Table Of ContentTEXTS IN COMPUTER SCIENCE
Editors
David Gries
Fred B. Schneider
Springer Science+Business Media, LLC
TEXTS IN COMPUTER SCIENCE
A/agarandPeriyasamy, Specification of Software Systems
Aptand O/derog, Verification of Sequential and Concurrent
Programs, Second Edition
Backand von Wright, Refinement Calculus
Beid/er, Data Structures and Aigorithms
Bergin, Data Structure Programming
Brooks, C Programming: The Essentials for Engineers and Scientists
Brooks, Problem Solving with Fortran 90
Dandamudi, Introduction to Assembly Language Programming
Fitting, First-Order Logic and Automated Theorem Proving,
Second Edition
Grillmeyer, Exploring Computer Science with Scheme
HomerandSe/man, Computability and Complexity Theory
/mmerman, Descriptive Complexity
Ja/ote, An Integrated Approach to Software Engineering, Second Edition
Kizza, Ethical and Social Issues in the Information Age
Kozen, Automata and Computability
Liand Vitanyi, An Introduction to Kolmogorov Complexity
and ItsApplications, Second Edition
(continuedafterindex)
Colin Stirling
MODAL AND TEMPORAL
PROPERTIES OF PROCESSES
With45 Illustrations
f springer
Colin Stirling
DivisionofInformatics
UniversityofEdinburgh
EdinburghEH93JZUK
[email protected]
SeriesEditors
DavidGries FredB.Schneider
DepartmentofComputerScience DepartmentofComputerScience
415BoydStudiesResearchCenter UpsonHall
TheUniversityofGeorgia CornellUniversity
Athens,GA30605, USA Ithaca,NY14853-7501,USA
LibraryotCongressCataloging-in-PublicalionData
Stirling,ColinP.
ModalandtemporalpropertlesofprocessesIColinSlirling.
p. cm.- (Textsincomputerseience)
tncludesbibliographicalreferencesandindex.
ISBN978-1-4419-3153-5 ISBN978-1-4757-3550-5(eBook)
DOI10.1007/978-1-4757-3550-5
1.Computerlogic. 2.Parallelprocessing(Electroniccomputers) I.Tille. 11.Series.
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Ta Sarah andSusie
Preface
In this book we examine modal and temporal logics for processes. First, we in
troduce concurrentprocesses astermsofanalgebraic language comprisingafew
basicoperators. Their behavioursaredescribed usingtransitions. Familiesoftran
sitions can be arranged as labelIed graphs, concrete summaries of the behaviour
ofprocesses.Variouscombinationsofprocesses and their resulting behaviour,as
determined by the transition mies, are reviewed. Next, simple modal logics are
introducedfordescribingthecapabilitiesofprocesses.
An important discussion point occurs when two processes may be deemed
to havethe same behaviour. Such anabstraction can bepresentedby defining an
appropriatebehaviouralequivalencebetweenprocesses.Amoreabstractapproach
istoconsiderequivalence interms ofhavingthesamepertinentproperties. There
isspecial emphasis withbisimulation equivalence,sincethediscriminatingpower
ofmodal logic istiedtoit.
More generally, practitioners have found it useful to be able to express tem
poral properties of concurrentsystems,especially Iivenessand safety properties.
A safety property amounts to "nothing bad ever happens," whereas a Iiveness
property expresses "somethinggood does eventually happen,"The crucial safety
property ofamutual exclusion algorithm isthatnotwoprocessesareeverintheir
critical sectionsconcurrently.Andanimportantlivenessproperty isthat,whenever
a process requests execution of its critical section, then eventually it is granted.
Cyclic propertiesof systemsare also salient: for instance,part of a specification
of a scheduler is that it must continually perform a particular sequence of ac
tions. A logic expressing temporal notions provides a framework for the precise
formalisation ofsuch specifications.
Forrnulas of the modal logic are not rich enough to express such temporal
properties,soanextra,fixedpointoperator,isadded.Theresultisaveryexpressive
viii Preface
temporal logic, modal mu-calculus.However, itisalso very importanttobe able
to verify that anagent has ordoes nothave aparticularproperty.
Thetextaims tobereasonably introductory,sothatparts ofthe bookcould be
used at undergraduate level, as weIl as at more advanced levels. I have used the
materialinthiswayatEdinburgh.Theextensive useofgamesforbothequivalence
and model checkingispartly pedagogical, since they are soconceptuallyclear.
Parts ofthe book have been presented previously at various summerschools
overtheyears,andIwishtothankalltheorganisersforallowing metopresentthis
material.Ishould also like tothank currentandprevious colleaguesatEdinburgh
forbuildingsuchaninteIlectuaIly stimulatingenvironmenttoworkin.Inparticular,
I wish to thank Julian Bradfield (a pioneer of infinite state model checking and
whoallowedmetousehisTeXtreeconstructorforbuilding derivation trees),Olaf
Burkart, Kim Larsen (who introduced me to modal mu-calculus), Robin Milner
(from whomIlearnt about process calculusandbisimulationequivalence),Perdita
Stevens and David Walker.
Colin Stirling
Edinburgh, UnitedKingdom
Contents
Preface vii
List ofFigures xi
1 Processes 1
1.1 First examples 1
1.2 Concurrentinteraction 8
1.3 Observabletransitions 17
1.4 Renamingandlinking 21
1.5 More combinationsofprocesses 25
1.6 Sets ofprocesses . 28
2 Modalitiesand Capabilities 31
2.1 Hennessy-Milnerlogic I . 32
2.2 Hennessy-Milnerlogic11 . 36
2.3 Algebraicstructureandmodal properties 39
2.4 Observablemodallogic . 42
2.5 Observablenecessity and divergence 47
3 Bisimulations 51
3.1 Processequivalences 51
3.2 Interactivegames 56
3.3 Bisimulationrelations 64
3.4 Modal propertiesand equivalences 69
3.5 Observablebisimulations 72
3.6 Equivalencechecking . 77
x Contents
4 TemporalProperties 83
4.1 Modal propertiesrevisited 83
4.2 Processesand their runs . 85
4.3 The temporal logic CTI.. . 89
4.4 Modal formulas with variables 91
4.5 Modal equationsand fixedpoints 95
4.6 Duality... . . . ..... . .. 100
5 Modal Mu-Calculus 103
5.1 Modallogic with fixedpoints .. . . . . 104
5.2 Macrosand normal formulas . 107
5.3 Observablemodallogic with fixedpoints 110
5.4 Preservationofbisimulationequivalence 112
5.5 Approximants . 115
5.6 Embedded approximants 121
5.7 Expressingproperties .. 128
6 VerifyingTemporalProperties 133
6.1 Techniquesfor verification 133
6.2 Propertycheckinggames 135
6.3 Correctnessofgames 144
6.4 CTL games .... . 147
6.5 Parity games . . . . . 151
6.6 Decidingparity games 156
7 Exposing Structure 163
7.1 Infinite statesystems 164
7.2 Generalisingsatisfaction 165
7.3 Tableaux I 168
7.4 Tableaux II . . . . . . . 173
References 183
Index 187
List of Figures
1.1 The transitiongraph for Cl 2
1.2 A vending machine 3
1.3 The transitiongraphfor Yen 3
1.4 Afamilyofcounters . . . . 4
1.5 The transitiongraph forCti 4
1.6 FlowgraphsofUserand Cop . 10
1.7 Flow graph ofCop IUser . . . . 11
1.8 Flow graph ofCop IUser IUser 11
1.9 Flow graph of(Cop IUser)\K 11
1.10 A levelcrossing 12
1.11 Flow graphsofthecrossingand itscomponents 13
1.12 Transitiongraph ofCrossing . . . . . . . . . . 14
1.13 Asimpleprotocol 14
1.14 Protocoltransitiongraph when there isone messagem. 15
1.15 Aslot machine . . . . . . . . . . . . . . . . . . . . . . . 15
1.16 Observabletransitiongraphsfor (CIU)\{in,ok}andUcop 20
1.17 Flow graph ofn instancesofB,and BI I...Ia.. 22
1.18 The flowgraph ofCy' 23
1.19 Flow graphofCy~ ICyzICy; ICy~ 24
3.1 Twoclocks . . . . . . . . . 52
3.2 Three vending machines . . 54
3.3 Gamegraphfor G(CI, CIs) 58
3.4 Reducedgame graph for G(CI,CIs) 59
3.5 Gamegraphfor (Ven2,Ven3) .. 60
3.6 Gameplay . 74
3.7 Asimplifiedslot machine . 79
