitution {h  → ¬h}. Similarly for belief change. Agent a prefers x to y, she changes her preference, and now she prefers y to x. Or suppose she reverses all her preferences. This can also be handled as a substitution, {a  → aˇ}. In [7], it was proposed to handle factual change by propositional substitution. Relational substitutions were proposed for belief change in [5], and it was shown in [11] that adding relational substitutions for preference change to epistemic PDL makes no difference for expressive power: the resulting system still reduces to PDL. A preference substitution is a map from agents to PROG expressions that can be represented by a finite set of bindings {a1  → π1 , . . . , an  → πn } where the aj are agents, all different, and where the πi are PROG expressions. It is assumed that each a that does not occur in the left hand side of a binding is mapped to a. Call the set {a ∈ Ag | ρ(a)  = a} the domain of ρ. If M = (W, P, V ) is a preference model and ρ is a preference substitution, then Mρ is the result of changing the preference map P of M to P ρ given by:  Pa for a not in the domain of ρ, ρ P (a) := [[[ρ(a)]]]M for a in the domain of ρ. Now extend our PDL language with a modality [ρ]φ for preference change, with the following interpretation: M, w |= [ρ]φ : ⇐⇒ Mρ , w |= φ. An important thing to note is that since there are constraints on the preferences Pa (they are linked), we need to ensure that the belief changing substitutions satisfy these constraints. We leave the general problem of finding a precise characterization of linking-preserving substitutions for future research. Here we focus on a particular kind of belief change. Consider the suggestive upgrade a φ discussed in Van Benthem and Liu [6]: a φ =def ?φ; a; ?φ ∪ ?¬φ; a; ?¬φ ∪ ?¬φ; a; ?φ. This is a variation on what is called lexicographic upgrade in the belief revision community (see e.g., [19]). The suggestive upgrade removes all relations from φ-worlds to ¬φ-worlds. Clearly, belief revisions with suggestive upgrade do not preserve linking of relations. For consider a case where wPa w1 and wPa∗ w2 and w1 Pa∗ w2 , with φ true in w1 but not in w and w2 . Then suggestive upgrade will remove the a-path from w1 to w2 (see Figure 4). However, if we revise the upgrade procedure so that it adds extra links instead of removing them, as follows, we get a variation that preserves linking: a φ =def ?φ; a∗ ; ?φ ∪ ?¬φ; a∗ ; ?¬φ ∪ ?¬φ; (a∗ ∪ aˇ∗ ); ?φ.

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w1 : φ

w1 : φ a∗

a

w : ¬φ

a∗

a

w2 : ¬φ

w : ¬φ

a∗

w2 : ¬φ

Fig. 4. Suggestive upgrade does not preserve linking

Thus, links from φ worlds x to ¬φ worlds y get reversed, and extra links to x get added to ‘support’ the new link from y to x. Moreover, φ to φ links and ¬φ to ¬φ links are strengthened by means of transitive closure, in order to deal with the problem of detours through worlds that assign a different truth value to φ. We can prove the following. Theorem 9. If M = (W, P, V ) is a preference model where Pa and Pb are linked, and φ is a PDL formula, then [[[a φ]]]M and Pb are also linked. Proof. Write a for Pa , b for Pb , and a φ for [[[a φ]]]M . First note that for any worlds x and y, if xa∗ y then either x(a φ)y or y(a φ)x. Suppose xby and x(a φ)∗ z. We will show that either wa∗ y or ya∗ w for all w on the path from x to z. Firstly let w = x. Since xby and xa∗ x, either xa∗ y or ya∗ x by linking of a and b. Now let w be the predecessor of w on the path, so x(a φ)∗ w and w (a φ)w. Suppose either ya∗ w or w a∗ y. Since w (a φ)w, either w a∗ w or wa∗ w . If ya∗ w and w a∗ w or wa∗ w and w a∗ y, then trivially ya∗ w or wa∗ y. Suppose w a∗ y and w a∗ w. By forward linking of a and Theorem 2, wa∗ y or ya∗ w. Suppose ya∗ w and wa∗ w . By backward linking of a and Theorem 2, ya∗ w or wa∗ y. So then for any w on the path wa∗ y or ya∗ w, so za∗ y or ya∗ z, so z(a φ)y or y(a φ)z. Suppose x(a φ)y and xb∗ z. Then either xa∗ y or ya∗ x. In the first case the result follows by Theorem 6. Suppose ya∗ x. We will show that for any w on the path from x to z, yb∗ w or wb∗ y. Firstly let w = x. ya∗ x and yb∗ y so by Theorem 6 the result holds. Suppose w is the predecessor of w on the path and the result holds for w . Suppose yb∗ w . Then since w bw, trivially yb∗ w. Suppose w b∗ y. Then the result holds by linkedness of b. Now call a substitution where all bindings are of the form a  → a φ a linked substitution. Then we get a complete logic for belief change with linked substitutions, by means of reduction axioms that ‘compile out’ the belief changes (see [11]): Theorem 10. The logic of epistemic preference PDL with belief change modalities for linked substitutions is complete.

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Proof. The preference change effects of [ρ] can be captured by a set of reduction axioms for [ρ] that make [ρ] act as identity on basic propositions, [ρ] ↔  [ρ]p ↔ p, make [ρ] commute with the boolean operators, [ρ]¬φ ↔ ¬[ρ]φ [ρ](φ1 ∧ φ2 ) ↔ [ρ]φ1 ∧ [ρ]φ2 and that handle formulas of the form [ρ][π]φ by means of reduction axioms of the form [ρ][π]φ ↔ [Fρ (π)][ρ]φ, with Fρ given by:

Fρ (a) := ρ(a) Fρ (?φ) := ?[ρ]φ, Fρ (π1 ; π2 ) := Fρ (π1 ); Fρ (π2 ), Fρ (π1 ∪ π2 ) := Fρ (π1 ) ∪ Fρ (π2 ), := (Fρ (π))∗ . Fρ (π ∗ )

It is easily checked that these reduction axioms are sound, and that for each formula of the extended language the axioms yield an equivalent formula in which [ρ] occurs with lower complexity, which means that the reduction axioms can be used to translate formulas of the extended language to PDL formulas. Completeness then follows from the completeness of PDL.

5

Analyzing Plenary Dutch Meetings

A plenary Dutch meeting (Dutch: ‘Vergadering’) is a simultaneous preference/belief change event where the following happens. Assume an epistemic situation M with actual world w, and assume proposition φ is on the agenda. – If a majority prefers φ to ¬φ, i.e., if |{i ∈ Ag | M |=w [→i ]φ}| > |{i ∈ Ag | M |=w [→i ]¬φ}| then simultaneous belief change {i  → i φ | i ∈ Ag} takes place. – If a majority prefers ¬φ to φ, i.e., if |{i ∈ Ag | M |=w [→i ]φ}| < |{i ∈ Ag | M |=w [→i ]¬φ}| then simultaneous belief change {i  → i ¬φ | i ∈ Ag} takes place. – If there is no majority either way, nothing happens.

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a

a, b

p¯q

p¯ q

b, c

c

b

c

Fig. 5. Preference model for the case of the three judges

In fact, Dutch meetings are procedures for judgement aggregation [17]. Let us return to our example of three judges a, b, c with a, b agreeing that p, and b, c agreeing that q, so that both p and q command a majority, but p ∧ q does not. Using our logic, we can picture the situation as a preference model. We assume that every agent has greater belief in worlds that match her beliefs in more propositions. Then we get the model of Figure 5. So a has the greatest belief in the world where p and not q hold, but after that she has more belief in a world where p and q both hold than in the world where q and not p hold, because in the first world at least her belief in p is right. Similarly for c. For b, she believes in the world where p and q hold, and values the other worlds equally plausible. In this model the following formulas hold: [→a ]p, [→b ]p, [→b ]q, [→c ]q, [→a ]¬(p ∧ q), [→c ]¬(p ∧ q). This shows that there are majority beliefs in p and in q, but there is also a majority belief in ¬(p ∧ q). If the judges decide to have a Dutch meeting about p, the result will be unanimous belief in p (see Figure 6). a pq

a, b, c

b, c

p¯ q

a, b, c

p¯q Fig. 6. Result of Dutch meeting about p

Multi-agent Belief Revision with Linked Preferences pq

p¯ q

a, b, c

a, b, c

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a, b, c

p¯q Fig. 7. Result of Dutch meeting about q after Dutch meeting about p

Now if the judges hold a subsequent Dutch meeting about q, the result will be unanimous belief in q (see Figure 7). Now the judges unanimously believe in p ∧ q, so the defendant will be judged guilty. pq

a

a, b, c

p¯q

p¯ q

a, b, c

b

c

Fig. 8. Result of Dutch meeting about p ∧ q

However, if a Dutch meeting about p ∧ q was held in the first place, the result would be belief in ¬(p ∧ q) (see Figure 8). Clearly, in this case the defendant would be acquitted. Experienced judges are of course familiar with this phenomenon. Procedural discussions about how to decompose a problem, and in which order to discuss the component problems may seem beside the point of a legal issue, but they turn out to be highly relevant for the outcome of the legal deliberations. Our logic can be used to model the various procedural approaches for arriving at a unanimous conclusion. See [20] for an alternative analysis of procedural aspects in processes of judgement aggregation, in terms of coalition logic.

6

Conclusion

We have shown how propositional dynamic logic with converse can be used as a basis for developing a very expressive system of multi-agent belief revision and

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belief change, and we have studied linking of beliefs as a natural requirement on multi-agent belief change. In future work, we would like to answer the open technical questions that were mentioned at the end of Section 2. Also on our research agenda is the problem of finding natural classes of substitutions that preserve relation linking. Next, since our logic provides a general mechanism for simultaneous belief change, it can be used to describe and analyse topics in judgement aggregation, the effects of agenda setting, the effects of subgroup meetings to create general belief, and many further issues of collective rationality. Finally, we would like to compare our logic to other proposals to give a modal analysis of judgement aggregation, such as [1], and we would like to find out if our logic can be put to work in practical social choice analysis.

References ˚gotnes, A., van der Hoek, W., Wooldridge, M.: Reasoning about Judgement and 1. A Preference Aggregation. In: Durfee, E.H., Yokoo, M., Huhns, M.N., Shehory, O. (eds.) AAMAS 2007, pp. 567–575. ACM Digital Library (2007) 2. Baltag, A., Moss, L.S., Solecki, S.: The Logic of Public Announcements, Common Knowledge, and Private Suspicions. In: Gilboa, I. (ed.) TARK 1998, pp. 43–56. Morgan Kaufmann, CA (1998) 3. Baltag, A., Smets, S.: Conditional Doxastic Models: A Qualitative Approach to Dynamic Belief Revision. Electron. Notes Theor. Comp. Sci. 165, 5–21 (2006) 4. Baltag, A., Smets, S.: A Qualitative Theory of Dynamic Interactive Belief Revision. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) LOFT 2006. Texts in Logic and Games, vol. 3, pp. 9–58. Amsterdam University Press, Amsterdam (2008) 5. van Benthem, J.: Dynamic Logic for Belief Revision. J. Appl. Non-Class. Log. 14(2), 129–155 (2007) 6. van Benthem, J., Liu, F.: Dynamic Logic of Preference Upgrade. J. Appl. NonClass. Log. 14(2), 157–182 (2007) 7. van Benthem, J., van Eijck, J., Kooi, B.: Logics of Communication and Change. Inform. Comput. 204(11), 1620–1662 (2006) 8. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001) 9. Board, O.: Dynamic Interactive Epistemology. Games Econ. Behav. 49, 49–80 (2002) 10. Boutilier, C.: Toward a Logic of Qualitative Decision Theory. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) KR 1994, pp. 75–86. Morgan Kaufmann, CA (1994) 11. van Eijck, J.: Yet More Modal Logics of Preference Change and Belief Revision. In: Apt, K.R., van Rooij, R. (eds.) New Perspectives on Games and Interaction. Texts in Logic and Games, vol. 4, pp. 81–104. Amsterdam University Press, Amsterdam (2008) 12. van Eijck, J., Wang, Y.: Propositional Dynamic Logic as a Logic of Belief Revision. In: Hodges, W., de Queiroz, R. (eds.) WoLLIC 2008. LNCS (LNAI), vol. 5110, pp. 136–148. Springer, Heidelberg (2008) 13. Goldblatt, R.: Logics of Time and Computation, Second Edition, Revised and Expanded. CSLI Lecture Notes, vol. 7. CSLI, Stanford (1992) (first edition 1987) 14. Grove, A.: Two Modellings for Theory Change. J. Philos. Log. 17, 157–170 (1988)

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15. Harel, D.: Dynamic Logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 497–604. Reidel, Dordrecht (1984) 16. Kozen, D., Parikh, R.: An Elementary Proof of the Completeness of PDL. Theor. Comput. Sci. 14, 113–118 (1981) 17. List, C., Pettit, P.: On the Many as One. Philos. Public Aff. 33(4), 377–390 (2005) 18. Lutz, C.: PDL with Intersection and Converse is Decidable. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 413–427. Springer, Heidelberg (2005) 19. Nayak, A.C.: Iterated Belief Change Based on Epistemic Entrenchment. Erkenntnis 41, 353–390 (1994) 20. Pauly, M.: Logic for Social Software. Ph.D. thesis, Universiteit van Amsterdam (2001); ILLC Publications DS-2001-10 21. Segerberg, K.: A Completeness Theorem in the Modal Logic of Programs. In: Traczyck, T. (ed.) Universal Algebra and Applications, pp. 36–46. Polish Science Publications (1982)

A Note on Assumption-Completeness in Modal Logic Jonathan A. Zvesper1 and Eric Pacuit2, 1

2

Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom [email protected] Center for Logic and Philosophy of Science, Universiteit van Tilburg, Postbus 90153, 5000 LE Tilburg, The Netherlands [email protected] Abstract. We study the notion of assumption-completeness, which is a property of belief models first introduced in [18]. In that paper it is considered a limitative result – of significance for game theory – if a given language does not have an assumption-complete belief model. We show that there are assumption-complete models for the basic modal language (Theorem 8).

1

Introduction

The literature on the epistemic foundations of game theory uses a variety of mathematical models to formalise talk about players’ beliefs about the game, beliefs about the rationality of the other players, beliefs about the beliefs of the other players, beliefs about the beliefs about the beliefs of the other players, and so on (see [16] for a recent survey). Examples include Harsanyi’s type spaces [25,26,27], interactive belief structures [15], knowledge structures [3] plus a variety of logic-based frameworks (see, for example, [9,24,14,13,5]). A recurring issue involves defining a space of all possible beliefs of the players and whether such a space exists. In this paper, we study one such definition: the notion of assumption-complete models. This notion was introduced in [15], where it is formulated in terms of “interactive belief models” (which are essentially qualitative versions of type spaces). Assumption-completeness is also explored in [18], where a number of significant results are found, and connections to modal logic are mentioned. A discussion of that paper, and a syntactic proof of its central result, are to be found in [32]. Within and between these different mathematical models, different epistemic notions can be formalised, one of which is the notion of an “assumption”, which is closely related to the only-knowing operator studied by Levesque [30] (cf. [23]). 

The first author was supported by a Marie Curie Early Stage Research fellowship in the project GLoRiClass (MEST-CT-2005-020841). He would like to thank the participants in the Amsterdam Dynamic Logic seminar for their feedback, especially Alexandru Baltag, Johan van Benthem and Daisuke Ikegami. Definition 12 was clarified in discussion with C´edric D´egremont.

G. Bonanno, B. L¨ owe, and W. van der Hoek (Eds.): LOFT 2008, LNAI 6006, pp. 190–206, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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Roughly speaking, a player’s assumption is defined as her strongest belief: the conjunction of all her beliefs (equivalently, a belief that implies all her other beliefs).1 Call the (two) players “Ann” and “Bob”. An interactive belief model (we shall use the shorter belief model in this paper) consists of states for Ann and for Bob. It specifies the beliefs, and some other information like the strategy chosen, of a player at any of his/her states. Each player’s beliefs are defined over the other player’s states: that is, beliefs of the players are given in terms of a set of the opponent’s states. Thus to each Ann state is associated a set of Bob states that Ann considers possible, and to each Bob state a set of Ann states that Bob considers possible. Bob’s assumption at a state is the set of Ann states to which that state is related. A belief model is assumption-complete for a language of Ann-states just when for every sentence of the language, there is some Bob-state (in the model) where Bob assumes (the set of states satisfying) that sentence. The idea of assumption-completeness is that the language of Ann-states should be accessible to Bob. And if it is accessible to Bob then he should be able to assume (in this artificial sense) any member of it. Brandenburger and Keisler prove the following impossibility result (cf. Theorem 4 and [18, Theorem 5.4]): There are no assumption-complete models for the first-order language. This was taken to be a limitative result, and one that should be of significance for game theory: [O]ur impossibility theorem says: If the analyst’s tools are available to the players, there are statements that the players can think about but cannot assume. The model must be [assumption-]incomplete. This appears to be a kind of basic limitation in the analysis of games. [18] As Brandenburger and Keisler point out, the existence of assumption-complete models is not only of theoretical interest [18]. It turns out to be relevant for the “epistemic program” in game theory. The goal of this program is to provide epistemic conditions on the players (e.g., common belief in rationality) that lead2 to various solution concepts (e.g., Nash equilibrium, iterated dominance, backward induction). Epistemic conditions sometimes involve such a notion of “completeness” of the underlying belief model, and conditions similar to 1

2

The formal details will follow in Section 2. This definition of “assumption” might seem strange, and we certainly do not claim that it, nor the formal definition that will follow, capture the common-sense meaning of the English word “assumption”. However, when we present the semantics of belief models we shall see that, formally speaking, assumption-completeness will be a natural enough property, albeit poorly named. Note that, while the term “assumption” is used in [18], there the notion that we shall study is called simply “completeness”. We prefer the more specific, if less wieldy, term “assumption-completeness”. More precisely representation theorems are proved stating that players satisfying such-and-such epistemic condition will play a particular solution concept and conversely if the players play according to some solution concept then there is an epistemic model where the players satisfy the epistemic conditions. See [19] for a critical survey of this line of reasoning.

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assumption-completeness occur in two analyses: in Battigalli and Siniscalchi’s analysis of extensive-form rationalisability [7] and Brandenburger, Friedenberg and Keisler’s analysis of iterated admissibility [17]. We return to the role that assumption-completeness plays in epistemic analyses of games in Section 6. Given the above interpretation of Theorem 4, a natural question3 is: can one define instead a restricted set of “tools” which can be “available” to the players, and which are also useful for the analyst? Theorem 4 shows that the first-order language is too powerful a tool to be available; what about weaker languages? We shall address this question from the perspective of modal logic, defining a modal language for belief models, giving a complete axiomatisation (Theorem 9), and obtaining, as a corollary of the completeness proof, that there are assumption-complete models for the basic modal language (Theorem 8). What about strengthening our possibility result? In Section 5 we shall look briefly at perspectives for doing exactly this. We shall conjecture that the bounded fragment of first-order logic has assumption-complete models. The bounded fragment is expressive enough to express some concepts that are very important for game theory. Notably, in the appropriate framework, it can express the proposition that a player is rational.4

2

Preliminaries

We shall work in the framework from [18], which we relate to the “possible worlds” approach in Section 4. In this framework, “belief models” are two-sorted first-order structures.5 Definition 1 ([18, Definition 3.1]). A belief model is a structure (U a , U b , Ra , Rb , {P α }α∈N ), where 1. U a = ∅ = U b , U a ∩ U b = ∅; 2. Ra  U a × U b , Rb  U b × U a ; 3. for every u ∈ U a there is a v ∈ U b such that uRa v, and similarly for every v ∈ Ub ; 4. each P α ⊆ U a ∪ U b . 3 4

5

This is also raised in [18, Section 2]. Here rationality is interpreted in the standard way: optimising given the agent’s current beliefs. Making the statement that a particular logic can/cannot express this notion of rationality precise is still ongoing work. This is also briefly discussed in [18, Section 2]. In [18], the definition of belief models is more general, allowing for almost arbitrary signatures for belief models. With condition (3), we restrict our attention to monadic predicates (the P α ’s) because it is natural to do so in the stated field of application of the belief models, viz. to games, in which the P α ’s represent choices made by the players. Furthermore, they allow arbitrary strategy sets, where we restrict our attention, for simplicity’s sake, to countable strategy sets.

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The elements of the domain U = U a ∪ U b are called “states”. Specifically those in U a are called “Ann states” and those in U b “Bob states”. We might also call them “types”, because each one specifies an epistemic type (in the sense of an Harsanyi type space) of the relevant player. The relations Ra and Rb specify those states considered possible by Ann and Bob respectively: uRa v means that the Ann-state u considers the Bob-state v to be possible. We write Ra (u) to mean {v ∈ U b | uRa v}, and we use similar terminology with b switched for a. We say that for u ∈ U a and E ⊆ U b , u believes E just when Ra (u) ⊆ E, and (stronger) that u assumes E just when Ra (u) = E. We shall write R for Ra ∪ Rb . The predicates P α are there to carry additional information about the states. When considering models for games, each P α could represent which strategy is chosen by each player state. That is, there would be one predicate P s for each strategy s of each player. We consider general models, in which the P α ’s do not necessarily represent strategies, but rather any fact that is non-epistemic, i.e. does not logically depend on the relations Ri . However, everything we say is equally valid for the special case in which the P α ’s do represent choice of strategy. The conditions imposed by Definition 1 are natural: (1) says that there are Ann states and Bob states, (2) that for both Ann and Bob there is at least one non-trivial belief state, i.e. at which some Bob or Ann state is ruled out as a possibility, and (3) that at every Ann or Bob state, some Bob or Ann state is taken to be possible. State-based models, and a definition of belief like the one given, are familiar from epistemic logic since the work of Hintikka [28]. In those single-sorted models that are standard in epistemic logic, states (sometimes called “possible worlds”) specify a type for each player, i.e. in the two-player case a state would specify an Ann-type and a Bob-type. (The connection between qualitative type spaces and single-sorted models is discussed in [6, Chapter 3].) We discuss in Section 4 how to define assumption-completeness for standard models. Modulo this different perspective given by the two-sortedness of the models, the notion of an assumption is basically the same as that of “only knowing”, which was introduced by Levesque [30], and axiomatised by Halpern and Lakemeyer ([23], cf. [8,29]). Fix a belief model (U a , U b , Ra , Rb , {P α }α∈N ). Following [18], a language for Ann (based on the fixed belief model), denoted La , is any subset of U a (similarly for Bob). For example, the powerset language for Ann (Bob) is La = p(U a ) (Lb = p(U b )), where we write p(X) to denote the power set of X. Another natural example is the first-order language defined to be the sets definable by sentence of first-order logic. Formally, a first-order formula is defined by the following recursion schema: ϕ ::= Ua x | Pα x | xRa y | xRb y | ¬ϕ | ϕ ∧ ϕ | ∃xa ϕ | ∃xb ϕ As usual, sentences are closed formulae (i.e., those without free occurrences of variables). Given a belief model, each first-order sentence defines a subset of that

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model in the standard way (with a caveat about two-sorted quantification: ∃xa quantifies only over Ann states). Then the first-order language La1 is the set of subsets of U a that are definable by a first-order sentence (similarly for Bob). We are now ready to define the notion that will be central to our concerns in this paper. Definition 2 ([18, Definition 4.2]). A belief model (U a , U b , Ra , Rb , . . .) is assumption-complete for the language L = La ∪ Lb just if for every ∅ = Eb ∈ Lb , there is an u ∈ U a such that Ra (u) = Eb , and for every ∅ = Ea ∈ La , there is a v ∈ U b such that Rb (v) = Ea .

(Ca)

That is, (Ca) ensures that every definable set of Ann (or Bob) states can be assumed by Bob (resp. Ann). Formally speaking, this is a natural condition to impose. We shall say of a language that it is assumption-complete just when there is some model that is assumption-complete for it, and that it is assumptionincomplete just when there is no such model.

3

Main Results

Intuitively, if a language is assumption-complete then it has a “big” model, which means, roughly, that any property of Bob states (expressible in the language) can be assumed by Ann (and vice versa for Bob). The following theorem demonstrates that some restriction on the language is needed for it to be assumption-complete: Theorem 3 ([15]). The powerset language (where La = p(U a ) and Lb = p(U b )) is assumption-incomplete. The proof of Theorem 3 is a straightforward application of Cantor’s Theorem that there is no surjection from a set onto its power set. But what about slightly weaker languages that still are stronger than the first-order language? (For example, first-order logic with fixed-points, or some second-order logic.) The following theorem states that the first-order language is already too expressive to be assumption-complete: Theorem 4 ([18, Theorem 5.4]). L1 is assumption-incomplete. Thus the standard tool first-order logic is too strong to have this formally natural property.6 The proof of Theorem 4 uses a generalised version of Russell’s paradox, and essentially relies on the same kind of diagonalisation argument used to prove Cantor’s Theorem. Although we refer to [18] for the proof, we 6

Of course, there may be some other language which is expressively incomparable with L1 , but we shall not pursue this line of reasoning here, although Conjecture 21 concerns such a fragment.

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note for later reference that it is important that the sentence (BK) is expressible in the first-order language L1 : Ann believes that Bob’s assumption is that Ann believes that Bob’s assumption is wrong.

(BK)

Theorem 4 rules out the first-order language; what then might we use in its stead? – What are our alternatives? We now mention very briefly one positive result, for the so-called “positive fragment ” (see [18, Definition 10.1]), which is essentially a mixture of first-order and modal languages without negation. Theorem 5 ([18, Theorem 10.4]). The positive fragment is assumptioncomplete. Related to this theorem, Mariotti, Meier and Piccione show that there exists a “universal possibility structure” [31]. We now begin an investigation of other fragments of first-order logic that may be assumption-complete. We start with the basic modal language ML. The basic modal formulae are those defined by the following schema: ϕ ::= ♀ | pα | ¬ϕ | ϕ ∧ ϕ | ϕ We write ♦ to abbreviate ¬¬, ϕ ⊃ ψ for ¬(ϕ ∧ ¬ψ), and ♂ for ¬♀. The basic modal language ML is the set of subsets that are definable by some basic modal formula, where ♀ defines the Ann states U a ; pα defines the set P α ; negation and conjunction work as usual; and ϕ defines the set where the stateowner believes ϕ. That is, where ϕ is the set defined by ϕ, ϕ defines the following set: {u ∈ U | R(u) ⊆ ϕ}, It was shown in [18, Section 9] that, since the basic modal language cannot express the assumption operator, there are belief models “complete in a weaker sense that every statement which is possible can be believed (instead of assumed) by the player.” What about the stronger statement, that if a language cannot express the assumption operator, then there are assumption-complete models? Certainly, being able to talk about Bob’s assumptions is essential in the particular proof of Theorem 4; however, the converse is open: if one cannot talk about the players’ assumptions in the language then as far as we know the possibility is open for there to be an assumption-complete model. We shall prove the following theorem: Theorem 6. ML is assumption-complete. In fact we can strengthen Theorem 6, by adding a property which is also in effect present in Brandenburger and Keisler’s positive result, Theorem 5. Definition 7. Say that a belief model is total just when for every P α , U a ∩ P α = ∅ and U b ∩ P α = ∅.

(Cb)

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Condition (Cb) means that every possible ‘basic configuration’ is present, so in the case where the P α ’s represent choice of strategy, it means that for each of player i’s strategies si , there is a state at which i chooses that si . (This is assuming the same strategy sets for both players; the case of disjoint strategy sets does not add any mathematical complication.) We shall therefore prove the following: Theorem 8. There are total assumption-complete belief models for the basic modal language. Although the totality condition is never made explicit in [18], it is implicit there. (Not only does their Theorem 5 actually show the existence of total complete belief models, but also if the requirement that the model be total is dropped, then [18, Problem 7.7] is trivially answerable [33].) We give now a complete axiomatisation for ML over belief models. The axioms consist of an axiomatisation of the propositional connectives ¬ and ∧, plus the following:  (p ⊃ q) ⊃ (p ⊃ q) K  p ⊃ ♦p D  ♂ ⊃ ♀ U1  ♦♀ ⊃ ♂ U2 The following are the rules of inference: ϕ

ϕ ϕ⊃ψ ϕ Sub MP N ec ψ  ϕ  ϕ[p → ψ]

If there is an inference (using only these axioms and rules) of ϕ then we write  ϕ. We write |= ϕ just when ϕ is valid, that is: when in every belief model, ϕ defines the whole set U . To show that a logic is (weakly) complete 7 is to show that |= ϕ ⇒ ϕ. Theorem 9.

|= ϕ ⇔  ϕ

Theorem 9 is proved in a standard way by building a “canonical model” (cf. [12]). The states in the canonical model are maximally-consistent sets, and the relation is defined as follows: R(Γ ) = {Δ | ∀ψ ∈ Δ, ♦ψ ∈ Γ } Theorem 8 can then be proved by observing that this canonical model is appropriately assumption-complete. We sketch now the proof of Theorem 8. Take any definable subset of the canonical model E ∈ MLa (without loss of generality: the same will hold with a switched for b). Then by definition of MLa , there is some modal formula ϕ such that ϕ = E. Furthermore, we must have  ϕ ⊃ ♀, 7

This notion of completeness, familiar from formal logic, has (usually!) nothing to do with assumption-completeness.

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because otherwise there would be some Bob-state Γ ∈ E. Then it remains to show that there is a state Γϕ that assumes ϕ in the canonical model. We use the following lemma: Lemma 10. The set Γϕ is consistent: Γϕ := {♦γ |  ¬(ϕ ∧ γ)} ∪ {ϕ} Proof. We appeal to completeness and invariance of basic modal formulae under disjoint unions and generated submodels ([12, Propositions 2.3 and 2.6]): Let Πϕ = {γ | ¬(ϕ ∧ γ)} be the set of formulae consistent with ϕ. For each such γ ∈ Πϕ , by completeness there is a model Mγ with state ωγ such that Mγ , ωγ  ϕ ∧ γ. Since the language is preserved under generated submodels and disjoint unions, we shall still have that each ωγ  ϕ when we take the disjoint union of all of the submodels generated by the ωγ ’s for every γ ∈ Πϕ . Now define a new model M by taking the disjoint union of the submodels generated by each ωγ , adding one new state ωϕ , and stipulating that R(ωϕ ) = {ωγ |γ ∈ Πϕ }. Notice that since by hypothesis ϕ  ♀, this is indeed a model, since every ωγ  ♀ (and so ωϕ  ♂). By construction we have ωϕ  ϕ, because for each γ ∈ Πϕ , ωγ  ϕ. Furthermore, for each γ ∈ Πϕ , we also have ωϕ  ♦γ, since ωϕ Rωγ and ωγ  γ. This simple construction is illustrated in figure 1. Mγ0

Mγ1

ωγ0

ωγ1

Mγi ωγi

ωϕ Fig. 1. The construction described in the proof of lemma 10

From Lemma 10, it follows that Γϕ can be extended to form a state Γϕ in the canonical model. We must show that R(Γϕ ) = ϕ: – Since ϕ ∈ Γϕ , R(Γϕ ) ⊆ ϕ; – Take any Δ ∈ ϕ. Then take any ψ ∈ Δ; we know that  ¬(ϕ ∧ ψ), since Δ is by hypothesis consistent. So by definition of Γϕ , ♦ψ ∈ Γϕ ⊆ Γϕ . So by definition of the canonical model, Δ ∈ R(Γϕ ). This concludes the proof of Theorem 8. This argument relies crucially on the locality of modal logic reasoning. In particular, Lemma 10 and its application rely (implicitly) on the fact that the language is invariant under “disjoint unions” and “generated submodels” (see [12]). This local form of reasoning is at odds with the first-order language used

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in [18], but we find it natural in the context of epistemic reasoning from the agent’s perspective. More specifically, if we want to say that the states to which an agent has epistemic access are all those that he considers possible, then we might not want to take other states into account when describing the agent’s beliefs in some language. We shall mention these properties again in Section 5, when we consider richer modal languages. In Section 6, we shall also remark that the definitions of “assumption” and “rationality” used in [17] do not have this local, modal quality to them. In the meantime, we devote the next section to explaining the connection between belief models as defined above, and possible-worlds based models that are familiar from the literature on epistemic logic.

4

Single-Sorted Models

The “belief models” of [18] are not standard in the epistemic logic literature. There is another more standard class of models used in epistemic logic, which are mentioned in [18] as “[a]nother approach”. The obvious way of defining assumption-completeness turns out to be much too strong: not even extremely un-expressive languages would have assumption-complete models in this sense, on the condition that the players are “introspective”.8 We shall point out that, on the other hand, if the players are not introspective, then the sentence (BK) can be satisfiable. So, since it is central to the proof of Theorem 4 that (BK) be unsatisfiable on the relevant class of models, the impossibility theorem might fail for non-introspective models. In any case, it is usually assumed that players are introspective, so we shall also look at a different definition of assumption-completeness for single-sorted models. This second definition is more faithful to the original definition of assumptioncompleteness for belief models: When we translate a belief model into a singlesorted model, the resulting single-sorted model is assumption-complete in the second sense if and only if the original belief model was assumption-complete. Single-sorted models consist of a non-empty set of “possible worlds” Ω and a relation Ri ⊆ Ω × Ω for each player. In the two-player case, a “frame” is a list (Ω, Ra , Rb ). In belief models there was additionally some information (in the P α ’s) about strategies and so forth; since we are now in a modal framework, we shall encode this using a set Φ = Φa ∪ Φb of proposition letters, and adding a valuation V : Φ → p(Ω): a single-sorted model is a frame equipped with a valuation. (We shall henceforth sometimes refer to single-sorted models simply as “models”.) The idea behind dividing the proposition letters into two sets is that some pertain to Ann (for example, those saying which strategy she plays), and some to Bob. So in the case when we want a model for an interactive model, we shall want a proposition letter for each player i and each strategy P α That is: 8

A player i is introspective just if when i believes ϕ she believes that she believes it, and when she does not believe it she believes that she does not believe it. Introspection is usually taken for granted (often tacitly) in formulating epistemic models used in game theory.

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Φi = {pα i | α ∈ N} We say that player i is consistent and introspective just if the following conditions hold: Ri (s) = ∅

(Di )

sRi t & tRi u ⇒ sRi u sRi t & sRi u ⇒ tRi u.

(4i ) (5i )

If a model (or frame) satisfies all of these properties for all players, then we say that it is “KD45” (following a standard classification from modal logic, cf. [12]). It is straightforward (though not entirely trivial) to translate in a meaningful way between belief models and single-sorted models. That is, to give a pair of functions (ρ, τ ) with ρ taking a single-sorted model and returning an “equivalent” belief model, and τ taking a belief model and returning an equivalent singlesorted model. We give the details of such a translation below: Definition 11 (Translation from Belief Models to Single-Sorted Models). τ (U a , U b , Ra , Rb , {P α }α∈N ) = (U a × U b , Ra , Rb , V ), where (ua , ub )Ri (ua , ub ) ⇔ (ui = ui & ui Ri uj ), α a b V (pα a ) = (P ∩ U ) × U , α a α b and V (pb ) = U × (P ∩ U ). We write s ∼Vi t to mean that s and t have the same propositional valuation with respect to Φi , i.e.: ∀p ∈ Φi , s ∈ V (p) ⇔ t ∈ V (p) Then we define for i ∈ {a, b} the equivalence relation ∼i : s ∼i t ⇔ (Ri (s) = Ri (t)&s ∼Vi t) Definition 12 (Translation from Single-Sorted Models to Belief Models). ρ(Ω, Ra , Rb , V ) = (U a , U b , Ra , Rb , {P α }α∈N ), where and

U i = Ω/∼i , [s]i R [u]j ⇔ ∃v ∈ [u] : sRi v, P α = {[s]i | s ∈ V (pα i )}. i

How are we to define assumption-completeness for models? The na¨ıve approach would be to say that a model S is assumption-complete for L ⊆ p(Ω) just if for any X ⊆ L, there is ω ∈ Ω such that Ra (ω) = X, and similarly for b. However, it is not difficult to see that this is not an innocent approach. For then even very simple languages are not assumption-complete:

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Fact 13. For the definition of assumption-completeness just proposed, any language that is closed under unions would be assumption-incomplete with respect to KD45 models. Furthermore, if we were to take this as a sign that we should not be working in KD45 models, here is another fact: Fact 14 ([33]). The formal translation of the sentence (BK) is satisfiable if Da or 4a or 5a does not hold. That is, if introspection fails then the sentence (BK) is consistent. It might seem puzzling that an informal argument is given in [18, Section 1] to the effect that (BK) is not satisfiable, an argument where the word “introspection” is never used, nor is any concept like it employed. It turns out that corners were cut in the informal argument; the threads of the argument are unpicked in [33]. (BK) cannot hold in belief models because in belief models there is an implicit assumption of introspection: Fact 15. For any belief model M, τ (M) is KD45. We shall now give a proper definition of assumption-completeness for models. Definition 16. A model S = (Ω, Ra , Rb , V ) is assumption-complete for a language L ⊆ p(Ω) just if for any X ∈ L, for {i, j} = {a, b}, there exists y ∈ Ω such that the following two conditions hold: – ∀x ∈ X, ∃v ∈ Ri (y) : Rj (v) = Rj (x) & v ∼Vj x; – ∀v ∈ Ri (y), ∃x ∈ X : Rj (v) = Rb (x) & v ∼Vj x. We say that S is assumption-complete tout court when it is assumption-complete for a and for b. Definition 16 might seem more long-winded, but it is equivalent to the definition for belief models, in the following sense: Theorem 17. Any belief model M is assumption-complete iff τ (M) is assumption-complete. And any model S is assumption-complete iff ρ(S) is assumptioncomplete. Thus we have found the “correct” definition of assumption-completeness for single-sorted models.

5

Beyond the Basic Modal Language

The basic modal language is natural, but is arguably not expressive enough a “tool” for the analyst, or for the players. It is not possible to give general results about the expressibility of game-theoretical concepts in terms of a language, since there are different ways of representing the various elements involved in a game. However, we get some indications from the fact that in [11] the authors deem it necessary to extend the basic modal language (with so-called “nominals”) in order to express Nash equilibrium, and in [1] the authors express rationality in

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a natural modal way using bounded quantification.9 (Again, in these papers the framework is one of single-sorted models; we have discussed the connection above in Section 4.) It is therefore natural to search for more expressive extensions of the basic modal language that nonetheless are assumption-complete; given Theorem 4, we know that we cannot go as far as the full modal language.10 In this section we briefly discuss perspectives for finding such fragments. Since we take the vocabulary of the basic modal language to be intuitively appealing, and of use to the analyst, we shall look at extensions of the basic modal language. One such extension that is of particular interest to us now is L↓ , known as “hybrid language with binder” (a number of such extended modal languages are studied model-theoretically in [20]). Let the binder formulae be those of the following form: ϕ ::= ♀ | pα | x | ¬ϕ | ϕ ∧ ϕ | ϕ |↓x.ϕ Binder formulae (↓ is know as the “binder”) define sets relative to a variable assignment, that is: a function σ : V ar → U .11 We write ϕσ for the subset of U defined by the formula ϕ relative to σ. And we write σ[x → u] for the assignment that agrees everywhere with σ except that it maps x to u. The semantics of the new connectives are as follows: – xσ = {σ(x)}; – ↓x.ϕσ = {u ∈ U | u ∈ ϕσ[x →u] } For a binder sentence ϕ (i.e. with no unbound variables), we write just ϕ (since it makes no difference which assignment we use). The language L↓ is a fragment of (i.e. a subset of) the first-order language. Indeed, it is expressively equivalent to the “bounded fragment” of the first-order language [2]. Feferman has studied the bounded fragment [22], which is model-theoretically characterised as the fragment of first-order logic that is invariant under generated submodels. In the details of the proof of Theorem 8, the only ‘modal’ behaviour we exploit, in showing that the set Γϕ is consistent, is that sentences are preserved under disjoint unions and generated submodels. Thus – although there is no canonical model construction for L↓ – we are still lead to suspect that this language also has assumption-complete models: Conjecture 18. There are (total) assumption-complete belief models for the bounded fragment. 9

10

11

Indeed, (instrumental) rationality essentially says that an agents choice ‘now ’ is optimal given her beliefs ‘now ’, something which is typically not expressible in a basic modal language, but which calls for the kind of “hybrid” language, i.e. using nominals. Though fixpoint logics for common knowledge go beyond first-order logic, and are obviously of relevance to the epistemic analysis of games; we do not examine those here, and mainly focus on languages between first-order and modal, though see Conjecture 21. The same is true of first-order formulae, but we skipped the details because they are more standard.

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An inspection of the proof of Theorem 4 reveals three conditions that together are sufficient to show that a language L is not assumption-complete. The first condition, satisfied by all of these modal languages, is that La be closed for the belief operator: X ∈ Lb ⇒ {ua ∈ Ua | Ra (ua ) ⊆ X} ∈ La .

(C0)

The second condition is that L contain the following set: / R(ω  )} Da := {ω ∈ Ua | ∀ω  ∈ (R(ω) ∩ Ub ), ω ∈ Da ∈ La .

(C1)

a

C1 says that Bob has “available” (i.e. in L , the language for defining sets of Ann states) the sentence “Ann believes that Bob’s assumption is wrong”. The third condition is that La be closed under the assumption operator, i.e., X ∈ La ⇒ {ub ∈ Ub | Rb (ub ) = X} ∈ Lb .

(C2)

If all of C0–C2 hold, then L is assumption-incomplete. In particular, L will allow sentence (BK) to be expressed. Note that Da is expressible by a binder sentence, so is in L↓ : Fact 19. Da = ♀∧ ↓x.¬x ∈ L↓ However, since the language is closed under generated submodels then, importantly, the assumption operator is not expressible: Fact 20. L↓ does not satisfy condition C2. Any extension of the modal language has C0; we’ve looked at a language with C1; so what about C2? Clearly, adding an assumption operator to L↓ will make the language assumption-incomplete. (So this would be a strictly weaker language than first-order logic, that is nonetheless assumption-incomplete.) Nonetheless, while we do not investigate the matter further here, we conjecture that adding an assumption operator into ML would not leave the happy realm of assumptioncompleteness. Another extension of the basic modal language would be to consider adding infinite conjunctions, up to some cardinality λ. There is a close connection between the formal of assumption-completeness and the impossibility of an unrestricted Comprehension axiom schema in set theory (cf. [21]). This connection with Russell’s paradox (which shows that the formula x ∈ / x cannot consistently be added as a Comprehension axiom, on pain of inconsistency) and the “paradox” of assumption-completeness is already remarked in [18]. Let us now render that connection a little more precisely: the consistency of a comprehension schema ∃x∀z(z ∈ x ≡ ϕ(z)), for all ϕ ∈ Ψ (with x not free in ϕ), is essentially a single-agent assumption-completeness for the language defined by Ψ . Thus since [4] shows the consistency, in the context of non-well-founded set theory, of an

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axiom of comprehension that is restricted to infinitary modal formulas, we are led to conjecture that for any weakly compact cardinal κ the infinitary modal language MLκ is assumption-complete, where MLκ is just like ML except that  rather than just binary conjunctions ϕ ∧ ψ, it contains conjunctions Φ where Φ is some set smaller than κ. (So ML in this notation is MLℵ0 .) Conjecture 21. MLκ is assumption-complete. [4] shows that this infinitary modal comprehension scheme entails apparently stronger such schemes, including for “generalised positive formulas”. If the assumption operator were expressible in MLκ this would make Conjecture 21 a strengthening of Theorem 5.

6

Discussion

We plan to explore a number of issues in the future. The most pressing issue is finding interesting languages that are assumption-complete, especially languages that can express concepts that are of interest to game-theorists, and that have a natural appeal in terms of being languages that we would want to attribute to the agents to capture their ability to think about the situation they are in (e.g., the bounded fragment). We also think that it is important to explicate the notion of assumption-completeness in terms of some of the other models that are used in the game-theory literature, for example those that we mentioned in the introduction. – If assumption-completeness is an important epistemic notion, then it is important to understand it also in terms of other epistemic models. Section 4 has addressed this issue, for the case of single-sorted epistemic models that are familiar from the epistemic logic literature since [28]. Another tack, in order to understand better the connection between interactive belief models and standard modal models, would be to use a “two-dimensional” modal logic. It is also natural to question the quoted interpretation of the Brandenburger and Keisler impossibility result: does Theorem 4, which is essentially a generalisation of Russell’s paradox, really point to “a kind of basic limitation in the analysis of games”? Why work in assumption-complete models? A natural reaction to the Brandenburger and Keisler impossibility result (Theorem 4) is to wonder what all the fuss is about. Assumption-completeness is a nice abstract property of a logical language, but what is the harm in working with languages without this property? One answer from the literature on the epistemic foundations of game theory is that assumption-complete belief models are needed to provide an epistemic analysis of certain solution concepts. Indeed, Battigalli and Siniscalchi argue convincingly that “analysing an extensive-form game in the framework of an incomplete type space12 introduces implicit and potentially undesirable 12

The notion of incomplete type from [7] is analogous to our notion of a belief model that is assumption-incomplete (with respect to some language). For the purposes of this paper, the differences are not important.

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restrictions on forward-induction reasoning” ([7, pg. 368], original italics). More broadly, Brandenburger, Friedenberg and Keisler point out that We think of a particular incomplete structure as giving the “context” in which the game is played. In line with Savage’s Small-Worlds idea in decision theory [...], who the players are in the given game can be seen as a shorthand for their experiences before the game. The players’ possible characteristics — including their possible types — then reflect the prior history or context. (Seen in this light, complete structures represent a special “context-free” case, in which there has been no narrowing down of types.) [17, pg. 319] So, we have two examples of epistemic analyses of solution concepts where it is crucial that the analysis takes place in assumption-complete models. In fact, [17] introduce a new form of irrationality where a player optimises, but does not consider all possibilities. In the “belief-based” approach (cf. [16]) of [17], “assumption” means something a little different: assuming ϕ means considering ϕ “infinitely more likely than” ¬ϕ. Beliefs are then represented using Lexicographic Probability Systems, allowing a distinction between considering something infinitesimally unlikely and ruling something out altogether. The constraint on rationality is then that a rational player should not rule out any possibility altogether, including the possibility that another player might (irrationally) rule out some possibilities altogether. A more modal notion of assuming ϕ in that context would be to say that amongst the states a player considers (infinitesimally) possible, she considers all ϕ states to be infinitely more plausible than ¬ϕ states. Finally, we note that there are other “epistemic” analyses of iterated admissibility based on the notion of “public announcement”, in which there is no need for assumption-completeness (see [10,1]). To what extent that approach can be said to give an epistemic foundation is questionable, and more generally comparing the different analyses to [17] and judging the need for assumption-complete models is difficult without a precise set of criteria (cf. [19]).

References 1. Apt, K.R., Zvesper, J.A.: Common Beliefs and Public Announcements in Strategic Games with Arbitrary Strategy Sets. Manuscript (2007); CoRR abs/0710.3536 2. Areces, C., Blackburn, P., Marx, M.: Hybrid Logic is the Bounded Fragment of First Order Logic. In: de Queiroz, R., Carnielli, W. (eds.) WoLLIC 1999, pp. 33– 50. Rio de Janeiro, Brazil (1999) 3. Aumann, R.J.: Agreeing to Disagree. Ann. Stat. 4(6), 1236–1239 (1976) 4. Baltag, A.: A Structural Theory of Sets. Ph.D. thesis, Indiana University (1998) 5. Baltag, A., Smets, S., Zvesper, J.A.: When All is Done but Not (Yet) Said: Dynamic Rationality in Extensive Games. In: van Benthem, J., Pacuit, E. (eds.) Proceedings of Workshop on Logic and Intelligent Interaction, ESSLLI (2008)

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6. Battigalli, P., Bonanno, G.: Recent Results on Belief, Knowledge and the Epistemic Foundations of Game Theory. Res. Econ. 53, 149–225 (1999) 7. Battigalli, P., Siniscalchi, M.: Strong Belief and Forward Induction Reasoning. J. Econ. Theory 106(2), 356–391 (2002) 8. van Benthem, J.: Minimal Deontic Logics. Bull. Section Log. 8(1), 36–42 (1979) 9. van Benthem, J.: Games in Dynamic Epistemic Logic. Bull. Econ. Res. 53(4), 219–248 (2001) 10. van Benthem, J.: Rational Dynamics and Epistemic Logic in Games. Int. Game Theory Rev. 9(1), 13–45 (2007) (Erratum reprint 9(2), 377–409) 11. van Benthem, J., van Otterloo, S., Roy, O.: Preference Logic, Conditionals, and Solution Concepts in Games. ILLC Publications PP-2005-28. Universiteit van Amsterdam (2005) 12. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001) 13. Board, O.: Dynamic Interactive Epistemology. Games Econ. Behav. 49, 49–80 (2002) 14. Bonanno, G.: Modal Logic and Game Theory: Two Alternative Approaches. Risk, Decision and Policy 7(3), 309–324 (2002) 15. Brandenburger, A.: On the Existence of a “Complete” Possibility Structure. In: Dimitri, N., Basili, M., Giboa, I. (eds.) Cognitive Processes and Economic Behavior, pp. 30–34. Routledge, London (2003) 16. Brandenburger, A.: The Power of Paradox: Some Recent Developments in Interactive Epistemology. Int. J. Game Theory 35(4), 465–492 (2007) 17. Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in Games. Econometrica 76(2), 307–352 (2008) 18. Brandenburger, A., Keisler, H.J.: An Impossibility Theorem on Beliefs in Games. Stud. Log. 84(2), 211–240 (2006) 19. de Bruin, B.: Explaining Games: On the Logic of Game Theoretic Explanations. Ph.D. thesis, Universiteit van Amsterdam (2004); ILLC Publications DS-2004-03 20. ten Cate, B.: Model Theory for Extended Modal Languages. Ph.D. thesis, Universiteit van Amsterdam (2005); ILLC Publications DS-2005-01 21. Devlin, K.: The Joy of Sets: Fundamentals of Contemporary Set Theory. Undergraduate Texts in Mathematics. Springer, Heidelberg (1993) 22. Feferman, S.: Persistent and Invariant Formulas for Outer Extensions. Compositio Math. 20, 29–52 (1968) 23. Halpern, J.Y., Lakemeyer, G.: Multi-agent Only Knowing. J. Log. Comp. 11(1), 41–70 (2001) 24. Halpern, J.Y., Moses, Y.: Characterizing Solution Concepts in Games Using Knowledge-based Programs. In: Veloso, M.M. (ed.) IJCAI 2007, pp. 1300–1307. Morgan Kaufmann, San Francisco (2007) 25. Harsanyi, J.C.: Games with Incompletete Information Played by ‘Bayesian’ Players. Part I: The Basic Model. Management Sci. 14(3), 159–182 (1967) 26. Harsanyi, J.C.: Games with Incompletete Information Played by ‘Bayesian’ Players. Part II: Bayesian Equilibrium Points. Management Sci. 14(5), 320–334 (1968) 27. Harsanyi, J.C.: Games with Incompletete Information Played by ‘Bayesian’ Players. Part III: The Basic Probability Distribution of the Game. Management Sci. 14(7), 486–502 (1968) 28. Hintikka, J.: Knowledge and Belief: an Introduction to the Logic of the Two Notions. Cornell University Press (1962)

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29. Humberstone, I.L.: The modal Logic of All and Only. Notre Dame J. Form. Log. 28, 177–188 (1987) 30. Levesque, H.J.: All I Know: a Study in Autoepistemic Logic. Artif. Intell. 42(2-3), 263–309 (1990) 31. Mariotti, T., Meier, M., Piccione, M.: Hierarchies of Beliefs for Compact Possibility Models. J. Math. Econ. 41, 303–324 (2005) 32. Pacuit, E.: Understanding the Brandenburger-Keisler Paradox. Stud. Log. 86(3), 435–454 (2007) 33. Zvesper, J.A.: The Brandenburger-Keisler Paradox in Normal Modal Logics (2007) (manuscript)

Author Index

Berwanger, Dietmar D´egremont, C´edric

1 151

Georgatos, Konstantinos Grossi, Davide 34 Hernich, Andr´e Lang, J´erˆ ome

61 86

Pacuit, Eric 190 Pucella, Riccardo 112

20

Sadrzadeh, Mehrnoosh 112 Schweikardt, Nicole 61 Sietsma, Floor 174 Spohn, Wolfgang 135 van Benthem, Johan 151 van der Torre, Leendert 86 van Eijck, Jan 174 Zvesper, Jonathan A.

190