Logic, Rationality, and Interaction - LORI 2011

Lecture Notes in Artificial Intelligence 6953 Edited by R. Goebel, J. Siekmann, and W. Wahlster Subseries of Lecture...

Author: Hans van Ditmarsch | Jerome Lang | Shier Ju

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Lecture Notes in Artificial Intelligence

6953

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

Hans van Ditmarsch Jérôme Lang Shier Ju (Eds.)

Logic, Rationality, and Interaction Third International Workshop, LORI 2011 Guangzhou, China, October 10-13, 2011 Proceedings

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 Hans van Ditmarsch University of Sevilla Camilo José Cela s/n, 41018 Sevilla, Spain E-mail: [email protected] Jérôme Lang Université Paul Sabatier, IRIT 118 Route de Narbonne, 31062 Toulouse Cedex 04, France E-mail: [email protected] Shier Ju Sun Yat-sen University Institute of Logic and Cognition Department of Philosophy Guangzhou, 510275, China E-mail: [email protected]

ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-24129-1 e-ISBN 978-3-642-24130-7 DOI 10.1007/978-3-642-24130-7 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011936231 CR Subject Classification (1998): F.4, G.2, I.2.6, F.3, I.2.3 LNCS Sublibrary: SL 7 – Artificial Intelligence

© Springer-Verlag Berlin Heidelberg 2011 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. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This volume contains the papers presented at LORI-3, the Third International Workshop on Logic, Rationality and Interaction held during October 10–14, 2011 in Guangzhou, P.R. China. There were 52 submissions. Each submission was reviewed by at least two programme committee members. The committee decided to accept 25 full papers and 12 abstracts for poster presentation. The First International Workshop on Logic, Rationality and Interaction (LORI-I) was initially proposed by Dutch and Chinese logicians; it took place in Beijing in August 2007, with participation by researchers from the fields of artificial intelligence, game theory, linguistics, logic, philosophy, and cognitive science. The workshop led to great advances in mutual understanding, both academically and culturally, between Chinese and non-Chinese logicians. The Second International Workshop on Logic, Rationality and Interaction (LORI-II) took place in Chongqing, China, during October 6–11, 2009. The programme of these previous LORI workshops can be found at http://www.golori.org/, a web portal for the research community on logic and rational interaction. The LORI-3 workshop followed the theme of previous LORI events and mainly centered on logical approaches to knowledge representation, game theory, linguistics, and cognitive science. More than half of the papers focused on modelling and reasoning about knowledge and belief, another one-third covered game theory and related matters. LORI-3 took place on the south campus of Sun Yat-sen University, Guangzhou, P.R. China, and was hosted by the Institute of Logic and Cognition (ILC). The ILC is one of the key research institutes of the Ministry of Education of China, and is dedicated to exploring the intersection between logic, cognitive science and computer science. The institute also contributed a lot to promoting cooperations and academic studies between Chinese and non-Chinese logicians, gaining increasing reputation in the international communities. The programme chairs of LORI-3 are greatly in debt to the support of ILC in making this workshop happen. In particular, we are indebted to ILC member Yuping Shen, who single-handedly produced these proceedings. We further wish to acknowledge the continuous support of LORI standing committee members Fenrong Liu and Johan van Benthem. Finally, we acknowledge the use of EasyChair, with its wonderful facility to check LNCS style file compliance and assist in the production of the proceedings. This greatly reduced our work in publishing the programme. July 2011

Hans van Ditmarsch Jérôme Lang Shier Ju

Organization

Programme Committee Guillaume Aucher Patrick Blackburn Richard Booth Mihir Chakraborty James Delgrande Cédric Dégremont Hans van Ditmarsch Jan van Eijck Ulle Endriss Nina Gierasimczuk Sven Ove Hansson Andreas Herzig Brian Hill John Horty David Janin Shier Ju Mamoru Kaneko Willem Labuschagne Jérôme Lang Fangzhen Lin Fenrong Liu Weiru Liu Emiliano Lorini Pierre Marquis Guo Meiyun Eric Pacuit Gabriella Pigozzi Francesca Poggiolesi Hans Rott Jeremy Seligman Yuping Shen Sonja Smets Leon van der Torre

University of Rennes 1/INRIA, France INRIA, Lorraine, France ILIAS, University of Luxembourg Jadavpur University, India Simon Fraser University, Canada University of Groningen, The Netherlands University of Seville, Spain CWI, The Netherlands ILLC, University of Amsterdam, The Netherlands University of Groningen, The Netherlands Royal Institute of Technology, Stockholm, Sweden IRIT-CNRS, France HEC Paris, France University of Maryland, USA LaBRI, Université de Bordeaux I, ENSEIRB, France Sun Yat-sen University, China University of Tsukuba, Japan University of Otago, New Zealand LAMSADE-CNRS, France Hong Kong University of Science and Technology Tsinghua University, China Queen’s University Belfast, UK IRIT-CNRS, France CRIL-CNRS and Université d’Artois, France South-West University, China Tilburg University, The Netherlands LAMSADE - Université Paris-Dauphine, France VUB, Belgium University of Regensburg, Germany The University of Auckland, New Zealand Sun Yat-sen University, China University of Groningen, The Netherlands ILIAS, University of Luxembourg

VIII

Organization

Minghui Xiong Tomoyuki Yamada Mingyi Zhang Beihai Zhou

Sun Yat-sen University, China Hokkaido University, Japan Guizhou Academy of Science, China Peking University, China

Additional Reviewers Enqvist, Sebastian Horty, John Kooi, Barteld Liu, Hu Ma, Jianbing Ma, Minghui Parent, Xavier Rodenhäuser, Ben

Schulte, Oliver Simon, Sunil Easaw Van Benthem, Johan Vesic, Srdjan Wang, Yanjing Wang, Yisong Wen, Xuefeng Wu, Maonian

Table of Contents

Logical Dynamics of Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johan van Benthem and Eric Pacuit

1

Dynamic Restriction of Choices: Synthesis of Societal Rules . . . . . . . . . . . Soumya Paul and R. Ramanujam

28

Agreeing to Disagree with Limit Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . Christian W. Bach and Jérémie Cabessa

51

A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ka Fat Chow

61

An Application of Model Checking Games to Abstract Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Davide Grossi

74

Schematic Validity in Dynamic Epistemic Logic: Decidability . . . . . . . . . . Wesley H. Holliday, Tomohiro Hoshi, and Thomas F. Icard III

87

Knowledge and Action in Semi-public Environments . . . . . . . . . . . . . . . . . . Wiebe van der Hoek, Petar Iliev, and Michael Wooldridge

97

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios with Off Equilibrium Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alistair Isaac and Tomohiro Hoshi

111

Update Semantics for Imperatives with Priorities . . . . . . . . . . . . . . . . . . . . Fengkui Ju and Fenrong Liu

125

A Measure of Logical Inference and Its Game Theoretical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mamoru Kaneko and Nobu-Yuki Suzuki

139

Partial Semantics of Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beishui Liao and Huaxin Huang

151

A Dynamic Logic of Knowledge, Graded Beliefs and Graded Goals and Its Application to Emotion Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emiliano Lorini

165

DEL Planning and Some Tractable Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedikt L¨ owe, Eric Pacuit, and Andreas Witzel

179

X

Table of Contents

Mathematics of Public Announcements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghui Ma

193

Logics of Belief over Weighted Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghui Ma and Meiyun Guo

206

Game Semantics for the Geiger-Paz-Pearl Axioms of Independence . . . . . Pavel Naumov and Brittany Nicholls

220

Algebraic Foundations for Inquisitive Semantics . . . . . . . . . . . . . . . . . . . . . . Floris Roelofsen

233

A Dynamic Analysis of Interactive Rationality . . . . . . . . . . . . . . . . . . . . . . . Eric Pacuit and Olivier Roy

244

Seeing, Knowledge and Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . Fran¸cois Schwarzentruber

258

Measurement-Theoretic Foundations of Probabilistic Model of JND-Based Vague Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satoru Suzuki

272

An Epistemic Logic with Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Levan Uridia and Dirk Walther

286

Minimal Revision and Classical Kripke Models: First Results . . . . . . . . . . Jonas De Vuyst

300

On Axiomatizations of PAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanjing Wang

314

Public Announcement Logic with Distributed Knowledge . . . . . . . . . . . . . Y`ı N. W´ ang and Thomas ˚ Agotnes

328

An Alternative Logic for Knowability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuefeng Wen, Hu Liu, and Fan Huang

342

Conditional Ought, a Game Theoretical Perspective . . . . . . . . . . . . . . . . . . Xin Sun

356

The Categorial Logic of Vacuous Components in Natural Language . . . . . Chongli Zou, Kesheng Li, and Lu Zhang

370

A Logic for Strategy Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Can Ba¸skent

382

Efficient Action Extraction with Many-to-Many Relationship between Actions and Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianfeng Du, Yong Hu, Charles X. Ling, Ming Fan, and Mei Liu

384

Table of Contents

XI

Reflections on Vote Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan van Eijck, Floor Sietsma, and Sunil Simon

386

Playing Extensive Form Negotiation Games: A Tool-Based Analysis (Abstract) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sujata Ghosh, Sumit Sourabh, and Rineke Verbrugge

388

The Dynamics of Peer Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhen Liang and Jeremy Seligman

390

On Logic of Belief-Disagreement among Agents . . . . . . . . . . . . . . . . . . . . . . Tian-Qun Pan

392

Algebraic Semantics and Model Completeness for Intuitionistic Public Announcement Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mehrnoosh Sadrzadeh, Alessandra Palmigiano, and Minghui Ma

394

Bayesianism, Elimination Induction and Logical Reliability . . . . . . . . . . . . Renjie Yang and Min Tang

396

A Logic of Questions for Rational Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . Zuojun Xiong and Jeremy Seligman

398

Capturing Lewis’s “Elusive Knowledge” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhaoqing Xu

400

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

403

Logical Dynamics of Evidence Johan van Benthem1 and Eric Pacuit2 1

ILLC, University of Amsterdam and Stanford University [email protected] 2 Tilburg Institute for Logic and Philosophy of Science [email protected]

Abstract. Evidence is the underpinning of beliefs and knowledge. Modeling evidence for an agent requires a more fine-grained semantics than possible worlds models. We do this in the form of “neighbourhood models”, originally proposed for weak modal logics. We show how these models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. This perspective leads to richer languages for neighborhood semantics, including modalities for new kinds of conditional evidence and conditional belief. Using these, we indicate how one can obtain relative completeness theorems for the dynamic logic of evidence-changing actions.1

1

Introduction

Logical studies of information-driven agency tend to use standard possible-worlds models as the vehicle for information change. Thus, knowledge is often based on what is true in the set of epistemically accessible worlds, the current information range. This set presumably resulted from some process of investigation, say, as an intersection of various information ranges, but these details have disappeared. Now, in a number of areas, the need has been recognized for more finely-grained notions of information structure, where we keep track of the “reasons”, or the evidence for beliefs and other cognitive attitudes.2 One might take reasons or evidence to have linguistic structure, that can be manipulated through deliberation, inference and argumentation. In this paper, however, we explore an intermediate level, viz. that of neighborhood semantics, where evidence is recorded as a family of sets of worlds. Neighborhood models have long been a technical tool for studying weak modal logics. But here, we show how they support a notion of evidence with matching languages for attitudes based on it, as well as an array of natural actions that transform evidence. 1

2

For further technical details and more elaborate proofs of definability and completeness results, we refer to J. van Benthem & E. Pacuit, “Dynamic Logics of EvidenceBased Beliefs”, to appear in Studia Logica. An extended preprint is available at http://www.illc.uva.nl/Research/Reports/PP-2011-19.text.pdf. Such more fine-grained ideas are found in the study of belief revision ([29], [18], [12]), conditionals ([17,40]), scientific theories ([35]), topological models for knowledge ([22]), and sensor-based models of information-driven agency in AI ( [33]).

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 1–27, 2011. c Springer-Verlag Berlin Heidelberg 2011

2

J. van Benthem and E. Pacuit

Our paper is a programmatic first step. We state the basics of neighborhood models for evidence and belief, and the logics that they support. Then we move to our main theme, showing how these models support natural actions of “evidence management”, from dealing with external new information to internal rearrangement. A dynamic analysis of recursion laws for such actions then feeds back into the design of new static neighborhood logics. For further perspective, we compare this picture with that provided by current plausibility models for belief. We indicate some further directions at the end.

2

Evidence Models

Our semantics of evidence is based on neighborhood models (cf. [6, Chapter 7], [14,24,15]). We use finite models, though most of our results generalize to infinite settings. For convenience, we will discuss single-agent models only, though evidence coming from many sources has a clear “social” character. 2.1

Neighborhood Models

Let W be a set of possible worlds, one of which represents the actual situation. An agent gathers evidence about the actual world from a variety of sources. To simplify things, we assume these sources provide evidence in the form of subsets of W , which may, but need not, contain the actual world. We impose a few intuitive constraints: – No evidence set is empty (evidence per se is never contradictory), – The whole universe W is an evidence set (agents know their ‘space’).3 Definition 1 (Evidence Model). An evidence model is a tuple M = W, E, V with W a non-empty set of worlds, E ⊆ W × ℘(W ) an evidence relation, and V : At → ℘(W ) a valuation function. A pointed evidence model is a pair M, w with “actual world” w. When E is a constant function, we get a uniform evidence model M = W, E, V , w with E the fixed family of subsets of W related to each state by E. We write E(w) for the set {X | wEX}. The above two constraints on the evidence function then become: (Cons) For each state w, ∅ ∈ E(w). (Triv) For each state w, W ∈ E(w). Example 1. To illustrate this structure, consider two worlds W = {w, v}, with p true at w and not at v. The following might be evidential states: 3

An additional often-found property is monotonicity: “If an agent i has evidence X and X ⊆ Y , then i has evidence Y .” To us, this is a property of propositions supported by evidence, not of evidence itself. We will model this feature differently later on, as a valid principle of our logic.

Logical Dynamics of Evidence

w

v

There is no evidence for or against p. w

w

v

There is evidence that supports p.

v

There is evidence that rejects p.

3

w

v

There is evidence that supports p and also evidence that rejects p.

In what follows, we shall mainly work with uniform evidence models. While this may seem very restrictive, the reader will soon see how much relevant structure can be found even at this level. Note that, even though evidence pieces are non-empty, their combination through the obvious operation of taking intersections need not yield consistent evidence: we allow for disjoint sets. But even though an agent may not be able to consistently combine all of her evidence, there will be maximal collections that she can safely put together: Definition 2 (Maximal consistent evidence). A family X of subsets of W has the finite intersection property (f.i.p.) if X = ∅. X has the maximal f.i.p. if X has the f.i.p. but no proper extension of X does. We will now develop the logic of this framework. Clearly, families of sets give us more detail than information states with just sets of (accessible) worlds. 2.2

A Static Logic of Evidence and Belief

To make a connection with familiar systems of reasoning, we first introduce a basic logic for reasoning about evidence and beliefs. Language of evidence and belief Definition 3 (Evidence and Belief Language). Let At be a set of atomic propositions. L0 is the smallest set of formulas generated by the grammar p | ¬ϕ | ϕ ∧ ψ | Bϕ | ϕ | Aϕ where p ∈ At. Additional propositional connectives (∧, →, ↔) are defined as usual, and the existential modality Eϕ is defined as ¬A¬ϕ.

4


The interpretation of ϕ is “the agent has evidence that implies ϕ” (the agent has “evidence for” ϕ”) and Bϕ says that “the agents believes that ϕ”. We include the universal modality (Aϕ: “ϕ is true in all states”) for convenience. One can also think of this as a form of knowledge. Having evidence for ϕ need not imply belief. In order to believe a proposition ϕ, an agent must consider all her evidence for or against ϕ. To model the latter scenario, we will make use of Definition 2. Semantics We now interpret this language on neighborhood models. Definition 4 (Truth). Let M = W, E, V be an evidence model. Truth of a formula ϕ ∈ L0 is defined inductively as follows: M, w |= p iff w ∈ V (p) (for all p ∈ At) M, w |= ¬ϕ iff M, w |= ϕ M, w |= ϕ ∧ ψ iff M, w |= ϕ and M, w |= ψ M, w |= ϕ iff there is an X with wEX and for all v ∈ X, M, v |= ϕ M, w |= Bϕ iff for each maximal f.i.p. family X ⊆ E(w) and for all worlds v ∈ X , M, v |= ϕ – M, w |= Aϕ iff for all v ∈ W , M, v |= ϕ – – – – –

The truth set of ϕ is the set of worlds [[ϕ]]M = {w | M, w |= ϕ}. The standard logical notions of satisfiability and validity are defined as usual. Various extensions to the above modal language make sense. For instance, our notion of belief is cautious, quantifying over all maximal f.i.p’s. But we might also say that an agent “boldly believes ϕ” if there is some maximal f.i.p. X in the current evidence set with X ⊆ [[ϕ]]. We will discuss such extensions below. 2.3

Conditional Belief and Conditional Evidence

Our language still lacks some basic features of many logics of belief. Anticipating the evidence dynamics of Section 4, we now introduce conditional belief and evidence: B ϕ ψ and ϕ ψ to obtain the language L1 .4 Conditional evidence. The interpretation of ϕ ψ is “the agent has evidence that ψ is true conditional on ϕ being true”. Now, when conditioning on ϕ one may have evidence X inconsistent with ϕ. Thus, we cannot simply intersect each piece of evidence with the truth set of ϕ. We say that X ⊆ W is consistent with ϕ if X ∩ [[ϕ]]M = ∅. Then we define: – M, w |= ϕ ψ iff there is an evidence set X ∈ E(w) which is consistent with ϕ such that for all worlds v ∈ X ∩ [[ϕ]]M , M, v |= ϕ. It is easy to see that ϕ ψ is not equivalent to (ϕ → ψ). No definition with absolute evidence modalities works, as can be shown by bisimulation methods. 4

We can define absolute belief and evidence: Bϕ := B ϕ and ϕ := ϕ.


5

Conditional belief. Conditional belief (B ϕ ψ) involves “relativizing” an evidence model to the formula ϕ. Some of the agent’s current evidence may be inconsistent with ϕ (i.e., disjoint with [[ϕ]]M ). Such evidence must be “ignored”: Definition 5 (Relativized maximal overlapping evidence). Let X ⊆ W . Given a family X of subsets of W , the relativization X X is the set {Y ∩ X | Y ∈ X }. We say thata family X has the finite intersection property relative to X (X-f.i.p.) if X X = ∅. X has the maximal X-f.i.p. if X has X-f.i.p. and no proper extension X of X has the X-f.i.p. When X is the truth set of formula ϕ, we write “maximal ϕ-f.i.p.” for “maximal [[ϕ]]M -f.i.p.” and so on. Now we define conditional belief: – M, w |= B ϕ ψ iff foreach maximal ϕ-f.i.p. family X ⊆ E(w), for each world v ∈ X ϕ , M, v |= ψ While this base language of evidence models looks rich already, it follows familiar patterns. However, there are further natural evidence modalities, and they will come to light through our later analysis of operations that change current evidence. The latter dynamics is in fact the main topic of this paper, but we first explore its static base logic a bit further.

3

Some Logical Theory: Axiomatization and Definability

Axiomatizing valid inference While complete logics for reasoning about evidence are not our main concern, we do note a few facts. Fact 1. (i) A satisfies all laws of modal S5, B satisfies all laws of KD, and satisfies only the principles of the minimal “classical” modal logic: the rule of upward monotonicity holds (“from a theorem ϕ → ψ, infer ϕ → ψ”.), but conjunction under the modality: (ϕ ∧ ψ) → (ϕ ∧ ψ) fails. (ii) The following operator connections are valid, but no other implications hold: Bϕ Aϕ

Eϕ ϕ

Verifying these assertions is straightforward. Over our special class of uniform evidence models, we can say much more. First note that the following are now valid: Bϕ → ABϕ

and

ϕ → Aϕ.

It follows easily that belief introspection is trivially true, as reflected in: ϕ ↔ Bϕ

and

¬ϕ ↔ B¬ϕ

These observations suggest the following more general observation:

6


Proposition 1. On uniform evidence models, each formula of L0 is equivalent to a formula with modal operator depth 1. Axiomatizing the complete logic of our models seems quite feasible, though the combination of a standard modality B and a neighborhood modality poses some interesting problems. As for the conditional variants, their logic shows analogies with logics of conditionals, and indeed, some of our later recursion axioms for effects of evidence change suggest interesting analogies (and dis-analogies) with principles of conditional logic. Model theory and definability. Moving from deductive power to expressive power, analyzing definability in our language requires a matching notion of bisimulation. A natural notion to start with is “monotonic bisimulation” for neighbourhood semantics [15,14] and game logics [25]. Definition 6 (Monotonic bisimulation). Let M1 = W1 , E1 , V1 and M2 = W2 , E2 , V2 be two evidence models. A non-empty relation Z ⊆ W1 × W2 is a bisimulation if, for all worlds w1 ∈ W1 and w2 ∈ W2 : Prop If w1 Zw2 , then for all p ∈ At, p ∈ V1 (w1 ) iff p ∈ V2 (w2 ). Forth If w1 Zw2 , then for each X ∈ E1sup (w1 ) there is a X ∈ E2sup (w2 ) such that for all x ∈ X , there is a x ∈ X such that xZx . Back If w1 Zw2 , then for each X ∈ E2sup (w2 ) there is a X ∈ E1sup (w1 ) such that for all x ∈ X , there is a x ∈ X such that xZx . We write M1 , w1 ↔ M2 , w2 if there is a bisimulation Z between M1 and M2 with w1 Zw2 . A bisimulation Z is total if every world in W1 is related to at least one world in W2 , and vice versa. The sublanguage of L0 without belief modalities is invariant under total bisimulations. Thus, with respect to statements about evidential states, two evidence models are the “same” if they are neighborhood bisimilar. But interestingly, beliefs are not invariant under this notion of bisimulation. Fact 2. The belief modality is not definable with only evidence modalities. Proof. Consider the following two evidence models:

q X

p E1 = {X, Y }

q

p Y

E1 = {X}

q Z


7

The dashed line is a total bisimulation between the two models. Still, Bp is true in the model on the left, but not in that on the right. Finding a notion of bisimulation respecting the whole language of evidence and belief, and their later conditionalized variants, seems a natural open problem for neighborhood modal logic.

4

Evidence Dynamics

Evidence is continually affected by new incoming information, and also by processes of internal re-evaluation. Our main new contribution is to show how this dynamics can be made visible on neighborhood models. Our methodology in doing so comes from recent dynamic logics of knowledge update [9,37] and belief revision [34,3], which model informational actions driving agency. Formally, these actions change current models, viewed as snapshots of an agent’s information and attitudes in some relevant process over time.5 Our neighborhood models of evidence and belief suggest a new scope for these methods in dealing with more finely-structured evidence dynamics.6 Deconstructing public announcement. For a start, consider the well-known operation of “public announcement” for a formula ϕ in a model M = W, E, V . Defining this is straightforward: remove all ¬ϕ-worlds, and intersect the old evidence sets with truthϕ when consistently possible. But from the more finegrained perspective of evidence, the event !ϕ can be naturally “deconstructed” into a combination of three distinct actions: 1. Evidence addition: the agent accepts that ϕ is an “admissible” piece of evidence (perhaps on par with the other available evidence). 2. Evidence removal: the agent removes any evidence for ¬ϕ. 3. Evidence modification: the agent incorporates ϕ into each piece of evidence gathered so far, making ϕ the most important piece of evidence. Our richer evidence models allows us to study these operations individually. 4.1

Public Announcements

Definition 7 (Public Announcement). Let M = W, E, V be an evidence model and ϕ a formula. The model M!ϕ = W !ϕ , E !ϕ , V !ϕ has W !ϕ = [[ϕ]]M , for each p ∈ At, V !ϕ (p) = V (p) ∩ W !ϕ , and for all w ∈ W , E !ϕ (w) = {X | ∅ = X = Y ∩ [[ϕ]]M for some Y ∈ E(w)}. 5

6

Examples range from “hard” information provided by public announcements or public observations [26,11] to softer signals encoding different policies of belief revision (cf. [28]) by radical or conservative upgrades of plausibility orderings. Other dynamic logics describe acts of inference or introspection that raise “awareness” [36,39], and of questions that modify the focus of a current process of inquiry [38]. Dynamic neighborhood methods have been used in game scenarios: [7,41].

8


There is a natural matching dynamic modality [!ϕ]ψ stating that “ψ is true after the public announcement of ϕ”: (PA)

M, w |= [!ϕ]ψ iff M, w |= ϕ implies M!ϕ , w |= ψ.

On evidence models, the standard recursion axioms for public announcement remain valid, yielding dynamic equations for evidence change under hard information. Here is the result, stated as a form of “relative completeness”: Theorem 3. The dynamic logic of evidence and belief under public announcement is axiomatized completely over the chosen static base logic, given the usual rulse of Necessitation and Replacement of Provable Equivalents, by (a) the minimal modal logic for the separate dynamic modalities, (b) the following set of recursion axioms: Table 1. Public Announcement Recursion Axioms (P A1)

[!ϕ]p

↔ (ϕ → p)

(p ∈ At)

(P A2)

[!ϕ](ψ ∧ χ) ↔ ([!ϕ]ψ ∧ [!ϕ]χ)

(P A3)

[!ϕ]¬ψ

↔ (ϕ → ¬[!ϕ]ψ)

(P A4)

[!ϕ]ψ

↔ (ϕ → ϕ [!ϕ]ψ)

(P A5)

[!ϕ]Bψ

↔ (ϕ → B ϕ [!ϕ]ψ)

(P A6)

[!ϕ]α ψ

↔ (ϕ → ϕ∧[!ϕ]α [!ϕ]ψ)

(P A7)

[!ϕ]B α ψ

↔ (ϕ → B ϕ∧[!ϕ]α [!ϕ]ψ)

(P A8)

[!ϕ]Aψ

↔ (ϕ → A[!ϕ]ψ)

Proof. We only verify P A6 as a typical example. Let M = W, E, V be an evidence model. Suppose for simplicity that M, w |= ϕ. Then we get M, w |= [!ϕ]α ϕ iff M!ϕ , w |= α ϕ iff there is X ∈ E !ϕ (w) compatible with [[α]]M!ϕ such that X ∩ [[α]]M!ϕ ⊆ [[ψ]]M!ϕ (note [[ψ]]M!ϕ = [[[!ϕ]ψ]]M and [[α]]M!ϕ = [[[!ϕ]α]]M )

iff there is X ∈ E !ϕ (w) compatible with [[[!ϕ]α]]M such that X ∩ [[[!ϕ]α]]M ⊆ [[[!ϕ]ψ]]M (note that X = Y ∩ [[ϕ]]M for some Y ∈ E(w))

iff there is Y ∈ E(w) compatible with [[ϕ ∧ [!ϕ]α]]M such that X ∩ [[ϕ ∧ [!ϕ]α]]M ⊆ [[[!ϕ]ψ]]M iff M, w |= ϕ∧[!ϕ]α[!ϕ]ψ.


4.2

9

Evidence Addition

Next consider the first component in our earlier deconstruction. Definition 8 (Evidence Addition). Let M = W, E, V be an evidence model, and ϕ a formula in L1 .7 The model M+ϕ = W +ϕ , E +ϕ , V +ϕ has W +ϕ = W , V +ϕ = V and for all w ∈ W , E +ϕ (w) = E(w) ∪ {[[ϕ]]M }.

This operation can be described explicitly with a dynamic modality [+ϕ]ψ stating that “ψ is true after ϕ is accepted as an admissible piece of evidence”: (EA)

M, w |= [+ϕ]ψ iff M, w |= Eϕ implies M+ϕ , w |= ψ.

Here, since evidence sets are non-empty, the precondition is that ϕ is true at some state. By contrast, public announcement required that ϕ be true. To capture evidence change, we want to find “dynamic equations” that describe the effect of its action on models. Here are a few preliminary notions: Definition 9 (Compatible/Incompatible). Let M = W, E, V be an evidence model, X ⊆ E(w) a family of evidence sets, and ϕ a formula: 1. X is maximally ϕ-compatible provided ∩X ∩ [[ϕ]]M = ∅ and no proper extension X of X has this property; and 2. X is incompatible with ϕ if there are X1 , . . . , Xn ∈ X such that X1 ∩ · · · ∩ Xn ⊆ [[¬ϕ]]M . Maximal ¬ϕ-compatibility need not imply incompatibility with ϕ. Next, we rephrase our definition of conditional belief, in a new notation: M, w |= B +ϕ ψ iff foreach maximally ϕ-compatible X ⊆ E(w), X ∩ [[ϕ]]M ⊆ [[ψ]]M But we also need a new conditional belief operator, based on incompatibility: M, w |= B −ϕ ψ iff for all maximal f.i.p., if X is incompatible with ϕ then X ⊆ [[ψ]]M Now, here is the axiom for belief after evidence addition that we are after: Lemma 1. [+ϕ]Bψ ↔ Eϕ → (B +ϕ [+ϕ]ψ ∧ B −ϕ [+ϕ]ψ) is valid. Proof. Let M = W, E, V be an evidence model and ϕ a formula with [[ϕ]]M = ∅. We first note the following facts: 1. X ⊆ E(w) is maximally ϕ-compatible iff X ∪{[[ϕ]]M } ⊆ E +ϕ (w) is a maximal f.i.p. family of sets. 7

Eventually, we can even allow formulas from our dynamic evidence logics themselves.

10


2. X ⊆ E(w) is a maximal f.i.p. that is incompatible with ϕ iff X ⊆ E +ϕ (w) is a maximal f.i.p. that does not contain [[ϕ]]M . The proof of both facts follows by noting that E(w) ⊆ E +ϕ (w), while any X that is a maximal f.i.p. in E +ϕ (w) but not in E(w) must contain [[ϕ]]M . Now suppose that M, w |= [+ϕ]Bψ. Then, (∗) for all maximal f.i.p. X ⊆ E +ϕ (w), we have X ⊆ [[ψ]]M+ϕ We must show M, w |= B +ϕ [+ϕ]ψ ∧ B −ϕ [+ϕ]ψ. To see that the left conjunct is true, let X ⊆ E(w) be any maximally ϕ-compatible collection of evidence. By (1), X ∪ {[[ϕ]]M } ⊆ E +ϕ (w) is a maximal f.i.p. set. Then, we have X ∩ [[ϕ]]M = (X ∪ {[[ϕ]]M }) ⊆ [[ψ]]M+ϕ = [[[+ϕ]ψ]]M where the inclusion comes from (∗). Since X was an arbitrary maximally ϕcompatible set, we have M, w |= B +ϕ [+ϕ]ψ. For the right conjunct, let X ⊆ E(w) be any maximal f.i.p. set incompatible with ϕ. By (2), X ⊆ E +ϕ (w) is a maximal f.i.p. (not containing [[ϕ]]M ). Again by (∗), X ⊆ [[ψ]]M+ϕ = [[[+ϕ]ψ]]M Hence, since X was an arbitrary maximal f.i.p. subset of E(w) incompatible with ϕ, we have M, w |= B −ϕ [+ϕ]ψ. This shows that [+ϕ]Bψ → B +ϕ [+ϕ]ψ ∧ B +ϕ [+ϕ]ψ is valid. Suppose now that M, w |= B +ϕ [+ϕ]ψ ∧ B +ϕ [+ϕ]ψ. Then A. For all maximally ϕ-compatible X ⊆ E(w), we have X ∩[[ϕ]] M ⊆ [[[+ϕ]ψ]]M ; B. For all maximally f.i.p. X ⊆ E(w) incompatible with ϕ, X ⊆ [[[+ϕ]ψ]]M . We must show M+ϕ , w |= Bψ. Let X ⊆ E +ϕ (w) be a maximal f.i.p. set. There are two cases to consider. First, [[ϕ]]M ∈ X . Then, by (1), X − {[[ϕ]]M } ⊆ E(w) is maximally ϕ-compatible. Furthermore, by (A) we have X = (X − {[[ϕ]]M }) ∩ [[ϕ]]M ⊆ [[[+ϕ]ψ]]M = [[ψ]]M+ϕ The second case is [[ϕ]]M ∈ X . Then by (2), X ⊆ E(w) is a maximal f.i.p. that is incompatible with ϕ. By (B), we have X ⊆ [[[+ϕ]ψ]]M = [[ψ]]M+ϕ In either case,

X ⊆ [[ψ]]M+ϕ ; hence, M+ϕ , w |= Bψ, as desired.

This proof will suffice to show that analyzing evidence changes is non-trivial. We had to come up with a new notion of conditional belief.8 8

In particular, the reader may verify that the new B −ϕ ψ is not the same as the conditional belief B +¬ϕψ.


11

Language Extension. But we are not yet done. We have now extended the base language, and hence, we need complete recursion axioms for the new conditional beliefs after evidence addition – hopefully, avoiding an infinite regress. Let L2 be the smallest set of formulas generated by the following grammar: p | ¬ϕ | ϕ ∧ ψ | ϕ | B ϕ,ψ χ | Aϕ where p ∈ At and ϕ is any finite sequence of formulas from the language.9 Definition 10 (Truth for L2 ). We only define the new modal operator: ϕ,ψ M, w |= B χ iff for all maximally ϕ-compatible sets X ⊆ E(w), if X ∩ [[ϕ]]M ⊆ [[ψ]]M , then X ∩ [[ϕ]]M ⊆ [[χ]]M

Note that we can define B +ϕ as B ϕ, and B −ϕ as B ,¬ϕ . Theorem 4. The dynamic logic of evidence addition is axiomatized completely by (a) the static base logic of evidence models for the extended language, (b) the minimal modal logic for each separate dynamic modality, and (c) the following set of recursion axioms: Table 2. Evidence Addition Recursion Axioms ↔ (Eϕ → p)

(p ∈ At)

(EA1)

[+ϕ]p

(EA2)

[+ϕ](ψ ∧ χ) ↔ ([+ϕ]ψ ∧ [+ϕ]χ)

(EA3)

[+ϕ]¬ψ

↔ (Eϕ → ¬[+ϕ]ψ)

(EA4)

[+ϕ]ψ

↔ (Eϕ → ([+ϕ]ψ ∨ A(ϕ → [+ϕ]ψ)))

(EA5)

[+ϕ]Bψ

↔ (Eϕ → (B +ϕ [+ϕ]ψ ∧ B −ϕ [+ϕ]ψ))

(EA6)

[+ϕ]α ψ

↔ (Eϕ → ([+ϕ]α [+ϕ]ψ ∨ (E(ϕ ∧ [+ϕ]α)∧ A((ϕ ∧ [+ϕ]α) → [+ϕ]ψ))))

(EA7)

[+ϕ]B ψ,α χ ↔ (Eϕ → (B ϕ∧[+ϕ]ψ,[+ϕ]α[+ϕ]χ∧ B [+ϕ]ψ,¬ϕ∧[+ϕ]α[+ϕ]χ))

(EA8)

[+ϕ]Aψ

↔ (Eϕ → A[+ϕ]ψ)

This result shows that the static and dynamic language of evidence addition are now in “harmony”. A proof is found in the extended version of this paper. Our dynamic logic of evidence addition with its natural modalities of conditional belief is an interesting extension of standard neighborhood logic. It also fits with our earlier analysis of public announcement: Fact 5. The following principle suffices for obtaining a complete dynamic logic of evidence addition plus public announcement: [!ϕ]B ψ,α χ ↔ B ϕ∧[!ϕ]ψ,ϕ→[!ϕ]α[!ϕ]χ 9

Absolute belief and evidence versions again arise by setting some parameters to .

12


4.3

Evidence Removal

With a public announcement of ϕ, the agent also agrees to ignore states inconsistent with ϕ. The latter attitude suggests an act of evidence removal as a natural converse to addition. While “removal” has been a challenge to dynamic-epistemic logics, our richer setting suggests a natural logic. Definition 11 (Evidence Removal). Let M = W, E, V be an evidence model, and ϕ a formula in L1 . The model M−ϕ = W −ϕ , E −ϕ , V −ϕ has W −ϕ = W , V −ϕ = V and for all w ∈ W , E −ϕ (w) = E(w) − {X | X ⊆ [[ϕ]]M }. This time, the corresponding dynamic modality is [−ϕ]ψ (“after removing the evidence that ϕ, ψ is true”), defined as follows: (ER)

M, w |= [−ϕ]ψ iff M, w |= ¬Aϕ implies M−ϕ , w |= ψ 10

Again, we look for a dynamic recursion axiom. As with evidence addition, the analysis is not purely a passive imposition of action superstructure. Finding a total dynamic language that is in harmony again affects the choice of the base language itself, and hence it is an instrument for discovering new logical structure concerning evidence. L For a start, let L− 1 extend the language L1 with the operator [−ϕ]. Proposition 2. L− 1 is strictly more expressive than L1 . Proof. Consider the two evidence models M1 = W, E1 , V and M2 = W, E2 , V : r

r

q

r

r

p

p

E1 10

q

E2

Removing the evidence for ϕ is weaker than the usual notion of contracting one’s beliefs by ϕ in the theory of belief revision [27]. It is possible to remove the evidence for ϕ and yet the agent maintains her belief in ϕ. Formally, [−ϕ]¬Bϕ is not valid. To see this, let W = {w1 , w2 , w3 } with p true only at w3 . Consider an evidence model with two pieces of evidence: E = {{w1 , w3 }, {w2 , w3 }}. The agent believes p and, since the model does not change when removing the evidence for p, [−p]Bp is true. The same is true for the model with explicit evidence for p, i.e., E = {{w1 , w3 }, {w2 , w3 }, {w3 }}.


13

The formula [−p](p ∨ q) of L− 1 is true in M1 but not in M2 . But no formula of L1 can distinguish M1 from M2 . To see this, note that E1sup = E2sup , while the agent has the same beliefs in both models. Adding compatibility. So far, we have looked at conditional evidence and beliefs, generalizing the usual notion to restriction and incompatibility versions. This time, we also need to look at evidence that is merely “compatible” with some relevant proposition. An agent had evidence that ψ conditional on ϕ if there is evidence consistent with ϕ such that restriction to the worlds where ϕ is true entails ψ. Our next conditional operator ϕ ψ drops the latter condition: it is true if the agent has evidence compatible with ϕ that entails ψ. In general, we include operators ϕ ψ where ϕ is a sequence of formulas. The intended interpretation is that “ψ is entailed by some admissible evidence compatible with each of ϕ”. Definition 12 (Compatible evidence). Let M = W, E, V be an evidence model and ϕ = (ϕ1 , . . . , ϕn ) a finite sequence of formulas. A subset X ⊆ W is compatible with ϕ if, for each ϕi , X ∩ [[ϕi ]]M = ∅. Truth of a matching new formula ϕ ψ is then defined as follows: M, w |= ϕ ψ iff some X ∈ E(w) compatible with ϕ has X ⊆ [[ψ]]M This new operator gives us a very natural reduction axiom for :

11

Fact 6. The formula [−ϕ]ψ ↔ (¬Aϕ → ¬ϕ [−ϕ]ψ) is valid. Proof. Let M = W, E, V be an evidence model with [[ϕ]]M = W (otherwise, for all w, E −ϕ (w) = ∅). We show that [−ϕ]ψ ↔ ¬ϕ [−ϕ]ψ is valid on M. Let w ∈ W . The key observation is that for all X ⊆ W , X ∈ E −ϕ (w) iff X ∈ E(w) and X is compatible with ¬ϕ. Then we get M, w |= [−ϕ]ϕ iff M−ϕ , w |= ϕ iff there is a X ∈ E −ϕ (w) such that X ⊆ [[ψ]]M−ϕ (note that [[ψ]]M−ϕ = [[[−ϕ]ψ]]M ) iff there is a X ∈ E(w) compatible with ¬ϕ such that X ⊆ [[[−ϕ]ψ]]M iff M, w |= ¬ϕ [−ϕ]ψ. Note how this principle captures the logical essence of evidence removal. But as before, we are not done yet. We also need a reduction axiom for our new operator ϕ . This can be stated in the same style. But we are not done even then. With the earlier conditional evidence present as well, we need an operator α ϕ ψ saying there is evidence compatible with ϕ and α such that the restriction of that evidence to α entails ψ. We also need one more adjustment: 11

The precondition is needed because the set of all worlds W is an evidence set.

14


Definition 13 (Compatibility evidence - set version). A maximal f.i.p. set X is compatible with a sequence of formulas ϕ provided for each X ∈ X , X is compatible with ϕ. Language and dynamic logic We are now ready to prceed. Let L3 be the set of formulas generated by the following grammar: p | ¬ϕ | ϕ ∧ ψ | Bϕα ψ | α ϕ ψ | Aϕ where p ∈ At and ϕ is any finite sequence of formulas from the language.12 Definition 14 (Truth of L3 ). We only define the new modal operators: – M, w |= α ϕ ψ iff there exists a set X ∈ E(w) compatible with ϕ, α such that X ∩ [[α]]M ⊆ [[ψ]]M . – M, |= Bϕα ψ iff for each maximal family α-f.i.p. X compatible with ϕ, w α X ⊆ [[ψ]]M . α We write α ϕ1 ,...,ϕn for (ϕ1 ,...,ϕn ) and ϕ, α for (ϕ1 , . . . , ϕn , α). Also, if ϕ = (ϕ1 , . . . , ϕn ), then we write [−ϕ]ϕ for ([−ϕ]ϕ1 , . . . , [−ϕ]ϕn ).

Theorem 7. The complete dynamic logic of evidence removal is axiomatized, over the complete logic of the static base language for evidence models as enriched above, by the following recursion axioms: Table 3. Evidence Removal Recursion Axioms

↔ (¬Aϕ → p)

(p ∈ At)

(ER1)

[−ϕ]p

(ER2)

[−ϕ](ψ ∧ χ) ↔ ([−ϕ]ψ ∧ [−ϕ]χ)

(ER3)

[−ϕ]¬ψ

↔ (¬Aϕ → ¬[−ϕ]ψ)

(ER4)

[−ϕ]α χ ψ

↔ (¬Aϕ → [−ϕ]ψ,¬ϕ [−ϕ]χ)

(ER5)

[−ϕ]Bψα χ

↔ (¬Aϕ → B[−ϕ]ψ,¬ϕ [−ϕ]χ)

(ER6)

[−ϕ]Aψ

↔ (¬Aϕ → A[−ϕ]ψ)

[−ϕ]α

[−ϕ]α

Proof. We only do axiom ER5. Let M = W, E, V be an evidence model, w ∈ W and [[ϕ]]M = W . First of all, the key observation in the proof of Fact 6 extends to sets of evidence sets (cf. Definition 13). That is, for all worlds w, X ⊆ E −ϕ (w) is compatible with ψ iff X ⊆ E(w) is compatible with [−ϕ]ψ, ¬ϕ. Next, for all states w, X ⊆ E −ϕ (w) is a maximal α-f.i.p. iff X ⊆ E(w) is a maximal [−ϕ]α-f.i.p. compatible with ¬ϕ.13 Then we calculate: 12 13

Absolute belief and evidence versions again arise by setting some parameters to . The compatibility with ¬ϕ is crucial: it is not true that every X ⊆ E −ϕ (w) that is a maximal α-f.i.p. corresponds to a maximal [−ϕ]α-f.i.p. subset of E(w).


15

M, w |= [−ϕ]Bψα χ iff M−ϕ , w |= Bψα χ −ϕ iff for each maximal α-f.i.p. α X ⊆ E (w) compatible with ϕ, X ⊆ [[χ]]M−ϕ = [[[−ϕ]χ]]M iff for each maximal [−ϕ]α-f.i.p. X ⊆ E(w) compatible with [−ϕ]ϕ and ¬ϕ, [−ϕ]α X ⊆ [[[−ϕ]χ]]M [−ϕ]α iff M, w |= Bψ,¬ϕ [−ϕ]χ. The above principles state the essence of evidence removal, as well as the beliefs one can still have after such an event. The additional insight is that removal essentially involves compatibility as well as implication between propositions – something of independent logical interest. Logics for evidence once more. This is a beginning rather than an end. Extending the base language in this manner will have repercussions for our earlier analyses. Using our style of analysis, it is possible to also find reduction axioms for our new evidence and belief operators under actions of evidence addition and public announcement. For example, for the compatible evidence operator ψ with ψ = (ψ1 , . . . , ψn ), we have the following validities: [+ϕ]ψ χ ↔ [Eϕ → ([+ϕ]ψ [+ϕ]χ ∨ (

E(ϕ ∧ ψi ) ∧ A(ϕ → [+ϕ]ψ)))]

i=1,...,n

[!ϕ]ψ χ ↔ (ϕ → ϕ [!ϕ]χ) [!ϕ]ψ We do not include all combinations here. The key point is that the analysis is in harmony, it does not lead to further extensions of the base language. Perhaps more challenging further problems have to do with the “action algebra” of combining our three basic actions on evidence so far. What happens when we compose them? Our guess is that we need to move to an “event model” version of our logics in the style of dynamic-epistemic logic. 4.4

Evidence Modification

We have analyzed the two major operations on evidence that we can see. Nevertheless, the space of potential operations on neighborhood models is much larger, even if we impose conditions of bisimulation invariance as in process algebra (cf. [23] and [15]). Instead of exploring this wide realm, we show one new operation that might make sense. So far, we added or removed evidence. But one could also modify the existing pieces of evidence. To see, the difference, here is a new way of making some proposition ϕ highly important: Definition 15 (Evidence Upgrade). 14 Let M = W, E, V be an evidence model and ϕ a formula in L1 . The model M⇑ϕ = W ⇑ϕ , E ⇑ϕ , V ⇑ϕ has W ⇑ϕ = W , V ⇑ϕ = V , and for all w ∈ W , 14

This operation is a bit like “radical upgrade” in dynamic logics of belief change.

16


E ⇑ϕ (w) = {X ∪ [[ϕ]]M | X ∈ E(w)} ∪ [[ϕ]]M . This is stronger than simply adding [[ϕ]]M as evidence, since one also modifies each admissible evidence set. But it is still weaker than publicly announcing ϕ, since the agent retains the ability to consistently condition on ¬ϕ. Fact 8. The following recursion principles are valid: 1. [⇑ϕ]ψ ↔ (Eϕ → A(ϕ → [⇑ϕ]ψ)) 2. [⇑ϕ]Bψ ↔ (Eϕ → A(ϕ → [⇑ϕ]ψ)) Proof. For the second law, note that in E ⇑ϕ (w), there is only one maximal f.i.p. whose intersection is [[ϕ]]M . The first law goes as with Fact 9 below. As these principles show, ⇑ϕ gives a very special status to the incoming information ϕ, blurring the distinction between evidence and belief. This suggests a weaker operation that modifies the evidence sets in favor of ϕ, but does not add explicit support for ϕ. Define M⇑w ϕ as in Definition 15 except for setting E ⇑w ϕ (w) = {X ∪ [[ϕ]]M | X ∈ E(w)}. A simple modification to Principle 2 in the above fact gives us a valid principle for our evidence operator. However, the case of belief poses some problems.15 Fact 9. The formula [⇑wϕ]ψ ↔ ([⇑wϕ]ψ ∧ A(ϕ → [⇑wϕ]ψ)) is valid. Proof. Let M = W, E, V be an evidence model with w ∈ W . Then, M, w |= [⇑wϕ]ψ iff M⇑ϕ , w |= ψ iff there is a X ∈ E ⇑ϕ (w) such that X ⊆ [[ψ]]M−ϕ (note that [[ψ]]M⇑ϕ = [[[⇑ϕ]ψ]]M ) iff there is X ∈ E(w) with X ∪ [[ϕ]]M = X ⊆ [[[⇑ϕ]ψ]]M iff there is X ∈ E(w) with X ⊆ [[[⇑ϕ]ψ]]M and [[ϕ]]M ⊆ [[[⇑ϕ]ψ]]M iff M, w |= [⇑ϕ]ψ ∧ A(ϕ → [⇑ϕ]ψ) 4.5

From External to Internal Actions: Evidence Combination

We have now brought to light a rich repertoire of evidence-modifying actions. Still, the operations discussed above all exemplify “external evidence dynamics” responding to some outside source, where the agent reacts appropriately, either by incorporating ϕ or removing ϕ from consideration. But our neighborhood models also suggest internal operations that arise from pondering the evidence, without external triggers. We will discuss only one such internal operation in this paper, be it a basic one. One natural operation available to an agent is to combine her evidence. Of course, as we have noted, an agent’s evidence may be contradictory, so she can only combine evidence that is not inconsistent. 15

The new complication is that, without adding ϕ to the evidence sets, intersections of maximal f.i.p. sets in the upgraded model may contain more than just ϕ states.


17

Definition 16 (Evidence combination). Let M = W, E, V be an evidence model. The model M# = W # , E # , V # has W # = W , V # = V and for all w ∈ W , E # (w) is the smallest family of sets of worls closed under (non-empty) intersection and containing E(w). The corresponding dynamic modal operator is defined as M, w |= [#]ϕ iff M# , w |= ϕ. A complete study of this operation will be left for future work, since it poses some challenges to our recursive style of analysis so far.16 Nevertheless, we can observe the following interesting facts: Fact 10. The following formulas are valid on evidence models: 1. 2. 3. 4.

[#]ϕ → [#]ϕ (combining evidence does not remove old evidence17 ) B[#]ϕ ↔ [#]Bϕ (beliefs are immune to evidence combination) Bϕ → [#]ϕ (beliefs are explicitly supported after combining evidence18 For factual ϕ, Bϕ → ¬[#]¬ϕ (if an agent believes ϕ then the agent cannot combine her evidence so that there is evidence for ¬ϕ)

Proof. The proof that the first three items are valid is left to the reader. For the fourth, note that ¬ϕ → ¬Bϕ is valid. The proof is as follows: First of all, in any evidence model M = W, E, V , every piece of evidence in X ∈ E(w) is contained in a maximal f.i.p. X ⊆ E(w) (models are finite, so simply find the maximal f.i.p. containing X which may be {X, W }). Suppose that ¬ϕ is true at a state w, then there is an X ∈ E(w) such that X ⊆ [[¬ϕ]]M . Let X be the maximal f.i.p. containing X. Hence, X ⊆ X ⊆ [[¬ϕ]]M . Therefore, Bϕ is not true at w. This shows that ¬ϕ → ¬Bϕ is valid, as desired. We can then derive Principle 3 by noting the following series of implications: Bϕ → [#]Bϕ → [#]¬¬ϕ → ¬[#]¬ϕ Here the first implication follows from the second principle applied to factual formulas ϕ (for which ϕ ↔ [#]ϕ is valid), the second implication follows from the fact that Bϕ → ¬¬ϕ is valid (as argued above) while [#] is a normal modal operator and the third implication follows from the fact that the evidence combination operation is functional. Finally, we note that a full account of combination dynamics seems to require an additional modality of “having evidence that ϕ”, but we forego details here. 16 17

18

The problem may be that standard modal languages are too poor, forcing us upward in expressive power to hybrid or first-order logics – but we suspend judgment here. Definition 16 assumed that always E(w) ⊆ E # (w). Thus, in the process of combination, an agent does not notice inconsistencies present in her evidential state. A deeper analysis would include acts of removing observed inconsistencies. The converse is not valid. In fact, one can read the combination [#] as an existential version of our belief operator. It is true if there is some maximal collection of evidence whose intersection implies ϕ. In plausibility models for doxastic logic, this says that ϕ is true throughout some maximal cluster. This notion of belief is much riskier then Bϕ, and again we encounter the variety of agent attitudes mentioned in Section 2.

18

5


Comparison with Plausibility Models

In this section, we will contrast our neighborhood models with another modal framework for belief change. This excursion (which can be skipped without loss of coherence) throws new light on our proposals. We merely state some main notions and results, referring to the extended version of this paper for proofs. Plausibility models. Originally used as a semantics for conditionals (cf. [19]), the following idea is wide-spread in modal logics of belief [34,37,3,12]. One endows epistemic ranges with an ordering w v of relative plausibility on worlds (usually uniform across epistemic equivalence classes): “according to the agent, world v is at least as plausible as w”.19 Plausibility orders are typically assumed to be reflexive and transitive, and often also connected, making every two worlds comparable. In our discussion, we will allow pre-orders with incomparable worlds. Definition 17 (Plausibility model). A plausibility model is a tuple M = W, , V where W is a finite nonempty set, ⊆ W × W is a reflexive and transitive ordering on W , and V : At → ℘(W ) is a valuation function. If is also connected (for each w, v ∈ W , either w v or v w) then we say M is a connected plausibility model. A pair M, w where w is a state is called a pointed (connected) plausibility model. Language and logic. Plausibility models interpret a standard doxastic language. Let L be the smallest set of formulas generated by the following language p | ¬ϕ | ϕ ∧ ψ | B ϕ ψ | []ϕ | Aϕ As usual, Bϕ is defined as B ϕ. For X ⊆ W , let M in (X) = {v ∈ X | v w for all w ∈ X } Given a set X, M in (X) is the set of most plausible worlds in X (minimal elements of X in the plausibility order). We only define the modal operators – M, w |= B ϕ ψ iff M in ([[ϕ]]M ) ⊆ [[ψ]]M – M, w |= []ϕ iff for all v ∈ W , if v w then M, v |= ϕ – M, w |= Aϕ iff for all v ∈ W , M, v |= ϕ. In particular, any pre-order forms a partial order of “clusters”, maximal subsets where the relation is universal. A finite pre-order has one or more final clusters, not having any proper successors. (Connected order have only one final cluster.) Belief means truth in all final clusters. The logic of this system is basically the minimal conditional logic over pre-orders that we encountered before. Instead of pursuing it, we make some comments on definability. Plausibility orders are binary relations supporting a standard modal language. Indeed, as was noted by [5], on finite models, belief and conditional belief are definable in the language with A and [] only: 19

In conditional logic, plausibility or “similarity” is a world-dependent ternary order.


19

Fact 11. Belief and conditional belief can be explicitly defined as follows: – Bϕ := A[]ϕ – B ϕ ψ := A(ϕ → (ϕ ∧ [](ϕ → ψ))) While the plausibility modality may look like a technical device, [3] interpret []ϕ as “a safe belief in ϕ”. Following [32], they show that this amounts to the beliefs the agent retains under all new true information about the actual world.20 This simple modal language over plausibility models will turn out to be a natural limit of expressive power. Dynamics on plausibility models. Plausibility models support a dynamics of informational action through model change. Belief change under hard information. One paradigmatic action was discussed in Section 4.1. “Hard information” reduces current models to definable submodels: Definition 18 (Public announcement - plausibility models). Let M = W, , V be a plausibility model. The model M!ϕ = W !ϕ , !ϕ , V !ϕ has W = [[ϕ]]M , for all p ∈ At, V !ϕ (p) = V (p) ∩ W !ϕ and !ϕ = ∩(W !ϕ × W !ϕ ). Dynamic logics exist that describe belief change under such events of new hard information, i.e., the logical laws governing [!ϕ]Bψ. The crucial recursion axioms for belief are the same as those for evidence models in Section 4.1: [!ϕ]Bψ ↔ (ϕ → B ϕ [!ϕ]ψ) ϕ∧[!ϕ]ψ

[!ϕ]Biψ χ ↔ (ϕ → Bi

[!ϕ]χ)

Public announcement assumes that agents treat the source of the new information as infallible. But in many scenarios, agents trust the source of the information only up to a point. This calls for softer announcements. Here are a few examples: [37, Chapter 7] and [4] have much more extensive discussion. Belief change under soft information. How to incorporate evidence that ϕ is true into some (epistemic-)doxastic model M? Soft announcements of a formula ϕ do not eliminate worlds, but rather modify the plausibility ordering that structures the current information state. The goal is to rearrange all states in such a way that ϕ is believed, and perhaps other desiderata are met. There are many “policies” for doing this (cf. [28]) – we only mention two basic ones. Example 2. The following picture illustrates soft update as plausibility change: B

E D

A

ϕ

C 20

For the same notion in the computational literature on agency (cf. [31]).

20


One policy that has been extensively studied is radical upgrade where all ϕ worlds are moved ahead of all other worlds, while keeping the order inside these two zones the same. In the above example, the radical upgrade by ϕ would result in the ordering A ≺ B ≺ C ≺ D ≺ E. More formally, the model transformation here is relation change: Definition 12 (Radical Upgrade.) Given an epistemic-doxastic model M = W, , V and a formula ϕ, the radical upgrade of M with ϕ is the model M⇑ϕ = W ⇑ϕ , ⇑ϕ , V ⇑ϕ with W ⇑ϕ = W , V ⇑ϕ = V , where ⇑ϕ is defined as follows: 1. for all x ∈ [[ϕ]]M and y ∈ [[¬ϕ]]M , set x ≺⇑ϕ y, 2. for all x, y ∈ [[ϕ]]M , set x ⇑ϕ y iff x y, and 3. for all x, y ∈ [[¬ϕ]]M , set x ⇑ϕ y iff x y. A logical analysis of this type of information change uses modalities [⇑ ϕ]ψ meaning “after radical upgrade of ϕ, ψ is true”, interpreted as follows: M, w |= [⇑ϕ]ψ iff M⇑ϕ , w |= ψ. The crucial recursion axiom for belief change under soft information is ([34]): [⇑ϕ]B ψ χ ↔ (E(ϕ ∧ [⇑ϕ]ψ) ∧ B ϕ∧[⇑ϕ]ψ [⇑ϕ]χ) ∨ (¬E(ϕ ∧ [⇑ϕ]ψ) ∧ B [⇑ϕ]ψ [⇑ϕ]χ) This shows how revision policies as plausibility transformations give agents not just new beliefs, but also new conditional beliefs. But radical upgrade is not the only way for an agent to accept incoming information. Equally important is conservative upgrade, which lets the agent only tentatively accept the incoming information ϕ by making the best ϕ the new minimal set and keeping the old plausibility ordering the same on all other worlds. In the above picture a conservative upgrade with ϕ results in the new ordering A ≺ C ≺ D ≺ B ∪ E. This, and many other revision policies can be analyzed in the same dynamic logic style. From plausibility models to evidence models. Here is an intuitive connection. Let M = W, , V be a plausibility model: the appropriate evidence sets are the downward -closed sets of worlds. To be more precise, we fix some notation: – Given a X ⊆ W , let X↓ = {v ∈ W | ∃x ∈ X and v x} (we write X↓ when it is clear which plausibility ordering is being used). – A set X ⊆ W is -closed if X↓ ⊆ X. Here is the formal definition for the above simple idea:


21

Definition 19 (Plausibility-Based Evidence Model). Let M = W, , V be a plausibility model. The evidence model generated from M is21 EV (M) = W, E , V with E as follows: E = {X | ∅ = X is -closed } Given a plausibility model M, the evidence model generated by the plausibility order of M satisfies the basic properties of Section 2: the sets are non-empty, and the whole universe is among them. But more can be said: Fact 13. The family of evidence sets of any generated model EV (M) is closed under taking intersections. Example 3. The following three plausibility models – with their induced evidence sets drawn in gray – highlight three key situations that can occur:

w

p

p

w

p

q

EV (M1 ), w |= ¬B(p ∧ q) EV (M1 ), w |= p ∧ q

q

EV (M2 ), w |= ¬B(p ∧ q) EV (M2 ), w |= ¬p ∧ ¬q

w p

p

p

p, q

p, q

EV (M3 ), w |= Bp ∧ ¬Bq EV (M3 ), w |= (p ∧ q) But not every evidence model comes from a plausibility model. Example 4. Let M be an evidence model with W = {w, v, x}, a constant evidence function with range E = {{w, v}, {v, x}} and the valuation function defined by V (p) = {w, v} and V (q) = {v, x}. Note that we have M, w |= B(p ∧ q) 21

Here the set of worlds and valuation function remain as in the model M.

22


(the agent believes p and q) but M, w |= p ∧ q ∧ ¬(p ∧ q) (even though there is evidence for p and evidence for q, there is no evidence for p ∧ q). Plausibility models represent a situation where the agent has already “combined” all of her evidence (cf. # in Section 3.4), as reflected in this property: If X, Y ∈ E and X ∩ Y = ∅ then X ∩ Y ∈ E . This connection between plausibility models and evidence models can be extended to a translation between their languages: Definition 20 (P -translation). The translation (·)P : L → L is defined as: – – – –

pP = p, (¬ϕ)P = ¬ϕP , (ϕ ∧ ψ)P = ϕP ∧ ψ P , (Aϕ)P = AϕP , (ϕ)P = E[]ϕP , (ϕ ψ)P = E(ϕP ∧ [](ϕP → ψ P )), P P P P P (ϕ γ ψ) = E( i γi ∧ (ϕ ∧ [](ϕ → ψ ))),

– (B ϕ ψ)P = A(ϕP → (ϕP ∧ [](ϕP → ψ P ))), – (B ϕ,α ψ)P = A(([]αP ∧ []ϕP ) → (ϕP ∧ [](ϕP → ψ P ))), and – (Bγϕ ψ)P = A((ϕP ∧ i γiP ) → ((ϕP ∧ i γiP ) ∧ []((ϕP ∧ i γiP ) → ψ P ))). Lemma 2. Let M = W, , V be a plausibility model. For any ϕ ∈ L1 and world w ∈ W , M, w |= ϕP iff EV (M), w |= ϕ From evidence models to plausibility models. Going in the opposite direction, we start with a family of evidence sets, and need to induce a natural ordering. Here one can use a ubiquitous idea, occurring in point-set topology, but also in theories of relation merge (cf. [2,20]), the so-called specialization (pre)-order: Definition 21 (Plausibility based evidence model). Suppose that M = W, E, V is an evidence model (with constant evidence function E whose range is E). The plausibility model generated by M is the structure ORD(M) = W, E , V where E is an ordering on W defined as follows: w E v iff ∀X ∈ E, v ∈ X implies w ∈ X 22 To make this definition more concrete, here is a simple illustration. 22

E is reflexive and transitive, so ORD(M) is indeed a plausibility model.


w1

w2

w3

w1

w4

w2

w3

w4

ORD(M)

M

w1 w1

w2

w3

23

w3

w4

w4 w2

M

ORD(M)

Our two representations are related as follows: Fact 14. (i) For all models plausibility models M, ORD(EV ((M)) = M, (ii) The identity EV (ORD(M)) = M does not hold for all evidence models M. (iii) For all evidence models M, EV (ORD(M)) = M# , where # is the combination operation of Definition 16. Translations and languages. The preceding connection again comes with a translation for modal languages, in particular for (conditional) beliefs on evidence neighborhood models and their induced plausibility models. But other notions are less easily reduced. The extended version of this paper will show how dealing with “safe belief” on plausibility orders requires a new notion of reliable evidence that extends our earlier modal evidence languages. This concludes our brief comparison of relational and neighborhood semantics for belief and evidence. We have clarified their relationship as one of generalization, where neighborhood models describe one more level of detail: the internal combination stages for evidence. Even so, many of the new operations that we have found in earlier sections would also make sense as definable operators in the natural modal logic of plausibility models, and we have shown how various interesting new questions arise at this interface.

6

Conclusion and Further Directions

We have shown that evidence dynamics on neighborhood models offers a rich environment for modeling information flow and pursuing logical studies. Here are some avenues for further research. Some are more technical, some increase coverage. We start with the former.

24


Exploring the static logic. We have found quite a few new evidence-based modalities of conditional belief. What is the complete logic of this system? This is a new question of axiomatization, that can be appreciated also outside of our dynamic perspective. One reason for its complexity may be that we are mixing a language of neighborhood-based modalities with normal operators of belief with a matching relational semantics. What is the right notion of bisimulation? Designing logical languages invites matching up with notions of structural invariance for evidence models. We have seen that standard bisimulation for neighborhood models matches modal logics with only evidence operators. But Fact 2 showed that this does not extend to the modal language with belief referring to intersections of maximally consistent families of evidence sets. And we introduced even stronger modal evidence languages in the dynamics of Section 4. What would be well-motivated stronger notions of bisimulation, respecting more evidence structure? Finally, there are some obvious technical generalizations to be made, to infinite models, and also to DEL-style product update mechanisms for rich input. Reliable evidence and its sources. But one can also use our setting for modeling further phenomena. For instance, there is a natural notion of “reliable” evidence, based only on sets containing the actual world. What is the complete logic of this operator? This suggests a broader study of types of belief based on reliable evidence, in line with current trends in epistemology. But eventually, we also want explicit modeling of sources of evidence and what agents know or believe about their reliability. Social notions. We have seen that interesting evidence structure arises in the single agent case. But multi-agent scenarios are also natural: e.g., argumentation is social confrontation of evidence, which may result in new group attitudes among participants. This raises a number of interesting issues of its own. The most pressing is to find natural notions of group evidence and belief. Here evidence structure soon takes us beyond the usual notions of group beliefs or group knowledge in the epistemic literature based on relational models. Priority structures. The evidence dynamics in this paper treats evidence sets on a par. As a consequence, removal may seem arbitrary and non-deterministic, since there is nothing in the structure of the evidence itself which directs the process. A next reasonable step would be to model levels of reliability of evidence. One natural format for this are the “priority graphs” of [2], which have already been used extensively in dynamic-epistemic logic [20,12]. These graphs provide much richer input to evidence management, and can break stalemates between conflicting pieces of evidence. It should be possible to extend the above framework to one with ordered evidence sets – and conversely, then, our logics may help provide something that has been missing so far: modal logics working directly on priority graphs.


25

Other logics of evidence. “Evidence” is a notion with many different aspects. Our proposal has been set-theoretic and semantic, while there are many other treatments of evidence for a proposition ϕ, in terms of proofs for ϕ, or using the balance of probability for ϕ versus ¬ϕ. What we find particularly pressing is a junction with more syntactic approaches making evidence something coded that can be operated on in terms of inference and computation. If finer operational aspects of inference and introspection enter one’s notion of evidence, then the methods of this paper should be extended to include dynamic logics of awareness and inference [10,1,36,39]. Related frameworks. But also, the style of analysis in this paper should, and can, be linked up with other traditions, including the seminal work by [8] and [30] on evidence, probabilistic logics of evidence [13], or the “topologic” of [22]. And one can add more, such as the “priority graphs” inducing preference orders in [21], or the “belief base” account of belief revision in (see [16] and references therein). We intend to clarify these connections in future work.

7

Conclusion

We have made a pilot proposal for using neighborhood models as fine-grained evidence structures that allow for richer representation of information than current relational models of belief. We have shown how these structures support a rich dynamics of evidence change that goes beyond current logics of belief revision. A number of relative completeness theorems identified the key dynamic equations governing this process, while also suggesting new static languages of evidence and belief. Finally, we discussed some of the interesting new issues that lie ahead, such as finding logics with priority structure and group evidence that exploit the more finely-grained neighborhood setting.

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Dynamic Restriction of Choices: Synthesis of Societal Rules Soumya Paul and R. Ramanujam The Institute of Mathematical Sciences CIT Campus, Taramani Chennai - 600 113, India {soumya,jam}@imsc.res.in

Abstract. We study a game model to highlight the mutual recursiveness of individual rationality and societal rationality. These are games that change intrinsically based on the actions / strategies played by the players. There is an implicit player - the society, who makes actions available to players and incurs certain costs in doing so. If and when it feels that an action a is being played by a small number of players and/or it becomes too expensive for it to maintain the action a, it removes a from the set of available actions. This results in a change in the game and the players strategise afresh taking this change into account. We study the question: which actions of the players should the society restrict and how should it restrict them so that the social cost is minimised in the eventuality? We address two variations of the question: when the players are maximisers, can society choose an order of their moves so that social cost is minimised, and which actions may be restricted when players play according to given strategy specifications.1

1

Motivation

In India, which is dependent greatly on oil imports for its energy needs, petrol prices are regulated but increase periodically. Every increase leads to a round of price rise of many commodities, and hence every announced oil price increase meets with great opposition and varied systemic reaction. In response, the government rolls back prices, not to the level before the announced increase but one a little higher, and another cycle of behaviour is initiated. We see this as an example, not of a game, but a dynamic game form: the structure of the game is preserved over time, but the set of options available to players changes dynamically, and the rationale for such change is dependent on player behaviour. Social choice theory concerns itself with aggregating individual choices. On the other hand, theories of mechanism design and market design seek to drive individual choices (based on players’ preferences over outcomes) towards desired social goals. This works well for one-shot games, but there are situations 1

This paper is a follow-up of [19].


Dynamic Restriction of Choices: Synthesis of Societal Rules

29

where societal interventions are not designed a priori but are required during play. Many social mechanisms (especially financial ones such as interest rates and taxes) depend on making available an initial set of options, and based on what strategies are selected by players, revising these options in the long run. Such revision of available moves is, in turn, anticipated by players, and this mutual interdependence of individual and social action influences game dynamics significantly. In principle, such dynamic game forms are unnecessary, since all possible changes can themselves be incorporated into one big game whereby these gamechanging moves are merely additional moves, and players strategize taking these possibilities into account. However, in the context of resource bounded players such reasoning may be impossible. In the societal contexts referred to above, it would be considered wiser to not rely on computations that undertake to include all possible effects but instead seek to observe player behaviour and apply course corrections. Economists are, of course, well aware of such phenomena. The availability of individual choices is, in general, determined by choices by the society as a whole, and in turn, social choices are influenced by patterns of individual choices. In this process, the set of choices may expand or contract over time. However, there is a political or philosophical value attached to availability of individual choices. A strategy sa may be justified by the presence of another option sb but if eventually sb is forced out, the rationale for sa may disappear, though sa is the only one present. In a world where all possible eventual consequences can be computed, the cost of such disappearance of choices can also be taken into account, but (as we see in the case of environment conservation) realisation typically comes post-facto. The general situation is as follows. At every stage, an individual has certain choices to make. But making a choice also comes with a cost which is associated with that choice and which the individual has to incur in making the choice. On the other hand, society also incurs a certain cost in making these choices available to individuals. This cost is a function of the choices being provided as well as the profile of choices made by individuals. From time to time, based on the history of choice profiles and predictions of the future, society revises the choices it provides to individuals as well as the cost individuals have to incur to make these choices. This in turn has an effect on individuals’ strategies, who switch between available choices. The dynamics of this back and forth process can be quite interesting and complicated. In game theoretic models of such social phenomena, social rules are considered as game forms, and individual behaviour is regulated using payoffs. Rule changes are considered to be exogenous, and correspond to change of payoff matrices. In evolutionary game theory, rules are considered as game equilibria: individuals following rules are players, and the desired properties of rules are given by equilibrium strategies, thus describing enforced rules. However what we discuss here is endogenous dynamics of these rules that takes into account the fact that individual behaviour and rules operate mutually and concurrently. In this sense,

30

S. Paul and R. Ramanujam

individual rationality and social rationality are mutually dependent, and what we seek to study are the patterns of reasoning that inform such dependence. In [19], we studied game forms that change dynamically according to prespecified rules stated in a formal logic, and asked which action choices are eventually stable (in the sense that no further game changes will eliminate them), and under what conditions. We showed that these questions are algorithmically solvable. In this paper, we address a dual question: if players were to play according to some strategy specifications, when should society intervene and in what form? We look at the quantitative aspect of the choice-restriction phenomenon. Since these are games with a large number of players, societal decisions are not influenced directly by individual choices but by distribution of strategies in the player population. Thus we consider anonymous games where the cost incurred by society in a particular round is given by a function of the action distribution of the players. The cumulative cost is the limit-average (mean-payoff) of these costs. We then ask whether it is possible for the society to synthesise rules for removal of actions of the players so that the eventual social cost is less than a certain threshold. We show that such synthesis is possible and that the rules require only finite memory. Related Work Dynamic game forms have been studied extensively in the context of evolutionary game theory ([23]). [18] studies a model where actions of players depend on the forecast outcome and behaviour switching of the players in effect causes the game itself to change in a dynamic fashion. In [20] Young defines and studies the long run dynamics of a model of how innovations spread in a social network. [5] looks at equilibrium selection by players who revise strategies by a learning process. They note that the stable behaviour of agents depend on the dynamics of the game itself and argue that it is important to incorporate these changes into the model. Switching behaviour of players has also been studied in dynamical system models of social interaction ([22], [16]). Going further, Hashimoto and Kumagai ([13]) even propose a model in which interaction rules of replicator equations change dynamically, and offer computer simulations of dynamic game changes. While this paper studies quantitative rule synthesis, our work is broadly situated in qualitative reasoning about games and is thus related to logical formalisms. Modal logics have been used in various ways to reason about games and strategies. Notable among these is the work on alternating temporal logic (ATL) and its extensions ([1], [15], [14], [17]): assertions are made on outcomes a coalition of players can ensure, and what strategy a player plays may depend on her intensions and epistemic attitudes. In [2,3] van Benthem uses dynamic logic to describe games as well as strategies. [12] presents a complete axiomatisation of a logic describing both games and strategies in a dynamic logic framework where assertions are made about atomic strategies. [21] studies a logic in which not only are games structured, but so also are strategies. [4] lists a range of issues to be studied in reasoning about strategies.


2

31

Dynamic Game Restriction

We study concurrent games on finite graphs. N = {1, 2, . . . , n} is the set of players and Ai is the set of actions of player i. An arena A = (V, E) is a finite graph with the vertex set V and edge relation E. For v ∈ V , let vE = {(v, u) ∈ E}, i.e., the set of edges outgoing from v. The edges of the arena are labelled with labels from A. For an edge label a = (a1 , . . . , an ) we let a(i) denote the ith component of a, i.e., a(i) = ai . Thus edges are labelled with tuples from the set A ∪ {}. The components give the default transitions in the arena. If the i i∈N ith component of an edge label is then that edge defines the transition of the game in a situation where player i plays an action a ∈ Ai but a is not available to her anymore, in that, the society has removed a from her available set of actions. Note that ∈ / Ai , that is is not a strategic choice for the players. We assume that for every a ∈ i∈N Ai ∪ {} and every v ∈ V , there exists an edge (v, v ) which is labelled with a. That is, the labelling is complete with respect to the labels at every vertex. An initial vertex v0 ∈ V is distinguished and the game G = (A, v0 ) consists of an arena A and the initial vertex v0 . A sub-arena A of the arena A is a graph (V , E ) such that V ⊂ V and E is the set of edges induced by V . The game proceeds as follows. Initially a token is placed at v0 . If the token is at some vertex v, then players 1 to n simultaneously choose actions a1 , . . . , an from their action sets A1 , . . . , An respectively. This defines a tuple a = (a1 , . . . , an ). If a is the label of the edge (v, u) then the token is moved to u. If a is not present among the labels of the outgoing edges then for all i : 1 ≤ i ≤ n such that the action a(i) is not available to player i, a(i) is replaced by in a to get a . If (v, u) is the edge with label a then the token is moved to u. This defines a path a a ρ = v0 →0 v1 →1 . . . in the arena. Such a path is called a play. A finite play is also called a history. The tree unfolding of the arena A at a node v0 is a subset TA ⊂ A∗ such that a a ∈ TA is the root and for all t = a0 a1 . . . ak ∈ TA such that v0 →0 . . . →k vk is the corresponding path in A, ta ∈ TA for all (vk , u) ∈ vk E such that (vk , u) is a a labelled with a. For a node t = a0 a1 . . . ak ∈ TA such that v0 →0 . . . →k vk is the corresponding path in A, we let λ(t) = vk . We also use the notation TG to denote the tree unfolding of the game G = (A, v0 ). A strategy of a player tells her how to play the game. In other words, it prescribes at every position which move to make. Formally a strategy si of player i is a function si : A∗ → Ai . Note that the codomain of si is Ai and not Ai ∪ {}. The empty action is not a strategic choice for a player; rather it is forced when the action she plays is not available. A strategy si can equivalently be thought of as a subtree Tsi , the strategy-tree, of TA with root corresponding to the position v0 such that: – For any node t = a0 a1 . . . ak if si (t) = a then the children of t in Tsi are exactly those nodes ta ∈ TA such that a(i) is equal to a. A strategy s is said to be bounded memory if there exists a finite state machine M = (M, g, h, mI ) where M is a finite set denoting the memory of the strategy,

32


mI is the initial memory, G : A × M → M is the memory update function, and h : A × M → Ai the output function which specifies the choice of the player such that if a0 . . . ak is a play and m0 . . . mk+1 is a sequence determined by m0 = mI and mi+i = g(ai , mi ) then s(a0 . . . ak ) = h(ak , mk+1 ). The strategy s is said to be memoryless if M is a singleton. The crucial elements for defining game restrictions are: when a restriction is to be carried out in the course of play, and what the effects of a restriction are. We choose a very simple answer to the latter, namely to eliminate a subset of choices at selected game positions, that is, to restrict the set of actions available to a player. The former is treated logically, to be defined below, by tests for logical conditions. Formally the restriction is triggered by a rule of the form r = pre ⊃ A where pre is a precondition which is interpreted on partial plays and A is a restriction of the arena. For an arena A and a partial (finite) play ρ ∈ A, we say that the rule r = pre ⊃ A is enabled at (A, ρ) if the following conditions hold. – The partial play ρ conforms to the precondition pre. – The arena A = (V , E ) is a sub-arena of A. – last(ρ) ∈ V . When the rule r = pre ⊃ A is applied to a partial play ρ, the game proceeds to the new arena A starting at the node last(ρ). 2.1

Induced Game Tree

The restriction rules are specified along with the initial game arena. Let R = {r1 , . . . , rm } be a finite set of restriction rules. For an arena A, let SA(A) denote the set of all subarenas of A. Given an initialised arena (A, v0 ) and a finite set of rules R, the extensive form game tree is the (infinite) tree TA (R) = (T, E) where T ⊆ (V × A∗ × SA(A)) and is defined inductively as follows: – t0 = (v0 , , A) is the root. – At any node t = (v, u, A ) of the tree, check if for a rule (rj = pre j ⊃ Aj ) ∈ R, it is the case that t |= rj . If more than one rule is enabled at t then choose any one of them, say pre j ⊃ Aj . Let Aj = (Vj , Ej ). • If pre j ⊃ Aj , then the subtree starting from t is the unfolding of the arena Aj from the vertex v. Note that v ∈ Vj since we have ensured that v = last(ρ(v0 , u)) ∈ Vj . • If there is no such rule then the children of t are the same as those of t in the unfolding of A and the edge labels are also the same. 2.2

Strategising by Players

In the presence of dynamic game restriction operations, the players can keep track of the restriction rules which are triggered by observing the history of play and adapt their strategies based on this information. A strategy specification for a player i would therefore be of the form pre ⊃ a where, as earlier, pre is a


33

precondition which is interpreted on partial plays and a ∈ Ai . The specification asserts that if a play ρ conforms to the precondition pre, then the action a is taken by the player. To formally define the structure of preconditions, we let P be a set of atomic propositions and bool (P) be the set of boolean formulas over P (i.e. built using the syntax p ∈ P | ¬β | β1 ∨ β2 ). We also use the following abbreviations:

≡ p∨¬p and ⊥ ≡ p∧¬p. Let val : V → 2P be a valuation function be given on the game arena. val can be lifted to TA in the natural way, i.e., val (t) = val (λ(t)). The strategy of players can in general depend on properties of the history of the play. These can therefore be specified as a collection of formulae of the form ϕ ⊃ a where ϕ ∈ Φ is given by the following syntax: − - ψ ϕ ∈ Φ ::= P ∈ P | ¬ψ | ψ1 ∨ ψ2 | a ψ |

A formula ϕ of player i is evaluated on the game tree TG . Then the truth of ψ at a node t of TG , denoted TG , t |= ψ is defined inductively in the standard manner: – – – – –

TG , t |= P iff p ∈ val (t). TG , t |= ¬ϕ iff TG , t ϕ . TG , t |= ϕ1 ∨ ϕ2 iff TG , t |= ϕ1 or TG , t |= ϕ2 . − TG , t |= a ϕ iff t = t a and TG , t |= ϕ . - ϕ iff for all prefixes t of t, TG , t |= ϕ . TG , t |=

Note that a strategy specification of this form is partial, since it does not constrain game positions at which the precondition does not hold; the player is free to choose any enabled action. Syntax and semantics of restriction precondition. A restriction precondition ψ comes from the syntax ϕ using which we can specify properties of the indefinite past and bounded future. ψ is evaluated on TA (R) as usual. - ψ makes assertion about the unbounded past, it specifies the The modality transitive closure of the one step past operator. We can define the corresponding construct for future, ψ which makes assertions about unbounded future. The technical results go through even with the addition of this construct. However, for the applications we have in mind, this construct is not required. 2.3

Capturing Costs in the Logical Formalism

Following a strategy induces a certain cost for the player. The distribution of strategies chosen by players carry a social cost. We first take an abstract view of costs associated with individual players and social costs associated with providing facilities. The social cost typically depends on the history of the choices made by players in the past. When the social cost crosses some pre-defined threshold, it might be socially optimal to make certain facilities part of the common infrastructure which reduces the individual costs.

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When the costs arise from a fixed finite set, they can be coded up using propositions in the logical framework on the lines of [8]. The cost c (say) can be represented using the proposition pc and orderings are inherited from the implication available in the logic. Furthermore, costs can be dependent on the actions enabled at a game position. This can also be easily represented in the logical formalism by making use of the one step future modality. Let (A, v0 ) be an initialised arena, R be a finite set of game restriction rules, {Σi }i∈N be a finite set of strategy specifications or each player i ∈ N . Let α be a formula from the syntax: α ::= α ∈ bool (P) | a+ α We say α is stable in (A, R, {Σi }i∈N ) if there exists a sub-arena A such that for all game positions t ∈ TA , we have: t |= α. Thus stability with respect to an observable property captures the existence of a subarena to which the game stabilises under the dynamics specified by R and {Σi }i∈N . For the applications we consider, we do not require the full power of temporal logic for α. In [19], we proved the following theorem. Theorem 1. Given an initialised arena (A, v0 ), a finite set of restriction rules R, a finite set of strategy specifications {Σi }i∈N and a formula α, the following question is decidable: – Is α stable in (A, R, {Σi }i∈N )?

3

Quantitative Objectives

In this section we change our model to one where the costs (both social and individual) are given by certain functions instead of being coded up as propositions as before. We then ask whether it is possible for the society to restrict the actions of the players in such a way that the social cost stays within a certain threshold. We first develop some preliminaries. As before we let N = {1, 2, . . . , n} be the set of players. However we assume that the players have a common action set A, that is, A1 = . . . = An = A. We study anonymous games [6,7,10,9] because in large games, the payoffs are usually dependent on the ‘distribution’ of the actions played by the players rather than the action profiles themselves. Moreover, in such games the payoffs are independent of the identities of the players. An action distribution is a tuple |A| y = (y1 , y2 , . . . , y|A| ) such that ∀i, yi ≥ 0 and i=1 yi ≤ n. Let Y be the set of all action distributions. Given an action profile a, we let y(a) be its corresponding action distribution, that is, y(a)(k) gives the number of players playing the kth action in A. Now as the payoffs are dependent on the action-distribution of the players, we convert the arena A to a new arena A[Y] so that the payoffs can be assigned to the vertices of the arena. A[Y] is defined as A[Y] = (V [Y], E[Y]) as follows:


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– V [Y] = V × Y a a – E[Y] ⊆ V [Y] × A × V [Y] such that (v1 , y1 ) → (v2 , y2 ) iff v1 → v2 and y(a) = y2 . – Delete all vertices in V [Y] that do not have any incoming or outgoing edges. As we shall exclusively deal with the expanded arena A[Y] in the entire development, we denote A[Y] = (V [Y], E[Y]) by just A = (V, E) and assure that it will not result in any confusion. A tuple (A, v0 ) where A is an arena and v0 is a distinguished vertex is called an initialised arena. Every player i has a function fi : Y → Q which can be seen as the payoff of i for a particular distribution. There is also a function f : Y → Q which can be viewed as the cost incurred by the society for maintaining the actions. These functions can be lifted to the vertices of V [Y] as f (v, y) = f (y). We now investigate if it is possible for the society to impose restrictions in such a way that the social cost stays within a certain threshold. We look at two variations of our model: a. At the beginning of each round the society chooses an order for the n players and makes it known to them. The players then choose their actions according to this order. b. The players play according to strategy specifications (as in the previous sections). The society, at any point, can restrict some action a ∈ A of the players, in that, it can make the action a unavailable. In (a), we wish to investigate if it is possible for the society to pick the orders of actions of the players in such a way that the eventual social cost is within a certain threshold. In (b), we want to find out if the society can restrict the action of the players based on certain rules so that the same effect is obtained. 3.1

Restriction of Order

The game proceeds in rounds. At the beginning of every round, the society chooses an order for the players to play and makes it known to them. The players choose actions in that particular order. These actions define a tuple a ∈ A and the play moves along the edge labelled a to the next vertex. This process goes on a a forever. Given an initial vertex v0 this defines an infinite play ρ = v0 →1 v1 →2 . . . in the arena. We study the discounted-payoff Player i gets: n

pi (ρ) = lim inf n→∞

1 fi (vj ). n j=1

Similarly the society incurs a cost of: n

1 c(ρ) = lim inf f (vj ). n→∞ n j=1 There is a threshold cost θ. The aim of each player i is to play in such a way that the quantity pi is maximised and the aim of the society is to choose the

36


orders in such a way that the quantity c(ρ) is always less than θ for every play ρ no matter what actions the players play according to the order it selects. We are interested in the following question: Question. What orders can the society choose so that the social cost c always remains less than the threshold θ? We first define a normalised version of the game where we subtract θ from the cost associated with every vertex of the arena A. In other words, we define a new function f˜ from f such that f˜(v) = f (v) − θ for every v ∈ V . For a play ρ in A we let: n

c˜(ρ) = lim inf n→∞

1˜ f(ρ(j)). n j=1

Note that for any play ρ in A, c(ρ) < θ iff c˜(ρ) < 0. Now, to answer the above question, we first define a tree unfolding TA of the initialised arena (A, v0 ). The unfolding takes into account the order in which the players choose their actions. TA is constructed inductively, the set of nodes being: T ⊆ (V × {soc}) ∪ (V × N × π(N )) where π(N ) is the set of permutations of 2N (the subsets of N ) such that (v, j, w) ∈ TA only if j = w(1).

Fig. 1. The unfolding

We now present the construction, see Figure 1 for an illustration. The root node is (v0 , soc). Suppose TA has been constructed till level i. Consider an unprocessed node at level i. – If this node is of the form (v, soc) and if (v, soc) has an identical ancestor already present in TA constructed so far, then we call (v, soc) a leaf node


37

and do not process it any further. Otherwise the set of children of (v, soc) correspond to all the permutations (orders) for the following round. In other words, the set of children of (v, soc) are of the form (v, j, w) ∈ T where w ∈ π(N ). – If the node is of the form (v, j, w) and |w| > 1 then its children correspond to all the possible actions that j can choose. That is, it has |A| many children of the form (v, k, w2 ) ∈ T . The edge from (v, j, w) to the th child is labelled with a ∈ A. If |w| = 1 then again (v, j, w) has |A| children, the th edge being labelled with a ∈ A such that the following holds. (v , soc) is a child of (v, j, w) if and only if the actions corresponding to the path from (v, soc) to (v , soc) in the tree give the action tuple a and v is the neighbour of v along the edge labelled a. The above procedure is repeated until all the branches have seen a leaf node and there are no more nodes to process. Note that as the set T is finite, the procedure does terminate. We then define a backward induction procedure on TA as follows. In the process, we construct another tree TA∗ which is a subtree of TA and which gives the orders that the society can choose so that the social cost always remains less than the threshold. Procedure 1. – Label the leaf nodes with tuples from Qn+1 as follows. For every leaf node (v, soc) there exists an identical ancestor. This means that on this branch, the game has settled down to a simple cycle involving the vertex v. Let a a a C = v0 →1 v1 →2 . . . →k vk where v0 = vk = v be this cycle. Label (v, soc) with (p0 (C), p1 (C), . . . , pn+1 (C)) where: p0 (C) =

k

f˜(vj )

j=1

and pi (C) =

k

fi (vj ), i ∈ N.

j=1

– For a non-leaf node, suppose all its children have been labelled. • If the non-leaf node is of the form (v, j, w) let L be the set of labels of its children. Let Lj = {pj | (p0 , . . . , pn ) ∈ L}. Let mj = max Lj and let (p0 , . . . , pn ) ∈ L be such that pj = mj . Label (v, soc) with (p0 , . . . , pn ). • If the non-leaf node is of the form (v, soc) let L be the set of labels of its children. Let L0 is unique, then LK(E) := limm→∞ K m (E) is the event that E is limit knowledge among the set I of agents. Accordingly, limit knowledge of an event E is constituted by – whenever unique – the limit point of the sequence of iterated mutual knowledge, and thus linked to both epistemic as well as topological aspects of the event space.

54

C.W. Bach and J. Cabessa

Limit knowledge can be understood as the event which is approached by the sequence of iterated mutual knowledge, according to some notion of closeness between events furnished by a topology on the event space. Thus, the higher the iterated mutual knowledge, the closer this latter epistemic event is to limit knowledge. Note that limit knowledge should not be amalgamated with common knowledge. Indeed, both operators can be perceived as sharing distinct implicative properties with regards to highest iterated mutual knowledge claims. While common knowledge bears a standard implicative relation in terms of set inclusion to highest iterated mutual knowledge, limit knowledge entertains an implicative relation in terms of set proximity with highest iterated mutual knowledge. Besides, limit knowledge also differs from Monderer and Samet’s [12] notion of common p-belief. Indeed, common p-belief – as an approximation of common knowledge in the sense of common almost-knowledge – is implied by common knowledge, whereas limit knowledge is not. Actually, it is possible to link limit knowledge to topological reasoning patterns of agents based on closeness of events. Indeed, agents satisfying limit knowledge of some event are in a limit situation arbitrarily close to entertaining all highest iterated mutual knowledge of this event, and the agents’ reasoning may be influenced accordingly. Note that a reasoning pattern associated with limit knowledge depends on the particular topology on the event space, which fixes the closeness relation between events. The operator limit knowledge is shown by Bach and Cabessa [7,8] to be able to provide relevant epistemic-topological characterizations of solution concepts in games. Despite being based on the same sequence of higher-order mutual knowledge claims, the distinguished interest of limit knowledge resides in its capacity to potentially differ from the purely epistemic operator common knowledge. Notably, it can be proven that such differing situations necessarily require an infinite event space as well as sequences of higher-order mutual knowledge that are strictly shrinking.1 In fact, the topologically amended epistemic framework enables agents with a common prior belief to agree to disagree on their posterior beliefs. Theorem 1. There exist an Aumann structure A = (Ω, (Ii )i∈I , p) equipped with a topology T on the ˆ ∈Ω event space P(Ω), an event E ⊆ Ω, and worlds ω, ω such that ω ∈ LK( i∈I {ω ∈ Ω : p(E | Ii (ω )) = p(E | Ii (ˆ ω ))}), as well as both p(E | Ii (ˆ ω )) = p(E | Ij (ˆ ω )) and p(E | Ii (ω)) = p(E | Ij (ω)) for some agents i, j ∈ I. Proof. Consider the Aumann structure A = (Ω, (Ii )i∈I , p), where Ω = {ωn : n ≥ 0}, I = {Alice, Bob}, IAlice = {{ω2n, ω2n+1 } : n ≥ 0}, IBob = {{ω0 }} ∪ 1 {{ω2n+1 , ω2n+2 } : n ≥ 0}, and p : Ω → R is given by p(ωn ) = 2n+1 for all1 n ≥ 0. Note that the common prior belief function p is well defined since n≥0 2n+1 = 1. Now, consider the event E = {ω2n : n ≥ 1}, and the world ω ∈ Ω. Besides, for 2 sake of notational convenience, let the event i∈I {ω ∈ Ω : p(E | Ii (ω )) = 1

Given some event E, the sequence of higher-order mutual knowledge (K m (E))m>0 is called strictly shrinking if K m+1 (E) K m (E) for all m ≥ 0.

Agreeing to Disagree with Limit Knowledge

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p(E | Ii (ω2 ))} be denoted by E . First of all, observe that p(E | IAlice (ω2 )) = 23 and p(E | IBob (ω2 )) = 13 . Moreover, {ω ∈ Ω : p(E | IAlice (ω )) = p(E | IAlice (ω2 )) = 23 } = Ω \ {ω0 , ω1 } and {ω ∈ Ω : p(E | IBob (ω )) = p(E | IBob (ω2 )) = 13 } = Ω \ {ω0 }, whence E = (Ω \ {ω0 , ω1 }) ∩ (Ω \ {ω0 }) = Ω \ {ω0 , ω1 }. Farther, the definitions of the possibility partitions of Alice and Bob ensure that K m (E ) = K m (Ω \ {ω0 , ω1 }) = Ω \ {ω0 , ω1 , . . . , ωm+1 }, for all m > 0. Consequently, (K m (E ))m>0 is strictly shrinking and the sequence CK(E ) = {ω ∈ Ω : i∈I Ii (ω) ⊆ E } = ∅. Now, consider the topology T on P(Ω) defined by T = {O ⊆ P(Ω) : {ω0 , ω1 , ω2 } ∈ O} ∪ {P(Ω)}. Then, the only open neighbourhood of the event {ω0 , ω1 , ω2 } is P(Ω), and all terms of the sequence (K m (E ))m>0 are contained in P(Ω). Thus (K m (E ))m>0 converges to {ω0 , ω1 , ω2 }. Moreover, for every event F ∈ P(Ω) such that F = {ω0 , ω1 , ω2 }, the singleton {F } is open, and since K m+1 (E ) K m (E ) for all m > 0, the strictly shrinking sequence (K m (E ))m>0 will never remain in the open neighbourhood {F } of F from some index onwards. Hence (K m (E ))m>0 does not converge to any such event F . Therefore the limit point {ω0 , ω1 , ω2 } of the strictly shrinking sequence (K m (E ))m>0 is unique, and LK(E ) = limm→∞ K m (E ) = {ω0 , ω1 , ω2 }. Next, consider the world ω1 . Note that ω1 ∈ LK(E ). Also, observe that p(E | IAlice (ω2 )) = 23 = 13 = p(E | IBob (ω2 )) as well as p(E | IAlice (ω1 )) = 0 = 13 = p(E | IBob (ω1 )). Finally, taking ω = ω1 and ω ˆ = ω2 concludes the proof.

The preceding possibility result counters Aumann’s impossibility theorem in the sense of showing that agents actually can agree to disagree. More precisely, agents may hold distinct actual posterior beliefs, while at the same time satisfying limit knowledge of their posteriors. Hence, agents may agree in the sense of satisfying limit knowledge of their posteriors, while at the same time disagree in the sense of actually entertaining different posterior beliefs. Generally speaking, the mere fact of topologically enriching the event space concurrently with replacing the purely epistemic operator common knowledge by the epistemic-topological operator limit knowledge enables our possibility result. In such an amended perspective, agents can now be seen to have access to a further dimension in their reasoning that remarkably permits them to agree to disagree on their posterior beliefs. In fact, the agents are in a limit situation of entertaining higher-order mutual knowledge of their posteriors, which, in connection with the particular notion of closeness furnished by the topology, leads them to actually possess different posterior beliefs.

3

A Representative Example

The extension of the standard set-based approach to interactive epistemology with a topological dimension has been shown to enable the possibility for agents to agree to disagree on their posterior beliefs. The question then arises whether agents can still agree to disagree in interactive situations furnished with topologies based on epistemic features. A topology describing a specific agents’ perception of the event space is now presented and is then shown to enable agreeing to disagree with limit knowledge.

56


Towards this purpose, suppose an Aumann structure A = (Ω, (Ii )i∈I , p) and an event E. Farther, for any world ω ∈ Ω, let Eω denote the event consisting of that induce the same agents’ posterior beliefs in E as ω, i.e. Eω = all worlds i∈I {ω ∈ Ω : p(E | Ii (ω )) = p(E | Ii (ω))}. Note that constancy of the agents’ posterior beliefs in E yields an equivalence relation on the set of possible worlds, and hence every Eω represents an equivalence class of worlds. Consequently, the collection C = {Eω : ω ∈ Ω} of all equivalence classes of worlds that induce a same posterior belief profile forms a partition of Ω. Given some event E and some index m∗ > 0, the epistemically-based topology TE,m∗ is defined as the topology on the event space P(Ω) generated by the subbase {{K m (Eω ) : m ≥ 0} : ω ∈ Ω} ∪ {P(Ω) \ {K m (Eω ) : m ≥ 0 and ω ∈ Ω}}

∪ {{K m (Eω )} : 0 ≤ m < m∗ and ω ∈ Ω} ∗ ∪ {{K m +j (Eω ) : 0 < j ≤ n} : n > 0 and ω ∈ Ω}. The topology TE,m∗ is illustrated in Figure 1, where the infinite sequence (K m (Eω ))m≥0 is represented by a horizontal sequence of points for each ω ∈ Ω, and open sets of the subbase by circle-type shapes around these points.

P(Ω)

Eω

K(Eω )

K 2 (Eω )

∗

K m (Eω )

Km

∗

+1

(Eω )

∗

Km

+2

(Eω )

∗

Km

+3

(Eω )

Fig. 1. Illustration of the topology TE,m∗

The topology TE,m∗ reveals a specific agent perception of the event space, according to which the agents entertain a more refined distinction between the m∗ first iterated mutual knowledge of their posterior beliefs in E than between the remaining ones. This specific perception is formally reflected by two separation properties satisfied by the topology TE,m∗ .


57

Firstly, given two events X and Y , if X and Y are two distinct terms of a same sequence (K m (Eω ))m>0 for some ω ∈ Ω, and both are iterated mutual knowledge of order strictly smaller than m∗ in this sequence, then X and Y are T2 -separable, and therefore also T0 -separable.2 Secondly, if X and Y are two different elements of a same sequence (K m (Eω ))m>0 for some ω ∈ Ω, and both are iterated mutual knowledge of order strictly larger than m∗ in this sequence, then X and Y are T0 -separable, yet not T2 -separable. According to these two separation properties, agents have access to a more refined distinction between the m∗ first iterated knowledge claims of their posterior beliefs in E than between the iterated mutual knowledge claims of order strictly larger than m∗ . In other words, iterated mutual knowledge claims are only precisely discerned up to a given amount of iterations, and thereafter the higher iterations become less distinguishable for the agents. Also, from a bounded rationality point of view, the agent perception of higher-order mutual knowledge furnished by the topology TE,m∗ reflects that people typically lose track from some iteration level onwards when reasoning about higher-order mutual knowledge. Farther, the topology TE,m∗ notably satisfies the following epistemic-topological property: for any event Eω , if the sequence (K m (Eω ))m>0 is strictly shrink∗ ing, then LK(Eω ) = K m (Eω ). Indeed, suppose that the sequence (K m (Eω ))m>0 is strictly shrinking. Then, by definition of TE,m∗ , the only open neighbour∗ hoods of K m (Eω ) are P(Ω) and {K m (Eω ) : m ≥ 0}. Since both sets contain ∗ all terms of the sequence (K m (Eω ))m>0 , it follows that K m (Eω ) is a limit point of the sequence (K m (Eω ))m>0 . To see that this limit point is actually ∗ unique, consider F ∈ P(Ω) such that F = K m (Eω ). Then either F = K m (Eω ) for some m < m∗ and some ω ∈ Ω, or F = K m (Eω ) for some m > m∗ ∗ and some ω ∈ Ω, or F = K m (Eω ) for some ω = ω, or F = K m (Eω ) for all m ≥ 0 and all ω ∈ Ω. These four mutually exclusive cases are now considered in turn. First of all, if F = K m (Eω ) for some m < m∗ and some ω ∈ Ω, then {K m (Eω )} is an open neighbourhood of F . Since the sequence (K m (Eω ))m>0 is strictly shrinking, it can then not be the case that the singleton open neighbourhood {K m (Eω )} of F contains all terms of the sequence (K m (Eω ))m>0 from some index onwards. Therefore F is not a limit point of the sequence (K m (Eω ))m>0 . Secondly, if F = K m (Eω ) for some m > m∗ and ∗ some ω ∈ Ω, then {K m +j (Eω ) : 0 < j ≤ m − m∗ } is an open neighbourhood m∗ +j of F . Since the set {K (Eω ) : 0 < j ≤ m − m∗ } is finite, F cannot be a ∗ limit point of the sequence (K m (Eω ))m>0 . Thirdly, if F = K m (Eω ) for some n ω = ω, then {K (Eω ) : n ≥ 0} is an open neighbourhood of F . Moreover, ∗ ∗ since K m (Eω ) = K m (Eω ) = F , it directly follows that Eω = Eω . Yet since C = {Eω : ω ∈ Ω} is a partition of Ω, it holds that Eω ∩ Eω = ∅. Moreover, as K m (Eω ) ⊆ Eω for all m ≥ 0, and K n (Eω ) ⊆ Eω for all n ≥ 0, as well as Eω ∩ Eω = ∅, it follows that K m (Eω ) = K n (Eω ) for all m, n ≥ 0. Thus the 2

Given a topological space (A, T ), two points in A are called T2 -separable if there exist two disjoint T -open neighbourhoods of these two points. Moreover, two points in A are called T0 -separable if there exists a T -open set containing precisely one of these two points. Note that T2 -separability implies T0 -separability.

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open neighbourhood {K n(Eω ) : n ≥ 0} of F contains no term of the sequence (K m (Eω ))m>0 whatsoever. Therefore, F is not a limit point of the sequence (K m (Eω ))m>0 . Fourthly, if F = K m (Eω ) for all m ≥ 0 and all ω ∈ Ω, then P(Ω) \ {K m (Eω ) : m ≥ 0 and ω ∈ Ω} is an open neighbourhood of F . Yet this set contains no term of the sequence (K m (Eω ))m>0 . Thus F is not a limit point of the sequence (K m (Eω ))m>0 . To summarize, there consequently exists no ∗ F = K m (Eω ) which is a limit point of the sequence (K m (Eω ))m>0 . Therefore, ∗ the limit point K m (Eω ) of the sequence (K m (Eω ))m>0 is unique, and thence ∗ LK(Eω ) = limm→∞ K m (Eω ) = K m (Eω ). Furthermore, since the sequence ∗ (K m (Eω ))m>0 is strictly shrinking, CK(Eω ) = m>0 K m (Eω ) K m (Eω ), and hence CK(Eω ) = LK(Eω ). Finally, the following example describes an interactive situation, in which the epistemically-based topology TE,m∗ provides a possibility for the agents to agree to disagree on their posterior beliefs with limit knowledge. Example 1. Consider the Aumann structure A = (Ω, (Ii )i∈I , p), where Ω = {ωn : n ≥ 0}, I = {Alice, Bob}, IAlice = {{ω0 }, {ω1 , ω2 }, {ω3 , ω4 , ω5 , ω6 }, {ω7 , ω8 , ω9 }} ∪ {{ω2n, ω2n+1 } : n ≥ 5}, IBob = {{ω0 , ω1 , ω2 , ω3 , ω4 }, {ω5 , ω6 , ω7 , ω8 }} ∪ {{ω2n+1 , ω2n+2 } : n ≥ 4}, and p : Ω → R is given by p(ωn ) = 1 for all n ≥ 0. Also, consider the event E = {ω1 , ω5 } ∪ {ω2n : n ≥ 1} 2n+1 and the world ω10 . Besides, for sake of notational convenience, let the event {ω ∈ Ω : p(E | Ii (ω )) = p(E | Ii (ω10 ))} be denoted by E . First of all, i∈I observe that the computation of the posterior beliefs of Alice and Bob gives a variety of distinct values for the first ten worlds {ω0 , ω1 , . . . , ω9 }, as well as p(E | IAlice (ωn )) = 23 and p(E | IBob (ωn )) = 13 , for all n ≥ 10. It follows that {ω ∈ Ω : p(E | IAlice (ω )) = p(E | IAlice (ω10 ))} = Ω \ {ω0 , ω1 , . . . , ω9 } and {ω ∈ Ω : p(E | IBob (ω )) = p(E | IBob (ω10 ))} = Ω \ {ω0 , ω1 , . . . , ω8 }, thus E = (Ω\{ω0 , ω1 , . . . , ω9 })∩(Ω\{ω0 , ω1 , . . . , ω8 }) = Ω\{ω0 , ω1 , . . . , ω9 }. Moreover, the definitions of the possibility partitions of Alice and Bob ensure that K m (E ) = Ω \ {ω0 , ω1 , . . . , ωm+9 }, for all m > 0. Consequently, the sequence (K m (E ))m>0 is strictly shrinking and CK(E ) = m>0 K m (E ) = ∅. Now, let m∗ > 0 be some index and suppose that P(Ω) is equipped with the topology TE,m∗ . Since the sequence (K m (E ))m>0 is strictly shrinking, the definition of this topology ∗ ensures that LK(E ) = K m (E ) = Ω \ {ω0 , ω1 , . . . , ωm∗ +9 }. Consequently, the computations of the posterior beliefs of Alice and Bob give p(E | IAlice (ω)) = 23 and p(E | IBob (ω)) = 13 , for all ω ∈ LK(E ). In other words, for all ω ∈ LK(E ), it holds that p(E | IAlice (ω)) = p(E | IBob (ω)).

4

Conclusion

In an epistemic-topoloigcal framework, agents have been shown to be able to agree to disagree. More precisely, if Bayesian agents entertain a common prior belief in a given event as well as limit knowledge of their posterior beliefs in the event, then their actual posterior beliefs may indeed differ. This possibility result also holds in interactive situations enriched by a particular epistemically-based topology revealing a cogent agent perception of the event space.


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The topological approach to set-based interactive epistemology, in which topologies model agent closeness perceptions of events, can be used to describe various agent reasoning patterns that do not only depend on mere epistemic but also on topological features of the underlying interactive situation. For instance, the event It is cloudy in London seems to be closer to the event It is raining in London than the event It is sunny in London. Now, agents may make identical decisions only being informed of the truth of some event within a class of close events. Indeed, Alice might decide to stay at home not only in the case of it raining outside, but also in the case of events perceived by her to be similar such as it being cloudy outside. Moreover, we envision the construction of a more general epistemic-topological framework – topological Aumann structures – comprising topologies not only on the event space but also on the state space. Such an extension permits an explicit consideration of a notion of closeness between events as well as between worlds, enabling to model common agent perceptions of the event and state spaces. In particular, it might be of distinguished interest to base topologies on first principles such as epistemic axioms or natural closeness properties. In line with this perspective, the topology provided in Section 3 reflects the natural agent perception for which iterated mutual knowledge becomes imprecise from some level onwards. Besides, in order to model subjective rather than common agent perceptions of the event and state spaces, the epistemic-topological framework envisioned here could be amended by assigning specific and potentially distinct topologies to every agent. A collective topology reflecting a common closeness perception could then be constructed on the basis of the particular agent topologies, and limit knowledge be defined in such a global topological context. For instance, by providing a topology that is coarser than each agent’s one, the meet topology could be used as a representative collective topology. Alternatively, an agent specific operator limit knowledge could be defined with respect to each particular topology, and mutual limit knowledge as their intersection then be considered. Finally, in a general epistemic-topological framework, various issues can be addressed. For example, the possibility of agents to agree to disagree with limit knowledge can be further analyzed for other epistemically-based as well as agent specific topologies. Furthermore, analogously to the epistemic program in game theory that attempts to provide epistemic foundations for solution concepts, an epistemic-topological approach could generate epistemic-topological foundations for solution concepts. In addition, it could be attempted to develop a theory of counterfactuals in set-based interactive epistemology founded on a notion of similarity of worlds or events furnished by topologies on the state or event space, respectively. Acknowledgement. We are highly grateful to Richard Bradley, Adam Brandenburger, Jacques Duparc and Andrés Perea for illuminating discussions and invaluable comments.

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References 1. Aumann, R.J.: Agreeing to disagree. Annals of Statistics 4(6), 1236–1239 (1976) 2. Aumann, R.J.: Correlated equilibrium as an expression of bayesian rationality. Econometrica 55(1), 1–18 (1987) 3. Aumann, R.J.: Backward induction and common knowledge of rationality. Games and Economic Behavior 8(1), 6–19 (1995) 4. Aumann, R.J.: Interactive epistemology i: Knowledge. International Journal of Game Theory 28(3), 263–300 (1999) 5. Aumann, R.J.: Interactive epistemology ii: Probability. International Journal of Game Theory 28(3), 301–314 (1999) 6. Aumann, R.J.: Musings on information and knowledge. Econ. Journal Watch 2(1), 88–96 (2005) 7. Bach, C.W., Cabessa, J.: Limit knowledge of rationality. In: Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge, TARK 2009, pp. 34–40. ACM, New York (2009) 8. Bach, C.W., Cabessa, J.: Common knowledge and limit knowledge. Theory and Decision (to appear) 9. Bonanno, G., Nehring, K.: Agreeing to disagree: A survey. Working paper series no. 97-18, Department of Economics, University of California, Davis (1997) 10. Geanakoplos, J.D., Polemarchakis, H.M.: We can’t disagree forever. Journal of Economic Theory 28(1), 192–200 (1982) 11. Milgrom, P., Stokey, N.: Information, trade and common knowledge. Journal of Economic Theory 26(1), 17–27 (1982) 12. Monderer, D., Samet, D.: Approximating common knowledge with common beliefs. Games and Economic Behavior 1(2), 170–190 (1989) 13. Samet, D.: Ignoring ignorance and agreeing to disagree. Journal of Economic Theory 52(1), 190–207 (1990)

A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory Ka Fat Chow The Hong Kong Polytechnic University [email protected]

Abstract. This paper introduces a semantic model for vague quantifiers (VQs) combining Fuzzy Theory (FT) and Supervaluation Theory (ST), which are the two main theories on vagueness, a common source of uncertainty in natural language. After comparing FT and ST, I will develop the desired model and a numerical method for evaluating truth values of vague quantified statements, called the Modified Glöckner’s Method, that combines the merits and overcomes the demerits of the two theories. I will also show how the model can be applied to evaluate truth values of complex quantified statements with iterated VQs. Keywords: vague quantifiers, Generalized Quantifier Theory, Fuzzy Theory, Supervaluation Theory, Modified Glöckner’s Method.

1

Introduction

Vagueness is a common source of uncertainty in natural language. No doubt vague quantifiers (VQs) constitute an important type of quantifiers, the target of study of the Generalized Quantifier Theory (GQT). However, since it is difficult to model vagueness under standard Set Theory, the study on VQs has remained a weak point of GQT. In GQT, the most typical approach of representing the truth condition of a VQ is to represent it as a comparison between an expression consisting of the VQ’s arguments and a context-dependent standard. For example, according to [11], there are three interpretations of “many”. The truth condition of “many2” is as follows: many2(A)(B) ↔ |A ∩ B| ≥ k|A|

(1)

where k ∈ (0, 1) is a context-dependent constant. This condition says that “Many As are B” is true whenever the proportion of those As that are B among all As is at least as great as a standard, i.e. k, representing the threshold of “many”. Since k is dependent on context, the above condition may yield different truth values for two different quantified statements “Many A1s are B1” and “Many A2s are B2” even if |A1| = |A2| and |A1 ∩ B1| = |A2 ∩ B2|. While this approach is most straightforward, what it genuinely reflects is the context dependence rather than the vagueness of VQs. In this paper, I will leave aside the issue of context dependence and concentrate on the vagueness of VQs. Moreover, I will only deal with VQs in a general manner and will not work out the detailed H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 61–73, 2011. © Springer-Verlag Berlin Heidelberg 2011

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semantics of any particular VQ. Since vague concepts are characterized by blurred boundaries and uncertain membership, we need to invoke theories that deal with such phenomena. In the next section, I will introduce two such theories: Fuzzy Theory and Supervaluation Theory. The former is further divided into two approaches: the Fuzzy Set Cardinality Approach and the Quantifier Fuzzification Mechanism Approach.

2

Basic Theories for VQs

2.1

Fuzzy Theory (Fuzzy Set Cardinality Approach)

Fuzzy Theory (FT) is a cover term for all those theories that are based on or derived from the Fuzzy Set Theory developed by [14]. Ever since [14], FT has become a new paradigm and is widely applied in many areas. Under FT, vague concepts are modeled by fuzzy sets, which differ from crisp sets (i.e. non-fuzzy sets) in one important aspect: instead of having sharp boundaries between members and non-members, every individual in the universe belongs to a fuzzy set to a certain degree ranging from absolute membership to absolute non-membership. By using ║p║ to denote the truth value of a proposition p, we can represent this degree by a membership degree function (MDF), ║x ∈ S║, which outputs a numerical value in [0, 1] representing the degree to which an individual x belongs to a fuzzy set S1. For example, ║j ∈ TALL║ = 0.7 means that John is tall to the degree 0.7. Sometimes, the MDF may take the form of a mathematical function that depends on a numerical value (henceforth called the “input” of the MDF). For example, as the tallness of a person depends on the person’s height, the aforesaid MDF for TALL may take the alternative form ║h ∈ TALL║, where h represents the height of a person. Fuzzy theorists have also defined certain crisp sets corresponding to each fuzzy set. Let X be a fuzzy set and α be a real number in [0, 1]. Then the α-cut (denoted X≥α), and strict α-cut (denoted X>α) of X are defined as follows (in what follows, U represents the universe): X≥α = {x ∈ U: ║x ∈ X║ ≥ α}

(2)

X>α = {x ∈ U: ║x ∈ X║ > α}

(3)

Another characteristic of FT is that it treats Boolean operators (BOs) as truth functions such as2: ║p ∧ q║ = min({║p║, ║q║})

(4)

║p ∨ q║ = max({║p║, ║q║})

(5)

║¬p║ = 1 – ║p║

1

2

(6)

In the literature, the MDF is often expressed as μS(x). In this paper I use ║x ∈ S║ instead for convenience. In the literature, there is a whole range of possible definitions of BOs. What follows are the “standard” definitions of the most commonly used BOs.

A Semantic Model for Vague Quantifiers Combining Fuzzy Theory

63

Inspired by GQT, fuzzy theorists also tried to formalize theories about VQs, with [15] and [12-13] being the earlier attempts. Since VQs can be seen as fuzzy sets of numbers, they can also be modeled by MDFs. For example, borrowing ideas from [1], we may represent the VQ “(about 10)” by the following MDF: ║(about 10)(A)(B)║ = T–4, –1, 1, 4(|A ∩ B| – 10)

(7)

There are several points to note concerning the above formula. First, I have adopted [8]’s notation that represents a quantified statement in the form of a tripartite structure “Q(A)(B)” where Q, A and B represent the quantifier and its two arguments, respectively3. Syntactically, these two arguments correspond to the subject (excluding the quantifier) and the predicate of the quantified statement. Second, the above formula makes use of a piecewise-defined function Ta, b, c, d(x) with the following definition:

Ta, b, c, d(x) =

0, (x – a) / (b – a),

if x < a if a ≤ x < b

1

if b ≤ x ≤ c

(d – x) / (d – c), 0,

if c < x ≤ d if x > d

(8)

The above function is named “T”, standing for “trapezoid”, because its graph has a trapezoidal shape. Figure 1 shows the graph of T–4, –1, 1, 4:

Fig. 1. T–4, –1, 1, 4

When the parameters are such that a = b or c = d, since there is no x such that a ≤ x < a or c ≤ x < c, the 2nd or 4th piece of (8) would disappear. In these cases, T becomes degenerate and its graph is shaped like half of a trapezoid. For example, Figure 2 shows the graph of T–0.25, –0.1, ∞, ∞: 3

In this paper, I only consider VQs that have two arguments. Using standard GQT notation, such kind of VQs belongs to type quantifiers, also called “determiners”.

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Fig. 2. T–0.25, –0.1, ∞, ∞

Note that the function T given in (8) gives just one possible example of MDFs that may be used for representing VQs. In fact, any function whose general shape is similar to T can also serve the same purpose. More specifically, this function should be a function whose domain can be partitioned into 5 parts such that the values at the 1st, 2nd, 3rd, 4th and 5th parts are constantly 0, increasing, constantly 1, decreasing and constantly 0, respectively4. Using the MDFs for VQs, one can then evaluate the truth values of sentences containing VQs. However, the evaluation of truth values of these sentences sometimes may involve some complications. For example, consider the following sentence: About 10 tall girls sang.

(9)

This sentence contains the VQ “(about 10)”. According to (7), the input of the MDF for “(about 10)(TALL-GIRL)(SING)” is the number |TALL-GIRL ∩ SING| – 10. However, since TALL-GIRL ∩ SING is fuzzy, its cardinality is not well defined. We now encounter the following problem: how can we evaluate the truth value of (9) if we cannot say for sure how many “tall girls” there are? The solution of the early fuzzy theorists is to generalize the notion of crisp set cardinality to fuzzy set cardinality, which may have different definitions. One definition (called the Sigma Count) is the sum of the membership degrees of all individuals in the universe with respect to the fuzzy set. For example, if the fuzzy set TALL-GIRL ∩ SING = {1/a, 0.7/b, 0.5/c, 0.2/d, 0.1/e}5, then the Sigma Count of this set is 1 + 0.7 + 0.5 + 0.2 + 0.1 = 2.5. Using this cardinality, the truth value of (9) is then equal to ║2.5 ∈ (about 10)║, which is equal to 0 according to (7). This shows that (9) is absolutely false with respect to the aforesaid fuzzy set TALL-GIRL ∩ SING. This is in accord with our intuition because according to that fuzzy set, there are only 2 members (i.e. a and b) who may be counted as singing tall girls with a relatively high certainty, and 2 absolutely falls short of being “about 10”. 4 5

In case the function becomes degenerate, then some of the aforesaid parts would disappear. Here I adopt a notation used by fuzzy theorists under which a fuzzy set S is represented in the form {r1/x1, r2/x2, …} where xis are individuals and ris are their respective membership degrees, i.e. ri = ║xi ∈ S║. In case the membership degree of an individual is 0, it is not listed.


2.2

65

Fuzzy Theory (Quantifier Fuzzification Mechanism Approach)

Later, some scholars (e.g. [1], [4-5], [10]) realized the demerits of the old approach, which was able to treat only certain types of VQs and could not be applied to more general types of VQs. Moreover, since different notions of fuzzy set cardinality were used for different VQs, there was not a uniform treatment for various types of VQs. Instead of using the concept of fuzzy set cardinality, they proposed the concept of quantifier fuzzification mechanisms (QFMs). This approach distinguishes two types of VQs: semi-fuzzy and fuzzy quantifiers. Semi-fuzzy quantifiers are those VQs that only take crisp sets as arguments; while fuzzy quantifiers are those VQs that may take either crisp or fuzzy sets as arguments. Note that the distinction between semi-fuzzy and fuzzy quantifiers has nothing to do with the meaning of the VQs. Thus, the same linguistic quantifier such as “(about 10)” may manifest either as a semi-fuzzy or a fuzzy quantifier, depending on the types of its arguments6. Under this approach, all VQs are initially modeled as semi-fuzzy quantifiers. This has the advantage of greatly simplifying the semantics of VQs. We only need to formulate an appropriate MDF or truth condition for each VQ without worrying about its inputs because all inputs are crisp. The evaluation of truth values of sentences involving semi-fuzzy quantifiers is easy: we only need to plug the crisp inputs into the appropriate MDFs or truth conditions. When it comes to a sentence involving fuzzy quantifiers with fuzzy inputs (such as (9)), we have to make use of a QFM, which is in fact a mapping that transforms a semi-fuzzy quantifier to a fuzzy quantifier. Among the QFM approach, [4]’s framework has certain merits compared with its competitors in that it proposes a number of axioms that an adequate QFM should satisfy 7 . These axioms guarantee that the QFM will preserve crisp arguments, the identity truth function and monotonicities of a VQ as well as its arguments, and that the QFM will commute with the operations of argument transposition, argument insertion, external negation, internal negation, internal meet (as well as other Boolean) operators and functional application. Note that the aforesaid properties / operations are crucial to the study of quantifiers under GQT. Next I introduce a QFM proposed in [4]8. First let X be a fuzzy set and γ be a real number in [0, 1] which is called the “cut level”. We can reduce X into two crisp sets Xγmin and Xγmax at the cut level γ using the following formulae: for γ > 0 9, Xγmin = X≥ 0.5 + 0.5γ; 6

7

8

9

Xγmax = X> 0.5 – 0.5γ

(10)

Since crispness can be seen as a special case of fuzziness, any crisp quantifier such as “every” can be seen as a semi-fuzzy or fuzzy quantifier, depending on the types of its arguments. Actually, Glöckner used the term “determiner fuzzification schemes” (DFSs) in [4]. In [5], he used QFMs as a general term for all mappings that map a semi-fuzzy quantifier to a fuzzy quantifier and used DFSs to refer to those QFMs that satisfy his axioms. To simplify notation, in what follows I will just use the umbrella term QFM. Glöckner has proposed a number of QFMs that satisfy all his axioms. This paper only discusses the simplest one. There are in fact separate definitions for Xγmin and Xγmax at γ = 0. But since a single point will only contribute the value 0 to a definite integral to be introduced below, we do not need to consider the case γ = 0 for computational purpose.

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Based on the above, we can then define a family of crisp sets associated with X: Tγ(X) = {Y: Xγmin ⊆ Y ⊆ Xγmax}

(11)

Then let Q be a semi-fuzzy quantifier and X1, … Xn be n fuzzy sets. Now for each of X1, … Xn we can define Tγ(X1), … Tγ(Xn). For each possible combination of Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn), we can evaluate ║Q(Y1, … Yn)║ by using a suitable MDF or truth condition because Y1, … Yn are crisp sets. Then we aggregate the various values of ║Q(Y1, … Yn)║ for all possible combinations of Y1, … Yn into ║Qγ(X1, … Xn)║10 by the following formula: ║Qγ(X1, … Xn)║ = m0.5({║Q(Y1, … Yn)║: Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn)})

(12)

where m0.5, called the “generalized fuzzy median”, is defined as follows. Let Z be a set of real numbers, then

m0.5(Z) =

inf(Z), sup(Z), 0.5, r,

if |Z| ≥ 2 ∧ inf(Z) > 0.5 if |Z| ≥ 2 ∧ sup(Z) < 0.5 if (|Z| ≥ 2 ∧ inf(Z) ≤ 0.5 ∧ sup(Z) ≥ 0.5) ∨ (Z = ∅) if Z = {r}

(13)

Now for each cut level γ, we have a corresponding value ║Qγ(X1, … Xn)║. Finally we need to combine all these values into one value. According to [4], there are various methods of combination, one such method (which is denoted by “M” in [4]) is to use the standard definite integral11: ║M(Q)(X1, … Xn)║ =

1



0

║Qγ(X1, … Xn)║dγ

(14)

Although the above formula appears as an integral, in practical calculation of linguistic applications involving finite universes, we often only need to consider a finite number of variations of γ and ║Qγ(X1, … Xn)║ is constant at each such γ, and so the integral above often reduces to a sum, which can be seen as a “weighted average” of ║Qγ(X1, … Xn)║ at the various γs. 2.3

Supervaluation Theory

The Supervaluation Theory (ST) for vagueness is a keen competitor of the FT. Some supervaluation theorists, such as [3] and [6-7], pointed out certain flaws of FT. The most serious one is that FT cannot correctly predict the truth values of certain statements that must be true / false by virtue of traditional logical laws or intuition with respect to a model (such statements are called “penumbral connections” in [3]). Consider the following model: M1

10 11

U = {j, m}; TALL = {0.5/j, 0.3/m}

Note that here Qγ should be seen as a fuzzy quantifier evaluated at the cut level γ. In the following formula, “M” should be seen as a QFM that transforms the semi-fuzzy quantifier Q to a fuzzy quantifier M(Q).


67

Intuitively, according to this model, the truth values of the following sentences should both be absolutely false (where John and Mary are represented by j and m above): John is tall and John is not tall.

(15)

Mary is tall and John is not tall.

(16)

But using the truth functions for BOs (4) – (6), the calculation results show that the above sentences are both true to a certain degree under FT: ║(15)║ = ║j ∈ TALL║ ∧ ║j ∉ TALL║ = min({0.5, 1 – 0.5}) = 0.5 ║(16)║ = ║m ∈ TALL║ ∧ ║j ∉ TALL║ = min({0.3, 1 – 0.5}) = 0.3 Supervaluation theorists point out that the above wrong predictions arise from the wrong assumption that BOs are truth functional when applied to vague concepts. Note that the aforesaid flaw does not hinge on the particular definitions of BOs. It is argued in [7] that the definitions of BOs are subject to various plausible constraints. For example, one may hope that the definitions will preserve p → q ≡ ¬p ∨ q, or that p → p is always true. But unfortunately, no set of definitions can satisfy all these plausible constraints under FT. Supervaluation theorists view vague concepts as truth value gaps and evaluate the truth values of vague sentences by means of complete specifications. A complete specification is an assignment of the truth value 1 or 0 to every individual with respect to the relevant vague sets in a sentence. In other words, a complete specification eliminates the truth value gaps and makes a vague sentence precise. Thus, this process is called “precisification”. If a sentence is true (false) on all admissible complete specifications, then we say that it is true (false)12. Otherwise, it has no truth value. The concept of “admissible” is very important in ST. Let’s use model M1 to illustrate this point. This model contains two individuals: j and m such that both are borderline cases of the vague set TALL with j taller than m. Here is a list of all admissible complete specifications for M1: (i) ║j ∈ TALL║ = 1, ║m ∈ TALL║ = 1; (ii) ║j ∈ TALL║ = 1, ║m ∈ TALL║ = 0; (iii) ║j ∈ TALL║ = 0, ║m ∈ TALL║ = 0. The above list does not include ║j ∈ TALL║ = 0, ║m ∈ TALL║ = 1 because it is inadmissible to assign a person to the set of TALL without at the same time assigning another person who is even taller to TALL. Having identified the admissible specifications, we can then evaluate ║(15)║ and ║(16)║. Since (15) and (16) are both false on all of (i) – (iii) above, we obtain ║(15)║ = ║(16)║ = 0, in conformity with our intuition. Thus, ST provides an alternative method that can deal with penumbral connections correctly. The same method can also be used to evaluate truth values of sentences containing VQs, although the precisification process may be more complicated. Using (9) as an example, the precisification process will involve two levels. At the first level, the vague concept “tall girl” will be precisified, after which we obtain a set 12

In [3] the terms “super-true” (“super-false”) were used to denote propositions that are true (false) on all admissible complete specifications. To simplify notation, I will just call such propositions “true” (“false”).

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TALL-GIRL ∩ SING whose cardinality is known. Then, at the second level, the VQ “about 10” will be precisified based on the aforesaid cardinality. The main weakness of ST is that it cannot distinguish different degrees of vagueness because it treats all borderline cases alike as truth value gaps. The evaluation of truth values of vague sentences under ST is uninteresting because all those vague sentences other than penumbral connections have no truth values. Moreover, in applied studies such as Control Theory, Artificial Intelligence, etc., the concept of membership degrees is of great use. That is why while FT has become very popular in applied studies, ST is only popular in theoretical studies. As a matter of fact, [6] has discussed how to develop a version of ST that incorporates the notion of degrees. More recently, [2] even showed that FT and ST, though often seen to be incompatible with each other, can in fact be combined. In the next section, I will propose such a combined theory.

3 3.1

Combining FT and ST The Modified Glöckner’s Method

Although all borderline cases can be treated as truth value gaps, they may behave differently in the process of precisification. For example, among all admissible complete specifications in which individuals are assigned to the set TALL, a taller person x is more likely to be assigned full membership of TALL than a shorter person y, because whenever y is assigned full membership of TALL in an admissible specification, x must also be so, but not vice versa. An individual x’s membership degree with respect to a vague set S may thus be seen as representing the likelihood of x being assigned to S in an admissible specification. By reinterpreting membership degrees in this way, we have established a link between FT and ST and the semantic model for VQs developed below will follow the tradition of FT by using MDFs as a measure of truth values of VQs. How are we to evaluate the truth values of vague sentences such as (15) and (16)? As mentioned ahove, the traditional FT approach of treating BOs as truth functions like (4) – (6) has to be abandoned. Neither can we use ST’s method because we now want to distinguish an infinite number of truth values. Fortunately, Glöckner’s method in [4] as introduced in Subsection 2.2 can meet our requirements. The essence of Glöckner’s method in [4] is to reduce a sentence with vague arguments to sentences with crisp arguments at different cut levels. The truth values of these sentences with crisp arguments are then evaluated using the MDFs or truth conditions and aggregated into the truth values of a vague quantified sentence at a cut level. Finally, the truth values at all cut levels are combined into a “weighted average”, which is then taken to be the truth value of the original sentence. Using the aforesaid method, there is no need to invoke (4) – (6). Moreover, the aforesaid reduction process can be seen as a precisification process and the family of crisp sets Tγ(X) as defined in (11) can be seen as a set of complete specifications of X. To guarantee that these are also admissible specifications, we need to modify the definition of Tγ(X) as shown below:


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Tγ(X) = {Y: Xγmin ⊆ Y ⊆ Xγmax ∧ Y represents an admissible complete specification of X}

(17)

Glöckner’s method with the above modification will henceforth be called the Modified Glöckner’s Method (MGM). With MGM, we can evaluate ║(15)║ and ║(16)║ with respect to M1. Since the result of ║(15)║ is obvious, I only show the evaluation of ║(16)║. In order to use MGM, we first need to express (16) as a conjoined quantified statement. One way is to make use of the quantifier “every” satisfying the truth condition every(A)(B) ↔ A ⊆ B: every({m})(TALL) ∧ ¬every({j})(TALL)

(18)

Now, for 0 < γ ≤ 0.4, we have by (10), TALLγ = ∅, TALLγ = {j}, By (17), Tγ(TALL) = {∅, {j}} since both ∅ and {j} represent admissible complete specifications. Then, we have min

max

║everyγ({m})(TALL) ∧ ¬everyγ({j})(TALL)║ =

m0.5({║every({m})(Y) ∧ ¬every({j})(Y)║: Y ∈ Tγ(TALL)})

=

m0.5({║every({m})(∅) ∧ ¬every({j})(∅)║, ║every({m})({j}) ∧ ¬every({j})({j})║})

=

m0.5({0})

=

0

by (12)

by (13)

For 0.4 < γ ≤ 1, TALLγmin = ∅, TALLγmax = {j, m}. By (17), Tγ(TALL) = {∅, {j}, {j, m}} since ∅, {j} and {j, m} represent admissible complete specifications. Note that although ∅ ⊆ {m} ⊆ {j, m}, {m} is not included in Tγ(TALL) because {m} represents the inadmissible complete specification ║j ∈ TALL║ = 0, ║m ∈ TALL║ = 1. Then, we have ║everyγ({m})(TALL) ∧ ¬everyγ({j})(TALL)║ =

m0.5({║every({m})(Y) ∧ ¬every({j})(Y)║: Y ∈ Tγ(TALL)})

=

m0.5({║every({m})(∅) ∧ ¬every({j})(∅)║, ║every({m})({j}) ∧ ¬every({j})({j})║, ║every({m})({j, m}) ∧ ¬every({j})({j, m})║})

=

m0.5({0})

=

0

by (12)

by (13)

Finally, by (14), ║(16)║ = ║(18)║

1

=



0

║everyγ({m})(TALL) ∧ ¬everyγ({j})(TALL)║dγ

= 0 × (0.4 – 0) + 0 × (1 – 0.4) =0

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which is as desired. Note that if we had included {m} as an admissible complete specification for 0.4 < γ ≤ 1, then we would have got ║(16)║ = 0.3, contrary to our intuition. The above computation shows that MGM is able to correct the flaw of FT. 3.2

Some Properties of MGM

The modification of the definition of Tγ(X) as shown in (17) may incur a cost in that some nice properties of Glöckner’s original theory may be lost. By scrutinizing the proofs of the various lemmas and theorems in [4], one can find that the important properties of the orginal theory introduced in Subsection 2.2 are not affected by the modification with two exceptions, namely under MGM the QFM represented by M does not commute with internal meet and functional application. This means, for example, that when we evaluate13 ║Qγ(X1 ∩ X2)║

(19)

for a particular γ, the result of first precisifying X1 and X2 and then intersecting the resultant crisp sets, i.e. m0.5({║Q(Y1 ∩ Y2)║: Y1 ∈ Tγ(X1), Y2 ∈ Tγ(X2)})

(20)

may be different from the result of first intersecting X1 and X2 and then precisifying the resultant fuzzy set, i.e. m0.5({║Q(Y)║: Y ∈ Tγ(X1 ∩ X2)})

(21)

because {Y1 ∩ Y2: Y1 ∈ Tγ(X1), Y2 ∈ Tγ(X2)} may not be equal to {Y: Y ∈ Tγ(X1 ∩ X2)}. The crux of the problem is that the intersection of two sets each representing an admissible complete specification may not be a set representing an admissible complete specification. For instance, while {a, b} represents an admissible complete specification for the set X1 = {1/a, 0.9/b, 0.8/c} and {b, c} represents an admissible complete specification for the set X2 = {0.5/a, 0.6/b, 0.7/c}, {a, b} ∩ {b, c} = {b} represents an inadmissible complete specification for X1 ∩ X2 = {0.5/a, 0.6/b, 0.7/c}. The same problem can be said of functional application for an arbitrary function. Is this a defect of MGM? Not necessarily. The essence of MGM is to deny the truth functionality of BOs and other arbitrary functions when applied to vague sets. Under MGM, when evaluating the truth value of a vague statement involving BOs or other arbitrary functions, we never apply the BOs or functions to the vague arguments directly because such application is undefined. Instead, we always proceed by first precisifying the vague arguments and then applying the BOs or functions to the resultant crisp arguments. This means, for example, that when evaluating (19) we always do (20), never (21), and so the problem that (20) ≠ (21) simply does not arise. Thus, we may say that MGM has preserved the essential nice properties of Glöckner’s original theory. Note that MGM also has another nice property. Suppose the membership degrees with respect to the vague sets X1, ... Xn in a model are restricted to {0, 1, 0.5} and the 13

To simplify notation, in what follows I use the same symbol “∩” to denote the intersection operation of crisp sets and vague sets. Under FT, the vague version of “∩” may be defined based on the BO “∧”.


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truth values of a semi-fuzzy quantifier Q applied to any n crisp arguments are also restricted to {0, 1, 0.5}. Then for 0 < γ ≤ 1, we must have {║Q(Y1, … Yn)║: Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn)} equal to any one of the following: {0}, {1}, {0.5}, {0, 1}, {0, 0.5}, {1, 0.5}, {0, 1, 0.5}. By (12) and (13), we have ║Qγ(X1, … Xn)║ equal to 0, 1 or 0.5 according as {║Q(Y1, … Yn)║: Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn)} contains only 0, only 1 or otherwise. Then by (14), we have ║M(Q)(X1, … Xn)║ also restricted to {0, 1, 0.5}. So in this case MGM gives us the same result as that obtained by the supervaluation method if we use 0.5 to represent the truth value gap. MGM is thus indeed a generalization of the supervaluation method and provides us with the flexibility in determining how we should model vagueness.

4

Iterated VQs

According to [9], a sentence containing both subject and object(s) can be viewed as containing a polyadic quantifier. There is an important type of polyadic quantifiers, called iterated quantifiers, that can be represented by a tripartite structure with one of its arguments containing another tripartite structure. For example, the sentence Every boy loves every girl.

(22)

may be seen as containing the iterated quantifier “(every … every)” and can be represented by the following tripartite structure: every(BOY)({x: every(GIRL)({y: LOVE(x, y)})})

(23)

which, in daily language, means “Every boy x is such that for every girl y, x loves y”. Based on the above expression, one can then evaluate the truth value of (22) with respect to any model according to the truth condition of “every”. MGM is readily applicable to iterated VQs14. Consider the following sentence: Almost every boy met about 10 girls.

(24)

with respect to the following model: M2

BOY = {a, b, c, d, e} x |GIRL ∩ {y: MEET(x, y)}|

a 10

b 9

c 11

d 13

e 8

For computational purpose, suppose we use (7) as the MDF for “(about 10)” and the following MDF for “(almost every)” (where ε represents an infinitesimal positive magnitude): ║(almost every)(A)(B)║ = T–0.4, –0.2, –ε, –ε(|A ∩ B| / |A| – 1)

(25)

To evaluate ║(24)║, we first write (24) as the following tripartite structure: (almost every)(BOY)({x: (about 10)(GIRL)({y: MEET(x, y)})})

14

(26)

For simplicity, here I only consider iterated VQs composed of 2 VQs. It is not difficult to generalize the theory to iterated VQs composed of more than 2 VQs.

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In the above, {x: (about 10)(GIRL)({y: MEET(x, y)})} denotes the set of those who met about 10 girls. For convenience, let’s call this set X. Since X is a vague set, we cannot evaluate ║(26)║ directly. To facilitate further computation, we need to determine this vague set first. According to (7), for each x, ║(about 10)(GIRL)({y: MEET(x, y)})║ depends on the input |GIRL ∩ {y: MEET(x, y)}| – 10. By substituting the data given in M2 into (7) for each x, we can determine the following vague set: X = {1/a, 1/b, 1/c, 0.33/d, 0.67/e}. We then use MGM to evaluate ║(26)║. For 0 < γ ≤ 0.33, we have Xγmin = {a, b, c, e}, Xγmax = {a, b, c, e} and Tγ(X) = {{a, b, c, e}} because {a, b, c, e} represents an admissible complete specification. Then, we have ║(almost every)γ(BOY)(X)║ = m0.5({║(almost every)γ(BOY)({a, b, c, e})║}) = m0.5({1}) = 1. For 0.33 < γ ≤ 1, we have Xγmin = {a, b, c}, Xγmax = {a, b, c, d, e} and Tγ(X) = {{a, b, c}, {a, b, c, e}, {a, b, c, d, e}} because {a, b, c}, {a, b, c, e} and {a, b, c, d, e} all represent admissible complete specifications (Note that {a, b, c, d} represents an inadmissible complete specification and is thus excluded). Then, we have ║(almost every)γ(BOY)(X)║ = m0.5({║(almost every)γ(BOY)({a, b, c})║, ║(almost every)γ(BOY)({a, b, c, e})║, ║(almost every)γ(BOY)({a, b, c, d, e})║}) = m0.5({0, 1}) = 0.5. Finally, by (14), ║(24)║ = ║(26)║

1

=



0

║(almost every)γ(BOY)(X)║dγ

= 1 × (0.33 – 0) + 0.5 × (1 – 0.33) = 0.67 Note that the above calculation has been greatly simplified because in (24), “boy”, “girls” and “met” are all represented by crisp predicates. In general, given a sentence with iterated VQs, we first express it in the following form: Q1(A1)({x: Q2(A2)({y: B(x, y)})})

(27)

Then for each possible x, we determine {y: B(x, y)}, which may be a vague set, and ||Q2(A2)({y: B(x, y)})|| by MGM. By doing so, we will obtain the following set: {x: Q2(A2)({y: B(x, y)})} = {||Q2(A2)({y: B(xi, y)})||/xi, …}, where xi ranges over all possible xs. Finally, we can evaluate ||Q1(A1)({x: Q2(A2)({y: B(x, y)})})|| by MGM.

5

Conclusion

In this paper, I have discussed the merits and demerits of the FT and ST approaches to vagueness and have proposed MGM as a model for VQs. This model inherits certain desirable properties of Glöckner’s framework in [4]. It is also able to distinguish different degrees of vagueness and is thus useful for practical applications. Moreover, this model has overcome a demerit commonly found in FT frameworks, i.e. it yields correct results for penumbral connections. I have also shown that Glöckner’s original method in fact includes a process reminiscent of the precisification process of ST. This provides a plausible way to combine FT and ST, the two main competing theories for vagueness.


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Nevertheless, this paper has just concentrated on one particular QFM represented by M. As a matter of fact, Glöckner and other scholars have proposed other possible QFMs in [1] and [4-5] which I have not had the chance to discuss in this paper. It would be instructive to consider how these QFMs can be modified to suit the requirement of ST and what properties of Glöckner’s original theory are preserved under the modification and would thus be a possible direction for future studies.

References 1. Díaz-Hermida, F., Bugarín, A., Barro, S.: Definition and classification of semi-fuzzy quantifiers for the evaluation of fuzzy quantified sentences. International Journal of Approximate Reasoning 34, 49–88 (2003) 2. Fermüller, C.G., Kosik, R.: Combining supervaluation and degree based reasoning under vagueness. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 212–226. Springer, Heidelberg (2006) 3. Fine, K.: Vagueness, Truth and Logic. Synthese 30, 265–300 (1975) 4. Glöckner, I.: DFS – An Axiomatic Approach to Fuzzy Quantification. Report TR97-06, Technical Faculty, University Bielefeld (1997) 5. Glöckner, I.: Fuzzy Quantifiers: A Computational Theory. Springer, Berlin (2006) 6. Kamp, H.: Two Theories about Adjectives. In: Keenan, E.L. (ed.) Formal Semantics of Natural Language, pp. 123–155. Cambridge University Press, Cambridge (1975) 7. Keefe, R.: Theories of Vagueness. Cambridge University Press, Cambridge (2000) 8. Keenan, E.L.: Some Properties of Natural Language Quantifiers: Generalized Quantifier Theory. Linguistics and Philosophy 25, 627–654 (2002) 9. Keenan, E.L., Westerståhl, D.: Generalized Quantifiers in Linguistics and Logic. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 837–893. Elsevier Science, Amsterdam (1997) 10. Losada, D.E., Díaz-Hermida, F., Bugarin, A.: Semi-fuzzy quantifiers for information retrieval. In: Herrera-Viedma, E., Pasi, G., Crestani, F. (eds.) Soft Computing in Web Information Retrieval: Models and Applications. Springer, Heidelberg (2006) 11. Westerståhl, D.: Quantifiers in Formal and Natural Language. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. IV, pp. 1–131. Reidel Publishing Company, Dordrecht (1989) 12. Yager, R.R.: Reasoning with Fuzzy Quantified Statements: Part I. Kybernetes 14, 233–240 (1985a) 13. Yager, R.R.: Reasoning with Fuzzy Quantified Statements: Part II. Kybernetes 15, 111–120 (1985b) 14. Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338–353 (1965) 15. Zadeh, L.A.: A Computational Approach to Fuzzy Quantifiers in Natural Languages. Computers and Mathematics with Applications 9(1), 149–184 (1983)

An Application of Model Checking Games to Abstract Argumentation Davide Grossi ILLC, University of Amsterdam Amsterdam, The Netherlands [email protected]

Abstract. The paper presents a logical study of abstract argumentation theory. It introduces a second-order modal logic, within which all main known semantics for abstract argumentation can be formalized, and studies the model checking game of this logic. The application of the game to the formalized semantics yields adequate game-theoretic proof procedures for all known extension-based semantics, in both their skeptical and credulous versions.

1

Introduction

Structures A = (A, ), where A is a non-empty set and ⊆ A2 is a binary relation on A, are the building blocks of abstract argumentation theory. Once A is taken to represent a set of arguments, and an ‘attack’ relation between arguments (so that a b means “a attacks b”), the study of these structures—called argumentation frameworks [1]—provides very general insights on how competing arguments interact and how collections of them form ‘tenable’ or ‘justifiable’ positions in an argumentation (cf. [2] for a recent overview). While the study of different formal definitions of the notion of ‘justifiability’— called extensions—constitutes the main body of abstract argumentation theory, many researchers in the last two decades have focused on ‘proof procedures’ for argumentation, i.e., procedures able to adequately establish whether a given argument belongs or not to a given extension. Many of such proof procedures have resorted to abstractions coming from game theory (cf. [3] for a recent overview) and have given rise to a number of different types of games, called dialogue or argument games. The present paper contributes to this line of research by showing that games commonly studied in logic—model checking games (cf. [4])—can be successfully used for this purpose. Outline. The paper builds on the simple idea—first put forth in [5]—that argumentation frameworks are, in fact, the sort of structures that modal logic [6] calls Kripke frames: arguments are modal states, and the accessibility relation is given by the attack relation. The paper formalizes all main Dung-style extension-based semantics in a second-order modal logic (Section 2). This logic is proven to have an adequate model checking game, which is then used to provide adequate games for all known extension-based semantics and, in general, for H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 74–86, 2011. c Springer-Verlag Berlin Heidelberg 2011

An Application of Model Checking Games to Abstract Argumentation

75

Table 1. Basic notions of abstract argumentation theory cA characteristic function of A iff X is acceptable w.r.t. Y in A iff X conflict-free in A iff X admissible set of A iff X complete extension of A iff X stable extension of A iff X grounded extension of A iff X preferred extension of A iff X semi-stable extension of A iff

cA (X) = {a | ∀b : [b a ⇒ ∃c ∈ X : c b]} X ⊆ cA (Y ) ∃a, b ∈ X s.t. a b X is conflict-free and X ⊆ cA (X) X is conflict-free and X = cA (X) X = {a ∈ A | ∃b ∈ X : b a} X is the minimal complete extension of A X is a maximal complete extension of A X is a complete extension and X ∪ {a | ∃b ∈ X : b a} is maximal

all such semantics (Section 3). To the best of our knowledge, no game-theoretic proof procedure for argumentation has ever achieved this level of generality. The paper concludes by briefly comparing our findings with the literature on dialogue games and indicates directions for future research at the interface of logic and argumentation (Section 4). The paper presupposes familiarity with both modal logic and abstract argumentation theory.1

2

Logical Languages for Argumentation Theory

The present section introduces a modal logic view on abstract argumentation theory—first proposed in [5]—and extends it to a rich logical language for the formalization of all known extension-based semantics. 2.1

Argumentation Models

Being Dung argumentation frameworks nothing but Kripke frames, the addition of a labeling (or valuation function) to an argumentation framework yields a Kripke model. Definition 1 (Argumentation models). Let P be a set of propositional atoms. An argumentation model, or labeled argumentation framework, M = (A, V) is a structure such that: A = (A, ) is an argumentation framework; V : P −→ 2A is an assignment from P to subsets of A.2 The class of all argumentation models is A. A pointed argumentation model is a pair (M, a) where M is a model and a an argument. Argumentation models are nothing but argumentation frameworks together with a way of ‘naming’ sets of arguments or, to put it otherwise, of ‘labeling’ arguments.3 So, the fact that an argument a belongs to the set V(p) in a given model 1 2 3

Some of the main argumentation-theoretic notions to which we will refer throughout the paper have been recapitulated in Table 1. Note that, although often assumed in researches on abstract argumentation, we do not impose a finiteness constraint on A. It might be worth noticing that this is a generalization of the sort of labeling functions studied in argumentation theory (cf. [2]).

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D. Grossi b

b

d

c

p

a

p

e

a

A

B

Fig. 1. Examples of labeled argumentation frameworks. Argument a is labeled with proposition p.

M reads in logical notation as (A, V), a |= p. By using the language of propositional logic we can then form ‘complex’ labels ϕ for sets of arguments stating, for instance, that “a belongs to both the sets called p and q”: (A, V), a |= p ∧ q. In order to formalize argumentation-theoretic statements one more linguistic ingredient is needed. Let us mention a couple of examples: “there exists an argument in a set named ϕ attacking argument a” or “for all attackers of argument a there exist some attackers in a set named ϕ”. These are statements involving a bounded quantification and they can be naturally formalized by a modal operator ♦ whose reading is: “there exists an attacking argument such that . . . ”. This takes us to modal languages. 2.2

Argumentation in Modal Logic

We turn to logic KU , an extension of the minimal modal logic K with the universal modality.4 As shown in [5], this logic, despite its simplicity, allows one to express a number of key notions of abstract argumentation theory. Syntax and semantics. Language LU is a standard modal language with two modalities: ♦ and U (the universal modality). It is built on the set of atoms P by the following BNF: LU (P) : ϕ ::= p | ⊥ | ¬ϕ | ϕ ∧ ϕ | ♦ϕ | Uϕ where p ranges over P. Standard definitions for the remaining Boolean operators and the duals and [U] are assumed. Definition 2 (Satisfaction for LU ). Let ϕ ∈ LU . The satisfaction of ϕ by a pointed argumentation model (M, a) is inductively defined as follows: M, a |= ♦ϕ ⇐⇒ ∃b ∈ A : a b and M, b |= ϕ 4

This logic is well-studied and well-behaved: it has a simple strongly complete axiomatics, a P-complete model checking problem and an EXPTIME-complete satisfiability problem (cf. [6, Ch. 7]).


77

M, a |= Uϕ ⇐⇒ ∃b ∈ A : M, b |= ϕ Boolean clauses are omitted. As usual, ϕ is valid in an argumentation model M iff it is satisfied in all pointed models of M, i.e., M |= ϕ. The truth-set of ϕ is denoted |ϕ|M . The set of LU -formulae which are true in the class A of all argumentation models is called (logic) KU . These are fully standard clauses for modal logic semantics, but let us see what their intuitive reading is in argumentation-theoretic terms. The first clause states that argument a belongs to the set called ♦ϕ iff some argument b is reachable via the inverse −1 of the attack relation and b belongs to ϕ or, more simply, iff a is attacked by some argument in ϕ. The second clause states that argument a belongs to the set called Uϕ iff there exists some argument b in ϕ, in other words, iff the set called ϕ is non-empty. So, to put it shortly, logic KU is endowed with modal operators of the type “there exists an argument attacking the current one such that . . . ”, i.e. ♦, and “there exists an argument such that . . . ”, i.e. U, together with their duals. Example 1. Consider the labeled frameworks A and B in Figure 1. Here are a few illustrative modal statements: MA , a |= ¬⊥ MA , a |= ♦p

MB , a |= U(♦♦p ∧ ♦♦♦p) MB , a |= U(p ∨ ♦p).

The two on the left state that argument a in framework A is not ‘unattacked’ and, respectively, that all its attackers are attacked by some argument in p (in this case a itself). The one at the top right corner states that in framework B there exists an argument (namely d) which has both a chain of two and three attackers ending in p. Finally, the one at the bottom right corner states that there exists an argument (namely c) such that all its attackers are either in p or are attacked by some argument in p. Argumentation theory in KU . Logic KU can express many key argumentationtheoretic notions such as: that a given set p is acceptable w.r.t. another set q, that it is conflict-free, that it is admissible, that it is a complete extension, and that it is a stable extension: Acc(p, q) := [U](p → ♦q)

(1)

CFr (p) := [U](p → ¬p) Adm(p) := CFr (p) ∧ Acc(p, p)

(2) (3)

Cmp(p) := CFr (p) ∧ [U](p ↔ ♦p))

(4)

Stb(p) := [U](p ↔ ¬p)

(5)

The adequacy of these definitions with respect to the ones in Table 1 is easily checked. Let us rather give an intuitive reading of some of these formulas. First, a set of arguments p is acceptable with respect to the set of arguments q

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if and only if all p-arguments are such that for all their attackers there exists a defender in q (Formula 1). A set of arguments p is conflict free if and only if all p-arguments are such that none of their attackers is in p (Formula 2). A set p is a complete extension if and only if it is conflict free and it is equivalent to the set of arguments all the attackers of which are attacked by some p-argument (Formula 4). Finally, a set p is a stable extension if and only if it is equivalent to the set of arguments whose attackers are not in p (Formula 5). Remark 1. It is worth noticing that only two modal patterns occur in Formulae 1-5: ¬ (“all attackers are not such that . . . ”) and ♦ (“all attackers are attacked by at least an argument such that . . . ”). By the semantics in Definition 2, both patterns specify functions from sets of arguments to sets of arguments. The first one yields, given ϕ, the set ¬ϕ of arguments that are not attacked by any argument in ϕ. The second one was shown in [5] to be the modal logic formulation of what in argumentation theory is called the characteristic function of a given framework (see Table 1). It yields, given ϕ, the set ♦ϕ of arguments defended by ϕ. 2.3

Argumentation in Second-Order Modal Logic

A quick inspection of Table 1 suffices to show that KU is not expressive enough to capture notions such as grounded, preferred, semi-stable extensions, or skeptical and credulous membership to extensions.5 In this section we add monadic second order quantification to KU . In doing this we are clearly not aiming at finding the right expressive power needed to express each of the notions in Table 1 in isolation, but we are rather fixing a language that can accommodate all at once and which, as we will see in Section 3, can still provide us with the sort of games we are after. Syntax and semantics. We expand language LU (P) by allowing quantification over monadic predicates (i.e., propositional variables). The resulting secondorder language is built by the following BNF: L2U (P) : ϕ ::= p | ⊥ | ¬ϕ | ϕ ∧ ϕ | ♦ϕ | Uϕ | ∃p.ψ(p) where by ψ(p) we indicate that p occurs free in ψ. In what follows we will make extensive use of the auxiliary symbol denoting inclusion between truth-sets: ϕ ψ := [U](ϕ → ψ). Strict inclusion is defined in the obvious way. 5

It is worth recalling that, given an extension type E (e.g., stable, grounded, etc.) and a framework A, an argument a is credulously (resp., skeptically) included in E in A iff there exists one extension (resp., for all extensions) of type E for A s.t. a is included in that extension (resp., those extensions).


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Definition 3 (Satisfaction for L2U ). Let ϕ ∈ L2U . The satisfaction of ϕ by a pointed argumentation model (M, a) is defined as follows (the clauses for LU formulae are omitted): M, a |= ∃p.ϕ(p) ⇐⇒ ∃X ⊆ A : Mp:=X , a |= ϕ(p) where Mp:=X denotes model M where V(p) is set to be X. The set of all L2U formulae which are satisfied by all argumentation model is called (logic) K2U . Intuitively, we are simply adding to KU the necessary expressivity to talk about properties involving quantification over sets of arguments. As the following remark shows in some detail, this takes us to the binary fragment of monadic second-order logic. Remark 2 (K2U , SOPML and MSO). Logic K2U is an extension of SOPML (secondorder propositional modal logic, [7]) with the universal modality. It coincides with the (one free variable) binary fragment of MSO (monadic second-order logic). To appreciate this, note that by a simple extension of the standard translation [6, Ch. 2.4] K2U can be easily proven to be a fragment of MSO. The translation in the other direction is less obvious, and is briefly sketched here. First let us recall the (constant-free) language L of the binary fragment of MSO: L : ϕ ::= x = y | Rxy | P x | ¬ϕ | ϕ ∧ ϕ | ∃x.ϕ | ∃P.ϕ The key idea of the translation is to treat first-order variables x, y as propositional variables which are interpreted as singletons. Now, that a propositional variable x is a singleton can be expressed in K2U by the formula: sing(x) := ∀q. (q x → (∀r.(q r) ∨ x q)). So, the desired translation is recursively defined as follows (Boolean cases omitted): tr(x = y) = x y ∧ sing(x) ∧ sing(y) tr(Rxy) = x ♦y ∧ sing(y) ∧ sing(x) tr(P x) = x p ∧ sing(x) tr(∃x.ϕ) = Ux ∧ sing(x) ∧ tr(ϕ) tr(∃P.ϕ) = ∃p.tr(ϕ) To give an example, here is a translation of the formula ∃P.∃x.(Rxy ∧ P (y)): tr(∃P.∃x.(Rxy ∧ P (y))) = ∃p.Ux ∧ sing(x) ∧ x ♦y ∧ sing(y) ∧ y p. It is then instructive to note that, as K2U coincides with (the binary fragment of) MSO, it includes the μ-calculus, which is known to be the bisimulation-invariant fragment of MSO.6 6

For a good presentation of the μ-calculus, the interested reader is referred to [8]. The μ-calculus will be of relevance later also in Remarks 4 and 5.

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Extensions formalized. We are now in the position to complete the formalization of the whole of Table 1 covering the cases of grounded, preferred and semi-stable extensions. Grn(p) := Cmp(p) ∧ ∀q.(Cmp(q) → p q)

(6)

Prf (p) := Cmp(p) ∧ ¬∃q.(Cmp(q) ∧ p q) SStb(p) := Cmp(p) ∧ ¬∃q.((p ∨ ♦p) (q ∨ ♦q))

(7) (8)

The adequacy of the above definitions w.r.t. Table 1 is easily checked, as the formalization is ‘literally’ following the wording of the table. Intuitively, an atom p is said to be: a grounded extension iff it is the smallest complete extension (Formula 6); a preferred extension iff it is a maximal complete extension (Formula 7); a semi-stable extension iff it is a complete extension which maximizes its union with the set of arguments it attacks, i.e., its range (Formula 8). Remark 3. One aspect worth noticing in the above formalization concerns the order theory of complete extensions. Formulae 6 and 7 express a minimality and, respectively, a maximality requirement. However, while Formula 6 expresses the minimality of p by stating that all other possible complete extensions lie above p, Formula 7 expresses the maximality of p by stating that no other complete extension exists which lies above p, rather than saying that all other complete extensions lie below p. This has to do with the fact that the set of complete extensions always form a pre-order with one minimal element—the grounded extension— and with a set of maximal elements—the preferred extensions—which is not necessarily a singleton (cf. [1]). Remark 4. As observed in [5], the grounded extension can be expressed by a simple formula of the μ-calculus: μp.♦p. This formula denotes the smallest fixpoint of the characteristic function, which, modally, corresponds to the modal pattern ♦ (Remark 1).7 The K2U formula Grn(p) (Formula 6) expresses something slightly different, namely that a given atom p denotes the grounded extension. The relation between the two is, formally, that for any argumentation model M: |q|M = |μp.♦p|M iff M |= Grn(q). In addition, K2U can express the two argumentation-theoretic variants of membership with respect to a given semantics: the skeptical and, respectively, the credulous membership. Here below we provide examples for the skeptically and credulously stable membership, and for the skeptically and credulously preferred membership.

7

SkStb := ∀p.Stb(p) → p CrStb := ∃p.Stb(p) ∧ p

(9) (10)

SkPrf := ∀p.Prf (p) → p CrPrf := ∃p.Prf (p) ∧ p

(11) (12)

Recall that the smallest fixpoint of the characteristic function coincides with the minimal complete extension [1].


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So, in general, for any extension E the skeptical membership w.r.t. E requires the argument at the point of evaluation to be included in some set which is an E-extension. Similarly, the credulous membership w.r.t. E requires that if some set is an E-extension, then it includes the argument at the point of evaluation in the argumentation model. It is finally worth stressing that other notions (e.g., ideal [9] extensions) are also formalizable in K2U since they are expressible in MSO (Remark 2).

3

Model-Checking Games for Argumentation

Model-checking games (sometimes also called evaluation games or Hintikka games) are two-player adversarial procedures for checking whether a formula from a given logical language is satisfied in a given structure (e.g., a pointed model). The proponent or verifier (∃ve) tries to prove that the formula holds at the given pointed model, while the opponent or falsifier (∀dam) tries to disprove it. The idea behind this section is very simple. As K2U allows one to formalize all key semantic notions of argumentation theory, a model checking game that can be proven adequate for K2U would automatically provide adequate game-theoretic proof procedures for all those notions. So we turn now to the presentation of the K2U -model checking game. 3.1

The Model Checking Game for K2U

Definition 4 (Model-checking game for K2U ). Let ϕ ∈ L2U (P) be in positive normal form,8 A be an argumentation framework and V a valuation. The K2U model checking game of ϕ on A is a tuple EU2 (ϕ, A) = (N, S, turn, move, win) where: – N := {∃, ∀}. An element of N , a player, is denoted P . – S := L2U × A × (2A )P , that is, the set of positions consists of the set of triples formula-argument-labeling.9 Sequences of elements of S are denoted s. – turn : S −→ N assigns players to positions as Table 2. – move : S −→ 2S assigns to each position a set of accessible positions (moves) as in Table 2. The set Play(EU2 (ϕ, A)) denotes the set of all finite sequences s (plays) which are consistent with function move. – win : Play (EU2 (ϕ, A)) −→ N is a partial function assigning winners to plays as in Table 3. A game EU2 (ϕ, A) is instantiated by pairing it with an initial position (ϕ, a, V) s.t. a ∈ A and V : P −→ 2A , in symbols: EU2 (ϕ, A)@(ϕ, a, V). 8 9

It must be clear that any formula of L2U can be easily translated into an equivalent one in positive normal form. Equivalently, positions could be pairs formula-model.

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D. Grossi Table 2. Rules of the model checking game for K2U Position Turn Moves (ϕ1 ∨ ϕ2 , a, V) ∃ {(ϕ1 , a, V), (ϕ2 , a, V)} (ϕ1 ∧ ϕ2 , a, V) ∀ {(ϕ1 , a, V), (ϕ2 , a, V)} (♦ϕ, a, V) ∃ {(ϕ, b, V) | (a, b) ∈} (ϕ, a, V) ∀ {(ϕ, b, V) | (a, b) ∈} (U ϕ, a, V) ∃ {(ϕ, b, V) | b ∈ A} ([U]ϕ, a, V) ∀ {(ϕ, b, V) | b ∈ A} (∃p.ϕ(p), a, V) ∃ {(ϕ, a, Vp:=X ) | X ⊆ A} (∀p.ϕ(p), a, V) ∀ {(ϕ, a, Vp:=X ) | X ⊆ A}

Position Turn Moves (⊥, a, V) ∃ ∅ (, a, V) ∀ ∅ (p, a, V) and a ∈ V(p) ∃ ∅ (p, a, V) and a ∈ V(p) ∀ ∅ (¬p, a, V) and a ∈ V(p) ∃ ∅ (¬p, a, V) and a ∈ V(p) ∀ ∅

So a model checking game is a finite two-player zero-sum extensive game with perfect information. The following features of the game are worth stressing. Positions consist of a formula, an argument, and a valuation. Each move reduces the syntactic complexity of the formula—hence the game is finite—and, apart from the Boolean positions, moves happen by either selecting another argument (possibly along the attack relation), or by picking an interpretation for a given atom. Finally, it should be noted that turn-taking, as well as the sort of moves available at a given position, are fully determined by the formula in that position. In other words, it is the logical structure of the to-be-checked formula which, together with the model at hands, dictates the rules of the game. The definition of winning strategy and position follows. Definition 5 (Winning strategies and positions). A strategy for player P in EU2 (ϕ, A)@(ϕ, a, V) is a function telling P what to do in any play from position (ϕ, a, V). Such a strategy is winning for P if and only if, for all plays in accordance with the strategy, P wins. A position (ϕ, a, V) in EU2 (ϕ, A) is winning for P if and only if P has a winning strategy in EU2 (ϕ, A)@(ϕ, a, V). The set of winning positions for P in EU2 (ϕ, A) is denoted Win P (EU2 (ϕ, A)). This brings us to the technical underpinnings of our contribution, consisting of the following simple theorem. Theorem 1 (Adequacy of the K2U -model checking game). Let ϕ ∈ L2U (P), and let A be an argumentation framework. Then, for all a ∈ A, and valuations V: (ϕ, a, V) ∈ Win ∃ (EU2 (ϕ, A)) ⇐⇒ (A, V), a |= ϕ. Table 3. Winning conditions for the K2U -model checking game Matches ∃ve wins ∀dam wins s ∈ Play(EU2 (ϕ, M)) turn(s) = ∀ and move(s) = ∅ turn(s) = ∃ and move(s) = ∅


83

Proof (Sketch of proof ). The proof is by induction on the syntax of ϕ. The base as well as the Boolean and modal cases in the induction step are taken care of by the adequacy of the model checking game for KU [5]. Here we focus on the two remaining cases. [ϕ = ∃p.ψ(p)] Left to right. Assume (∃p.ψ(p), a, V) ∈ Win ∃ (EU2 (∃p.ψ(p), A)). As, according to Definition 4 it is ∃ve’s turn to play, by Definition 5 it follows that there exists X ⊆ A s.t. (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)), from which, by IH, we conclude that (A, Vp:=X ), a |= ψ and, by Definition 3, that (A, V), a |= ∃p.ψ. Right to left. Assume (A, V), a |= ∃p.ψ. This means, by Definition 3, that there exists X ⊆ A s.t. (A, Vp:=X ), a |= ψ. By IH it follows that (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)) for some X, and from the fact that it is ∃ve’s turn to play, by Definition 5 we can thus conclude that (∃p.ψ(p), a, V) ∈ Win ∃ (EU2 (∃p.ψ(p), A)). [ϕ = ∀p.ψ(p)] Left to right. Assume (∀p.ψ(p), a, V) ∈ Win ∃ (EU2 (∀p.ψ(p), A)). As, according to Definition 4 it is ∀dam’s turn to play, by Definition 5 it follows that for all X ⊆ A we have that (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)). From this, by IH we obtain that for all X ⊆ A (A, Vp:=X ), a |= ψ and by Definition 3 we can conclude that (A, V), a |= ∀p.ψ. Right to left. Assume (A, V), a |= ∀p.ψ. This means, by Definition 3, that for all X ⊆ A s.t. (A, Vp:=X ), a |= ψ. By IH it follows that (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)) for all X, and from the fact that it is ∀dam’s turn to play, by Definition 5 we can conclude that (∀p.ψ(p), a, V) ∈ Win ∃ (EU2 (∀p.ψ(p), A)). The K2U -model checking game in argumentation. Theorem 1 bears immediate relevance for argumentation. It states that for any ϕ ∈ L2U and any framework A, the K2U -model checking game for ϕ on A is adequate. But, as shown in Section 2.3, L2U can express all known argumentation semantics (e.g., Formulae 9-12) and, more generally, any MSO-definable semantics. It follows that Theorem 1 actually guarantees that all extension-based semantics get an adequate game, namely their K2U -model checking game. We discuss now an example of definite relevance for argumentation. Example 2 (The game for skeptical preferred). For a given framework A, the game for skeptically preferred extensions is EU2 (SkPrf , A). Take A to be the argumentation framework B in Figure 1, and let the game be played from argument d, i.e., be initialized at position (SkPrf , d, V) with V arbitrary. Part of the resulting game tree is depicted in Figure 2. By Theorem 1 ∃ve has a winning strategy in this game, as {d} is skeptically preferred since it belongs to both the preferred extensions of the framework, namely: {a, d} and {b, d}. In fact, a visual inspection of the tree in Figure 2 reveals that ∃ve can force the game to end up in a position where she wins independently of ∀dam’s choices. It is now important to stress the natural argumentation-theoretic reading of these games. To do so, consider again the game in Figure 2 and the play indicated by ‘∗’. In this branch ∃ve plays wrongly—not in accordance to her winning strategy—and ∀dam wins. The game starts with ∃ve claiming that d is skeptically preferred. ∀dam challenges the claim, and to do so, he selects a valuation for p (in this case the set {e}). Now it is up to ∃ve to show that if {e} is a preferred

84

D. Grossi (∀p.Prf (p) → p, V, d)

(Prf (p) → p, Vp:={a,d} , d) (p, Vp:={a,d} , d)

∀

(Prf (p) → p, Vp:={e} , d)

∃

(¬Prf (p), Vp:={e} , d)

∃ (p, Vp:={e} , d)

∃

∀ wins

∃ wins (∃q.(Cmp(q) ∧ p q), Vp:={e} , d)

∃

(¬Cmp(p), Vp:={e} , d)

(U((p ∧ ♦¬p) ∨ (¬p ∧ ♦p)), Vp:={e} , d)

((p ∧ ♦¬p) ∨ (¬p ∧ ♦p), Vp:={e} , e) (p ∧ ♦¬p, Vp:={e} , e) (p, Vp:={e} , e)

∃ wins

(♦¬p, Vp:={e} , e)

∀ ∃

(¬p, Vp:={e} , e)

∃

∃

∃

(¬CFr (p), Vp:={e} , d)

(p ∧ ♦p, Vp:={e} , e)

∃ ∀

(♦p, Vp:={e} , e) ∃

∗ (p, V

p:={e} , d)

∀ wins

∃ wins

Fig. 2. Partial game tree of the KU -model checking game for CrPrf run at d in framework B of Figure 1. Turn-taking is given through the ∃ and ∀ labels.

extension, then d belongs to it. Rightly, she chooses to show that {e} is not a preferred extension, rather then trying to show that d belongs to {e}, which would lead her to sure defeat. So she goes on proving that {e} is not even a complete extension. But here she makes a mistake, and she chooses to prove that {e} is not conflict-free by claiming that e belongs to p and is attacked by some argument in p. ∀dam capitalizes on this mistake and challenges her to show which argument belonging to p attacks e. To this ∃ve cannot reply and loses. Remark 5 (Alternative games for grounded). As observed in Remark 4, the fact that an argument belongs to the grounded extension can be expressed with either a K2U -formula, i.e., Cmp(p) ∧ ∀q.(Cmp(q) → p q)), or with a μ-formula, i.e., μp.♦p. Now, also the μ-calculus has an adequate model checking game (cf. [4] or [8]). It turns so out that the grounded extension has two adequate logicbased proof procedures. Interestingly, these model checking games are radically different, and to appreciate this consider the 2-cycle depicted in part A of Figure 1, whose grounded extension is ∅. The μ-calculus game run on that framework at argument a generates an infinite game consisting of one single play won by ∀dam. On the contrary, the K2U model-checking game generates a finite game whose tree has maximum branching factor equal to |2A | = 4 (i.e., the cardinality of the set of all possible valuations of variable q).


4

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Discussion and Conclusions

4.1

Model-Checking Games vs. Dialogue Games

The choice of Example 2 to illustrate our games has not been casual, as a game theoretic proof procedure for skeptical preferred is still, to the best of our knowledge, an open problem in the literature (cf. [3]). In general, while adequate games for some extensions are known (typically grounded and credulous preferred), many are still lacking (e.g., skeptical preferred, skeptical stable, semi-stable, etc.) even in highly systematized settings such as [10]. The K2U -model checking game provides adequate games for all of them. What are the key differences between our logic-based approach, and the dialogue games studied in the argumentation theory literature, (which we will call here dialogue games)? A systematic comparison falls out of the scope of the present paper, however, there are key differences worth stressing here. First, the structure of the games is, at least at first sight, very different. Positions in dialogue games consist of arguments alone and the game proceeds along the (inverse of) the attack relation (cf. [3]). In our games, positions consist of arguments plus a formula and a valuation. This allows the game to host much more structure as players can ‘move’ in the game not only by navigating along the attack relation but also by selecting sets of arguments.10 Second, while in our games the rules of play follow directly from the logical structure of the formula formalizing the extension at hands, in dialogue games the rules are exogenously determined. This might be related to the fact that adequate dialogue games seem harder to find for some semantics, while in our case we could easily obtain, via logic, a general adequacy theorem. 4.2

Future Work

The results presented, we hope, show the feasibility and usefulness of the importation of methods and techniques from logic to argumentation. At the same time, they open the door to research questions at the interface of the two disciplines. Let us mention two of them. First, grounded extensions have two types of adequate model checking games (Remark 5), and one adequate dialogue game (with some variants). The question is whether any significant structural relations exist between these different types of games that could provide us with some form of inter-definability between them and, in particular, between dialogue and model checking games. This would provide a first systematic comparison of the two types of games at which we hinted above. Another way to systematically approach such comparison could be by analyzing the complexity of each model checking game with respect to its dialogue game counterparts. Second, the paper has used a very expressive logic (essentially, the binary fragment of MSO) to provide a common language for the formalization of all 10

In the literature on dialogue games, this latter feature can be found only in the type of games studied in [10].

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main Dung-style skeptical and credulous semantics. The question now arises of the extent to which weaker languages can deal with, if not all, at least some of those semantics. For instance, in the paper we have constantly referred to the μ-calculus (Remarks 2, 3 and 5). However, this logic turns out to be unable to express even credulous conflict-freeness, ∃p(CFr (p) ∧ p), which is clearly not invariant under bisimulation (consider a reflexive point vs. its unraveling). In short, what extensions of the μ-calculus are needed to cope with Formulae 9-12? Answering this question would provide an accurate picture of the sort of logical languages needed to handle significant fragments of abstract argumentation theory, without explicitly resorting to second-order quantification. Acknowledgments. We are grateful to the anonymous reviewers of LORI3 for their helpful comments. This work has been supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek under the NWO VENI grant 639.021.816.

References 1. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77(2), 321–358 (1995) 2. Baroni, P., Giacomin, M.: Semantics of abstract argument systems. In: Rahwan, I., Simari, G.R. (eds.) Argumentation in Artifical Intelligence. Springer, Heidelberg (2009) 3. Modgil, S., Caminada, M.: Proof theories and algorithms for abstract argumentation frameworks. In: Rahwan, I., Simari, G. (eds.) Argumentation in AI, pp. 105–132. Springer, Heidelberg (2009) 4. Gr¨ adel, E.: Model checking games. In: de Queiroz, R., Pereira, L.C., Haeusler, E.H. (eds.) Proceedings of WOLLIC 2002. Electronic Notes in Theoretical Computer Science, vol. 67, pp. 15–34. Elsevier, Amsterdam (2002) 5. Grossi, D.: On the logic of argumentation theory. In: van der Hoek, W., Kaminka, G., Lespérance, Y., Sen, S. (eds.) Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2010), IFAAMAS, pp. 409–416 (2010) 6. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) 7. Fine, K.: Propositional quantifiers in modal logic. Theoria 36, 336–346 (1970) 8. Venema, Y.: Lectures on the modal μ-calculus. Renmin University in Beijing, China (2008) 9. Dung, P.M., Mancarella, P., Toni, F.: A dialectic procedure for sceptical assumption-based argumentation. In: Proceedings of the 1st International Conference on Computational Models of Argument (COMMA 2006), pp. 145–156. IOS Press, Amsterdam (2006) 10. Dung, P.M., Thang, P.M.: A unified framework for representation and development of dialectical proof procedures in argumentation. In: Proceedings of the TwentyFirst International Joint Conference on Artificial Intelligence (IJCAI-2009), pp. 746–751 (2009)

Schematic Validity in Dynamic Epistemic Logic: Decidability Wesley H. Holliday1,2 , Tomohiro Hoshi1,2,3 , and Thomas F. Icard, III1,2 1

Logical Dynamics Lab, Center for the Study of Language and Information, Stanford, CA, USA 2 Department of Philosophy, Stanford University 3 Educational Program for Gifted Youth, Stanford University {wesholliday,thoshi,icard}@stanford.edu

Abstract. Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. The classic example is Public Announcement Logic (PAL), an extension of epistemic logic based on the idea of information acquisition as elimination of possibilities. In this paper, we address the open question of whether the set of schematic validities of PAL, the set of formulas all of whose substitution instances are valid, is decidable. We obtain positive answers for multi-agent PAL, as well as its extension with relativized common knowledge, PAL-RC. The conceptual significance of substitution failure is also discussed. Keywords: modal logic, dynamic epistemic logic, Public Announcement Logic, schematic validity, substitution core, decidability.

1

Introduction

The schematic validities of a logic are those formulas all of whose substitution instances are valid [3]. Typically the set of schematic validities of a logic, its substitution core, coincides with the set of validities, in which case the logic is closed under uniform substitution. However, many dynamic epistemic logics axiomatized using reduction axioms [8,1,4,16] are not substitution-closed.1 The classic example is Public Announcement Logic (PAL) [17,10]. In this paper, we consider the schematic validity problem for PAL and its extension PAL-RC with relativized common knowledge [4]. We answer positively the open question [3,2,4] of whether the substitution cores of multi-agent PAL and PAL-RC are decidable. The conceptual significance of substitution failure is also discussed. 1

Dynamic epistemic logics are not the only modal logics to have been proposed that are not closed under substitution. Other examples include the modal logic of “pure provability” [6], ˚ Aqvist’s two-dimensional modal logic as discussed by Segerberg [18], and an epistemic-doxastic logic proposed by Halpern [11]. For each of these logics there is an axiomatization in which non-schematically valid axioms appear.


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1.1

Review of Public Announcement Logic

Let us briefly recall the details of PAL. The language LPAL is defined as follows, for a countable set At of atomic sentences and a finite set Agt of agent symbols: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Ki ϕ | ϕϕ, where p ∈ At and i ∈ Agt. We denote the set of atoms in ϕ by At(ϕ) and define [ϕ]ψ as ¬ϕ¬ψ. As in epistemic logic, we take Ki ϕ to mean that agent i knows or has the information ϕ. For the “announcement” operator, we take ϕψ to mean that ψ is the case after all agents publicly receive the true information ϕ. We interpret LPAL using standard relational structures of the form M = M, {∼i | i ∈ Agt}, V , where each ∼i is an equivalence relation on M . We use the notation ∼i (w) = {v ∈ W | w ∼i v} to denote the set of possibilities consistent with the knowledge or information of agent i in world w. Each Ki is the universal modality for the associated ∼i relation, and each ϕ is a dynamic modality corresponding to a model-relativization, with the following truth definitions: M, w Ki ϕ iff ∀v ∈ W : if w ∼i v then M, v ϕ; M, w ϕψ iff M, w ϕ and M|ϕ , w ψ, where M|ϕ = M|ϕ , {∼i|ϕ | i ∈ Agt}, V|ϕ is the model obtained by eliminating from M all worlds in which ϕ was false, i.e., M|ϕ = {v ∈ M | M, v ϕ}, each relation ∼i|ϕ is the restriction of ∼i to M|ϕ , and V|ϕ (p) = V (p) ∩ M|ϕ for all p ∈ At. We denote the extension of ϕ in M by ϕM = {v ∈ M | M, v ϕ}. In essence, the semantics of PAL is based on the intuitive idea of information acquisition as elimination of possibilities, as illustrated by Example 2 below. An axiomatization of PAL is given by the S5 axioms for each Ki modality, the rule of replacement of logical equivalents (from α ↔ β, derive ϕ(α/p) ↔ ϕ(β/p)), and the following reduction axioms [17]: (i) (ii) (iii) (iv)

ϕp ↔ (ϕ ∧ p); ϕ¬ψ ↔ (ϕ ∧ ¬ϕψ); ϕ(ψ ∧ χ) ↔ (ϕψ ∧ ϕχ); ϕKi ψ ↔ (ϕ ∧ Ki (ϕ → ϕψ)).

Using (i) - (iv) and replacement, any LPAL formula can be reduced to an equivalent formula in the basic modal language. Completeness and decidability for PAL are therefore corollaries of completeness and decidability for multi-agent S5. The language of PAL-RC [4], LPAL-RC , extends LPAL with relativized common knowledge operators C ϕ ψ with the truth definition: M, w C ϕ ψ iff every path from w through ϕM along any ∼i relations ends in ψM . The standard notion of common knowledge, that everyone knows ψ, and everyone knows that everyone knows that ψ, etc., is defined as Cψ := C ψ. Using the reduction axiom (v) ϕC ψ χ ↔ (ϕ ∧ C ϕψ ϕχ), every LPAL-RC formula can be reduced to an equivalent formula without dynamic operators. Therefore, an axiomatization for PAL-RC may be obtained from (i) (v) plus an axiomatization for multi-agent S5 with relativized common knowledge [4]. Since the latter system is decidable, so is PAL-RC by the reduction.

Schematic Validity in Dynamic Epistemic Logic: Decidability

1.2

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Conceptual Significance of Substitution Failure

Reduction axiom (i) reflects an important assumption of PAL: the truth values of atomic sentences p, q, r, . . . are taken to be unaffected by informational events. It is implicitly assumed that no atomic sentence is about the epistemic or informational states of agents. Hence an atomic sentence in PAL is not a propositional variable in the ordinary sense of something that stands in for any proposition. For there is an implicit restriction on the atomic sentence’s subject matter. Purists may protest that the atomic sentences of a real logic are supposed to be “topic-neutral.” Our reply is practical: it is useful for certain applications to use the atomic p, q, r, . . . to describe stable states of the external world, unaffected by informational events, while using modal formulas to describe the changeable states of agents’ knowledge or information. As we show in other work [14], it is possible to develop a variant of PAL, which we call Uniform Public Announcement Logic (UPAL), in which atomic sentences are treated as genuine propositional variables. Which way one goes is a modeling choice. Given the special treatment of atomic sentences in PAL, it is perhaps unsurprising that uniform substitution should fail. For example, the substitution instance pKi p ↔ (p∧Ki p) of reduction axiom (i) is not valid. Since we take Ki p to mean that agent i knows or has the information p, if pKi p ↔ (p ∧ Ki p) were valid, it would mean that an agent could learn p only if the agent already knew p. Since PAL is designed to reason about information change, the non-schematic validity of reduction axiom (i) is a feature of the system, not a bug. Although substitution failures are to be expected in PAL, the specific failures illuminate subtleties of information change. Example 1 provides the classic example. Example 2 shows that some substitution failures are not at all obvious. Example 1 (Moore Sentence). The formula [p]Ki p is valid, for when agent i acquires the information p, agent i comes to know p. Yet this formula is not schematically valid, and neither is the valid formula [p]p. Simply substitute the famous Moore sentence p ∧ ¬Ki p for p. The non-schematic validity of [p]p is the well-known issue of “unsuccessful formulas” [9,8,15], which is also at the heart of the Muddy Children puzzle [9, §4]. In these cases, the failure of schematic validity for a valid PAL principle shows that the principle does not hold for all types of information—in particular, for information about agents’ own information. Not only is the substitution instance [p∧¬Ki p](p∧¬Ki p) of [p]p invalid, but also [p ∧ ¬Ki p]¬(p ∧ ¬Ki p) is valid. Is the latter also schematically valid? Informally, is there a ϕ such that if you receive the true information that “ϕ but you don’t know ϕ,” it can remain true afterward that ϕ but you don’t know ϕ? As Hintikka [12] remarks about sentences of the Moorean form, “If you know that I am well informed and if I address the words . . . to you, these words have a curious effect which may perhaps be called anti-performatory. You may come to know that what I say was true, but saying it in so many words has the effect of making what is being said false” (p. 68f). Surprisingly, this is not always so. Example 2 (Puzzle of the Gifts [13]). Holding her hands behind her back, agent i walks into a room where a friend j is sitting. Agent j did not see what if anything

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i put in her hands, and i knows this. In fact, i has gifts for j in both hands. Instead of the usual game of asking j to “pick a hand, any hand,” i (deviously but) truthfully announces: (G) Either I have a gift in my right hand and you don’t know that, or I have gifts in both hands and you don’t know I have a gift in my left hand. Let us suppose that j knows i to be an infallible source of information on such matters, so j accepts G. Question 1: After i’s announcement, does j know whether i has a gift in her left/right/both hand(s)? Question 2: After i’s announcement, is G true? Question 3: After i’s announcement, does j know G? Finally, Question 4: If ‘yes’ to Q2, then what happens if i announces G again? Let l stand for ‘a gift is in i’s left hand’ and r stand for ‘a gift is in i’s right hand’. Before i’s announcement, j has not eliminated any of the four possibilities represented by the model M in Fig. 1. (Reflexive arrows are not displayed.) w1 l, r

M

w2

w1

r

l, r

w1

w2 r

M|G|G

l, r

l w3

M|G

w4

Fig. 1. models for the Puzzle of the Gifts

We can translate G into our language as (G) (r ∧ ¬Kj r) ∨ (l ∧ r ∧ ¬Kj l). Clearly GM = {w1 , w2 }. Hence after i’s announcement of G, j can eliminate possibilities w3 and w4 , reducing j’s uncertainty to that represented by the model M|G in Fig. 1. Inspection of M|G shows that the answer to Question 1 is that M|G , w1 Kj r ∧ ¬(Kj l ∨ Kj ¬l). Observe that GM|G = {w1 }, which answers Questions 2 (‘yes’) and 3 (‘no’). It follows that the principle ϕϕ → ϕKj ϕ is not schematically valid. One can fail to come to know what is (true and remains true after being) announced by a source whom one knows to be infallible! Suppose that instead of initially announcing G, i announces (H) G ∧ ¬Kj G.2 2

“The following is true but you don’t know it: either I have a gift in my right hand and you don’t know that, or I have gifts in both hands and you don’t know I have a gift in my left hand.”


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Given GM = {w1 , w2 } above, clearly HM = {w1 , w2 }. It follows that M|G = M|H , so given GM|G = {w1 } above, clearly HM|H = {w1 }. It follows that M, w1 HH. Hence [p ∧ ¬Kp]¬(p ∧ ¬Kp) is valid but not schematically valid. Announcements of Moore sentences are not always self-refuting! We leave the answer to Question 4 to the reader (see M|G|G in Fig. 1). There are many other examples of valid but not schematically valid PAL principles. Noteworthy instances include Ki (p → q) → (qKi r → pKi r) and (pKi r ∧ qKi r) → p ∨ qKi r. Example 2 shows that discovering that there is an invalid substitution instance of a valid PAL formula can be a non-trivial task. A natural question is whether we can give an effective procedure to make such discoveries. The rest of the paper addresses this technical question. 1.3

The Problem of the Substitution Core

Let us now state precisely the problem to be solved. For a language L whose set of atomic sentences is At, a substitution is any function σ : At → L, and σ ˆ : L → L is the extension such that σ ˆ (ϕ) is obtained from ϕ by replacing each p ∈ At(ϕ) by σ(p). Abusing notation, we write σ(ϕ) for σ ˆ (ϕ). A formula ϕ is schematically valid iff for all such σ, σ(ϕ) is valid. The substitution core of PAL is the set {ϕ ∈ LPAL | ϕ schematically valid} and similarly for PAL-RC. In van Benthem’s list of “Open Problems in Logical Dynamics” [3], Question 1 is whether the substitution core of PAL-RC is decidable. We answer this question positively for PAL and PAL-RC in the following section.

2

Decidability

The idea of our proof is to provide a procedure for constructing a finite set of substitution instances for a given formula ϕ, such that if ϕ is not schematically valid, then there is a falsifiable substitution instance in the finite set. Suppose that for some substitution σ and model M, we have M, w σ(ϕ). From σ and M, we will construct a special substitution τ such that τ (ϕ) is false at w in a suitable extension (on the valuation function) of M. The construction reveals that τ is in a finite set of substitutions determined solely by the structure of ϕ. Therefore, to check whether ϕ is schematically valid, we need only check the validity of finitely many substitution instances of ϕ, which is a decidable problem for PAL and PAL-RC. We begin with a preliminary definition and result. Definition 1. The set of simple formulas is defined as the smallest set such that: all p ∈ At are simple; if ϕ is simple, so are ¬ϕ, Ki ϕ, and ϕ ± p, where ±p is either p or ¬p for p ∈ At; if ϕ and ψ are simple, so are ϕ ∧ ψ and C ϕ ψ. Proposition 1. For every formula ϕ ∈ LPAL-RC , there is an equivalent simple formula ϕ . Proof : By induction on ϕ, using the schematic validities (ii) - (v) in §1 and the schematic validity pqr ↔ pqr [7].

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2.1

Transforming Substitutions

Fix a formula ϕ in LPAL or LPAL-RC . By Proposition 1, we may assume that ϕ is simple. Suppose that for some substitution σ and M = M, {∼i | i ∈ Agt}, V , we have M, w σ(ϕ). We will now provide a procedure to construct a special substitution τ from σ and a model N from M, as discussed above, such that N , w τ (ϕ). Whether ϕ is in LPAL or LPAL-RC , the resulting formula τ (ϕ) will be in LPAL-RC . However, in §2.2 we will obtain substitution instances in LPAL . To construct τ (p) for a given p ∈ At, let B1 , . . . , Bm be the sequence of all Bi such that [Bi ] ± p occurs in ϕ, and let B0 := . For 0 ≤ i, j ≤ m, if σ(Bi )M = σ(Bj )M , then delete one of Bi or Bj from the list (but never B0 ), until there is no such pair. Call the resulting sequence A0 , . . . , An , and define s(i) = {j | 0 ≤ j ≤ n and σ(Aj )M ⊂ σ(Ai )M }. Extend the language with new variables p0 , . . . , pn and a0 , . . . , an , and define τ (p) = κ1 ∧ · · · ∧ κn such that

κi := pi ∨

0≤j≤n, j=i

Caj ∧

¬Cak .

0≤k≤n, k∈s(j)

Without loss of generality, we assume that M is generated by {w} [5, Def. 2.5], so the C operator in κi functions as the global modality in M. Having extended the language for each p ∈ At(ϕ), extend the valuation V to V such that for each p ∈ At(ϕ), V (p) = V (p), and for the new variables: (a) V (pi ) = σ(p)M|σ(Ai ) ; (b) V (ai ) = σ(Ai )M . Let N = M, {∼i | i ∈ Agt}, V be the extension of M with the new V . We will show that τ (p) has the same extension as σ(p) after relativization by any σ(Ai ), which has the same extension as τ (Ai ). It will follow that N , w τ (ϕ) given M, w σ(ϕ). Fact 1. For p ∈ At(ϕ), σ([Ai ] ± p)M = ±pi N . Proof : By basic definitions, σ([Ai ] ± p)M = [σ(Ai )] ± σ(p)M = ±σ(p)M|σ(Ai ) = ±pi N , where the last equality holds by (a) and the definition of N .


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Lemma 1. For p ∈ At(ϕ) and 0 ≤ i ≤ n, τ (p)N|ai = pi N . Proof : We first show that for 0 ≤ i, j ≤ n, i = j: 1. κi N|ai = pi N|ai ; 2. κj N|ai = ai N|ai (= M|ai ). For 1, we claim that given i = j, Caj ∧

¬Cak N|ai = ∅.

0≤k≤n, k∈s(j)

By construction of the sequence A0 , . . . , An for p and (b), aj N = ai N . If ai N ⊂ aj N , then Caj N|ai = ∅. If ai N ⊂ aj N , then by (b) and the definition of s, i ∈ s(j). Then since ai is propositional, ¬Cai N|ai = ∅. In either case the claim holds, so κi N|ai = pi N|ai given the structure of κi . For 2, κj contains as a disjunct:

Cai ∧

¬Cak .

0≤k≤n, k∈s(i)

Since ai is propositional, Cai N|ai = M|ai . By definition of s and (b), for all k ∈ s(i), ak N ⊂ ai N , which gives ¬Cak N|ai = M|ai . Hence κj N|ai = M|ai . Given the construction of σ, 1 and 2 imply: τ (p)N|ai = κi N|ai ∩ κj N|ai = pi N|ai ∩ ai N|ai = pi N . j=i

The last equality holds because pi N ⊆ ai N , which follows from (a) and (b). Lemma 2. For all simple subformulas χ of ϕ, τ (χ)N = σ(χ)M . Proof : By induction on χ. For the base case, we must show τ (p)N = σ(p)M . By construction of the sequence A0 , . . . , An for p ∈ At(ϕ), there is some Aj = , so σ(Aj )M = M . Then by (b), aj N = M , and hence N

τ (p)N = τ (p) |aj = pj N M = σ(p) |σ(Aj ) = σ(p)M .

by Lemma 1 by (a)

The boolean cases are straightforward. Next, we must show τ (Kk ϕ)N = σ(Kk ϕ)M . For the inductive hypothesis, we have τ (ϕ)N = σ(ϕ)M , so

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τ (Kk ϕ)N = Kk τ (ϕ)N = {w ∈ M | ∼k (w) ⊆ τ (ϕ)N } = {w ∈ M | ∼k (w) ⊆ σ(ϕ)M } = Kk σ(ϕ)M = σ(Kk ϕ)M . Similar reasoning applies in the case of C ϕ ψ. Finally, we must show τ ([Bi ] ± p)N = σ([Bi ] ± p)M . For the inductive hypothesis, τ (Bi )N = σ(Bi )M . By construction of the sequence A0 , . . . , An for p ∈ At(ϕ), there is some Aj such that σ(Bi )M = σ(Aj )M . Therefore, τ (Bi )N = σ(Aj )M = aj N

by (b),

and hence τ ([Bi ] ± p)N = [τ (Bi )] ± τ (p)N = [aj ] ± τ (p)N = ±τ (p)N|aj = ±pj N = σ([Aj ] ± p)M = σ([Bi ] ± p)M

by Lemma 1 by (a) given σ(Bi )M = σ(Aj )M .

The proof by induction is complete.

Fact 2. N , w τ (ϕ). Proof : Immediate from Lemma 2 given M, w σ(ϕ).

2.2

Proof of Decidability

Given M, w σ(ϕ), using the procedure of §2.1, we can construct a special substitution τ and an extended model N with N , w τ (ϕ). It is clear from the procedure that we need M, σ, and ϕ to construct τ . For each p ∈ At(ϕ), given the subformulas A0 , . . . , An of ϕ, we defined τ (p) = κ1 ∧ · · · ∧ κn , where Caj ∧ κi := pi ∨ ¬Cak . 0≤j≤n, j=i

0≤k≤n, k∈s(j)

Since we defined s(i) = {j | 0 ≤ j ≤ n and σ(Aj )M ⊂ σ(Ai )M } for i ≤ n, we required information from σ and M in order to construct τ . However, there are only finitely many functions s : n + 1 → ℘(n + 1), and n is bounded by |ϕ|. Hence ϕ induces a finite set of substitution instances, one for each s function (for each p ∈ At(ϕ)), in which at least one formula is falsifiable if ϕ is not schematically valid. This observation yields a decision procedure for the substitution core of PAL-RC. For a given ϕ, construct the finite set of substitution instances as described. Check the validity of each formula in the set


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by the standard decision procedure for PAL-RC. If ϕ is schematically valid, then all of its substitution instances in the set will be valid. If ϕ is not schematically valid, then one of the substitution instances will be falsifiable by Fact 2. Theorem 1 (Decidability for PAL-RC). The substitution core of multi-agent PAL-RC is decidable. Suppose that we have obtained from the PAL-RC procedure a substitution instance τ (ϕ) and a model M for which M, w τ (ϕ). Since the C operator appears in the definition of τ (p), we have τ (ϕ) ∈ LPAL-RC . If ϕ ∈ LPAL , we may now obtain a substitution τ with τ (ϕ) ∈ LPAL and a model M for which M , w τ (ϕ). If there is a Kj modality that does not occur in ϕ, we may modify τ to τ by replacing all occurrences of C in τ (ϕ) by Kj ; then modify M to M by setting the ∼j relation equal to the transitive closure of the union of all ∼i relations. It is straightforward to verify that M , w τ (ϕ) given M, w τ (ϕ). If all Kj modalities occur in ϕ, then we use the fact that for any finite model M, we can define the formula Cα in M by E |M| α, where Ki α and E n+1 α := EE n α. E 1 α := i∈Agt

By the finite model property for PAL-RC [4], we may assume that the model M, for which M, w τ (ϕ), is finite. Hence we modify τ to τ by replacing all occurrences of Cα in τ (ϕ) by E |M| α. It is straightforward to verify that M, w τ (ϕ) given M, w τ (ϕ). Theorem 2 (Decidability for PAL). The substitution core of multi-agent PAL is decidable.

3

Conclusion

In this paper, we have answered positively the open question [3,2,4] of whether the substitution cores of multi-agent PAL and PAL-RC are decidable. In a continuation of this work [14], we will show that our approach to proving decidability applies not only when interpreting the languages of PAL and PAL-RC in models with equivalence relations, but also when allowing models with arbitrary relations. We will also present axiomatizations of the substitution cores of PAL and PAL-RC in a system of Uniform Public Announcement Logic (UPAL). Acknowledgements. We wish to thank Johan van Benthem for stimulating our interest in the topic of this paper and two anonymous referees for comments.

References 1. Baltag, A., Moss, L., Solecki, S.: The Logic of Public Announcements, Common Knowledge and Private Suspicions. In: Gilboa, I. (ed.) Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 1998), pp. 43–56. Morgan Kaufmann, San Francisco (1998)

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2. van Benthem, J.: One is a Lonely Number: Logic and Communication. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium 2002, pp. 96–129. ASL & A.K. Peters (2006) 3. van Benthem, J.: Open Problems in Logical Dynamics. In: Gabbay, D., Goncharov, S., Zakharyashev, M. (eds.) Mathematical Problems from Applied Logic I, pp. 137–192. Springer, Heidelberg (2006) 4. van Benthem, J., van Eijck, J., Kooi, B.J.: Logics of communication and change. Information and Computation 204(11), 1620–1662 (2006) 5. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) 6. Buss, S.R.: The modal logic of pure provability. Notre Dame Journal of Formal Logic 31(2), 225–231 (1990) 7. van Ditmarsch, H.: The Russian cards problem. Studia Logica 75, 31–62 (2003) 8. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2008) 9. van Ditmarsch, H., Kooi, B.: The Secret of My Success. Synthese 151, 201–232 (2006) 10. Gerbrandy, J., Groenevelt, W.: Reasoning about Information Change. Journal of Logic, Language and Information 6(2), 147–169 (1997) 11. Halpern, J.Y.: Should Knowledge Entail Belief? Journal of Philosophical Logic 25, 483–494 (1996) 12. Hintikka, J.: Knowledge and Belief: An Introduction to the Logic of the Two Notions. Cornell University Press, Ithica (1962) 13. Holliday, W.H.: Hintikka’s Anti-Performatory Effect and Fitch’s Paradox of Knowability (2011) (manuscript) 14. Holliday, W.H., Hoshi, T., Icard, T.F.: A Uniform Logic of Information Update (2011) (manuscript) 15. Holliday, W.H., Icard III, T.F.: Moorean Phenomena in Epistemic Logic. In: Beklemishev, L., Goranko, V., Shehtman, V. (eds.) Advances in Modal Logic, vol. 8, pp. 178–199. College Publications (2010) 16. Kooi, B.: Expressivity and completeness for public update logics via reduction axioms. Journal of Applied Non-Classical Logics 17(2), 231–253 (2007) 17. Plaza, J.: Logics of public communications. In: Emrich, M., Pfeifer, M., Hadzikadic, M., Ras, Z. (eds.) Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216. Oak Ridge National Laboratory (1989) 18. Segerberg, K.: Two-dimensional modal logic. Journal of Philosophical Logic 2(1), 77–96 (1973)

Knowledge and Action in Semi-public Environments Wiebe van der Hoek, Petar Iliev, and Michael Wooldridge University of Liverpool, United Kingdom {Wiebe.Van-Der-Hoek,pvi,mjw}@liverpool.ac.uk

Abstract. We introduce and study the notion of a Public Environment: a system in which a publicly known program is executed in an environment that is partially observable to agents in the system. Although agents do not directly have access to all variables in the system, they may come to know the values of unobserved variables because they know how the program is manipulating these variables. We develop a logic for reasoning about Public Environments, and an axiomatization of the logic.

1

Introduction

Our primary concern in the present paper is the following issue: Suppose that a number of agents are engaged in a commonly known algorithmic activity in some environment. What can be said about (the evolution of ) their knowledge if they cannot make a complete and correct observation of their environment? To investigate this issue, we introduce a computational model for epistemic logic known as Public Environments (pes), and we then investigate the notions of knowledge that arise in pes. pes build upon the well-known interpreted systems model of knowledge [6], in which the knowledge of an agent is characterised via an “epistemic indistinguishability” relation over system states, whereby two states s and t are said to be indistinguishable to an agent i if the variables visible to i have the same values in s and t . In pes, as in the interpreted systems model of knowledge [6], agents have access to a local set of variables. But, crucially, their knowledge can be based on a refinement of the indistinguishability relation that derives from these variables: agents may observe the occurrence of an action, and from this be able to rule out some states, even though the local information in such states is the same as in the current state. Moreover, the protocol in a pe is a “standard” pdl program, and it is commonly known that the program under execution is “visible” to all agents in the pe. To better understand this idea, suppose there is some pdl program π that is being executed, and that π manipulates the values of the variables in the system. The program π is assumed to be public knowledge. Suppose agent i sees the variables y and z , and this is commonly known. So, it is public knowledge that i knows the value of y and z , at any stage of the program. There is a H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 97–110, 2011. c Springer-Verlag Berlin Heidelberg 2011

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W. van der Hoek, P. Iliev, and M. Wooldridge

third variable, x , which is not visible to i. Now, suppose that the program π is actually the assignment x := y. After this program is executed, it will be common knowledge that i knows the (new) value of x , even though the value of x is not visible to i, because i sees the value of y and knows that the program assigned this value to x . If the program y := x is executed instead, then i will come to learn the (old and current) value of x . Were the assignment y := x + z , then i would learn the value of x as well. Thus agents can ‘learn’ the values of variables through the execution of certain publicly known programs, even though those variables are not visible to the agents. The kind of questions that we want to investigate using pes then typically relate to, for example, how the process of executing a publicly known program in a pe will affect the knowledge of agents who are able to see only some subset of the variables that the program is manipulating. Using pe, it is possible to analyse problems like the Russian Cards [4] and the Dining Cryptographers [3], and indeed many game-like scenarios, by incorporating the protocol (the outcome of the tossing of a coin, the passing of a card from one player to another) explicitly in the object language: we are not restricted to modelling every action by an information update, as in DEL. So, pe’s address the problem of Dynamic Epistemic Logic with factual change [9]. [5] also studies Dynamic Epistemic Logic with assignments, but where our assignments are restricted to program variables, assignments in [5] regard the assignment of the truth of an arbitrary formula to a propositional atom.

2

Motivation

Recall our motivating concern. Suppose that a number of agents are engaged in a commonly known algorithmic activity. What can be said about (the evolution of ) their knowledge if they cannot make a complete and correct observation of their environment? To motivate this question we present two well-known, but seemingly unrelated, scenarios and show that the solutions to the problems described in these scenarios require an answer to particular instances of our question above. We begin with a well known story, first described in [3]. Three cryptographers are sitting down to diner at their favorite threestar restaurant. Their waiter informs them that arrangements have been made with the maitre d’hotel for the bill to be paid anonymously. One of the cryptographers might be paying for the dinner, or it might have been NSA. The three cryptographers respect each other’s right to make an anonymous payment, but they wonder if NSA is paying. They resolve their uncertainty fairly by carrying out the following protocol: Each cryptographer flips an unbiased coin behind his menu, between him and the cryptographer on his right, so that only the two of them can see the outcome. Each cryptographer then states aloud whether the two coins

Knowledge and Action in Semi-public Environments

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he can see-the one he flipped and the one his left-hand neighbor flippedfell on the same side or on different sides. If one of the cryptographers is the payer, he states the opposite of what he sees. An odd number of differences uttered at the table indicates that a cryptographer is paying; an even number indicates that NSA is paying (assuming that the dinner was paid for only once). Yet if a cryptographer is paying, neither of the other two learns anything from the utterances about which cryptographer it is. What are the essential features of the problem and its solution? The obvious answer is that we are looking for a cryptographic protocol that is provably correct with respect to some formal model of the task. The search for such a protocol is not related to our main question. However, we find the proposed solution to be very relevant to the problem we study. Note that in order to be convinced of the correctness of this solution, we must be able to reason about the evolution of the knowledge of the cryptographers during the execution of a commonly known protocol; what is more, the essential feature of this protocol is that the cryptographers cannot make complete and accurate observations of their environment. Our second example comes from [2]. Before describing the problem, we would like to remind the reader about the main idea of this paper. Roughly, the authors want to formalise the following intuition. Suppose that we have a task that must be performed by a robot. Then we can decide if the robot can perform the task as follows. First, we try to determine the minimal knowledge required to perform the task and then we try to determine if the robot’s sensors and “reasoning abilities” allow it to gain that knowledge. If we have a match, then the robot is potentially able to perform the task. Their formal treatment is based on the following example. Two horizontal, perpendicular robotic arms must coordinate as follows. The first arm must push a hot object lengthwise across a table until the second arm is able to push it sideways so that it falls into a cooling bin. The length of the table is marked in feet, from 0 through 10. The object is initially placed at position 0 on the table. The second arm is able to push the object if it is anywhere in the region [3, 7]. . . . We consider two variants of the problem: 1. The arms share a controller. The controller has access to a sensor reporting the position of the object with error no greater than 1, i.e., if the object’s current location is q then the reading can be anywhere in [q − 1, q + 1]; 2. Same as above, except that the error bound is 4 rather than 1. It is not hard to see that in the second case, there is no protocol that performs the task, whereas in the first case there is. For example, a centralized protocol that deals with 1 is the following (where r is the current reading): If r ≤ 4 then Move(arm1 ) else Move(arm2 ).

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If we apply the authors’ intuition to this problem, we may say that the minimal knowledge necessary to perform the task is to know when you are in a certain location (namely the region [3, 7]) within some reasonable error bound. It is obvious that in the first case the robot is able to aquire this knowledge while in the second it is not. And again, it is obvious that in order to see that this answer is correct, we must say something about (the evolution of) the knowledge of an agent engaged in an algorithmic activity while not being able to make complete and correct observations of its environment. We show that the protocols described in both examples can be modeled in an uniform way. Let’s first model the solution to the dining cryptographers problem proposed by David Chaum. We name the three cryptographers 1, 2 and 3 respectively. Suppose that instead of coins each different pair of cryptographers (i, j ), where i, j ∈ {1, 2, 3} and i = j , is assigned a different variable c(i,j ) that can take the boolean values 0 and 1. These values model the two sides of the coin shared between i and j and the current value of the variable c(i,j ) is visible only to i and j . Let each cryptographer i be assigned a private variable pi that is not visible to the rest. These three variables can take only the values 0 and 1. If a certain variable is set to 1, then the respective cryptographer is paying. We will assume that at most one of the variables p1 , p2 , p3 is set to 1. Next we model the announcement made by each cryptographer in the following way. Let us associate a variable ai with each cryptographer i. Each ai is visible to all the cryptographers and holds the value c(i,j ) ⊕ c(i,k ) if pi = 0 or the value 1 − c(i,j ) ⊕ c(i,k ) otherwise, where ⊕ stands for the “exclusive or ” or xor of the two values and j = k . In this way we model the two types of announcement a cryptographer can make depending on the fact whether he is paying or not. Finally, we introduce a variable r which holds the result of a1 ⊕ a2 ⊕ a3 , i.e., the number of differences uttered at the table. Then r = 1 if and only if the number of differences is odd. Let us summarise. We have a set of variables V = {p1 , p2 , p3 , c(1,2) , c(1,3) , c(2,3) , a1 , a2 , a3 , r }. We associate a subset V (i) ⊆ V with every cryptographer i, where i ∈ {1, 2, 3}. Each set V (i) represents the variables visible to the respective cryptographer. Therefore, we have V (1) = {p1 , c(1,2) , c(1,3) , a1 , a2 , a3 , r }; V (2) = {p2 , c(1,2) , c(2,3) , a1 , a2 , a3 , r }; V (3) = {p3 , c(1,3) , c(2,3) , a1 , a2 , a3 , r }. All variables range over the set B = {0, 1}. Now we can represent the protocol described in [3] as an (non-deterministic) algorithm that changes the values of the variables in the appropriate way. For example, 1. if p1 := 0 then a1 := c(1,2) ⊕ c(1,3) ; else a1 := 1 − ( c(1,2) ⊕ c(1,3) ); 2. if p2 = 0 then a2 := c(1,2) ⊕ c(2,3) ; else a2 := 1 − ( c(1,2) ⊕ c(2,3) );

4. if p3 = 0 then a3 := c(1,3) ⊕ c(2,3) ; else a3 := 1 − (c(1,3) ⊕ c(2,3) ); 5. r := a1 ⊕ a2 ⊕ a3 ;

The first 4 lines of the program say that if cryptographer i is not paying, i.e., the variable pi is set to 0, then i truthfully announces whether c(i,j ) and c(i,k )


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are different or not, i.e., ai is set to c(i,j ) ⊕ c(i,k ) . If i is paying, i.e., pi is set to 1, then ai is set to the opposite value of c(i,j ) ⊕ c(i,k ) . Line 5 sets the value of r to xor of a1 , a2 , a3 . Note that we assume that not more than one of the variables p1 , p2 , p3 is set to 1. This is common knowledge among the agents. We do not assume that they have no other knowledge beside the knowledge obtained from the observable variables. For example, they must be convinced that an odd number of differences implies that one of them is paying, i.e., they must know some of the properties of the xor function and they implicitly know that not more than one of them is paying, no one is lying etc. In short, in order to model the dynamics of the relevant knowledge (in this case this means the knowledge obtained from observation), we may have to assume a lot of relevant background knowledge. The only requirement we have is: – The knowledge gained from observation does not contradict the background knowledge of the agents, i.e., in this case no one of the agents can consider it possible (based on the values of the observable variables only) that more than one of the variables pi can have the value 1. Now, we show how to model our second example in this setting. The essential problem here is that the robot’s sensor can detect its current position within some error bound. Therefore, the problem consists in the relation between the position read by the sensor, and the real position, i.e., between the current value of some observable variable that is within some error bound with respect to the value of some unobservable variable. So, let x be a variable observed by an agent (intuitively, this variable stores the sensor readings), y be a variable that is not observable (intuitively y stores the real position) and e be another visible variable (intuitively, e has some constant value that represents the error bound). This means that we assume that the robot “knows” what its error bound is. We have the following simple program. 1. x := y − k ∪ x := y + k ; where k ≤ e . That is, x is non-deterministically assigned some value that differs from y within the error bound e. Now, we may ask the question what a robot observing x can know about the value of y. Having established that the essential features of the scenario from [2] can be modeled in our framework, we will concentrate on the dining cryptographers problem to motivate the assumption we make. Of course, these assumptions will be true for our second example. Let us address the evolution of the knowledge gained from observation of the three cryptographers during the execution of this program. Each step of the program induces some change in the knowledge of each cryptographer by changing the value of the relevant program variables. We can model the knowledge of all the cryptographers at a given point during the execution of the algorithm using the standard machinery of Kripke models. The underlying set of such a model consists of: – (W ) All the assignments of values to the program variables considered possible by at least one of the cryptographers at this particular program step.

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The epistemic relation modelling the knowledge of a cryptographer i must be such that – If i cannot distinguish between two valuations then they must coincide on the values assigned to the variables in V (i); – If a cryptographer i can distinguish two valuations at a particular step of the computation, then s(he) can distinguish their updated images at the next step of the computation. If M is a model for our description of the protocol, then we would like it to satisfy properties like the following. Let n = (¬p1 ∧ ¬p2 ∧ ¬p3 ) (i.e., the NSA paid) Ki n) ∧ (¬n → [π] Ki ¬n) M |= (n → [π] i=1,2,3

i=1,2,3

Saying that if the NSA paid, all cryptographers will know it afterwards, and if the NSA did not pay but a cryptographer did, this will become known as well. Of course, on top of this we need: M |= pi → [π] (¬Kj ¬pi ) j =i

Note that we do not say that if two valuations coincide on the values of the variables visible to i, then i cannot differentiate between them. This assumption would lead to counterintuitive scenarios. Let us summarise. A group of agents is engaged in an algorithmic activity. These agents can observe only a part of their environment which is affected by this activity. Based on their knowledge of the algorithm and on the values of the observable variables, they can draw some conclusions and update their knowledge at each step of the program. We assume that each “variable” can be assigned only finitely many values. To make things easier, we further assume that the possible values can be just 0 and 1. During its execution, the algorithm can act on only finitely many variables of the agents’ environment. The basic algorithmic steps are assignments of the form x := t , where t is a term of the language to be defined soon and tests. The agents’ knowledge at each particular step of the algorithm is modelled using Kripke models. The dynamics of the knowledge is modelled using suitably defined updates of the Kripke models.

3

Language and Semantics

Let Ag = {a1 , . . . , am } be a set of agents and Var = {x1 , . . . xn } a set of variables. We define ϕ ∈ L: ϕ := ϕ0 | Vi x | ¬ϕ | ϕ ∧ ϕ | [τ ]ϕ | Ki ϕ | Oi ϕ | 2ϕ where i ∈ Ag. Vi x says that agent i sees the value of x . Oi ϕ denotes that i observes that ϕ holds. Ki is the knowledge operator, and 2 will be a universal


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modal operator: it facilitates us to express properties like 2[τ ](Ki (x = 0) ∧ ¬Kj (x = 0): ‘no matter what the actual valuation is, after execution of τ , agent i knows that the value of x is 0, while j does not know this. To define a Boolean expression ϕ0 , we first define terms t . In this paper, terms will have values over {0, 1}, this is a rather arbitrary choice, but what matters here is that the domain is finite. Terms are defined as t := 0 | 1 | x | t + t | t × t | −t VAR(t ) denotes the set of variables occurring in t . Boolean expressions over terms are: ϕ0 := t = t | t < t | ¬ϕ0 | ϕ0 ∧ ϕ0 Finally, we define programs: τ := ϕ0 ? | x := t | τ ∪ τ | τ ; τ where x ∈ Var and ϕ0 is a Boolean expression. A valuation θ : Var → {0.1} assigns a value to each variable. Let the set of valuations be Θ. We assume that v is extended to a function v : T er → {0, 1} in a standard way, i.e., we assume a standard semantics |=B for which v |=B ϕ0 is defined in terms of giving a meaning to +, −, ×, m Vi xj ). Then Oi ϕ is equivalent to the following schema, where each ci ranges over the set {0, 1}. Ψi → (x1 = c1 ∧ ×2 = c2 ∧ . . . ∧ xm = cm → 2(x1 = c1 ∧ x2 = c2 ∧ . . . ∧ xm = cm → ϕ))

Example 1. Figure 1 (left) shows a public environment for two variables x and y where V1 = {x } and V2 = {y}. The program that is executed is (x = 1 ∨ y = 1)?; x := 0; y := 1. Note that the final epistemic model it is common knowledge that x = 0∧y = 1. Note however that we cannot identify the three states despite the fact that the valuations of the variables are the same.

01

11 2

1

1 2

00

10

(x = 1 v y = 1)? 01

11 2 1

10

x:=0

00 01

01 1, 3

01 2

2

2

1

1, 3 10

11

00

x := y y:=1

01

00

01

11 3

2 1

01

00

3 11

Fig. 1. Executing (x = 1 ∨ y = 1)?; x := 0; y := 1 (left) and x := y (right)

Example 2. Consider the model M from Figure 1 (right). Assume V1 = {x }, and V2 = V3 = ∅. The following table summarises the change in knowledge from M , 00 (first row) to M , 00 (second row) while executing the program x := y. ¬K1 (y = 0) ¬K2 (x = 0) ¬K3 (x = y) K3 (x = 0) K1 (y = 0) K2 (x = 0) K3 (x = y) ¬K3 (x = 0) Through the assignment x := y agent 1 learns the value of y (because he reads x ), agent 2 learns the value of x (because he know the value of y) and agent 3, like all other agents, comes to know that x = y.

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Example 3. We show under which conditions an agent i can learn the value of a variable. Consider the following program, where x and y are different variables: α = ((y = 0?; x := t ) ∪ (y = 1?; x := u)) The following are valid: 1. (Ki (y = 0) → [α]Ki (x = t )) ∧ (Ki (y = 1) → [α]Ki (x = u)) Knowing the condition for branching implies knowing which program is executed; 2. (Vi x ∧ Ki (t = u)) → ([α]Ki y = 0 ∨ [α]Ki y = 1) If an agent can read a variable x , and the value of the variable depends on a variable y, then the agent knows retrospectively what y was. 3. (¬Vi y ∧ ¬Vi x ∧ ¬Ki (y = 0) ∧ ¬Ki (y = 1)) → [α](¬Ki (y = 0) ∧ ¬Ki (y = 1)) An agent who cannot see x nor y cannot deduce y’s value from α. Example 4. We want to swap the value of two variables in a public environment, where the only spare variable is visible to an observer. Can we swap the variables without revealing the variables we want to swap? Formally, let Var = {x1 , x2 , x3 }, Ag = {1}, and V (1) = {x3 }. Informally, the designer of the program wants to ensure that agent 1 never learns the value of x1 or x2 . Formally, if i ∈ {1, 2}, we can capture this in the following scheme: χ = 2[π](¬K1 (xi = 1) ∧ ¬K1 (xi = 0)) Consider the following program π: x3 := x1 ; x1 := x2 ; x2 := x3 Clearly, if M is the epistemic model that formalises this, we have M |= ¬χ. But of course, π above is not the only solution to the problem of swapping variables. Now consider the following program π: x1 := x1 + x2 ; x2 := x1 − x2 ; x1 := x1 − x2 ; In this case, with M the epistemic model, we have M |= χ, as desired. Learning and Recall The principle of recall (wrt. a program τ ) is Ki [τ ]ϕ → [τ ]Ki ϕ

(2)

It is straightforward to verify that assignment and test satisfy (2), and moreover, that it is preserved by sequential composition and choice. However, now consider the converse of (2), which is often referred to as no-learning: [τ ]Ki ϕ → Ki [τ ]ϕ

(3)

This principle is not valid, as we can see from Example 2. In that example, we have P , M , 00 |= M1 x := y (x = 1), but not P , M , 00 |= x := y M1 (x = 1) (and hence [x := t ]Ki ϕ → Ki [x := t ]ϕ is not valid). Semantically, no learning is ∀w , t (Ri vw & Rτ wt ⇒ ∃s(Rτ vs ∧ Ri st ))

(4)

We can phrase failure of (4) in Example 2 as follows: In 00, agent 1 considers the state 01 possible, which with the assignment x := y would map to 11, however,


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after the assignment takes the state 00 to 00, since 1 sees variable x , he does not consider the state 11 as an alternative any longer. Loosely formulated: through the assignment x := y in 00, agent 1 learns that 01 was not the current state. From the definition of assignment, it is easy to see that the following is valid: ˆ i ϕ ∧ Mi x := t ϕ x := t Mi ϕ ↔ x := t O

(5)

Also, the test operator fails no learning: again, in Example 2, we have M , 00 |= M1 y = 1? , but we do not have P , M , 00 |= y = 1? M1 : by the fact that 00 does not ‘survive’ the test y = 1, agent 1 ‘learns’ that it was not the real state. Before revisiting the dining cryptographers, we mention some validities. • |= [x := t ]Ki (x = t ) ∧ [x = t ?]Ki (x = t ) ∧ x := t Ki (x = t ) Agents see the programs and are aware of their effects. An assignment x := t can always be executed, which is not true for a test x = t ?. • |= Ki (t = u) → [x := t ]Ki (x = u) Knowing the value of a term implies knowing the value of a variable if it gets assigned that term. Dining cryptographers revisited. We will now indicate how to model this scenario in such a way that it is possible to reason about lying participants, a situation where more than one of the cryptographers is paying etc. We introduce just three new variables h1 , h2 , h3 and modify slightly our previous algorithm. 1. if h1 = 1 then else p1 := 1 − p1 2. if h2 = 1 then else p2 := 1 − p2 3. if h3 = 1 then else p3 := 1 − p3

p1 := p1 ; ; p2 := p2 ; ; p3 := p3 ; ;

4. if p1 = 0 then a1 := else a1 := 1 − ( c(1,2) ⊕ 5. if p2 = 0 then a2 := else a2 := 1 − ( c(1,2) ⊕ 6. if p3 = 0 then a3 := else a3 := 1 − ( c(1,3) ⊕ 7. r := a1 ⊕ a2 ⊕ a3

c(1,2) c(1,3) c(1,2) c(2,3) c(1,3) c(2,3)

⊕ c(1,3) ; ); ⊕ c(2,3) ; ); ⊕ c(2,3) ; );

Notice the way we have modelled lying in the first three lines of the program. If hi = 0 then the agent Ai will, in effect, behave contrary to what his paying variable indicates. In the original treatment of the problem, it is implicity assumed that all the cryptographers are honest and at most one of them is paying. This (extremely) strong assumption can be made explicit in our framework. Let ϕ stand for the formula: (h1 ∧h2 ∧h3 )∧{(p1 ∧¬p2 ∧¬p3 )∨(¬p1 ∧p2 ∧¬p3 )∨(¬p1 ∧¬p2 ∧p3 )∨(¬p1 ∧¬p2 ∧¬p3 )} If we stipulate that our initial epistemic models satisfy ϕ then the original assumptions of [3] will become common knowledge among the agents. And for the epistemic model M associated with our assumptions, we have the properties we wanted to confirm in Example 1, namely: Ki ¬n) ∧ (¬a4 → [π] Ki n) M |= (a4 → [π] i=1,2,3

i=1,2,3

If we chose, however, to concentrate on the general case where the motivation and honesty of the participants is unknown then many of the shortcomings of

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this protocol can be easily seen. If two participants behave as if they have paid then we will have a collision and the final result displayed in r will be 0 making this case indistinguishable from the case where no one is paying. Notice the words ’behave as if they have paid’. This behaviour can result from the fact that they may be lying, or only one of them is lying or both have paid. As long as the variable hi , is visible only to Ai , these cases cannot be distinguished by the other two agents. Similar observations can be made in the case where only one or all the cryptographers behave as if they have paid.

4

Axiomatization

Let L0 be the language without modal operators. The usual ([8]) dynamic axioms apply, although not to the full language: exceptions are the axioms Assign, and test (τ ). Propositional and Boolean Component Prop ϕ if ϕ is a prop. tautology if B ϕ0 Bool ϕ0

Rules of Inference Modus Ponens ϕ, (ϕ → ψ) ⇒ ψ Necessitation ϕ ⇒ 2ϕ

Epistemic and Universal Component IK the S 5 axioms for knowledge UB the S 5 axioms for 2 BO 2ϕ → Oi ϕ OK Oi ϕ → Ki ϕ KV (x = c ∧ Vi x ) → Ki (x = c)

Visibility VD1 VD2 VB1 VB2 VK

Dynamic Component Assign [x := t]ϕ0 ↔ ϕ0 ([t/x ]) Func [x := t]ϕ ↔ x := t ϕ K (τ ) [τ ](ϕ → ψ) → ([τ ]ϕ → [τ ]ψ) union(τ ) [τ ∪ τ ]ϕ ↔ ([τ ]ϕ ∧ [τ ]ϕ) comp(τ ) [τ ; τ ]ϕ ↔ [τ ][τ ]ϕ test(τ ) [ϕ?]ϕ0 ↔ (ϕ → ϕ0 )

Dynamic and Epistemic RL x := t Mi ϕ ↔ ˆ i ϕ ∧ Mi x := t ϕ x := t O Dynamic and Universal 3 τ 3ϕ → 3τ ϕ x := t 3 3x := t ϕ → x := t 3ϕ ϕ0 ? 3 3ϕ0 ? ϕ ↔ (ϕ ∧ ϕ0 ? 3ϕ)

Vi x → [τ ]Vi x ¬Vi x → [τ ]¬Vi x Vi x → 2Vi x ¬Vi x → 2¬Vi x (Vi x ∧ x = c) → Ki (x = c), c ∈ {0, 1}

Fig. 2. Axioms of kppe

Axiom Assign is not valid for arbitrary ϕ, as the following example shows. First of all, we need to exclude formulas of the form Vi x . It is obvious that [x := y]Vi x does not imply Vi y. But even if we made an exception for Vi formulas, we would leave undesired equivalences: Take y for t . If [x := y]K1 (y = 0) holds and Ass were valid, this would imply that K1 y = 0, which it should not (it might well be that agent 1 learned the value of y since he sees that x becomes 0). Note that VK is only valid for values 0 and 1, and not for arbitrary terms: we do not have |= (Vi x ∧ x = t ) → Ki (x = t ) (take for instance t = z for a counterexample). Our completeness proof uses Theorem 1, of which the proof immediately follows from the following equivalences:

Knowledge and Action in Semi-public Environments [x := t]Vi x [x := t]ϕ0 [α](ϕ ∧ ψ) [ϕ0 ?]¬ϕ [ϕ0 ?]2ϕ [ϕ0 ?]Ki ϕ

↔ Vi x ↔ ϕ0 [t/x ] ↔ ([α]ϕ ∧ [α]ψ) ↔ (ϕ0 → ¬[ϕ0 ?]ϕ ↔ (ϕ0 → 2[ϕ0 ?]ϕ ↔ (ϕ → Ki [ϕ0 ?]ϕ)

[ϕ0 ?]Vi x ↔ [ϕ0 ?]ψ0 ↔ [x := t]¬ϕ ↔ [x := t]2ϕ ↔ [x := t]Ki ϕ ↔

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(ϕ0 → Vi x ) (ϕ0 → ψ0 ) ¬[x := t]ϕ 2[x := t]ϕ (Ki [x := t]ϕ ∨ [x := t]Oi ϕ)

Theorem 1. Every formula is equivalent to one without dynamic operators. Theorem 1 implies that every effect of a program is completely determined by the ‘first’ epistemic model. More precisely, it implies that for every formula ϕ there is an equivalent epistemic formula (using Vi and 2) ϕ which is provably equivalent. Theorem 2. The logic kppe is sound and complete with respect to public environments.

5

Related Work and Conclusions

We have introduced a framework for reasoning about programs and valuations where knowledge is based on a hybrid semantics using notions from interpreted systems and general S 5 axioms. This is only a first step in doing so. Possible extensions are manyfold. First, we think it is possible to include repetition ( ) as an operator on programs and still obtain a well-behaved logical system, although the technical details for doing so can become involved. There are several restrictions in our framework that may be worthwhile relaxing, like allowing a richer language for tests, and not assuming that it is common knowledge which variables are seen by whom, or what the program under execution is. Those assumptions seem related, and removing them may well be a way to reason about Knowledge-based Programs ([7]), where the programs are distributed over the agents, and where it would be possible to branch in a program depending on the knowledge of certain agents.

References 1. Baltag, A., Moss, L.S., Solecki, S.: The logic of common knowledge, public announcements, and private suspicions. In: Gilboa, I. (ed.) Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 1998), pp. 43–56 (1998) 2. Brafman, R., Halpern, J.Y., Shoham, Y.: On the knowledge requirements of tasks. Artificial Intelligence 98, 317–349 (1998) 3. Chaum, D.: The dining cryptographers problem: Unconditional sender and recipient untraceability. Journal of Cryptology 1(1), 65–75 (1988) 4. van Ditmarsch, H.: The russian cards problem. Studia Logica 75, 31–62 (2003) 5. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic epistemic logic with assignments. In: AAMAS 2005, pp. 141–148 (2005)

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6. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (1995) 7. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Knowledge-based programs. Distributed Computing 10(4), 199–225 (1997) 8. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. The MIT Press, Cambridge (2000) 9. Sietsma, F.: Model checking for dynamic epistemic logic with factual change. Tech. rep., UvA and CWI, Amsterdam (2007), http://homepages.cwi.nl/~ sietsma/papers/mcdelfc.pdf

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios with off Equilibrium Play Alistair Isaac1,3, and Tomohiro Hoshi2,3 1

2

Department of Philosophy, University of Michigan, Ann Arbor, MI [email protected] Educational Program for Gifted Youth, Stanford University, Stanford, CA [email protected] 3 Logical Dynamics Lab, Stanford University, Stanford, CA

Abstract. This paper investigates strategies for responding rationally to opponents who make mistakes. We identify two distinct interpretations of mistakes in the game theory literature: trembling hand and risk averse mistakes. We introduce the concept of an EFG Scenario, a game plus strategy profile, in order to probe the properties of these different types of mistake. An analysis of equivalence preserving transformations over EFG Scenarios reveals that risk averse mistakes are a form of rational play, while trembling hand mistakes are equivalent to moves by nature.

1

Introduction: Two Types of Mistake

People make mistakes, and these mistakes affect the structure of strategic interactions. But what is the rational response to an opponent’s mistake? Suppose, for example, an agent knows an opponent has committed a mistake, but is uncertain which amongst a set of possible mistakes has been made. In such a situation, the agent needs to place a probability distribution over possible scenarios in order to calculate rational action. But this requires him to reason about the mistake: to “guess” what mistake has been made. How does rationality constrain the probability distribution over possible mistakes? At a first pass, one might assume that mistakes are completely random. In this case, any probability distribution over possible mistakes may be rationally justifiable. Consider typos, for example. If I know you’ve mistyped a letter on your keyboard, but I don’t know which letter was substituted, what probability distribution should I place over letters? A flat distribution? A distribution peaked around letters on keys close to the intended letter? But what if you are using a non-standard keyboard layout? Many different probability distributions seem rationally justifiable here, and the mere fact that a mistake has occurred alone does not decide between them. We call mistakes for which any probability distribution over outcomes is justifiable, “trembling hand” mistakes.

Supported by a postdoctoral fellowship from the McDonnell Foundation Research Consortium on Causal Learning.


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A more sophisticated analysis of mistakes is possible if we know more about the agent. For example, you are much more likely to misspell a word while typing than to get into an accident while driving. Why is that? One possible explanation is that a car crash has a far greater cost for you than a misspelling, and consequently you are more careful while driving than you are while typing. If this is a general feature of mistakes, we might analyze it by treating the probability of a mistake as inversely proportional to the cost to the agent. We call mistakes for which only probability distributions over outcomes inversely proportional to cost are justifiable, “risk averse” mistakes. These two types of mistakes are relevant to the theory of games when one considers how to calculate off equilibrium play. In a strategic interaction, a (Nash) equilibrium is a pattern of moves such that no agent can do better against the others by changing his move. In the case of an extensive form game, we can think of an equilibrium as a path through the tree, or, more generally, as a probability distribution over branches. If an agent finds himself at a decision point assigned zero probability by an equilibrium, how should he play? Since the point is off equilibrium, he can only have reached it if his opponent made a mistake. Nash’s solution concept is unhelpful here because it allows the agent to make a completely arbitrary move; this is equivalent to abandoning all rationality when an opponent makes a mistake. [5] defined the perfect equilibrium to characterize rational play against opponents who make trembling hand mistakes. [3] defined the proper equilibrium to characterize rational play against opponents who make risk averse mistakes. But what exactly are we committed to when we adopt one of these analyses of mistake making? Section 2 argues that there is a close conceptual connection between the treatment of off equilibrium play as communicative (i.e. deliberate) and the risk aversion interpretation of mistakes. The remainder of the paper is devoted to demonstrating the conceptual connection between trembling hand mistakes and moves by nature. Section 3 reviews basic game definitions and Sect. 4 introduces our fundamental formal object, game scenarios. In Sect. 5, we extend equivalence relations for extensive form games to game scenarios and argue that trembling hand mistakes can best be studied by analyzing their behavior across the “coalescing of moves” transformation. Section 6 implements this project and proves our main result: trembling hand mistakes are equivalent to moves by nature.

2

Proper Equilibria and Wishful Thinking

Consider the extensive form games in Fig. 1. ΓA and ΓB both share the same reduced normal form representation. This is because in each game player I has three non-redundant strategies. In ΓA , I’s strategies are {a, c, d}; in ΓB his strategies are {ac, ad, bc, bd}, but ad and ac are equivalent, so removing redundancies leaves three strategies, {a, bc, bd}. Since ΓA and ΓB share reduced normal form, they are related by a Thompson transformation, in this case coalescing of moves ([6], see Sect. 5 for details). Since proper equilibria are preserved across Thompson transformations, both games have the same (in this case unique) proper equilibrium, which takes players to the payoff 6,6.

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios

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Fig. 1. Two extensive form games which share a normal form, but not perfect equilibria

However, as discussed by [2], there is a perverse perfect equilibrium for ΓA which does not appear in ΓB . In this perverse scenario, I plays a while II plays y. A perfect equilibrium is the limiting best response to a sequence of completely probabilistic strategies. To see that (a, y) is such a limit, consider the perturbed strategy for player I which assigns probabilities (1 − 10, , 9) and the strategy for player II which assigns (, 1 − ). As shrinks, Player II’s rational probability assignment for the lower node of his information set converges on .9. Player II is thus rationally justified in playing y because his expected payoff is then .9(2)+.1(0) = 1.8, while his expected payoff for playing x is only .1(6)+.9(0) = .6. Consequently, the limiting strategy (a, y) is a perfect equilibrium. This perturbation of strategies assumes any deviation player I makes from his choice of a is brought about by a trembling hand error, i.e. assignment of small probabilities to c and d is completely unconstrained. This analysis is ruled out in the proper equilibrium case by the restriction to risk averse mistakes. If I intends to play a but makes a risk averse mistake, then I is much more likely to mistakenly play c than d (because I’s expected payoff is greater after c than after d). This implies that no matter what I plans, II places higher probability at his top node, and consequently moves x. Once I reasons that II will move x, his best response move becomes c. The same reasoning applies to the equilibrium (bc, x) in ΓB (for full discussion, see [4], Ch. 5). There is another way to reason about off equilibrium play which produces this same result: [P]layer II knows that I will never choose d, which is strictly dominated by a . . . so if II sees he has to play, he should deduce that I, who was supposed to play a and was sure to get 4 in this way, certainly did not choose d, where he was sure to get less than 4; player II should thus infer that I had in fact played c, betting on a chance to get more than 4 (and on the fact that II would understand this signal); and so player II should play x. ([2], 1007, labels replaced appropriately throughout) Notice that in this pattern of reasoning, we started with two of the same assumptions (that I would play a and that II (against expectation) would find himself

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able to make a move) and reasoned to the same conclusion (that II should play x and thus I should switch his intentions to c) as in the risk aversion analysis. An important difference, however, is that we dropped the interpretation of (supposed) off equilibrium play as produced by a mistake, and instead treated it as a form of communication. According to this “forward induction” style of reasoning, player II infers from his ability to play that player I has sent him a “signal” that he is at his top node (the one best for player I). Forward induction and risk averse reasoning won’t always coincide. However, we think the reason they coincide here is rather deep. Forward induction arguments assume players do not make mistakes at all. If a player finds himself at an unexpected information set, he can infer which node he is at by assuming that his very presence at that information set is a signal from an earlier player. This assumes a kind of “wishful thinking” on the part of the earlier player—he takes a risk under the (hopeful) assumption that the later player will understand his signal and respond accordingly. This (rational, but risky) play is mimicked by risk averse mistakes. The player who makes a risk averse mistake acts “rationally” even while making a mistake, since his mistaken act (probabilistically) minimizes risk (hence the name “risk averse” for these types of mistakes). This allows players that find themselves at surprising information sets to reason as if they arrived there via a rational choice, since even the “errors” of a risk averse agent are self interested. This also explains why ΓA and ΓB are strategically equivalent for risk averse agents. In ΓA , the choice between c and d may be determined by mistake, whereas in ΓB it must be determined deliberately. Yet for the risk averse player, deliberate and accidental play coincide in relative probability (c is more likely than d, whether the choice is made by accident or deliberation). If these considerations are correct, then the risk averse interpretation of mistakes is deeply unsatisfying. Mistakes are not being taken seriously here, just treated as a domain to which rationality can be extended. The powerful concept of proper equilibrium which is thus produced assumes complete rationality. If we want to think seriously about how to play against an agent who makes mistakes, we’ll need to assume a weaker equilibrium concept. If that concept is perfect equilibrium, what exactly are we committing ourselves to in our assumptions about mistakes? The remainder of the paper addresses this question.

3

Review: Extensive Form Games and Their Equilibria

This section reviews basic definitions, and may be omitted without loss of continuity. Our presentation will closely follow [4], to which readers are referred for explanation and motivation. Definition 1 (Extensive Game). An extensive form game (EFG) is a tuple Γ = (T , N , {Ak }k∈N , {Ik }k∈N , {uk }k∈N ) where: 1. T = (W, ≺) is a finite tree with a disjoint union of vertices W = k∈N Vk ∪Z where Vk denotes the set of player k’s decision points and Z is the set of


2. 3.

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terminal vertices. The set of immediate successors of w ∈ W is defined by Succ(w) = {v | w ≺ v ∧ ¬∃x : w ≺ x ∧ x ≺ v}. N is a set of players. Ak maps (w, w ), where w ∈ Succ(w), to the action that k can play at w which leads to w . It is assumed that u = v implies Ak (w, u) = Ak (w, v). The set of available moves at w is defined by Ak (w, ·) = {Ak (w, v)|v ∈ Succ(w)}. Ik partitions Vk and induces the function fk that maps w ∈ Vk to k’s information set fk (w) containing w. I denotes the set of all information sets. It is assumed that, if w, w are in the same information set, Ak (w, ·) = Ak (w , ·). The set of available moves at a point in an information set I may be thus denoted by AI . uk : Z → R is the payoffs for the player k on terminal nodes.

For the purposes of this paper, we assume that EFG’s satisfy perfect recall, i.e. players cannot be uncertain about their own past moves (see Sect. 5). Let Γ be an EFG. A pure strategy sk of a player k ∈ N in Γ is a function that assigns to every information set I ∈ Ik an action in A I . We define Sk to be the set of all pure strategies for k ∈ N so that Sk= I∈Ik AI . The set of pure strategy profiles for Γ is then defined by S = k∈N Sk . The outcome O(s) of s ∈ S is the terminal node that results when each player k plays the corresponding game by following sk , i.e. O(s) = w1 . . . wn such that, for each m (1 ≤ m ≤ n), wm ∈ Vk for some k ∈ N and Ak (wm , wm+1 ) = sk (fk (wm )). Definition 2 (Strategic Form). The strategic form of an extensive form game Γ is a strategic game Γ s = (N , S, {vk }k∈N ), where vk : S → R is such that vk (s) = uk (O(s)) for s ∈ S. Definition 3 (Multiagent Form). The multiagentform of an EFG Γ is a strategic game Γ a = (I, I∈I AI , (vI )I∈I ), where vI : I∈I (AI ) → R is defined in the following way: for any (aI )I∈I ∈ I∈I (AI ), if (sk )k∈N ∈ S is the strategy profile such that si (J) = aJ for all i ∈ N and J ∈ Ik , then vI ((aI )I∈I ) = uk ((si )i∈N ) for k such that I ∈ Ik . Intuitively, the multiagent form of an EFG is a strategic game which treats every information set in Γ as a separate player. Each new agent receives the same payoffs as the player in the EFG who plays at that information set. A mixed strategy for k ∈ N in Γ is a probability distribution over Sk . Let Δ(X) be the set of probability distributions over X for a given set X. The set of all mixed strategies k ). The set of all mixed strategy for k is then denoted by Δ(S profiles of Γ is k∈N Δ(Sk ). Thus, each σ ∈ k∈N Δ(Sk ) can be written as σ = (σk )k∈N where each σk is a probability distribution in Δ(Sk ). We also denote the set of mixed strategy profiles that assign only non-zero probabilities to all elements in Sk by k∈N Δ0 (Sk ). Finally, for a given set X and a function f : X → R, we define argmax f (x) = {y ∈ X | f (y) = max f (x)}. x∈X

x∈X

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And for a sequence x = (xm )m∈X , we write x−n for (xm )m∈X−{n} . Definition 4 (Perfect Equilibrium). Let Γ a = (N, (Sk )k∈N , (u k )k∈N ) be the multiagent form of an extensive game Γ . A mixed strategy σ ∈ k∈N Δ(Sk ) is a perfect equilibrium of Γ iff there exists a sequence (ˆ σ m )∞ m=1 such that 1. σ ˆ m ∈ k∈N Δ0 (Sk ) for all m ∈ N, 2. lim σ ˆkm (sk ) = σk (sk ) for all k ∈ N and sk ∈ Sk , and k→∞

m 3. σ ∈ argmax uk (ˆ σ−k , τk ) for all k ∈ N . τk ∈Δ(Sk )

A perfect equilibrium is a game solution in which each strategy is the best response to the limit of a sequence of purely probabilistic strategies. This ensures that there exists a small perturbation in an opponent’s strategy to which any off equilibrium move is a best response. Definition 5 (-Proper Equilibrium). Let Γ s = (N, (Sk )k∈N , (uk )k∈N ) be the strategic form of an extensive game Γ . For any positive number , a mixed strategy profile σ is an -proper equilibrium iff 1. σ ∈ k∈N Δ0 (Sk ) 2. for every k ∈ N and sk , tk ∈ Sk , uk (σ−k , [sk ]) < uk (σ−k , [tk ]) implies σk ([sk ]) < σk ([tk ]) Where (σ−k , [xk ]) is exactly like σ except that player k’s mixed strategy is replaced with the pure strategy xk ∈ Sk , i.e. k plays xk with probability 1. Definition 6 (Proper Equilibrium). Let Γ s = (N, (Sk )k∈N , (uk )k∈N ) be the strategic form of an extensive game Γ . A mixed-strategy profile σ ¯ is a proper equilibrium iff there is a sequence ((m), σ m )∞ such that k=1 1. for all k ∈ N and sk ∈ Sk , (a) lim (m) = 0 m→∞

(b) lim σkm (sk ) = σ ¯k (sk ) m→∞

2. for every m, σ m is an (m)-proper equilibrium. A proper equilibrium is a perfect equilibrium in which only strategies which assign move probabilities proportional (modulo ) to payoff are permitted in the limiting sequence of purely probabilistic strategies.

4

Defining Mistakes in EFG Scenarios

What is the appropriate representation of a game for analyzing mistakes? Mistakes occur in time—a mistake made before a move may affect play, while a mistake made after a move may not. As demonstrated in Sects. 1 and 2, the beliefs of a rational player about what mistake has occurred influence his response to it. So, if we want to explore the conceptual commitments behind the trembling


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hand interpretation of mistakes, we’ll need to consider game structures which represent both temporal and epistemic structure. This motivates investigating EFG Scenarios: intuitively, EFG’s with probability distributions over actions at each information set. An EFG Scenario represents a player’s beliefs about how play progresses, including where, if at all, mistakes occur. Let Γ be an EFG. A behavioral strategy for an information set I is a probability distribution over AI . A behavioral strategy for a player k is an ele ment of Δ(A ). The I I∈I k set of behavioral strategy profiles in Γ is defined by k∈N I∈Ik Δ(AI )(= I∈I Δ(AI )). Based on this definition, for a behavioral strategy profile σ, we may write σ = (σk )k∈N = (σk,I )I∈I,k∈N . Definition 7 (EFG Scenario). An extensive form game scenario (EFGS) is a pair G = (Γ, σ), where Γ is an extensive form game and σ is a behavioral strategy profile in Γ . EFGS’s induce a probability distribution over nodes in the tree. Let G = (Γ, σ). An action path A(w, v) is the sequence of actions (edges) which leads from w to v. Suppose that A(w, v) = a1 . . . an and let ki be the player of the action ai . Define Pσ (v | w) in G to be the probability that a node v is reached in Γ via σ, given w is reached, i.e. n σki (ai ). Pσ (v | w) = i=1

The evaluation node nI of an information set I is the “most recent” decision point in an EFG Γ such that all nodes in I fall under it: ∀w[∀v ∈ I(w ≺ v) → w nI ]. The evaluation node defines a subgame in the game tree with respect to which play at an information set may be evaluated for rationality. We define the conditional evaluation probability Pσ (w | I) of w ∈ I as Pσ (w | nI ) . w ∈I Pσ (w | nI )

Pσ (w | I) =

Suppose that a = AI (w, v) (a is played at I from w to v). Let Zv be the set of endpoints reachable from v and k ∈ N the agent who moves at I. The normalized expected value of a ∈ AI at w is ev(σ, w, a) = uk (z)Pσ (z | v). z∈Zv

The normalized expected value of action a weights the payoffs of reachable end nodes by the probabilities that they will be reached. Definition 8 (Rational Play at I). Given an EFGS G, a player k ∈ N plays rationally at I if 1. there exists w ∈ I such that Pσ (w | nI ) = 0 and σI = argmax Pσ (w | I) τ (a)ev(σ−I τ, w, a) τ ∈Δ(AI ) w∈I

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2. or for all w ∈ I, Pσ (w | nI ) = 0 and there exists some P ∈ Δ0 (I) such that P (w) τ (a)ev(σ−I τ, w, a) σI = argmax τ ∈Δ(AI ) w∈I

a∈AI

Definition 9 (Mistake at I). Given an EFGS G, a player k ∈ N makes a mistake at I if k does not play rationally at I. Intuitively, a mistake is a failure to maximize utility. For example, consider ΓB of Fig. 1; if σ takes play to any endpoint other than 6, 6 in the subgame below Player I’s second decision point, then either Player I makes a mistake, or Player II makes a mistake, even if σI (b) = 0. The only move for Player I which maximizes his expected payoff at his second decision node is c, so if he plays d he makes a mistake. Likewise, since Player I’s second decision point is the evaluation node for Player II’s information set, a positive probability is assigned to at least one of the nodes in Player II’s information set, and clause 1 applies (unlike in ΓA , where clause 2 applies, and y is a rational move for Player II). A mixed strategy is a probability distribution over pure strategies. A behavioral strategy is a probability distribution over actions at each information set. Clearly, behavioral strategies over EFG’s are equivalent to mixed strategies over multiagent representations of EFG’s. This motivates the following definition. Definition 10 (In Perfect Equilibrium). An EFGS G = (Γ, σ) is in perfect equilibrium if σ is a perfect equilibrium of the EFG Γ . The following is a straightforward consequence of the above definitions. Proposition 1. If EFGS G = (Γ, σ) is in perfect equilibrium, then for all I ∈ I, play is rational at I. The relationship between mixed strategies and behavioral strategies is not unique, but many–many. Since EFGS’s begin with a behavioral strategy, we are interested here in only one direction, from behavioral strategies to mixed strategies. Given an EFGS G = (Γ, σ), we would like to identify a unique member σ ¯ ∈ k∈N Δ(Sk ) to use as the mixed strategy representation of σ = (σk )k∈N . For each k, define σ ¯k such that for all sk ∈ Sk , σ ¯k (sk ) = σk,I (sk (I)) I∈Ik

Then σ ¯ = (¯ σk )k∈N is the mixed strategy representation of σ. Definition 11 (In Proper Equilibrium). An EFGS G = (Γ, σ) is in proper equilibrium if the mixed representation σ ¯ of σ is a proper equilibrium of the normal form of EFG Γ . The following is an immediate consequence of Proposition 1. Proposition 2. If EFGS G = (Γ, σ) is in proper equilibrium, then for all I ∈ I, play is rational at I.


5

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The Problem is Coalescing of Moves

EFGS’s allow us to probe the nature of trembling hand mistakes. One strategy for doing this is to examine the effect on perfect equilibria of “strategically irrelevant” game transformations. The example of Fig. 1 demonstrates that perfect equilibria are not preserved over one such transformation, coalescing of moves. Coalescing of moves is one of four transformations proposed by [6] which are sufficient to take any EFG into any other EFG with the same strategic form. Unfortunately, they do not preserve perfect recall. [1] demonstrated that three transformations are sufficient to take any EFG with perfect recall into any other with the same strategic form, while preserving perfect recall. Since we assume perfect recall, we will examine the transformations of [1]. Two of these transformations, interchange of moves and coalescing of moves, are unchanged from those of [6]. The third transformation, addition of decision nodes, generalizes [6]’s addition of a superfluous move. Here, we will discuss interchange of moves and addition of a decision node informally, arguing that they preserve perfect equilibria. Consequently, coalescing of moves is the only transformation which does not preserve perfect equilibria, motivating a closer investigation of the relationship between coalescing of moves and mistakes in the following section. For the full definitions of these transformations, see [1].

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Fig. 2. Interchange of moves over EFG Scenarios

Interchange of moves is illustrated in Fig. 2. This transformation reverses the order of play of two contiguous information sets when the latter player is unaware of the former’s move. Extending interchange of moves to EFGS’s is trivial since the change in the ordering of moves does not affect the actions available to each player, consequently, the behavioral strategy profile with which the game is labeled may remain unchanged. Thus, we have the following result. Proposition 3. If G = (Γ, σ) and G = (Γ , σ ) are EFG Scenarios such that Γ is derived from Γ by interchange of moves and σ = σ, then G is in perfect equilibrium iff G is in perfect equilibrium. Addition of decision nodes is illustrated in Fig. 3. The basic idea here is to add an information set for Player II which spans the entire subtree without affecting

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Fig. 3. Addition of decision nodes over EFG Scenarios

perfect recall. In order to do this, copies of information sets below it will need to be duplicated. So, if Γ is derived from Γ by addition of decision nodes, it will not in general have the same number of information sets, and thus will not share a multiagent representation with Γ . Nevertheless, there is a natural transformation over strategy profiles that preserves perfect equilibrium across addition of decision nodes. We illustrate it with an example. Let Γ, Γ be the left and right games in Fig. 3. Let X and Y be the left and right information sets for player II in Γ , and X , Y1 , Y2 , his topmost and left and right lower information sets in Γ . Now, suppose that σ is a perfect equilibrium in Γ . Then there is a sequence (ˆ σm )∞ m=1 of strategies in the multiagent form of Γ satisfying the conditions in Definition 4. Construct a sequence (ˆ τ m )∞ ˆm in Γ in the follow way: m=1 of behavioral strategies τ m m Îm , if I ∈ {X , Y1 , Y2 }; τˆX ˆX ; and τˆYm1 = τˆYm2 = σ ˆYm . τÎm = σ = σ

τ m )∞ Given that (ˆ σ m )∞ m=1 converges to σ, (ˆ m=1 clearly converges to τ defined by: τI = σI , if I ∈ {X , Y1 , Y2 }; τX = σX ; and τY1 = τY2 = σY . It is easy to check that τ and (ˆ τ m )∞ m=1 satisfy the other conditions in Definition 4. The other direction is similar once we note that any sequence converging on τY1 will also converge on τY2 , so, without loss of generality, one of them may be chosen for constructing a sequence which converges in Γ . This strategy can easily be extended to the general case. Therefore we have the following. Proposition 4. Let G = (Γ, σ) and G = (Γ , τ ) be EFG scenarios. Suppose that Γ is derived from Γ by addition of decision nodes. Further suppose that τ is constructed from σ as above (and vice versa). Then G is in perfect equilibrium iff G is in perfect equilibrium.

6

Trembling Hands and the Will of Nature

Any breakdown in the preservation of perfect equilibria across games which share strategic form must be due to coalescing of moves. The basic form of coalescing


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Fig. 4. Coalescing of moves over EFG Scenarios

of moves is to remove (introduce) a redundant decision node for a single player (Fig. 4). In this section, we show that equilibrium can always be preserved when coalescing of moves occurs on a positive path of play. When coalescing of moves occurs off the equilibrium path, we can preserve equilibrium by identifying the new node with a move by nature. This demonstrates a deep equivalence between trembling hand mistakes and moves by nature. COA We write Γ −−→ Γ if coalescing of moves transforms Γ into Γ ; we call Γ the pre-coalesce game and Γ the post-coalesce game. The definition of the coalesce transformation is relatively straightforward; essentially a node or nodes in Γ (J2 below) is dropped in Γ while ≺ stays the same. Definition 12 (Coalescing of Moves). Let Γ and Γ be EFG scenarios. We COA write Γ −−→ Γ , if there is a player l ∈ N who plays at information sets J1 = {w1 , . . . , wn } and J2 = {v1 , . . . , vn } that satisfy the following conditions, where Γ = (T , N , {Ak }k∈N , {Ik }k∈N , {uk }k∈N ): 1. vi ∈ Succ(wi ) for 1 ≤ i ≤ n (vi is an immediate successor of wi .) 2. Al (wi , vi ) = Al (wj , vj ) for 1 ≤ i, j ≤ n. 3. Γ = (T , N , {Ak }k∈N , {Ik }k∈N , {uk }k∈N ), where (a) N = N and {uk }k∈N = {uk }k∈N (b) T is the restriction of T to W \ J2 , where W is the domain of T . (c) Ak = Ak for all k ∈ N such that k = l. For all (x, y), Al is defined by: i. if x = wi for all i, Al (x, y) = Al (x, y), and ii. if x = wi for some i, Al (x, y) = Al (vi , y) for all y ∈ Succ(x). (d) Ik = Ik for all k ∈ N such that k = l, and Il = Il \ {J2 }. We refer the reader to [6] or [1] for further details. COA Suppose G = (Γ, σ) and G = (Γ , σ ) are such that Γ −−→ Γ ; what trans formation between σ and σ might preserve equilibrium? In Fig. 4, an obvious demand is that α = ζδ and β = ζ. Unfortunately, this constraint produces a unique strategy assignment in only one direction, from left to right. So, from left to right we can show a preservation theorem.

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A. Isaac and T. Hoshi COA

Definition 13. Let G = (Γ, σ) and G = (Γ , σ ) be EFGS’s such that Γ −−→ Γ . COA We define G −−→ G (using the notation in Definition 12) as follows: 1. for all players k in Γ such that k = l, σk = σk , and 2. σl (Al (wi , x)) = σl (Al (wi , vi )) · σl (Al (vi , x)) for 1 ≤ i ≤ n coa

We say σ coalesces σ and write σ −−→ σ when σ and σ satisfy 1 and 2. COA

Proposition 5. Given G = (Γ, σ) and G = (Γ , σ ) such that G −−→ G , if G is in perfect equilibrium, then G is in perfect equilibrium. Proof. Given σ is a perfect equilibrium in Γ (G is in perfect equilibrium), there is a sequence (ˆ σ m )∞ m=1 of strategies in the multi-agent form of Γ satisfying the conditions in Definition 4. Construct a sequence (ˆ τ m )∞ m=1 in the following way: – τÎm = σ Îm , if I = J1 , m – τÎ (Al (wi , x)) = σ ˆJm1 (Al (wi , x)), if I = J1 and x ∈ Succ(vi ), and m ˆJm1 (Al (wi , vi )) · σ ˆJm2 (Al (vi , x)), if I = J1 and x ∈ Succ(vi ), – τÎ (Al (wi , x)) = σ where we use the same notation as in Definition 12. Given (ˆ σm )∞ m=1 converges to σ, (ˆ τ m )∞ converges to σ . Also it is straightforward to check that (ˆ σm )∞ m=1 m=1 satisfies conditions 1–3 in Definition 12. Hence G is in perfect equilibrium.

This demonstrates that rational play is preserved across coalescing of moves in the left to right direction. So what about the right to left direction? Consider again Fig. 4 and note that if either α or β are not equal to zero, the constraint coa that σ −−→ σ uniquely specifies a probability distribution in the right to left direction. In the example of Fig. 4 it is ζ = α + β, δ = α/(α + β), and = β/(α + β). The same consideration applies in the general case. COA

Proposition 6. Given G = (Γ, σ) and G = (Γ , σ ) such that G −−→ G and G is in perfect equilibrium, if σI (AI (wi , x)) = 0 for some wi ∈ J1 and x ∈ Succ(vi ) in Γ (as defined in Definition 12), then G is in perfect equilibrium. Proof. Given the (ˆ σ m )∞ m=1 which converges to σ by the assumption that σ is a perfect equilibrium, we can construct a (ˆ τ m )∞ which converges to σ by m=1 reversing the construction in the proof of Proposition 5. The constraint that σJ 1 (AJ1 (wi , x)) = 0 for some wi ∈ J1 ensures the uniqueness of σJ1 (AJ1 (vi , ·)), which in turn is needed to check that (ˆ τ m )∞

m=1 in fact converges to σ.

The real problem for the right to left direction arises when α and β are both zero. coa Then the constraint that σ −−→ σ does not uniquely determine the strategy in the pre-coalesce game. In particular, for a fixed σ there may be members of coa {σ : σ −−→ σ } such that Player I or Player II (or both) play irrationally (as in Fig. 1). This blocks a general preservation theorem. However, we can show a preservation theorem over a stronger transformation. Reversible coalescing of moves ensures the rationality of play below the coalesced node.


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Definition 14. Let G = (Γ, σ) and G = (Γ , σ ) be EFG scenarios such that COA R-COA G −−→ G . Using the notation in Definition 12, we define G ←−−− G and say that G reversibly coalesces G , if for all vi ∈ J2 and all information sets J(⊆ Succ(vi )) reached from vi : τ (a) σJ (b)ev(σ−J2 τ, xa , b) σJ2 = argmax τ ∈Δ(AJ2 ) a∈A

J

b∈AJ

where AJ is the set of actions at J and xa is defined by a = Al (vi , xa ). r-coa coa We say σ reverse coalesces σ , and write σ ←−−− σ , if σ −−→ σ and this condition is satisfied. Remark 1. By Propositions 1, 5, and 6, if G is in perfect equilibrium and coa r-coa σI (AI (wi , x)) = 0 for some wi ∈ J1 , then σ −−→ σ implies σ ←−−− σ . Now we can ask, when is Player I’s play at the added node in Γ rational? Of course, it is rational if ζ = 0, since we have already demonstrated that equilibrium is preserved in those cases, and play is rational at all nodes when games are in equilibrium. It will also be rational in situations where ζ = 0 and Player II’s move below constitutes a proper equilibrium. Remember our discussion in Sect. 2: the rationality of the risk averse player propagates through his mistakes, in the sense that players respond to him as if he plays rationally, even when he makes mistakes. Consequently, a move introduced like this to replace a mistake will also satisfy the constraints of rationality. Conversely, introducing a move to replace the mistake of a trembling hand player will in general look irrational, in particular, completely random. This is because we placed no constraint on the probability of different errors for such a player. What does completely random mean here? Just that a trembling hand player is indistinguishable from a move by nature if we isolate his mistake node R-COA from the rest of his play. Let G = (Γ, σ) and G = (Γ , σ ) such that G ←−−−− G . Consider the precoalesced nodes in Γ , J2 = {v1 , . . . , vn } (by the notation in Definition 12). Define G J2 = (Γ J2 , σJ2 ) by – Γ J2 is the EFG with the nodes in J2 replaced with chance nodes in which nature take actions AJ2 according to the strategy σJ2 . – Use σ J2 is a behavioral strategy profile of Γ J2 in which a strategy of J2 is dropped, i.e. σJ2 = σ−J2 R-COA

Proposition 7. Let G = (Γ, σ) and G = (Γ , σ ) such that G ←−−− G . If G is in perfect equilibrium, then G J2 is in perfect equilibrium. Proof. If σ is in perfect equilibrium in Γ , then there exists a sequence of strategies (ˆ σ m )∞ τ m )∞ m=1 as in Defintion 4. Let (ˆ m=1 be a sequence of behavioral strateJ2 m gies in Γ such that each τÎ is identical to σ Îm for all I (= J1 , J2 ). For I = J1 , σ Îm (Al (wi , x)). τÎm (Al (wi , vi )) = x∈Succ(vi ) It is clear that (ˆ τ m )∞ σm )∞

m=1 converges on σ−J2 , given (ˆ m=1 converges on σ .

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Remark 2. The results of Propositions 3–7 can be extended to the case of proper equilibrium with varying degrees of difficulty. In the case of Proposition 3, for example, it is straightforward to check that the conditions for -proper equilibrium are unchanged by the transformation. Not so for Proposition 7; the additive construction of the τÎm ’s will not in general preserve the ratios necessary to ensure -proper convergence. A much more elaborate procedure is needed to construct the appropriate converging sequence. Note, however, that Proposition 7 is uninteresting in the case of proper equilibrium. As discussed above, if G is in proper equilibrium, play at the pre-coalesced node in G will always be rational. So, the “nature” move introduced in G J2 will replace a rational move by player l. We have shown that trembling hand mistakes are equivalent to moves by nature. More specifically, a player who reasons about an opponent as if he may make trembling hand mistakes produces the same rationalizable path of action as a player who faces a combination of rational opponents and moves by nature.

7

Conclusion: Taking Mistakes Seriously

In order to calculate a rational response to a player who makes mistakes, we need a model of mistake making. We discussed two such models implicit in solution concepts in the game theory literature: trembling hand mistakes and risk averse mistakes. We saw that players who make risk averse mistakes act rationally even as they make mistakes. By examining transformations over EFG Scenarios, we found that trembling hand mistakes are equivalent to moves by nature. These two analyses of mistaken play represent extremes: either mistakes are made rationally, or randomly. But do either of these approaches take mistakes seriously? Realistic mistake-making agents may lie somewhere between these extremes. Investigating the strategic consequences of more nuanced analyses of mistake making is a project for future research.

References 1. Elmes, S., Reny, P.J.: On the strategic equivalence of extensive form games. Journal of Economic Theory 62(1), 1–23 (1994) 2. Kohlberg, E., Mertens, J.F.: On the strategic stability of equilibria. Econometrica 54(5), 1003–1037 (1986) 3. Myerson, R.B.: Refinements of the nash equilibrium concept. International Journal of Game Theory 7(2), 73–80 (1978) 4. Myerson, R.B.: Game Theory: Analysis and Conflict. Harverd University Press, Cambridge (1991) 5. Selten, R.: Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4(1), 25–55 (1975) 6. Thompson, F.B.: Equivalence of games in extensive form. Rand Corporation Research Memorandum 789 (1952)

Update Semantics for Imperatives with Priorities Fengkui Ju1, and Fenrong Liu2 1

Department of Philosophy, Beijing Normal University, Beijing, China [email protected] 2 Department of Philosophy, Tsinghua University, Beijing, China [email protected]

Abstract. Imperatives occur ubiquitously in our social communications. In real life we often get conflicting orders issued by different speakers whose authorities are ranked. We propose a new update semantics to interpret the meaning of imperatives with priorities and illustrate what changes they bring about in the addressee’s cognitive state. The general properties of the semantics, as well as its core philosophical ideas are discussed extensively in this paper. Keywords: imperatives, conflicts, update semantics, priorities.

1

Introduction

Imperatives occur ubiquitously in our social communications. To act successfully in a society, we have to fully understand their meaning, as imperatives regulate actions. Logical studies on imperatives have been carried out for some time, and deontic logics belong to such a tradition. From the 1990s, several prominent new frameworks have been proposed. Following the slogan “you know the meaning of a sentence if you know the change it brings about in the cognitive state of anyone who wants to incorporate the information conveyed by it”, update semantics ([9]) was proposed to deal with information update, and it was later applied to imperatives in [8], [6] and [10]. On the basis of deontic logics, [1] made a proposal to study actions that are typically expressed by STIT-sentences “see to it that...”, bringing actions with choices made by agents together. Other recent works in this line are [4], [2] and [3]. Adopting dynamic epistemic logic (DEL) approach, [11] and [12] introduced a new dynamic action of “commanding” to deontic logic, and dealt with imperatives in the framework of dynamic deontic logics. So far, the main purpose of those approaches has been to understand the meaning of one single imperative. Not much attention has been paid to conflicting orders, which were simply taken to be absurd, thus resulting very trivial facts in the existing frameworks. In addition, though agency was introduced to the research agenda, the focus has been always on the addressee, not on the addressors. However, in

The order of the authors’ names is alphabetical, and both authors contributed equally to this work.


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real life, we often get conflicting orders issued by different authorities. Facing such situations, consciously taking the ranks of addressors into account, we form an ordering over possible actions and resolve the difficulties successfully. The ideas in this paper combine two sources: natural language semantics and logics of agency. We hope to show that this is a fruitful mixture of traditions. Let us start immediately with an example: Example 1. A general d, a captain e and a colonel f utter the following sentences, respectively, to a private. (1) The general: Do A! Do B ! (2) The captain: Do B ! Do C ! (3) The colonel: Don’t do A! Don’t do C ! Clearly, these are conflicting orders, w.r.t actions A and C. Intuitively, instead of getting stuck, the private will come up with the following plan after a deliberation: She should do A, do B, but not do C. Her reasoning rests on the following fact: The authorities of d, e and f are ranked as e < f < d. According to this, she can make her decision on which orders to obey, and which ones to disobey. 1 The aforementioned existing frameworks cannot systematically handle such examples. The aim of this paper is to take some recently developed ideas from preference logic, in particular, priority-based preference models for agency (cf. [5] and [7]) to deal with the problem. To simply phrase our ideas in the language of preference, agent’s preference over alternatives is derived from some ordered priorities (or better call it ranked authorities in this context). On our way of developing this idea, we will retain the tradition of update semantics and hope to make a new contribution to natural languages studies. The remaining sections are organized as follows. We will first introduce the basic definitions and techniques of update semantics for imperatives in Section 2. In Section 3, we present our new proposal and study its general properties. In Section 4, we show that introducing ranks of authorities can solve the difficulties we had in Example 1. We also state our background philosophical ideas. We end the paper with some conclusions and possible future directions in Section 5.

2

Force Structures and Track Structures

An update system is a triple L , Σ, [·], where L is a language, Σ a set of information states, and [·] a function from L to Σ → Σ, which assigns each sentence φ an operation [φ]. For any φ, [φ] is called an update function, which is intended to interpret the meaning of φ. The meaning of a sentence lies in how it updates information states — the core idea of update semantics. In his recent work ([10]), Veltman has presented a new semantics for imperatives based on the update semantics, and argued that the meaning of imperatives is an update function on plans. Inspired by [10], [6] interpreted the meaning of 1

Note that [12] also discussed conflicting commands from different authorities.

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imperatives as an update function on force structures. In this section, we introduce an equivalent version of the semantics given in [6] in a different way, and then extend it in the next section and make it work for our problems. Definition 1 (languages). Let Φ be a set of propositional variables, and p ∈ Φ. The standard language of propositional logic Y is defined as follows:2 φ := p | ¬φ | φ ∧ ψ | φ ∨ ψ The language L of imperatives is defined as the set {!φ | φ ∈ Y }. Each finite set T of literals of Y is called a track. A track T is consistent if and only if it does not contain both p and ¬p for any p. Information states are identified with track structures, as defined below. Definition 2 (track structures). A finite set L of tracks is a track structure iff (1) Each T ∈ L is consistent; (2) Any T, T ∈ L contain the same variables. Example 2. The following picture represents a track structure. p4 , ¬p2 , p3 , p1 p4 , p2 , p3 , p1 p4 , p2 , p3 , ¬p1 ¬p4 , ¬p2 , p3 , p1

The reading of track structures is this: For any track structure L, the agent has to choose a track of L and make all literals in it true, but she may freely choose which one. If the agent makes all literals of some track of L true, we say that L is performed. There are two special track structures: {∅} and ∅. The former can always be trivially performed, which is called the minimal track structure. The later can never be performed, which is called the absurd track structure. In what follows we define a procedure which recursively outputs a track structure for any given imperative !φ. To do that, we first introduce the notion of force structures. Definition 3 (force structures). Each finite set K of finite sets of literals of Y is called a force structure. Example 3. {{p4 , ¬p2 }, {p3 }, {p2 , p1 }} is a force structure. 2

We do not consider the connective → as a primitive symbol. The reason is that we will use the language Y to express propositional content of imperatives. In natural languages, imperatives do not take implications as propositional content.

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Definition 4 (tracks of force structures). Let K = {X1 , . . . , Xn } be any force structure. For any Xi , let Xi be the smallest set such that both p and ¬p are in Xi for any p occurring in Xi . T = X1 ∪ . . . ∪ Xn is a track for K iff (1) Xi ⊆ Xi and Xi ∩ Xi = ∅; (2) For any p occurring in Xi , one and only one of p and ¬p is in Xi . Example 4. The picture in Example 2 represents the set of all consistent tracks of the force structure in Example 3. Let K be any force structure, we define functions T + and T − as follows: Definition 5 (T + and T − ). (a) T + (K, p) = T − (K, p) =

{{p}} {X∪{p} | X∈K}

{{¬p}} {X∪{¬p} | X∈K} −

if K = ∅ otherwise if K = ∅ otherwise

(b) T + (K, ¬φ) = T (K, φ) T − (K, ¬φ) = T + (K, φ) (c) T + (K, φ ∧ ψ) = T + (K, φ) ∪ T + (K, ψ) T − (K, φ ∧ ψ) = T − (T − (K, φ), ψ) (d) T + (K, φ ∨ ψ) = T + (T + (K, φ), ψ) T − (K, φ ∨ ψ) = T − (K, φ) ∪ T − (K, ψ)

For any imperative !φ, T + (∅, φ) is called the force structure of it. We see that imperatives correspond to force structures in a recursive way. Example 5. The force structure of the imperative !((p1 ∧ p2 ) ∨ (p3 ∧ p4 )) is {{p1 , p3 }, {p1 , p4 }, {p2 , p3 }, {p2, p4 }}. Let U (φ) be the set of all consistent track of T + (∅, φ).3 U (φ) is the track structure of !φ. Definition 6 (compatibility of track structures). Track structures L1 and L2 are compatible iff (1) For any track T1 ∈ L1 , there is a track T2 ∈ L2 such that T1 ∪ T2 is consistent; (2) For any track T2 ∈ L2 , there is a track T1 ∈ L1 such that T1 ∪ T2 is consistent. Compatibility is used to characterize conflicts among imperatives. Example 6. Two speakers respectively utter these two imperatives to an agent: (a) Close the door or the window! (b) Close the window! 3

Readers may realize that T + (∅, φ) corresponds to a conjunctive normal form (CNF) of φ in propositional logic. That is true. However, please note that the new notions (e.g. compatibility, validity) to be defined on the basis of track structures have very different meaning in this context.


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Intuitively, there is some conflict between these two commands, although they are consistent from the propositional logic point of view. It is easy to verify that the track structures corresponding to the two imperatives are not compatible. Definition 7 (merge of track structures). 4 L1 L2 = {T1 ∪T2 |T1 ∈ L1 , T2 ∈ L2 , and T1 ∪ T2 is consistent} The semantics for imperatives is defined as follows: Definition 8 (update of track structures with imperatives). L U (φ) if L and U (φ) are compatible L!φ = ∅ otherwise

5

Basically, updating a track structure L with an imperative !φ is the merge of L and the track structure of !φ. The exceptional cases are those that L and the track structure of !φ are not compatible. When this case takes place, the result of the update is the absurd track structure ∅. 6 At this point, we would like to return to Example 1. We take the minimal force structure {∅} as the beginning point. After the general’s and captain’s commands, the track structure of the private becomes {{A, B, C}}. This means that the agent has to make A, B and C true. The track structure of the imperative “don’t do A” is {{¬A}}, which is not compatible with {{A, B, C}}. Therefore, after the colonel’s first command, the track structure of the private becomes {∅}. Intuitively, this means that the agent gets stuck, and he will be stuck forever. We see that the semantics given in Definition 8 does not work for Example 1. Similarly, regarding this example, no satisfying solution has been provided in [10] either. This is the starting point of the current work. In our view, to handle such difficulties, we should take the ranks of the speakers into account. Our attempt will follow in the next section.

3

Update with Priorities

3.1

Introducing Authorities

A new update system is a tuple L , Σ, [·], A, ≤, where A is a finite set of speakers, and ≤ is a preorder on A. For any a, b ∈ A, a ≤ b means that b has a rank at least high as what a has. Now we formulate the semantics based on the new update system, incorporating the authorities in the framework presented in the preceding. 4 5

6

This terminology is from [10]. [10] defines meaning of imperatives as an update function on plans. Intuitively, a plan is a set of free choices, and a track structure is also a set of free choices. In this sense, the update defined here is similar to [10]. Their main difference lies in what are viewed as free choices. The notion of validity by the invariance of track structures can solve Ross’s paradox.

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Definition 9 (agent-oriented language). The language L of imperatives is defined as the set {!aφ|φ ∈ Y , a ∈ A}, where Y is the language given in Definition 1. One can see that all imperatives are relative to specific speakers now. Let L be the set of literals of Y . Let L = {la |l ∈ L, a ∈ A}. Each finite set T of L is called a track. We define three properties of tracks below. Definition 10 (resolvability of tracks). A track T is resolvable iff for any pa and pb , if both pa and ¬pb are in T , then either a < b or b < a. Definition 11 (succinctness of tracks). A track T is succinct iff there are no pa and pb such that (1) a < b; (2) Either both pa and pb are in T or both ¬pa and ¬pb are in T . Definition 12 (consistency of tracks). A track T is consistent iff (1) T is succinct; (2) There are no pa and pb such that both pa and ¬pb are in T . The property of consistency is not just stronger than succinctness, but also stronger than resolvability: For any track T , if T is consistent, then it is resolvable, but this might not be the case the other way around. Example 7. Suppose a < b. The track T1 = {pa , qc , rd , ¬pb } is resolvable and succinct, but not consistent. Compared with the ordinary notion of consistency in logic, the notion of consistency defined here seems somewhat heavy, as it contains the notion of succinctness. We do this for technical reasons, which will be explained in Section 4. Intuitively, consistent tracks are “good” ones, while inconsistent tracks are not. Definition 13 (track structures with authorities). A finite set L of tracks is a track structure iff (1) Each T ∈ L is resolvable; (2) For any T, T ∈ L, T and T contain the same variables. Definition 2 defines track structures without authorities, where each track of a track structure is required to be consistent. This requirement gets relaxed for track structures with authorities, namely, each track of a track structure is resolvable. Finally, if all tracks of a track structure are consistent, we call it a consistent one. In the previous section, we describe a procedure by which an imperative !φ corresponds to a track structure, where authorities are not considered. Similarly, we can build up a correspondence between an imperative !a φ and a track structures with authorities. Here we don’t go through the details. We simply use U (!a φ) to denote the track structure of !a φ. 3.2

Update Function and Some Properties

First, again, some technical notions. For any two tracks T and T , we call T a sub-track of T if T ⊆ T . If T is consistent, it is a consistent sub-track.


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Definition 14 (maximally consistent sub-tracks). A track T is a maximally consistent sub-track of a track T iff T is a consistent sub-track of T and for any sub-track T of T , if T ⊂ T , then T is not consistent. Example 8. Suppose b < a. The track {pa , qc , rd } is a maximally consistent subtrack of the track {pa , pb , qc , rd , ¬pe }. Note that the track {pa , pb , qc , rd } is not a maximally consistent sub-track of {pa , pb , qc , rd , ¬pe }, as {pa , pb , qc , rd } is not succinct. Proposition 1. For any track T , all of its maximally consistent sub-tracks contain the same variables as what T has. Proof. Let T be any track. Let T1 be any maximally consistent sub-track of T . Suppose that T and T1 do not contain the same variables. Since T1 ⊆ T , there is a variable, say p, such that T contains p but T1 does not. Then there is a literal li containing p such that li ∈ T but li ∈ / T1 . Since T1 does not contain p, then T1 ∪ {li } ⊃ T1 is consistent and a sub-track of T . Therefore, T1 is not a maximally consistent sub-track of T , which is strange. Hence, T1 and T contain the same variables. Definition 15 (preorder on maximally consistent sub-tracks). Let T be any track. Let T and T be any maximally consistent sub-tracks of T . T T iff for any la ∈ T , there is a lb ∈ T such that lb contains the same variable as what la has, moreover, a ≤ b. It is easy to see that is reflexive and transitive, so it’s a preorder. may not be antisymmetric. Here is a simple counter-example. Let T = {pa , ¬pa }, T = {pa} and T = {¬pa }. Both T and T are maximally consistent sub-tracks of T . T T and T T , but T = T . Hence, might not be a partial order. Definition 16 (strict partial order ≺ on maximally consistent subtracks). Let T be any track. Let T and T be any maximally consistent subtracks of T . T ≺ T iff T T but T T . Lemma 1. Let T be any resolvable track containing only one variable. Let X be the set of its maximally consistent sub-tracks. X has a greatest element under the relation ≺. Proof. We consider two cases. First, we suppose that T is consistent. Then X is a singleton. Clearly, X has a greatest element. Next, we suppose that T is not consistent. Again, there are two possible cases: (1) There are no pa and ¬pb such that both of them are in T ; (2) there are such pa and ¬pb . In the first case, T = {pa1 , . . . , pam } or T = {¬pb1 , . . . , pbm }.7 we can verify that X is a singleton, no matter whether T = {pa1 , . . . , pam } or {¬pb1 , . . . , pbm }. Therefore, X has a greatest element. We consider the second case. Let T = {pa1 , . . . , pam , 7

Note that {pa1 , . . . , pam } and {¬pb1 , . . . , ¬pbn } might not be consistent, because they might not be succinct.

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¬pb1 , . . ., ¬pbn }, where 1 ≤ m, n. T has two maximally consistent sub-tracks: T1 = {pam1 , . . . , pamk }, T2 = {¬pbn1 , . . . , ¬pbnl }, where k ≤ m and l ≤ n. Hence, X = {T1 , T2 }. Suppose that X does not have a greatest element under ≺, then T1 ≺ T2 and T2 ≺ T1 . We can get that for any bj , there is an ai such that ai < bj , because otherwise T1 ≺ T2 . Since T is resolvable, we have that for any bj , there is an ai such that bj < ai . Similarly, for any ai , there is a bj such that ai < bj . Now we can obtain an infinite decreasing chain, say bh < ai < bj < . . .. This is impossible. Therefore, X has a greatest element. Proposition 2. Let T be any resolvable track. Let X be the set of its maximally consistent sub-tracks. X has a greatest element under the relation ≺. Proof. Suppose that T contains n different variables. Let T = T1 ∪ . . . ∪ Tn such that for any Ti , all literals in it contain the same variables. In fact, T ∈ X if and only if T = T1 ∪ . . . ∪ Tn , where each Ti is a maximally consistent sub-track of Ti . By Lemma 1, each Ti has a greatest maximally consistent sub-track under ≺. Let T = T1 ∪ . . . ∪ Tn , where Ti is the greatest maximally consistent sub-track of Ti . We see that T ∈ X. It can be easily verified that T is a greatest element of X under ≺. Example 9. Suppose a < d. The track {¬pd , qb , rc } is a greatest maximally consistent sub-track of {pa , qb , rc , ¬pd } under the relation ≺. The conflict in the track {pa, qb , rc , ¬pd } can be resolved according to the given authority rank a < d and the conflict-free result is {¬pd , qb , rc }. Definition 17 (sub-structures). Let L be any track structure. A track structure L is a sub-structure of L iff (1) For any T ∈ L , there is a T ∈ L such that T ⊆ T ; (2) For any T ∈ L, there is a T ∈ L such that T ⊆ T . If L is consistent, we say that L is a consistent sub-structure of L. Definition 18 (sufficient consistent sub-structures). Let L = {T1 , . . . , Tn} be any track structure. A track structure L is a sufficient consistent sub-structure of L iff L = {T1 , . . . , Tn }, where each Ti is the greatest maximally consistent sub-track of Ti under the relation ≺. By Proposition 1, it can be verified that for any structure L = {T1 , . . . , Tn }, the set {T1 , . . . , Tn }, where each Ti is the greatest maximally consistent sub-track of Ti , is always a consistent sub-structure of L. Therefore, any track structure has a sufficient consistent sub-structure. Furthermore, any track structure has only one sufficient consistent sub-structure. Example 10. Suppose c < e. L is the sufficient consistent sub-structure of L.


pa , qb , rc , ¬sd , ¬re

pa , qb , ¬sd , ¬re

¬pa , qb , rc , sd , ¬re

¬pa , qb , sd , ¬re

pa , ¬qb , rc , ¬sd , ¬re

pa , ¬qb , ¬sd , ¬re

¬pa , ¬qb , rc , ¬sd , ¬re

¬pa , ¬qb , ¬sd , ¬re

L

133

L

Track structure L is not consistent. The sufficient consistent sub-structure of L is the result of making L consistent while respecting the authorities in L to the greatest extent. Definition 19 (merge of track structures). L1 L2 = {T1 ∪T2 |T1 ∈ L1 , T2 ∈ L2 , T1 ∪ T2 is resolvable} Definition 20 (compatibility of track structures). Track structures L1 and L2 are compatible iff (1) For any T1 ∈ L1 , there is a T2 ∈ L2 such that T1 ∪ T2 is resolvable; (2) For any T2 ∈ L2 , there is a T1 ∈ L1 such that T1 ∪T2 is resolvable. We can verify that L1 and L2 are compatible if and only if both L1 and L2 are sub-structures of L1 L2 . For any track structure L, we use V (L) to denote the sufficient consistent sub-structure of L. The semantics for imperatives which takes authorities into account is defined as follows. Definition 21 (update of track structures with authorities). V (L U (!a φ)) if L and U (!a φ) are compatible L!a φ = ∅ otherwise Meaning of imperatives is an update function on track structures. Let L be any track structure and !a φ be any imperative. If L and the track structure L corresponding to !a φ are compatible, the result of updating L with !a φ is the sufficient consistent sub-structure of the merge of L and L , otherwise the result is ∅, which is an absurd track structure.

4 4.1

Illustrations and Background Ideas Illustrations

We illustrate some properties of the semantics defined above. First, let us look at Example 1 again. Recall that a general d, a captain e and a colonel f utter the following sentences, respectively, to a private. (1) The general: Do A! Do B ! (2) The captain: Do B ! Do C ! (3) The colonel: Don’t do A! Don’t do C !

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Suppose that the starting track structure of the private is L0 = {∅}, which means that the private does not bear any imperative force. According to our new semantics, these imperatives update the track structures of the private in the following way. !d A

→

L0 !e C

→

!d B

Ad

→

L1

Ad , Bd , Ce

!f ¬A

→

!e B

Ad , Bd

→

L2 !f ¬C

Ad , Bd , Ce

L4

→

L3 Ad , Bd , ¬Cf

L5

Ad , Bd

L6

Ad , Bd , Ce , ¬Af

Ad , Bd , Ce , ¬Cf

L5

L6

After the imperative !d A, the track structure L0 changes to L1 , and after !d B, L1 changes to L2 , and so on. L6 is the final track structure, according to which the private should do A, B, but should not do C. This is what we expect. The track structures L5 and L6 in the dash rectangles are auxiliary for us to understand the update process, and they are not results of any update of this process. When !f ¬A is uttered, L4 is updated to L5 . Since L5 is not consistent, it changes to L5 after a deliberation of the private. Actually, L5 is equal to L4 . L5 is the sufficient consistent sub-structure of L5 and respects the authority of the general. The similar case also happens to L6 . This example shows how our semantics works in practice. Next, let us consider an example from [10] which involves free choices. Example 11. John is ill and goes to see doctors c and d respectively. (1) The doctor c: Drink milk or apple juice! (2) The doctor d : Don’t drink milk! Suppose that the original track structure of John is L0 = {∅}. First, we suppose that John trusts d more than c. The update process of John’s track structures is illustrated by the following picture. Mc , A c !c (M ∨ A)

→ L0

¬Mc , Ac

!d ¬M

→

¬Md , Ac

Mc , ¬Ac

¬Md , ¬Ac

L1

L2


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L2 is the final result of this update process. According to L2 , John should not drink milk, and he may and may not drink apple juice. As d has a higher authority, this result is perfectly fine. Now we suppose that c has a higher authority than d has. With this constraint, John’s track structures are updated in the following: Mc , A c !c (M ∨ A)

→

¬Mc , Ac

Mc , Ac !d ¬M

→

¬Mc , Ac

Mc , ¬Ac

Mc , ¬Ac

L1

L2

L0

L2 is the final result of this update process. The imperative uttered by d does not essentially make much sense to John, because L1 = L2 . According to L2 , John should drink milk or apple juice, and he may only drink milk, only drink apple juice and drink both. We see that drinking milk is allowed. This result seems not plausible. It seems practically reasonable to think that John should drink apple juice but not drink milk, as if he does so, both of the imperatives could be performed. In other words, only drinking apple juice seems to be safer than only drinking milk or drinking both. However, we think that even not to drink milk is more practically reasonable than to drink milk, John is still allowed to drink milk in this case. We show this point by an example. Example 12. A general and a captain utter the following to a private respectively. (1) The general: You may have a rest. (2) The captain: Move! These two sentences are conflicting. Suppose that the private chooses to have a rest, then normally, he will not be punished. This implies that the private is allowed to stop, even to move is safer for him. Hence, we think that the result mentioned above is plausible. Actually, the following claim seems reasonable: An agent a has a higher authority than what b has in giving commands if and only if a has a higher authority than b in giving permissions. The semantics given in Definition 8 is a special case of the semantics given in Definition 21: When restricted to singleton of agents, the latter collapses to the former. Note that the semantics defined in Definition 8 does not satisfy the property of commutativity.8 Therefore, the semantics given in Definition 21 does not have that property either. Previously, we have taken tracks which are succinct and do not contain any conflict as “good” tracks. The reason why we require succinctness is out of the following consideration: Without succinctness, some thing “bad” could happen, in which commutativity plays a role. Here is an example. 8

See [6] for examples.

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Example 13. Consider some agent a, his grandmother has a higher authority than his parents, and his father and mother have the same authority. Here are two sequences. (1) Grandmother: Stop! Father: Don’t stop! Mother: Stop! (2) Grandmother: Stop! Mother: Stop! Father: Don’t stop! If we keep all other things unchanged and just drop the succinctness requirement, the second sequence makes the agent get stuck, while the first one does not. This is weird. 4.2

Imperative Forces and Authorities

In this subsection we are going to state our intuitions and main ideas in understanding imperatives in natural languages. In practice, any kind of sentences are uttered by specific speakers. We think that speakers contribute to the meaning of sentences. We first explain our ideas in terms of indicatives. According to the meaning theory of dynamic semantics, the meaning of a sentence lies in how it changes an agent’s information state. However, one same indicative may change an agent’s information state very differently, if it is uttered by different speakers. For an example, consider the situation in which the sentence uttered conflicts with the agent’s knowledge. Following the same philosophy, we think that meaning of an imperative also lies in how it changes an agent’s cognitive state, more specifically this time, imperative force state. Imperative force states are states of imperative forces which agents bear. Imperatives have propositional content. Imperatives produce imperative forces, which tend to “push” the agent to make their propositional content true. Uttering an imperative may change the imperative force state of the agent. Similarly, one same imperative may cause different changes to an agent’s imperative force state, if the imperative is uttered by different speakers. This is the reason that we introduce authorities into the semantics for imperatives. Technically, we use track structures to characterize states of imperative forces, as explained in Section 2. We do not distinguish state of imperative force and imperative force conceptually, and they are considered to be the same. Imperative forces produced by imperatives are also characterized as track structures. An imperative is in force if and only if (1) The agent has to make the propositional content of the imperative true; (2) The agent may make the propositional content of it true in any way. Particularly, the imperative “drink milk or apple juice” is in force if and only if the agent has to drink milk or apple juice, he may drink milk but not drink apple juice, he may drink apple juice but not drink milk, and he may drink both. About imperative force, there is one more thing which we want to emphasize. Whether an agent is bearing some imperative force is not objective, but determined by the agent’s mind. The word “imperative force” might be misleading, as it reminds us of physical forces. Instead of saying that an agent is bearing some imperative force, we should say that, the agent thinks that he is bearing some imperative force. Consider the following example.


137

Example 14. A general and a captain utters the following to a private. (1) The general: Move! (2) The captain: Stop! The private would think that the imperative of the captain is not in force. Finally, some comments on authority order. First of all, authority order is relative to specific agents. Two speakers might be ranked differently from one agent to another. For example, two doctors might be in different authority relations for different patients. Secondly, authority order is not fixed universally. More specifically, it depends on specific contexts. Speaker a might have a higher authority than speaker b for agent c in one context, but b might have a higher authority than a in another context. For instance, suppose that a is c’s father, and b is c’s mother. Suppose that a is a general, b is a colonel and c is a private in the same army. In army, a has a higher authority than b for c, but b might have a higher authority than a in family.

5

Conclusions and Future Work

Motivated by the realistic examples that involves conflicting orders, we have introduced authorities explicitly into the logical language, and proposed a new semantics for imperatives. It combined ideas from natural language semantics and logics for agency. We think that the resulting picture of “force” and “authority” is more realistic, both in understanding imperative discourse and commands driven human action. We have applied the new semantics to analyzing many examples in the paper. Some general properties of the new semantics have been proved. In addition, we have discussed extensively our background ideas. Nevertheless, our investigation is still on the side of semantics, next on our agenda are the following issues: (i) Thinking syntactically, we would like to see whether there is a complete logic for this update system with priorities. (ii) The issue studied in our paper seems to be closely related to normative conflicts in the literature of deontic logics, and we would like to compare our proposal and those in that tradition. (iii) Finally, we would like to extend our framework to the situation in which knowledge plays a role in understanding imperatives, shown by the example: Example 15. A general and a captain utter the following imperatives to a private respectively in some circumstance. (1) The general: Save the boy! (2) The captain: Don’t make him get hurt! This is considered conflict-free in our framework. But suppose the reality is that the private can not save the boy without making him get hurt to some extent. In this circumstance, the private can not perform these two imperatives, as there is

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a conflict after all. What is going on here? In fact what the agent knows about the world matters. We would like to take the agent’s knowledge into account, and model the interaction between the agent’s knowledge and imperative forces. Acknowledgement. The authors would like to thank Maria Aloni, Johan van Benthem, Frank Veltman, Tomoyuki Yamada, the anonymous LORI referees, and the audience of the Workshop on Modal Logic, Language and Logical Dynamics in June 2011 in Beijing, for their useful comments. Fengkui Ju is supported by The Major Bidding Project of the National Social Science Foundation of China (NO. 10&ZD073). Fenrong Liu is supported by the Project (NO. 09YJC7204001) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References 1. Belnap, N., Perloff, M., Xu, M.: Facing the Future. Oxford University Press, Oxford (2001) 2. Broersen, J., Herzig, A., Troquard, N.: Embedding alternating-time temporal logic in strategic STIT logic of agency. Journal of Logic and Computation 16, 559–578 (2006) 3. Herzig, A., Troquard, N.: Knowing how to play: Uniform choices in logics of agency. In: Stone, P., Weiss, G. (eds.) Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 209–216. ACM, New York (2006) 4. Horty, J.: Agency and Deontic Logic. Oxford University Press, Oxford (2001) 5. de Jongh, D., Liu, F.: Preference, priorities and belief. In: Grune-Yanoff, T., Hansson, S. (eds.) Preference Change: Approaches from Philosophy, Economics and Psychology. Theory and Decision Library, pp. 85–108. Springer, Heidelberg (2009) 6. Ju, F.: Imperatives and logic. Studies in Logic 3(2), 361–379 (2010) 7. Liu, F.: A two-level perspective on preference. Journal of Philosophical Logic 40, 421–439 (2011) 8. Mastop, R.: What Can You Do? Imperative Mood in Semantic Theory. Ph.D. thesis, ILLC, University of Amsterdam (2005) 9. Veltman, F.: Defaults in update semantics. Journal of Philosophical Logic 25, 221–261 (1996) 10. Veltman, F.: Imperatives at the borderline of semantics and pragmatics (2010) (manuscript) 11. Yamada, T.: Acts of commands and changing obligations. In: Inoue, K., Satoh, K., Toni, F. (eds.) Proceedings of the 7th Workshop on Computational Logic in Multi-Agent Systems, pp. 1–19. Springer, Heidelberg (2006) 12. Yamada, T.: Logical dynamics of some speech acts that affect obligations and preferences. Synthese 165(2), 295–315 (2008)

A Measure of Logical Inference and Its Game Theoretical Applications Mamoru Kaneko1 and Nobu-Yuki Suzuki2 1

Institute of Policy and Planning Sciences, University of Tsukuba, Japan [email protected] 2 Faculty of Science, Shizuoka University, Shizuoka, Japan [email protected]

Abstract. This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculus. The definition of the measure takes two steps: First, we measure the width of a given proof. Then the measure of inference assigns, to a given sequent, the minimum value of the widths of its possible proofs. It counts the indispensable cases for possible proofs of a sequent. This measure expresses the degree of difficulty in proving a given sequent. Although our problem is highly proof-theoretic, we are motivated by some general and specific problems in game theory/economics. In this paper, we will define a certain lower bound function, with which we may often obtain the exact value of the measure for a given sequent. We apply our theory a few game theoretical problems and calculate the exact values of the measure. Keywords: Classical Logic, Intuitionistic Logic, Gentzen-style Sequent calculus, Game Theoretic Decision Making.

1

Introduction

This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculus (Gentzen [2], [3]). The definition of the measure takes two steps: For each proof (tree) P , we measure the width, i.e., the number of leaves, of P. Then the measure of inference assigns, to a given sequent σ = Γ → Θ, the minimum value in the widths of possible proofs of σ, if σ is provable, and if not, the assigned value is +∞. It counts the indispensable cases for possible proofs of σ. By this measure, we represent the degree of difficulty in proving a given sequent. Although our problem is highly proof-theoretic, we are motivated by problems in game theory/economics. Here, we explain, first, our motivation; and second, the contribution of this paper. Lastly, we present one game theoretical example, to which our theory will be applied in Section 5. The aim of game theory/economics is to study human behavior and decisionmaking in a game/social situation. It is more directly related to human activities than mathematics. The importance of bounded rationality has been emphasized in the economics literature since Simon [11]. Simon himself criticized the assumption of super-rationality for economic agents’ decision making, but touched only H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 139–150, 2011. c Springer-Verlag Berlin Heidelberg 2011

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a particular perceptual form of bounded rationality. Since then, only scattered approaches have been given. The aim of this paper is to provide a conceptual tool for a study bounded rationality from the viewpoint of logic. It is also related to the problem of logical omniscience/omnipotence, which will be mentioned below. One important aspect of bounded rationality is the logical inferential ability of a player (person). As soon as, however, we consider provability of a logical system, we effectively assume super-rationality and logical omnipotence. To discuss bounded rationality, we should consider how complex inferences are required for a given problem. Game theoretical problems are mathematically formulated, and players involved there are implicitly assumed to have mathematical inferential abilities. Such mathematical inferences are explicitly studied in proof theory. In this sense, proof theory is a suitable field for a study of bounded rationality. Our approach looks related to the computational complexity in computer sciences. This is formulated as a question of how the required time and memory size increase as the length of input data increases. The approach of proof complexity (the lengths of proofs) is along this line; the literature has focused on the size of a required algorithm - - see Krajiˇcek [9] and Pudl´ ak [10]. In these approaches, algorithms are compared by their limiting behaviors, while we focus on measuring inferences required for each single instance of a sequent but not on the performance of an algorithm. Our approach is well understood from the viewpoint of bounded rationality in game theory/economics. For this, we should mention two related literatures: epistemic logics of shallow (interpersonal) depths (Kaneko-Suzuki [6]); and inductive game theory (Kaneko-Kline [4]). We will discuss the first in several places i n this paper , and the second only in Section 7. Let us see our approach briefly. The measure of inference, denoted by η L∗ , is defined in classical and intuitionistic logics L = CL or IL in Gentzen’s [2], [3] sequent calculus. We have four types of those measures, depending upon L = CL or IL, and with or without cuts, i.e., η CLw , η CLf , η ILw and η ILf . We are interested in giving a method to calculate the exact value η L∗ (σ) for an arbitrary given sequent σ = Γ → Θ. Finding a proof P of σ is not enough for the calculation of η L∗ (σ), since it gives only its upper bound for η L∗ (σ). We give the lower bound method (LB-method) to calculate η L∗ (σ). We present Theorem 2 that this function β L gives a lower bound β L (σ) of η L∗ (σ) for any sequent σ. This β L gives often the exact value of η L∗ (σ). We adopt classical and intuitionistic logics as the environments for our study. Classical logic is the basic reference point, but intuitionistic logic is also important for us since it is of constructive nature and game theoretical decision making is directly related to constructiveness of a choice. Also, proofs with cuts will be important for further developments of our study in various manners, though the cases of cut-free proofs are easier than those with cuts. In this paper, we do not directly touch epistemic logic of shallow depths of Kaneko-Suzuki [6], but the main result of this paper (Theorem 2) can be extended to it. Here, we provide a small game theoretic example to motivate ourselves more. Consider the situation where a large store, 1 (a supermarket), and a small store,

A Measure of Logical Inference and Its Game Theoretical Applications

141

2 (a minimart) are competing. Store 2 has the subjective understanding of the situation: Store 1 is large enough to ignore store 2, but store 2’s profits are influenced by 1’s choice. 2’s understanding is described by Tables 1 and 2. Store 1 has only three alternative actions, and his payoff is determined by his own choice. On the other hand, 2 has 10 alternative actions, and the resulting payoffs are determined by the choices of both 1 and 2. Table 1; g1 a1 6000 a2 2000 a3 1000

s1 1 5 5

Table 2; g2 s2 s3 · · · s9 s10 2 3 · · · 9 10 6 7 · · · 13 14 7 9 · · · 21 23

In Table 2, store 2 has a dominant action, s10 , which gives the highest payoff whatever 1 chooses. To achieve this knowledge, he compares the payoff from s10 with those from s1 , ..., s9 in all the three cases of a1 , a2 and a3 ; hence it needs at least 9 × 3 = 27 comparisons. In this consideration, he thinks only about Table 2 but not about Table 1. Store 2 has an alternative decision criterion: First, he predicts the choice by store 1, and, using his prediction he chooses an action. For this criterion, he needs 2 comparisons to predict that 1’s choice would be a1 , and then he needs to make at least 9 comparisons to verify that s10 is the best toward a1 . Here, the minimum number of required comparisons is 11. There is a trade-off: The concentration on his own payoff matrix does not require interpersonal inferences, but interpersonal considerations may simplify his decision-making with respect to the number of payoff comparisons. This argument will be described in terms of the measure ηL∗ of inference in Section 5. The above example shows two important aspects of bounded rationality. In the game theory literature, it is regarded as necessary to assume that the game structure is common knowledge between the players. However, no common knowledge is required, but only interpersonal thinking of very shallow depths are required, which is captured in epistemic logic of shallow (interpersonal) depths (KanekoSuzuki [6]). This paper takes one step further to measure intrapersonal complexity for decision making in such a situation. Thus, the trade-off mentioned above can be discussed in a meaningful manner. In this sense, the contribution of the paper is to put one step forward in the direction of a study of bounded rationality.

2

Classical and Intuitionistic Logics

We present classical and intuitionistic logics CL and IL. We adopt the following list of primitive symbols: countably infinite number of propositional variables: p0 , p1 , ...; logical connective symbols: ¬ (not), ⊃ (implies), ∧ (and), ∨ (or); parentheses: ( , ); comma: , ; and braces { , }.

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We define formulae inductively: (o): any propositional variable p is a formula; (i): if C, D are formulae, so are (C ⊃ D) and (¬C); (ii): if Φ is a finite set of formulae with its cardinality |Φ| ≥ 2, then (∧Φ) and (∨Φ) are formulae. We denote the set of all formulae by P. Note that (ii) is not standard; this simplifies our game theoretical arguments since conjunctions and disjunctions consisting of many formulae appear often in game theoretical applications. This will change slightly the rules for ∧ and ∨ in the formulation of sequent calculus. Let Γ, Θ be finite (possibly empty) sets of formulae in P. Using auxiliary symbol →, we introduce a new expression Γ → Θ, which we call a sequent. We abbreviate (set-theoretical) braces, for example, {A} ∪ Γ → Θ ∪ {B} is written as A, Γ → Θ, B, and also, Γ ∪ Δ → Θ ∪ Λ is abbreviated as Γ, Δ → Θ, Λ. The logical inferences are governed by one axiom schema and various inference rules. Axiom Schema (Initial Sequents): A → A, where A is any formula. Structural Rules: The following inference rules are called the thinning and cut : Γ →Θ (th) Δ, Γ → Θ, Λ Γ → Θ, A A, Δ → Λ (cut) Γ, Δ → Θ, Λ In (th), the sets Δ and Λ may be empty. Operational Rules: Γ → Θ, A (¬ →) ¬A, Γ → Θ Γ → Θ, A B, Δ → Λ (⊃→) A ⊃ B, Γ, Δ → Θ, Λ A, Γ → Θ (∧ →) where A ∈ Φ ∧Φ, Γ → Θ {A, Γ → Θ : A ∈ Φ} (∨ →) ∨Φ, Γ → Θ

A, Γ → Θ (→ ¬) Γ → Θ, ¬A A, Γ → B, Θ (→⊃) Γ → A ⊃ B, Θ {Γ → Θ, A : A ∈ Φ} (→ ∧) Γ → Θ, ∧Φ Γ → Θ, A (→ ∨) where A ∈ Φ. Γ → Θ, ∨Φ

In (→ ∧) and (∨ →), the multiple upper sequents indexed with A ∈ Φ are assumed to be proved. This change from the standard formulation is needed by adopting the applications of ∧ and ∨ to a set of formulae Φ. Since we measure the complexity of a proof, we need an explicit definition of it. A proof P in CL is defined as a triple (X,

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