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The Dynamics of Imperfect Information The Dynamics of Imperfect Information Pietro Galliani The Dynamics of Imperfect Information ILLC Dissertation Series DS-200X-NN For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Science Park 904 1098 XH Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Copyright c 2012 by Pietro Galliani Printed and bound by GVO Drukkers & Vormgevers B.V. ISBN: 978–90–9026932–0 The Dynamics of Imperfect Information Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op vrijdag 21 september 2012, te 13.00 uur door Pietro Galliani geboren te Bologna, Itali¨e. Promotor: Prof. dr. J. V¨a¨an¨anen Overige leden: Prof. dr. S. Abramsky Prof. dr. J.F.A.K. van Benthem Dr. A. Baltag Dr. U. Endriss Prof. dr. E. Gr¨adel Prof. dr. D.H.J. de Jongh Prof. dr. B. L¨owe Faculteit der Natuurwetenschappen, Wiskunde en Informatica to Dino and Franca Davoli thanks for everything v Contents Acknowledgments xi 1 Introduction 1 2 Logics of Imperfect Information 9 2.1 From Branching Quantifiers to Dependence Logic . 9 2.1.1 BranchingQuantifiers . 9 2.1.2 IndependenceFriendlyLogic . 11 2.1.3 DependenceLogic .. .. .. .. .. .. .. .. .. 13 2.2 DependenceLogicanditsExtensions. 14 2.2.1 TeamSemantics ....................... 14 2.2.2 SomeKnownResults. 17 2.2.3 GameTheoreticSemantics . 20 2.3 SensibleSemantics .. .. .. .. .. .. .. .. .. .. .. 25 2.3.1 The Combinatorics of Imperfect Information . 25 2.3.2 Sensible Semantics of Imperfect Information . 28 2.4 ExtensionsofDependenceLogic . 33 2.4.1 IndependenceLogic . 34 2.4.2 Intuitionistic and Linear Dependence Logic . 35 2.4.3 TeamLogic .......................... 36 3 Announcement Operators 39 3.1 SomeStrangeOperators . 39 3.1.1 1, 1 and δ1 ......................... 39 ∃ ∀ 3.1.2 κ and δκ ........................... 43 ∀ 3.2 A Game Theoretic Semantics for Announcement Operators . .. 46 vii 3.2.1 Game Theoretic Semantics for (δ1)............ 46 D 3.2.2 Game Theoretic Semantics for δκ .............. 48 3.3 Some Properties of Public Announcements . 50 3.3.1 An Ehrenfeucht-Fra¨ıss´egame for ( , κ)......... 50 D ⊔ ∀ 3.3.2 UniformDefinability . 55 4 Dependencies in Team Semantics 59 4.1 ConstancyLogic ........................... 59 4.2 Multivalued Dependence Logic is Independence Logic . ... 63 4.3 InclusionandExclusioninLogic . 66 4.3.1 Inclusionand ExclusionDependencies . 66 4.3.2 InclusionLogic .. .. .. .. .. .. .. .. .. .. 72 4.3.3 EquiextensionLogic . 80 4.3.4 ExclusionLogic.. .. .. .. .. .. .. .. .. .. 81 4.3.5 Inclusion/ExclusionLogic . 84 4.4 GameTheoreticSemanticsforI/ELogic. 91 4.5 Definability in I/E Logic (and in Independence Logic) . ... 94 4.6 Announcements, Constancy Atoms, and Inconstancy Atoms ... 99 5 Proof Theory 103 5.1 GeneralModels............................103 5.2 EntailmentSemantics . 111 5.3 TheProofSystem ..........................116 5.4 AddingMoreTeams . .. .. .. .. .. .. .. .. .. .. .124 5.5 Conclusions ..............................126 6 Transition Dynamics 129 6.1 OnDynamic Game Logicand FirstOrderLogic . 129 6.1.1 DynamicGameLogic . 129 6.1.2 TheRepresentationTheorem . 132 6.2 TransitionLogic ...........................133 6.2.1 A Logic for Imperfect Information Games Against Nature 133 6.2.2 A Representation Theorem for Dependence Logic . 138 6.2.3 TransitionDependenceLogic . 144 6.3 DynamicSemantics. 146 6.3.1 DynamicPredicateLogic . 146 6.3.2 DynamicDependenceLogic . 149 6.3.3 Game Theoretic Semantics for Dynamic Dependence Logic151 viii 7 The Doxastic Interpretation 159 7.1 BeliefModels .............................159 7.2 AtomsandFirstOrderFormulas . 162 7.3 BeliefUpdates ............................169 7.4 Adjoints................................173 7.5 Minimalupdates ...........................175 7.6 Quantifiers ..............................178 8 Conclusions 183 Bibliography 185 Index 193 Samenvatting 197 Abstract 199 ix Acknowledgments First of all, I would like to thank my supervisor Jouko V¨a¨an¨anen: his advice and encouragement during all these years has been nothing short of invaluable. Furthermore, I would also like to thank all the members of the LINT (Logic for Interaction) LogICCC subproject for many stimulating discussions, and the Eu- ropean Science Foundation as a whole for this fascinating and highly successful project. The Dependence Logic research community as a whole has been extremely welcoming, and an endless source of inspiration and suggestions. In particular, I wish to thank Samson Abramsky, Dietmar Berwanger, Denis Bonnay, Fredrik Engstr¨om, Erich Gr¨adel, Lauri Hella, Jarmo and Juha Kontinen, Antti Kuu- sisto, Allen Mann, Jonni Virtema, and Fan Yang for many, many interesting and fruitful discussions. I also thank Alexandru Baltag, Johan van Benthem, Dietmar Berwanger and Dag Westerst˚ahl for many useful suggestions and com- ments on my work. Furthermore, I thank Karin Gigengack, Tanja Kassenaar, Peter van Or- mondt and Marco Vervoort for helping me in many practical matters, and Stefanie Kooistra for translating the abstract of this thesis into Dutch. Moreover, I thank Fausto Barbero, Vincenzo Ciancia, C´edric D´egremont, Sujata Ghosh, Nina Gierasimvzuk, Umberto Grandi, Lauri Keskinen, Lena Kurzen, S¸tefan Minic˘a, Daniele Porello, Federico Sangati, Jakub Szymanik, Paolo Turrini, Fernardo Vel´azquez-Quesada and Joel and Sara Uckelman; and finally, I thank my mother Chiara and my father Pierluigi for sending me many packages of excellent Italian cheeses and sausages, a fundamental source of in- spiration and motivation for me during all of my studies. Amsterdam Pietro Galliani April, 2012 xi Chapter 1 Introduction Logics of imperfect information are extensions of First Order Logic1 in which very general patterns of dependence and independence between logical opera- tions and/or between variables are allowed. Among these logics, Dependence Logic is particularly suitable for the study of the very notion of dependence, be- cause it represents dependence of variables directly by means of special atomic formulas. After the introduction of Dependence Logic in 2007, a considerable amount of results (some of which we will summarize in Chapter 2) have been ob- tained about it and its extensions. In this thesis we solve several open problems of the area and suggest new ways to think about this family of logics. Logics of imperfect information admit a Game Theoretic Semantics, an im- perfect information generalization of the Game Theoretic Semantics for First Order Logic; and furthermore, they also admit an equivalent Team Semantics (also referred to in the literature as Hodges Semantics or Trump Semantics), which instead generalizes Tarski’s semantics for First Order Logic. Team Se- mantics extends Tarski’s semantics by defining the satisfaction relation not in terms of single assignments but in terms of sets of assignments, called teams. This thesis is a Team Semantics-centered exploration of the properties of variants and extensions of Dependence Logic. Our two principal claims, for which we will build gradually support through this whole work and which will find their most general formulations in Chapters 6 and 7, are the following: 1. Teams represent information states; 2. Formulas in Dependence Logic and its variants can be interpreted in terms of transitions between information states. 1Or, more rarely, of other logics: see for example the Modal Dependence Logic of [67], or the Independence-Friendly Modal Logic(s) of [64, 6]. 1 2 Chapter 1. Introduction The first claim is not new, and in a way it is already implicit in Hodges’ proof of the equivalence between Team Semantics and Game Theoretic Semantics. However, what is (to the knowledge of the author) new is the idea that the teams-as-information-states interpretation of Team Semantics can (and, in the opinion of the author, should) be used as the main driving impulse towards the further development of this fascinating area of research, as we try to do in this work. Chapter 2 The second chapter is a brief introduction to the study of logics of imperfect information. First, in Section 2.1, we recall the history of the development of such logics, from the early days of Branching Quantifiers Logic until the creation of Dependence Logic. This account is neither complete nor impartial: more could certainly be said about the development of Independence Friendly Logic, for example, during which many of the salient peculiarities of logics of imperfect information were first isolated. Furthermore, we will say nothing about modal logics of imperfect information such as IF Modal Logic or Modal Dependence Logic: indeed, even though such formalisms are certainly of no small interest, the present work will be exclusively concerned with first order logics of imperfect information. Then, in Section 2.2, we introduce formally Dependence Logic, its Team Semantics, its Game Theoretic Semantics, and some of its main properties. Our presentation here is essentially a summarized and updated version of the introduction to Dependence Logic contained in [65]. The principal differences between our approach and the one of V¨a¨an¨anen’s book (to which we encourage the reader to refer for a more in-depth introduction to the field) are the following: 1. We assume that all formulas
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