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Comparative Metric Semantics of Programming Languages: Nondeterminism and Recursion PDF

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Progress in Theoretical Computer Science Editor Ronald V. Book, University of California Editorial Board Erwin Engeler, ETH Zentrum, Zurich, Switzerland Jean-Pierre Jouannaud, Universite de Paris-Sud, Orsay, France Robin Milner, Cambridge University, England Martin Wirsing, UniversiHit Passau, Passau, Germany Franck van Breugel Comparative Metric Semantics of Programming Languages N ondeterminism and Recursion Birkhauser Boston • Basel • Berlin Franck van Breugel School of Computer Science McGill University Montreal, Quebec H3A 2A7 Canada Library of Congress Cataloging-in-Publication Data Breugel, Franciscus Cornelus van, 1966- Comparative metric semantics of programming languages: nondeterminism and recursion / Franciscus Cornel us van Breugel. p. cm. -- (Progress in theoretical computer science) Based on the author's thesis (Ph.d.) ISBN -l3: 978-1-4612-8680-6 e-ISBN-l3: 978-1-4612-4160-7 DOl: 10.1007/978-1-4612-4160-7 1. Programming languages (Electronic computers )--Semantics. I. Title. II. Series. QA76.7.B73 1997 005.1' 31--dc21 97-37052 CIP AMS Subject Classifications: 68Q55 ACM Subject Classifications: D.3.!., F.3.2 Printed on acid-free paper $ © 1998 Birkhauser Boston Birkhiiuser ® Copyright is not claimed for works of U.S. Government employees. 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 of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (Ccq, provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts A venue, Cambridge, MA 02139, U.S.A. ISBN -l3: 978-1-4612-8680-6 Typeset by the author in LATEX. 9 8 7 6 5 4 3 2 1 Contents Notat~on ix Preface xiii History ..... . xiv Overview .... . xvi Acknowledgements . xviii Introd uction 1 No ndeterminism and recursion 1 Operational semantics . 3 Denotational semantics . 5 Metric spaces . . . . . . 6 Comparative semantics. 6 Bibliographic notes . . . 7 Part I 9 1 Domain equations 11 1.1 Building domain equations 13 1.2 Solving domain equations 15 1.3 Bibliographic notes .... 16 2 Linear and branching domains 19 2.1 Two linear domains ..... 20 2.2 Comparison of the linear domains 22 2.3 Three branching domains . . . . . 23 2.4 Comparison of the branching domains 32 2.5 Relating linear and bran.ching domains 44 2.6 Bibliographic notes .......... . 48 Part II 51 3 Operational semantics 53 3.1 Labelled transition systems 55 3.2 Operational semantics ... 57 3.3 Linear semantics transformations 62 3.3.1 Compactness preserving .. 64 3.3.2 Closedness preserving ... 66 3.4 Branching semantics transformations 68 vi Contents 3.4.1 Compactness preserving .. 68 3.4.2 Closedness preserving . . . 71 3.5 Relating semantics transformations 73 3.6 Bibliographic notes ........ . 77 4 Nondeterministic choice 79 4.1 Language definition . 80 4.2 Operational semantics . 82 4.3 Denotational semantics 87 4.4 Relating 0 and 1) . 93 4.5 Bibliographic notes 95 5 Random assignment 97 5.1 Language definition 98 5.2 Operational semantics. 99 5.3 Denotational semantics 105 5.4 Relating 0 and 1) . 112 5.5 Bibliographic notes . . 114 Part III 117 6 Generalized finiteness conditions 119 6.1 Metric labelled transition systems 120 6.2 Operational semantics ...... . 125 6.3 Linear semantics transformations 134 6.3.1 Compactness and nonexpansiveness preserving 135 6.3.2 Closedness and nonexpansiveness preserving 137 6.4 Branching semantics transformations 139 6.5 Relating semantics transformations 141 6.6 Bibliographic notes ........ . 147 7 Dense choice 149 7.1 Language definition 150 7.2 Operational semantics . 152 7.3 Denotational semantics 159 7.4 Relating 0 and 1) . 162 7.5 Bibliographic notes .. 164 8 Second order communication 165 8.1 Language definition .. 166 8.2 Operational semantics . 167 8.3 Denotational semantics 174 8.4 Relating 0 and 1) . . . 178 Contents vii 8.4.1 Intermediate semantics 178 8.4.2 Relating I and V 183 8.4.3 Relating 0 and I 185 8.5 Bibliographic notes 190 A Metric spaces 193 A.1 Metrics .............. . 193 A.2 Completeness and contractiveness 194 A.3 Hyperspaces . . . . . . 197 A.4 Nonexpansive functions 200 A.5 Bibliographic notes 200 Bibliography 201 Author index 217 Subject index 219 To Mirjam Notation ~.X multiplication of metric by ~ 13 Pk powerspace of compact subsets 14 Pc powerspace of closed subsets 14 Aoo metric space of finite and infinite sequences 20 E empty sequence 20 IL linear space 22 ILk linear domain 22 ILc linear domain 22 2in size of sets classifier 23 Bk branching domain 24 Be branching domain 24 Bj branching domain 24 Pj powerset of finite subsets 27 B (a) action quotient of branching process 33 lin linearize operator 44, 46 a·L action prefixing of linear process 45 FB finitely branching 56 IF image finite 57 Atom atomic actions 80 SVar statement variables 80 Stat statements 80 sequential composition 80 SiS s+s nondeterministic choice 80 J-Lx[s) J-L-recursion 80 GStatx guarded statements for X 81 GStat guarded statements 81 svar free statement variables 81 CStat closed statements 81 CGStat closed and guarded statements 81 E empty statement 82 CGStatE extended closed and guarded statements 82 c --70 c zero-transition 82 s{s/x} substitution 82 S;E S extended sequential composition 83 comp complexity function 85 x Notation Var variables 98 Exp expressions 98 BExp boolean expressions 98 Stat statements 99 v:= e assignment statement 99 v:= ? random assignment statement 99 skip skip statement 99 if b then s else s fi if statement 99 while b do s ad while statement 99 StatE extended statements 99 Val values 100 State states 100 da/v} state update 100 £ expression evaluation 100 B boolean expression evaluation 100 comp complexity function 103 B branching domain 105 CB compactly branching 121 IC image compact 123 lin linearize operator 142, 146 TVar time variables 150 lR~ nonnegative real numbers 150 Stat statements 150 a[r] timed atomic action 150 LtE[r, rJ s dense choice 150 GStat guarded statements 151 tvar free time variables 151 CGStat closed and guarded statements 151 TAtom timed atomic actions 152 s{r/t} substitution 152 comp complexity function 156 Chan channels 167 Stat statements 167 c!s send statement 167 c?x receive statement 167 call (x) call statement 167 s lis parallel composition 167 SynStore syntactic stores 167 B{s/x} syntactic store update 167 SynCom syntactic communications 168

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