The Semantics and Proof Theory of the Logic of Bunched Implications

323 Pages·2002·8.523 MB·English

by  David J. Pym

The Semantics and Proof Theory of the Logic of Bunched Implications
APPLIED LOGIC SERIES
VOLUME 26
Managing Editor
Dov M. Gabbay, Department 0/ Computer Science, King's College, London,
U.K.
Co-Editor
Jon Barwiset
Editorial Assistant
Jane Spurr, Department o/Computer Science, King's College, London, U.K.
SCOPE OF THE SERIES
Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects
of philosophy and mathematics to the more recent disciplines of cognitive science, compu
ter science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject.
Kluwer, through its Applied Logic Series, seeks to provide a home for outstanding books and
research monographs in applied logic, and in doing so demonstrates the underlying unity and
applicability of logic.
The titles published in this series are listed at the end of this volume.
The Semantics and
Proof Theory of the
Logic of Bunched
Implications
by
DAVIDJ. PYM
University of Bath, U.K.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6072-3 ISBN 978-94-017-0091-7 (eBook)
DOI 10.1007/978-94-017-0091-7
Printed on acid-free paper
AlI Rights Reserved
© 2002 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover I st edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfiIming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Contents
List of Figures IX
List of Tables Xl
Preface xiii
Acknowledgments xv
Foreword XVll
Dov M. Gabbay
Introduction XXI
David J. Pym
Part I PROPOSITIONAL BI
1. INTRODUCTION TO PART I 3
1 A Proof-theoretic Introduction 3
2 A Semantic Introduction 6
2.1 Algebraic and Topological Semantics 6
2.2 Categorical Semantics 6
2.3 Kripke Semantics 7
3 Towards Classical Propositional BI 10
4 Logical Relations 11
5 Computational Models 11
2. NATURAL DEDUCTION FOR PROPOSITIONAL BI 13
1 Introduction 13
2 A Natural Deduction Calculus 13
3 The aA-calculus 19
4 Normalization and Subject Reduction 25
5 Structural Variations on BI and aA 28
v
VI THE SEMANTICS AND PROOF THEORY OF BI
5.1 Affinity and Relevance 28
5.2 Dereliction 30
5.3 Non-commutativity 30
5.4 More Combinators 31
3. ALGEBRAIC, TOPOLOGICAL, CATEGORICAL 33
1 An Algebraic Presentation 33
2 A Topological Presentation 35
3 A Categorical Presentation 36
3.1 Day's Construction 45
3.2 Conservativity 46
3.3 Structural Variations 47
4. KRIPKE SEMANTICS 51
1 Kripke Models of Propositional BI 51
2 Soundness and Completeness for BI without l.. 55
3 Kripke Models Revisited 65
5. TOPOLOGICAL KRIPKE SEMANTICS 67
1 Topological Kripke Models of Propositional BI with l.. 67
2 Soundness and Completeness for BI with l.. 71
3 Grothendieck Sheaf-theoretic Models 76
6. PROPOSITIONAL BI AS A SEQUENT CALCULUS 89
1 A Sequent Calculus 89
2 Cut-elimination 89
3 Equivalence 93
4 Other Proof Systems 95
7. TOWARDS CLASSICAL PROPOSITIONAL BI 97
1 Introduction 97
2 An Algebraic View 98
3 A Proof-theoretic View 100
4 A Forcing Semantics 102
5 Troelstra's Additive Implication 103
Contents Vll
8. BUNCHED LOGICAL RELATIONS 107
1 Introduction 107
2 Kripke (lA-models 107
2.1 Kripke (lA-models and DCCs 115
3 Bunched Kripke Logical Relations 116
9. THE SHARING INTERPRETATION, I 121
1 Introduction 121
2 Proof-search and (Propositional) Logic Programming 122
3 Interference in Imperative Programs 129
4 Petri Nets 134
5 CCS-like Models 136
6 A Pointers Model 138
Part II PREDICATE BI
10. INTRODUCTION TO PART II 147
1 A Proof-theoretic Introduction to Predicate BI 147
2 Kripke Semantics for Predicates and Quantifiers 151
3 Fibred Semantics and Dependent Types 154
4 Computational Interpretations 156
11. THE SYNTAX OF PREDICATE BI 157
1 The Syntax of Predicate BI 157
2 Variations on Predication 162
12. NATURAL DEDUCTION & SEQUENT CALCULUS 163
1 Propositional Rules 163
2 Quantifier Rules 168
3 Strong Normalization and Subject Reduction 172
4 Predicate BI as a Sequent Calculus 174
13. KRIPKE SEMANTICS FOR PREDICATE BI 179
1 Predicate Kripke Models 179
2 Elementary Soundness and Completeness for Predicate
BI 186
viii THE SEMANTICS AND PROOF THEORY OF BI
14. TOPOLOGICAL KRlPKE SEMANTICS FOR PREDICATE
BI WI
1 Topological Kripke Models of Predicate BI with .1 201
2 Soundness and Completeness for predicate BI with .1 202
15. RESOURCE SEMANTICS, TYPE THEORY & FIBRED
CATEGORlES 207
1 Predicate BI 207
2 Logical Frameworks 209
3 The >'A-calculus 213
4 Context Joining 219
5 Multiple Occurrences 219
6 Variable Sharing 221
7 Equality 223
8 Basic Properties 223
9 The Propositions-as-types Correspondence 225
10 Kripke Resource Semantics for >'A 227
11 Kripke Resource >'A-structure 228
12 Kripke Resource E->'A-model 234
13 Soundness and Completeness 243
14 A Class of Set-theoretic Models 253
15 Towards Systematic Substructural Type Theory 256
16. THE SHARlNG INTERPRETATION, II 263
1 Logic Programming in Predicate BI 263
2 ML with References in RLF 267
Bibliography 271
Index 283
List of Figures
2.1 ,81]-reductions 21
2.2 Term Context 22
2.3 ( -reductions 22
9.1 A Search 'free 127
9.2 A Variation on the Search 'free 128
9.3 The Sharing Interpretation for Logic Programming Goals129
9.4 The Sharing Interpretation for Logic Programming
Clauses 130
9.5 The Sharing Interpretation for Imperative Programming 131
9.6 Net for a Buffer 135
9.7 Pointers and Aliases 139
12.1 Substitution and Contraction 171
15.1 Fibred Models 208
15.2 Representing Object-logics in a Meta-logic 210
15.3 Fibred Kripke Models of Dependent Types 228
15.4 Fibred Models of >'A 230
15.5 Dependent Bunches 258
15.6 Fibred Models of Bunched Types 258
15.7 Kripke Models of Bunched Types 261
IX
List of Tables
2.1 Propositional NBI 16
2.2 NBI for the a>.-calculus 20
3.1 Hilbert-type BI 35
4.1 Kripke Semantics 54
5.1 Semantics in Sheaves 72
5.2 Semantics in Grothendieck Sheaves 78
6.1 LBI for Propositional BI 90
7.1 Some Sequential Rules for (Boolean, De Morgan) BI 101
7.2 Some Sequential Implicational Rules for (Boolean,
De Morgan) BI 102
7.3 Clauses for Classical Additives 103
7.4 Clauses for Classical Multiplicatives 104
7.5 The CLLif Sequent Calculus 104
7.6 The CLif Sequent Calculus 105
7.7 The CLif Sequent Calculus 106
11.1 NBI for a>. 160
11.2 Rules for Well-formed Propositions 161
12.1 Predicate NBI 166
12.2 Quantifier Rules 169
13.1 Predicate Kripke Semantics 183
14.1 Predicate Semantics in Sheaves 203
15.1 >'A-calculus 216
15.2 >'A-calculus (continued) 217
15.3 Parallel Nested Reduction 224
Xl