Handbook of Epistemic Logic

674 Pages·2015·45.072 MB·English

by  Hans van Ditmarsch (editor), Joseph Y. Halpern (editor), Wiebe van der Hoek (editor), Barteld Kooi (editor)

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Handbook of Epistemic Logic
Edited by
Hans van Ditmarsch
Joseph Y. Halpern
Wiebe van der Hoek
Barteld Kool
© Individual authors and College Publications 2015. All rights reserved.
ISBN 978-1-84890-158-2
College Publications
Scientific Director: Dov Gabbay
Managing Director: Jane Spurr
http:/Avww.collegepublications.co.uk
Cover produced by Laraine Welch
Printed by Lightning Source, Milton Keynes, UK

All rights reserved, No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise
without prior permission, in writing, from the publisher.
Contents
Preface xiii
Contributors XV
1 An Introduction to Logics of Knowledge and Belief 1
Hans van Ditmarsch, Joseph Y. Halpern, Wiebe van der Hoek and Barteld
Kooi
1.1 Introduction to the Book... ...............0.0.4
er
1.2. Basic Concepts and Tools .................04 N
O
1.2.1 Language .. 2.
O
1.2.2 Semantics .. 2... 2... ee C
1.2.3 Expressivity and Succinctness .. 2... 0.00004 21
1.2.4 Reasoning problems ...............004 26
1.2.5 Axiomatisation .... 0.0.000.0.. ee.e ee 32
13 Overview of the Book 2... 0.0... . 0... 0.000 A2
L4 Notes . 2... 45
References 00. 49
I Informational Attitudes 53
2 Only Knowing 55
Gerhard Lakemeyer and Hector J. Levesque
2.1 Introduction. 2...
2.2 Single-Agent Only Knowing ...........00.000.
2.2.1 Syntax and Semantics ........0.....00..
2.2.2 Some properties of OL 2... ee
2.2.3 Proof Theory... 0...
2.24 Only Knowing and Autoepistemic Logic .......
2.2.5 Honesty 2.
vi CONTENTS
2.3 Multi-Agent Only Knowing ............0.0.04 65
2.3.1 The Canonical-Model Approach. ........... 66
2.3.2 Proof Theory .. 2.0... 0. ee 68
| 2.3.3 Honesty Revisited ©... 0.0.0.0... 0020004 69
24 Notes 2... 72
References... 0 74
3 Awareness 77
Burkhard C. Schipper
3.1 Introduction. 2... 77
3.2 Preliminary discussion .. 0... 0.0ee0 ee. 79
| 3.2.1 Some properties of awareness in natural languages . 79
| 3.2.2. Some properties of awareness in a formal language . 80
3.3 Awareness and knowledge ...........0....00004 82
| 3.3.1 Awareness structures 2.0.0.0e.e .ee.e 82
| 3.3.2 Impossibility of unawareness in Kripke frames... . 91
| 3.3.3 Unawareness frames 2... 0. ee 94
: 3.3.4 Unawareness structures . 1... 0..e.e e es 100
3.3.5 Generalized standard models ............. 104
3.3.6 Product models... 0... 0.0.0.0 .0..000.40 , 107
3.4 Awareness and probabilistic belief... ......0.000., 112
3.4.1 Type spaces with unawareness ............, 112
3.4.2 Speculative trade and agreement .....,..... 116
3.5 Awareness of unawareness .. 0... 0. ee 118
3.5.1 Propositional quantifiers and extended awareness
structures ©. ao 118
3.5.2 First-order logic with unawareness of objects .... 121
3.5.3 Neighborhood semantics and first-order logic with a-
WATENESS 127
3.5.4 Awareness of unawareness without quantifiers .... 129
3.6 Notes 2... es 133

References... 00. Ds 141
4 Epistemic Probabilistic Logic 147
Lorenz Demey and Joshua Sack
4,1 Introduction... 0... ee 148
4,2 Probabilistic Propositional Logic .... 0.0.0.0... 050. 150
4.2.1 Probabilistic Models 2... 151
4.2.2 Language and Semantics. 2... 0... 0. 152
4.2.3 The Expressivity of Linear Combinations ...... 155
CONTENTS vii
4.2.4 ProofSystem .. 0... ee 158
4.2.5 Decidability and Complexity .........0.... 160
4.3 Probabilistic Epistemic Logic .........0.0, 0.0004 161
4.3.1 Probabilistic Relational Models ..........2.. 161
4.3.2 Language and Semantics... 1... ee 162
4.3.3 Expressivity of Linear Combinations ......... 168
4.3.4 ProofSystem . 2... ee 169
4.3.5 Decidability and Complexity ............. 170
4.4 Probabilistic Interpreted Systems ............00. 171
4.4.1 Adding Probabilities to Interpreted Systems ..... 171
44.2 Language and Semantics... 2.0.0.0..00.0. 173
4.4.3. Temporal and Dynamic Epistemic Logic ....... 174
4.5 Probabilistic Public Announcement Logic .......... 176
4.5.1 Language and Semantics... ..........005. 176
4.5.2 Proof System .. 0... 0. 181
4.5.3 Decidability and Complexity .........0.., 182
4.5.4 Higher-Order Information in Public Announcements 182
4.6 Probabilistic Dynamic Epistemic Logic. .........0.. 185
4.6.1 Probabilistic Product Update .......0..0.020. 185
4.6.2 Language and Semantics... . 2.0.0.2. .0005 189
4.6.3 ProofSystem ... 2. 191
4,7 Further Developments and Applications ........... 193
4.8 Conclusion. 2... 194
4.9 Notes . 2... 195
References. 0 0 198
II Dynamics of Informational Attitudes 203
5 Knowledge and Time 205
Clare Divon, Cléudia Nalon and Ram Ramanujam
5.1 Introduction. 2... 206
5.1.1 A temporal logic of knowledge ............ 206
5.1.2 Structural investigations... 207
0.2 Language 2... 209
B.2.1 Syntax... 209
5.2.2 Semantics 6... 0. ee 210
5.3 Complexity 2... 213
5.4 Expressiveness 2.00 217
5.4.1 First order fragments. 2... 217
5.4.2 Quantified extensions ..........0. 0004 219
vill CONTENTS
5.5 Axiomatisation . 2... 219
5.6 Reasoning... 221
5.6.1 Resolution... 0.0... 223
5.6.2 Tableau... 227
5.7 Knowledge and communication ............000.4. 233
5.7.1 The history based model... .........00.., 234
5.8 Levels of knowledge... 0. 0 235
5.9 Knowledge based semantics ........0..0. 0.00040, 237
5.10 Automata theory of knowledge .............004 239
5.10.1 Epistemic automata ........2..0.... .00 4 240
5.10.2 Finite state automata . 2... 2... 0.00e.ee 240
5.10.3 Distributed alphabets ................ ., 241
5.10.4 Epistemic transition systems ...........04 242
5.10.56 Behaviours 2... ee 245
5.11 Applications... 0. ee 247
5.12 Inconclusion ©. 249
5.13 Notes 2 ee 250
References. .... ee 254
6 Dynamic Epistemic Logic 261
Lawrence S. Moss
6.1 Introduction... 0. ee 262
6.1.1 Scenarios 2... 263
6.1.2 General notes on this chapter ........... .. 267
6.2 Public Announcements and PAL .............0.4 268
6.2.1 Public Announcement Logic PAL ..... eas 271
6.2.2 Announcing a true fact might falsify io. 273
6.2.3 Sound and complete logical systems ......... 275
6.2.4 Succinctness and complexity. ............0. 282
6.2.5 The substitution core ..........0 0.0005 284
6.3 AdditionstoPAL 2.. 00.00..00.. e.ee 284
6.3.1 Common knowledge and relativized common knowledge285
6.3.2 Iterated announcements ........., as 288
6.3.3 Arbitrary announcement logic... .....0.004, 290
6.4 More General Epistemic Actions ............04., 291
6.4.1 Private announcements and more general epistemic
ACTIONS . 291
6.4.2 The update product .......0..... 0.00 . 293
6.4.3 Slower version of the update product ......... 296
6.4.4 DEL based on epistemic actions ........... 299
6.5 Conclusions... 0 ee 305
CONTENTS ix
6.6 Notes... 307
References... ......0.. ee 310
7 Dynamic Logics of Belief Change 313
Johan van Benthem and Sonja Smets
7.1 Basics of belief revision. . 2... ee 314
7.1.1 The AGM account of belief revision ........., 314
7.1.2 Conditionals and the Ramsey Test .......... 317
7.2 Modal logics of belief revision... . 0.0. ee ee 318
7.3 Static base logics 2... 322
7.3.1 Static logic of knowledge and belief... ......, 322
7.3.2 Plausibility models... 2.0.2...... 0.000 4 323
7.4 Belief revision by model transformations ........... 330
7.4.1 From knowledge to belief, hard and soft information 330
7.4.2 Belief change under hard information. ........ 331
7.4.3 From dynamics to statics: safe and strong belief... 332
7.4.4 Belief change under soft information: radical upgrade 333
7.4.5 Conservative upgrade and other revision policies .. 336
7.5 General formats for belief revision... . 0... 00.0000. 337
7.5.1 Relation transformers as PDL programs........ 337

7.5.2 Product update in general dynamic-epistemic logic . 338
7.5.3 Priority update... 2... ee 342
7.6 Belief revision and probability dynamics .........., 345
7.7 Time, iterated update, and learning... ........... 349
7.7.1 Epistemic temporal logic. 2... 2... 350
7.7.2 Protocols in dynamic epistemic logic ........., 353
7.7.3 Representing DEL inside temporal logic... ..... 355
7.74 Beliefs over time ... 0.0.0e.e 0.ee 356
7.7.5 Iterated belief upgrades and limit behavior ..... 357
7.7.6 From belief revision to formal learning theory .... 360
7.8 Belief, games, and social agency... ........0,00 . 363
7.8.1 erated beliefs in games... 0...e e, 363
7.8.2 Belief change in social networks. ........... 366
7.8.3 Informational cascades... .. 0.0.0.. .0 . 366
7.8.4 Influence in social networks ...........0... 370
7.9 Further directions... 0.0. 372
7.9.1 Postulational and constructive approaches ...... 372
7.9.2 Belief revision versus nonstandard consequence ... 375
7.9.3 Belief revision and evidence ..............4 375
7.9.4 Combining information and preference ........ 376
7.9.5 Group belief and merge ............0004 377
FI

x CONTENTS
7.9.6 Belief revision and social choice ............ 378
7.10 Notes 2... 379
! References... ee 385
III Applications 395
8 Model Checking Temporal Epistemic Logic 397
Alessio Lomuscio and Wojciech Penczek
8.1 Introduction... .... ee 398
8.2 Syntax and Semantics . 0... 0... 00.ee0e .ee n 400
8.2.1 Syntax... 400
8.2.2 Interpreted systems semantics ............, A401
8.2.3 Interleaved interpreted systems ............ 402
8.2.4 ‘Temporal-epistemic specifications ........., .. 403
8.3 OBDD-based Symbolic Model Checking ........... 405
8.3.1 State space representation and labelling ....... 406
8.3.2 MCMAS... 0.0.00... cee 410
8.4 SAT-based Symbolic Model Checking. ............ Ail
8.4.1 Bounded Model Checking .........0....00, 412
| 8.4.2 Unbounded Model Checking. ............. 415
| B43 VelCS. 0. eee, 418
8.5 Extensions to Real-Time Epistemic Logic .......... 418
8.5.1 Example... 0... ee = 420
8.6 Partial Order Reductions .............0.0, oe ee AQ]
8.6.1 Evaluation... .. 0.0.000 0.. 0.ee e ew 423
8.7 Notes . 2... ee ee Lee ee ey 424
References... ee 433
9 Epistemic Foundations of Game Theory 443
Giacomo Bonanno
9.1 Introduction... .....0.....00, ee 444
9.2 Epistemic Models of Strategic-Form Games ......... 445
9.3 Common Belief of Rationality: Semantics ........., 449
9.4 Common Belief of Rationality: Syntax ...........0, 453
9.5 Common Belief versus Common Knowledge ......... 458
9.6 Probabilistic Beliefs and Payoffs................ 466
9.7 Dynamic Games with Perfect Information .......... 469
9.8 The Semantics of Belief Revision ..........,.... A73
9.9 Common Belief of Rationality ...............0., ATA
9.10 Notes 2... 480