Derivation and Computation: Taking the Curry-Howard Correspondence Seriously

412 Pages·2000·12.447 MB·English

by  Harold Simmons

ISBN 0-521-77173-0
I I
9 780521 771733 .
Derivation and Computation
Cambridge Tracts in Theoretical Computer Science
Editorial Board
S. Abramsky, Department of Computing Science, Edinburgh University
P. H. Aczel, Department of Computer Science, University of Manchester
J. W. de Bakker, Centrum voor Wiskunde en Informatica, Amsterdam
Y. Gurevich, Department of Electrical Engineering and Computer Science, University of Michigan
J. V. Thcker, Department of Mathematics and Computer Science, University College of Swansea
Titles in the series
1. G. Chaitin Algorithmic Information Theory
2. L. G. Paulson Logic and Computation
3. M. Spivey Understanding Z
5. A. Ramsey Formal Methods in Artificial Intelligence
6. S. Vickers Topology via Logic
7. J.-Y. Girard, Y. Lafont & P. Taylor Proofs and Types
8. J. Clifford Formal Semantics & Pragmatics for Natural Language Processing
9. M. Winslett Updating Logical Databases
10. K. McEvoy & J. V. Thcker (eds) Theoretical Foundations of VLSI Design
11. T. H. Tse A Unifying Framework for Structured Analysis and Design Models
12. G. Brewka Nonmonotonic Reasoning
14. S. G. Hoggar Mathematics for Computer Graphics
15. S. Dasgupta Design Theory and Computer Science
17. J. C. M. Baeten (ed) Applications of Process Algebra
18. J. C. M. Baeten & W. P. Weijland Process Algebra
19. M. Manzano Extensions of First Order Logic
21. D. A. Wolfram The Clausal Theory of Types
22. V. Stoltenberg-Hansen, I. Lindstrom & E. Griffor Mathematical Theory of Domains
23. E.-R. Olderog Nets, Terms and Formulas
26. P. D. Mosses Action Semantics
27. W. H. Hesselink Programs, Recursion and Unbounded Choice
28. P. Padawitz Deductive and Declarative Programming
29. P. Gardenfors (ed) Belief Revision
30. M. Anthony & N. Biggs Computational Learning Theory
31. T. F. Melham Higher Order Logic and Hardware Verification
32. R. L. Carpenter The Logic of Typed Feature Structures
33. E. G. Manes Predicate Transformer Semantics
34. F. Nielson & H. R. Nielson Two Level Functional Languages
35. L. Feijs & H. Jonkers Formal Specification and Design
36. S. Mauw & G. J. Veltink (eds) Algebraic Specification of Communication Protocols
37. V. Stavridou Formal Methods in Circuit Design
38. N. Shankar Metamathematics, Machines and Godel's Proof
39. J. B. Paris The Uncertain Reasoner's Companion
40. J. Dessel & J. Esparza F'ree Choice Petri Nets
41. J.-J. Ch. Meyer & W. van der Hoek Epistemic Logic for AI and Computer Science
42. J. R. Hindley Basic Simple Type Theory
43. A. Troelstra & H. Schwichtenberg Basic Proof Theory
44. J. Barwise & J. Seligman Information Flow
45. A. Asperti & S. Guerrini The Optimal Implementatiq",of Functional Programming Languages
46. R. M. Amadio & P.-L. Curien Domains and Lambda'P~lP!~i
47. W.-P. de Roever & K. Engelhardt Data Refinement
48. H. Kleine Biining & T. Lettman Propositional Logic. . ,
49. L. Novak & A. Gibbons Hybrid Graph Theory and Network· Analysis
Derivation and Computation
Taking the Curry-Howard correspondence seriously
Harold Simmons
University of Manchester
CAMBRIDGE
UNIVERSITY PRESS
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PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
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CAMBRIDGE UNIVERSITY PRESS
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© Cambridge University Press 2000
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2000
Printed in the United Kingdom at the University Press, Cambridge
Typeset by the author in Computer Modern 10/13pt, in 1l'"IEX2e [EPC)
A catalogue record of this book is available from the British Library
Library of Congress Cataloguing in Publication data
Simmons, Harold.
Derivation and computation: taking the Curry~Howard correspondence seriously:
derivation systems, substitution algorithms, computation mechanisms / Harold Simmons.
p. cm.
Includes bibliographical references and index.
ISBN 0 521 77173 0 (hb)
1. Proof theory. 2. Lambda calculus. 3. Type theory. I. Title.
QA9.54 S55 2000
511.3 21-dc21 99-044953
ISBN 0 521 77173 0 hardback
:.('.' ::*
. J 1~.-.
CONTENTS
INTRODUCTION xi
PREVIEW XV
I DEVELOPMENT AND EXERCISES 1
1 DERIVATION SYSTEMS 3
1.1 Introduction ... 3
Exercises. 8
1.2 Generalities' . . . 8
Exercises. 14
1.3 The systems H and N . 14
Exercises .... 18
1.4 Some algorithms on derivations 19
Exercises ..... 24
2 COMPUTATION MECHANISMS 27
2.1 Introduction . . . . 27
Exercises .. 28
2.2 Combinator terms . 29
Exercises .. 31
2.3 Combinator reduction 31
Exercises. 35
2.4 .A-terms 37
0 ••••
Exercises. 40
2.5 .A-reduction ... 40
Exercises. 43
2.6 Intertranslatability 43
Exercises .. 45
2.7 Confluence and normalization 45
Exercises ........ 47
3 THE TYPED COMBINATOR CALCULUS 48
3.1 Introduction ... 48
Exercises .... 50
0 •• 0 •••
1295895
Vl Contents
302 Derivation 0 0 0 0 0 0 50
Exercises 0 0 0 0 52
303 Annotation and deletion 53
Exercises 0 57
3.4 Computation 57
Exercises 0 60
305 Subject reduction 60
Exercises 0 65
4 THE TYPED A-CALCULUS 67
401 Introduction 0 0 67
Exercises 0 68
402 Derivation 0 0 0 0 68
Exercises 0 70
403 Annotation and deletion 71
Exercises 0 75
4.4 Substitution 0 0 0 75
Exercises 0 77
405 Computation 0 0 78
Exercises 0 80
406 Subject reduction 80
Exercises 0 81
5 SUBSTITUTION ALGORITHMS 82
501 Introduction 0 0 0 0 82
Exercises 0 0 0 85
502 Formal replacements 86
Exercises 0 0 0 89
503 The generic algorithm 89
Exercises 0 0 0 91
5.4 The mechanistic algorithm 92
Exercises 0 0 0 0 0 0 95
505 Some properties of substitution 95
Exercises 0 0 0 0 0 0 0 0 0 98
6 APPLIED A-CALCULI 100
601 Introduction 0 0 100
Exercises 0 102
602 Derivation 0 0 0 0 102
Exercises 0 105
603 Type synthesis 106
Exercises 0 111
6.4 Mutation 0 111
••• 0
Exercises 0 120
605 Computation 122
Exercises 0 125
Contents vii
6.6 Type inhabitation . 125
Exercises. 128
6.7 Subject reduction 129
Exercises. 135
7 MULTI-RECURSIVE ARITHMETIC 137
7.1 Introduction ..... 137
Exercises ... 140
7.2 The specifics of )..G . 140
Exercises ... 143
7.3 Forms of recursion and induction 144
Exercises .... 148
7.4 Small jump operators . . . . . . 150
Exercises ......... 158
7.5 The multi-recursive hierarchies . 159
Exercises .. 163
7.6 The extent of )..G . 164
Exercises. 165
7.7 Naming in AG 166
Exercises. 169
8 ORDINALS AND ORDINAL NOTATIONS 170
8.1 Introduction .... 170
Exercises .. 170
8.2 Ordinal arithmetic 170
Exercises .. 174
8.3 Fundamental sequences . 174
Exercises ..... 177
8.4 Some particular ordinals 177
Exercises. 183
8.5 Ordinal notations 183
Exercises. 188
9 HIGHER ORDER RECURSION 189
9.1 Introduction ... 189
Exercises. 192
9.2 The long iterator 193
Exercises. 196
9.3 Limit creation and lifting . 196
Exercises ...... 198
9.4 Parameterized ordinal iterators 198
Exercises ........ 201
9.5 How to name ordinal iterates 202
Exercises. 205
9.6 The GODS ... 206
Exercises. 211
viii Contents
II SOLUTIONS 213
A DERIVATION SYSTEMS 215
A.1 Introduction . . . . 215
A.2 Generalities .... 216
A.3 The systems H and N . 219
A.4 Some algorithms on derivations 225
B COMPUTATION MECHANISMS 234
B.1 Introduction . . . . . . 234
B.2 Combinator terms . . . 237
B.3 Combinator reduction 237
B.4 .A-terms ...... 242
B.5 .A-reduction ...... 244
B.6 Intertranslatability .. 246
B.7 Confluence and normalization 247
c
THE TYPED COMBINATOR CALCULUS 249
C.l Introduction . . . . . . . 249
C.2 Derivation . . . . . . . . 249
C.3 Annotation and deletion 252
C.4 Computation .. 256
C.5 Subject reduction .. 258
D THE TYPED A-CALCULUS 264
D.1 Introduction . . . . . 264
D.2 Derivation . . . . . . 264
D.3 Annotation and deletion 267
D.4 Substitution . . . 269
D.5 Computation .. 271
D.6 Subject reduction 275
E SUBSTITUTION ALGORITHMS 279
E.1 Introduction . . . . . . 279
E.2 Formal replacements .. 281
E.3 The generic algorithm 281
E.4 The mechanistic algorithm 283
E.5 Some properties of substitution 287
F APPLIED A-CALCULI 290
F.1 Introduction . . 290
F.2 Derivation . . . 290
F.3 Type synthesis 292
FA Mutation .... 297
F.5 Computation 306
F.6 Type inhabitation . 308
F.7 Subject reduction . 314