Topology in Process Calculus: Approximate Correctness and Infinite Evolution of Concurrent Programs

225 Pages·2001·10.76 MB·English

by  Mingsheng Ying (auth.)

Topology
In
Process Calculus
Springer Science+ Business Media, LLC
Mingsheng Ying
Topology
I• n
Process Calculus
Approximate Correctness
and Infinite Evolution of
Concurrent Programs
, Springer
Mingsheng Ying
Department of Computer Science and Technology
State Key Laboratory of Intelligent Technology, and Systems
Tsinghua University
Beijing 100084
China
[email protected]
Library of Congress Cataloging-in-Publication Data
Ying, Mingsheng.
Topology in process calculus : approximate correctness and infinite evolution of
concurrent programs / Mingsheng Ying.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4612-6522-1 ISBN 978-1-4613-0123-3 (eBook)
DOI 10.1007/978-1-4613-0123-3
1. Parallel processing (Electronic computers) 2. Aigebraic topology. 1. Title.
QA76.58 .Y56 2000
004'. 35-dc21 00-059583
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
OriginalIy published by Springer-Verlag New York, Inc in 2001
Softcover reprint of the hardcover lst edition 2001
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Production managed by Jenny Wolkowicki; manufacturing supervised by Erica Bresler.
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9 8 765 432 1
ISBN 978-1-4612-6522-1
Preface
Concurrency is one of the most challenging research areas in computer
science, and process algebras are among the most successful models ofcon
current systems. Various behavior equivalences between processes are cen
tral notions in R. Milner's Calculus of Communicating Systems (CCS for
short, or processcalculus, inotherwords) and some other process algebras.
Inthe actual applications of process algebras, specification and implemen
tation are expressed as two processes, and correctness of programs is seen
as a certain behavior equivalence between them. In many situations, how
ever, implementations at best approximate the formal specifications. So
formal methods are needed to describe the mechanism of approximation
among processes. Unfortunately, the behavior equivalences in the previous
literature are unable to serve for this purpose.
Roughly speaking, there are two different modes of approximation, the
static and the dynamic. In the realm of programming methodology, they
lead to the notions of approximate correctness and infinite evolution of
programs, respectively. The objective of this small book is to establish
a preliminary theory of approximate correctness and infinite evolution of
concurrent programs byemployingsome mathematicaltools from point-set
topology. This book is restricted to R.Milner's CCS forsimplicity, but the
main idea also applies to many other process algebras and to even a wider
area of formal methods. I introduce the concept of bisimulation limits and
expose a certain compatibilityofthe limit behaviorsofprocesseswith their
transitional semantics. This is very useful for the understanding and anal
ysis of infinite evolution of concurrent programs. I propose the notions of
near bisimulation and bisimulation index,which are suitable for describing
approximate correctness of concurrent programs. Some applications are
presented to illustrate how bisimulation indexes can be used to describe
approximate satisfaction of real-time systems to their specifications with
the allowance of a certain time error. To the best of my knowledge, this
is the first attempt ever made in formalizing the notions of approximate
correctness and infinite evolution of concurrent programs. The material
presented is entirely new and draws on my most recent work. Therefore,
this book may contain some novel ideas that need further development. I
would highly appreciate criticisms and comments from readers.
vi Preface
This book isintended as a research monograph, and I tried to make it as
self-contained as possible. As mentioned, my approach is presented within
the framework of CCS. Therefore, complete familiarity with process alge
bra is essential to understand the material presented. For those who are
unfamiliar with it, Chapter 2 provides a brief introduction. Mathematical
tools used in this book come mainly from point-set topology, so a certain
knowledge ofpoint-set topology isalso required ofthe reader. I shall briefly
recall some basic definitions and results on topology before using them, but
I would suggest that readers unfamiliar with them consult some standard
textbooks fora betterunderstanding.This book presents four formal meth
ods ofattackingthe problemofapproximation ofprocesses. Thesemethods
are motivated by four different profiles of the same problem, and they are
elaborated respectively in Chapters 3 through 6. After finishing the first
two chapters, Chapters3through6may be read more or lessindependently
ofeach other. For example, readers who hope to find some practical formal
methods for describing approximate implementations of systems can read
Chapter 6 first; in particular, they might be interested in the examples in
Sections 6.6 and 6.7.
Thefinal revisions to thematerial ofthis book weremadewhen Iwas vis
itingtheInstitutfiirInformatik, Ludwig-Maximilians-UniverstatMiinchen,
as an Alexandervon Humboldt researchfellow.Iam very gratefulto Profes
sor Martin Wirsing for his stimulating discussion and invaluable comments
and suggestions, and for providing the excellent working enviroment.
I would like to express my sincere thanks to the anonymous referees for
their kind and invaluable suggestions for improving the manuscript. I also
owe a great deal to Wayne Yuhasz and Wayne Wheeler, computer and
information science editors and Jenny Wolkowicki, production editor at
Springer-Verlag New York, who piloted this book to publication. Indeed,
this book would not have been possible without their effort and expertise
in the review and editing process.
This work was supported by the National Scientific Foundation for Dis
tinguished Young Scholars (Grant No. 69725004), Key Project for Basic
Research (Grant No. 1998030509), Foundation of Natural Sciences (Grant
No. 196710038) and Research and Development Project of High Technol
ogy (Grant No.863-306-05-05-3B) ofChina, and Fok Ying-TungEducation
Foundation.
Munich, April 2000 Mingsheng Ying
Contents
Preface
v
1 Introduction
1
2 Process Calculus
11
2.1 Transition Systems 11
2.2 Hennessy-Milner Logic 15
2.3 Basic Language 19
2.4 Transitional Semantics 21
2.5 Strong Bisimulations 24
2.6 Bisimulations 29
3 Bisimulation and 'Trace Limits of Agents
37
3.1 Strong Bisimulation Limits 37
3.1.1 Strong Limit Bisimulations 38
3.1.2 Strong Bisimulation Limits 46
3.1.3 Strong Bisimulation Topology 53
3.1.4 Strong Limit Bisimulations Up to 58
rv
3.1.5 Continuity of Combinators 60
3.1.6 Limit Representations of Solutions of Equations 66
3.1.7 Determinacy Preserved by Strong Bisimulation 68
Limits
3.2 *-Strong Bisimulation Limits 69
3.3 Bisimulation Limits 78
3.4 Trace Limits 86
4 Limit Behavior of Agents
95
4.1 Modifications of Topologies on Actions 96
4.2 Topologies on Agents 100
4.3 Several Limit Theorems 102
viii Contents
5 Near Bisimulations Defined by Closures 113
5.1 Near Bisimulations in Transition Systems 113
5.2 Near Strong Bisimulations 125
5.3 Near (Weak) Bisimulations 131
6 Bisimulation Indexes Induced by Metrics
on Actions 139
6.1 Bisimulation Indexes in Transition Systems 140
6.2 Hennessy-Milner Logical Characterization of 154
Bisimulation Indexes
6.3 Strong Bisimulation Indexes 158
6.4 (Weak) Bisimulation Indexes 169
6.5 Approximate Communication 177
6.6 Some Applications:Two Examples in Timed CCS 182
6.7 Some Applications:An Example in Real Time ACP 198
7 Conclusion 207
References 213
Index 219
1
Introduction
Communication and concurrency are essential in understanding complex
dynamic systems, and there have been many theories for dealing with
them, such as Petri nets [Petri 1962;Reisig 1985],CSP [Hoare 1978, 1985;
Brookes, Hoare and Roscoe 1984] and ACP [Bergstra and Klop 1985;
Baeten and Weijland 1990]. Among them, CCS proposed by Milner [1980,
1989],or process calculus as it is called in [Milner 1989],isone ofthe most
important and mathematically developed models of communication and
concurrency.
Various behavior equivalences between agents (or processes), such as
(strong and weak) bisimilarity, observation congruence, trace equivalence,
testing equivalence and failure equivalence, are central notions in process
calculus. In the real applications of process calculus, specification and im
plementationare described astwo agents,correctnessofprogramsistreated
as a certain behavior equivalence between specification and implementa
tion, and thenthe proofofcorrectnessofprograms isa task forestablishing
some behavior equivalence between them. For some examples ofsuccessful
applications, see Sections 5.4, 5.5 and 5.6 and Chapter 6 in [Milner 1989].
On the other hand, N. Francez [1992] pointed out: "Sometimes,the 'very
idea' of program verification, using any mathematical or logical method,
is criticized. The main argument is that, when programs are actually exe
cuted on electronic computers, the prerequisites for successful application
(listed in [Francez 1992],pages 4 and 5) cannot be assumed to hold. Com
puters,being physical devices, cannot be assumed to behave reliably.Inad
dition,standardimplementationsat best approximate the formal definition
of semantics. Thus, no logical conclusion can be drawn about the real-life
behavior of programs, no more than about any other natural phenomena,
with absolute certainty." As iswell known, the reliability of computer sys
tems, both hardware systems and software systems, has been thoroughly
investigated [Lyu 1996].However,the problemofdescribingthe mechanism
thatimplementationsapproximatetheirspecificationsisstill untouched. As
the first repercussion of this problem, wemay reasonably envisage that al
though an exact correctness ofsome complex programs isdifficult to draw,
we still can obtain some looser versions of their correctness, and in many
cases the looser correctness may be enough for our purpose. Thus, the
problem may be explicitlyrestated as follows:What are the looser versions
M. Ying, Topology in Process Calculus
© Springer-Verlag New York, Inc. 2001
2 1. Introduction
of correctness of programs? In my opinion, there are at least two ways to
loosen the notion of program correctness:
1. We are given a program and a certain small error between it and
its specification is allowed. In this case, the program may be seen as
an approximate implementation of the specification, and it can be
thought that an approximate correctness ofthis program is derived.
This is a static approach.
2. Alternatively, we may use a dynamic approach. Suppose that a se
quence of programsisgiven and the programsapproximate the speci
fication closer and closer. Then this sequence ofprograms can be seen
as an evolution toward the specification.
Asfar asI know,few formal descriptions oflooser correctness forsequential
or concurrent programs have been proposed. Concretely in the framework
of process calculus, various behavior equivalences require that the behav
iors of processes - to be thought of as being equivalent - must match
eachotherexactly.In addition, they do not dealwithevolutionofprocesses.
Thusthese concepts ofbehavior equivalence cannot be used to describe the
mechanisms ofapproximation ofprocesses. The purpose of this small book
is to provide some suitable and useful concepts and tools for the under
standing and analysis of approximate correctness and infinite evolution of
concurrent programs. Throughout this book we focus on R. Milner's pro
cess calculus [Milner 1989], and the main idea is to construct some natural
and reasonable topological structuresthat can suitably reveal a mechanism
of approximate computation in process calculus and to work out various
relationships among these topological structures and fundamental proper
ties of processes that are compatible with them. We shall construct two
different classes of topological structures: one is determined by behaviors
of processes and so is completely extensionaland observable; the other in
volves somewhat intensional factors in the sense that a certain topological
structure on actions is presumed and is employed to induce some topologi
calstructureson processes.In accordance withthe twodifferent approaches
to looser correctness ofprograms,on the other hand,we have two different
kinds of methods to construct our intended topological structures: one is
dynamic-we consider thetopologies possessed by sequences (or more gen
erally, nets) ofprocesses; the other is static- we introduce some weakened
versions of behavior equivalences in a topological way. In summary, our
overall problem is the description of approximation of processes. We shall
introduce several approaches to attack this problem.These approaches are
inspired by the different profiles ofthe problem.
This book is organized into seven chapters. In the remaining part of this
chapter, the motivations of the notions to be introduced are explained in
detail and the content of each chapter is surveyed. In Chapter 2 we re
call some fundamental notions and results in process calculus needed in