Table Of ContentProgress 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
$
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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
