Nonmonotonic Logic: Context-Dependent Reasoning

424 Pages·1993·13.251 MB·English

by  V. Wiktor Marek, Mirosław Truszczyński (auth.)

Artificial Intelligence
Editors: S. Amarel A. Biermann L. Bole P. Hayes A. Joshi
D. Lenat D. W. Loveland (Managing Editor) A. Mackworth
D. Nau R. Reiter E. Sandewall S. Shafer Y. Shoham
J. Siekmann W. Wahlster
v.
W Marek M. Truszczynski
Nonmonotonic Logic
Context-Dependent Reasoning
Foreword by Ray Reiter
Springer-Verlag
Berlin Heidelberg GmbH
V. Wiktor Marek
Miroslaw Truszczynski
Department of Computer Science, University of Kentucky
Lexington, KY 40506-0027, USA
CR Subject Classification (1991): 1.2.0, 1.2.3-4, FA.O-1
With 14 Figures
ISBN 978-3-662-02908-4
Library of Congress Cataloging-in-Publication Data
Marek. V. Wiktor. 1943- . Nonmonotonic logic: context-dependent reasoning / V. Wiktor
Marek, M. Truszczynski. p. cm. - (Artificial intelligence) (Berlin, Germany)
ISBN 978-3-662-02908-4 ISBN 978-3-662-02906-0 (eBook)
DOI 10.1007/978-3-662-02906-0
I. Artificial intelligence. 2. Reasoning. 3. Optoelectronics. I. Truszczynski, Miroslaw. II.
Title. III. Series. Q335.M37 1993 006.3-dc20 93-31084
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© Springer-Verlag Berlin Heidelberg 1993
Originally published by Springer -Verlag Berlin Heidelberg New York in 1993
Softcover reprint of the hardcover 1st edition 1993
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
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Typesetting: Camera-ready by authors
45/3140 -5 4 3 2 I 0 -Printed on acid-free paper
To Ela, Malgosia and Natalia
W.M.
To Lena, Ania and Na talka
M.T.
Foreword
When I first participated in exploring theories of nonmonotonic reasoning in
the late 1970s, I had no idea of the wealth of conceptual and mathematical
results that would emerge from those halting first steps. This book by Wiktor
Marek and Miroslaw Truszczynski is an elegant treatment of a large body of
these results. It provides the first comprehensive treatment of two influen
tial nonmonotonic logics - autoepistemic and default logic - and describes a
number of surprising and deep unifying relationships between them. It also
relates them to various modal logics studied in the philosophical logic litera
ture, and provides a thorough treatment of their applications as foundations
for logic programming semantics and for truth maintenance systems.
It is particularly appropriate that Marek and Truszczynski should have
authored this book, since so much of the research that went into these results
is due to them. Both authors were trained in the Polish school of logic and
they bring to their research and writing the logical insights and sophisticated
mathematics that one would expect from such a background. I believe that
this book is a splendid example of the intellectual maturity of the field of
artificial intelligence, and that it will provide a model of scholarship for us
all for many years to come.
Ray Reiter
Department of Computer Science
University of Toronto
Toronto, Canada M5S 1A4
and
The Canadian Institute for
Advanced Research
Table of Contents
1 Introduction ......... 1
1.1 The subject of this book 1
1.2 What do we do in this book? 4
1.3 To whom do we address this book? 5
1.4 The contents of this book 6
1.5 Acknowledgments ..... 8
2 Rudiments of logic and set theory 9
2.1 Syntax of propositional logic 9
2.2 Semantics of propositional logic 11
2.3 Proof theory for propositional logic 17
2.4 Completeness and compactness of propositional logic 20
2.5 Relations, partial orderings 21
2.6 Elements of predicate logic 23
2.7 Well-orderings, ordinal numbers. 31
2.8 Monotone operators ....... 34
3 General default theories .......... 37
3.1 Monotone rules in propositional logic . 38
3.2 Default logic - motivation 46
3.3 Default logic - syntax. . . 50
X Table of Contents
3.4 Extensions for default theories 51
3.5 Two more characterizations of extensions 61
3.6 Properties of default theories 66
3.7 Extensions and well-orderings 71
3.8 Prioritized default theories. 82
3.9 Computing extensions . . . 84
3.10 Other structures associated with default theories 87
3.11 Partial extensions .... 94
3.12 Comments and remarks 101
4 Normal default theories . . . 105
4.1 Normal default theories 105
4.2 Properties of normal default theories 109
4.3 The Closed World Assumption 111
4.4 Comments and remarks . . . . 123
5 Representation theory for default logic 125
5.1 Representability of default theories 125
5.2 Semi-representability of default theories 132
5.3 A representation result for weak extensions 137
5.4 Comments and remarks .......... . 139
6 Logic programming and nonmonotonic reasoning 141
6.1 Logic programs - basic definitions 141
6.2 Horn programs . . . . . . . . . . . 147
6.3 Stable models of general logic programs 149
6.4 Supported models of logic programs .. 152
6.5 Logic programming with classical negation. 159
6.6 Clausal logic programming. . . . ..... . 170
6.7 Stratified and locally stratified logic programs 172
Table of Contents XI
6.8 Proof schemes and stable models 182
6.9 Comments and remarks 186
7 Modal logic . . . . . . . . . . . . . . . . . . . . . 189
7.1 Modal logics - syntax and basic properties 189
7.2 General Kripke semantics . . . . . 200
7.3 Kripke semantics for normal logics 210
7 A Index logics . . . . . . . 218
7.5 Comments and remarks 222
8 Stable theories ...... . . 223
8.1 Stable theories and their elementary properties 223
8.2 Characterizations of stable theories 228
8.3 Algorithms ...... . 238
804 Comments and remarks 247
9 Modal nonmonotonic logics . . . . . . . . . . . . 249
9.1 Context-dependent proofs in a modal logic . 250
9.2 S-expansions and their elementary properties 252
9.3 Minimal model semantics ........ . 259
904 Consistency with respect to introspection 271
9.5 Characterization of S-expansions 279
9.6 Comments and remarks ..... 287
10 Nonmonotonic logic of pure necessitation and autoepistemic logics 289
10.1 Nonmonotonic logic N . . . . . . . . 290
10.2 Nonmonotonic logics kd45 and Sw5 296
10.3 Autoepistemic logic; context semantics 308
lOA Comments and remarks . . . . . . . . . 315
11 Topics in modal nonmonotonic logic 317
11.1 Algorithms to compute S-expansions 317
XII Table of Contents
11.2 Monotonic modal logics with the same nonmonotonic counter-
part . . . . . . . . . . . . . . . . . 325
11.3 Adding new definitions to a theory 337
11.4 Bounding the depth of introspection 341
11.5 Ground S-expansions. . . . . . . . . 343
11.6 Bounding introspection to £ U {.....,Lcp: cp E £} 346
11.7 Comments and remarks . . . . . . . . 350
12 Relations among nonmonotonic formalisms 351
12.1 Expressing default reasoning in modal nonmonotonic logics 351
12.2 Another semantics for default logic . . . . . . . . . . . . . 357
12.3 Interpreting logic programs in modal nonmonotonic logics 362
12.4 More characterizations of stable answer sets . . . . . . . . 364
12.5 Weak extensions, stable expansions and supported models 368
12.6 Interpreting logic programs in autoepistemic logic. 374
12.7 Truth maintenance and nonmonotonic systems 376
12.8 Comments and remarks . . . . . . . . . . . . . 380
13 Complexity of some forms of nonmonotonic reasoning 383
13.1 Basic notions of the complexity theory 383
13.2 Complexity of default reasoning. . . . 385
13.3 Complexity of reasoning with modal nonmonotonic logics 391
13.4 Complexity issues for finite logic programs . . . . . . . . 396
13.5 Issues in complexity of models of infinite logic programs 399
13.6 Comments and remarks 404
References 405
Index ... 411
List of Figures
3.1 Algorithm for computing BDjw(W) .......... 44
3.2 Algorithm for computing extensions of default heories 85
3.3 Algorithm for computing a candidate theory for an extension,
given an ordering of defaults. . . . . . . . . . . . . . . . 86
3.4 Algorithm for computing extensions of default theories . . .. 87
3.5 Algorithm for computing a partial extension of a default theory
given an ordering of defaults. . . . . . . . . . . . . . . . . .. 101
4.1 Algorithm for computing all extensions of a normal default
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107
7.1 Axiomatizations of some modal logics 197
8.1 Algorithm InStabA 240
8.2 Algorithm InStabB 241
8.3 Algorithm InStabC 246
8.4 Algorithm InStabD 247
11.1 Algorithm IC .... 319
11.2 Algorithm AIL.ExpsA . 320
11.3 Algorithm AILExpsB . 321