Process Algebras for Petri Nets: The Alphabetization of Distributed Systems

303 Pages·2017·2.69 MB·English

by  Roberto Gorrieri

Roberto Gorrieri
Process Algebras
for Petri Nets
The Alphabetization of Distributed Systems
Roberto Gorrieri
Dipartimento di Informatica - Scienza e Ingegneria
Università di Bologna
Bologna, Italy
ISSN 1431-2654 ISSN 2193-2069 (electronic )
Monographs in Theoretical Computer Science. An EATCS Series
ISBN 978-3-319-55558-4 ISBN 978-3-319-55559-1 (eBook)
DOI 10.1007/978-3-319-55559-1

Library of Congress Control Number: 2017936136

© Springer International Publishing AG 2017
This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Foreword
Distributed,open-endedsystemsareubiquitousintoday’sinformationandcommu-
nicationtechnology:inmostcasestheyexhibitanindependent,concurrentbehavior.
Theprocessofspecification,designanddevelopmentrequiresformalmodelsable
toexpressinterestingpropertiesandtoallowforefficientvalidationandverification
procedures.InadditiontotheirroleasdesigntoolsinICTengineering,distributed,
concurrentmodelsareverysuccessfulinformalizingkeyconceptsofbiology,eco-
nomics,complexsystemsandsoon.
In many cases it is convenient to develop specialized models for the classes of
systemsweareinterestedin,equippedwithconvenientfeatures.Forinstance,while
in all cases a formal semantics is required, there is a conflict between model ex-
pressivenessandcomplexityofverificationalgorithms.Anotherimportantchoiceis
betweenagraphicalmodelandatext-basedmodel,bothequippedwithobservation
criteria and consequent semantics: the former is often more perspicuous, but less
convenientforspecifyinglarge,modularsystemsinacompositionalway.
The main achievement of this book is to present a hierarchy of six models of
increasing expressiveness, each of them described both in graphical form and in
textual,term-likeform.Plentyofresultsguaranteethatthereisaone-to-onecorre-
spondencebetweenthetworepresentations,andthatcorrespondingformshavethe
sameinterleavingandstepsemantics.
In reality, graphical and textual forms are not equivalent: the former is more
expressive since its semantics can be enriched with observational criteria to make
certainrelevantpropertiesevident,whicharepresent,buthidden,inthelatter.For
instance, the presentation in graphical form of the solution of the dining philoso-
phersproblemcanbeshowntobecorrect,whilethetextualversion,oncethechoice
ismadeofobservingthebehaviorsequentially,hastoorestrictedobservationcapa-
bilities. In other words, as soon as the textual form is translated into the graphical
form, new observation and analysis methods become applicable. Thus the transla-
tion has the flavor of associating with the language a more expressive concurrent
semantics.
In the book, a clear choice is made about which models to handle: process de-
scriptionlanguageCCSandPlace/Transition(P/T)Petrinets,withvarioussyntactic
restrictions to yield the expressiveness hierarchy. Only the top level, Nonpermis-
sive nets, is an extension of P/T nets: the extension is needed to achieve Turing-
completeness.
Ifullyagree.Thereareseveralreasonsforthechoice.CCSisequippedwitharich
theory,astestifiedbyseveraltextbooksandmonographs,someofthemveryrecent.
Also,CCSisthestartingpointofthemostpopularlanguageformobilesystems,i.e.
π-calculus,andofitsextensionsforsecurityandservice-orientedarchitectures.P/T
Petrinetswerethefirstconcurrencymodel(CarlPetri’sthesisisdated1962),and
theyaremaybethemodelwiththerichesttheory,thelargestapplicationexperience
andthemostextensivecollectionoftools.Furthermore,(unsafe)finiteP/Tnetshave
aneasy-to-understandcyclicbehaviorwithapossiblyinfinitesetofstates,butsome
importantproperties,e.g.reachability,arestilldecidable.
Thetechnicalcontentofthebookisofextremelyhighquality.Actually,theau-
thor personally contributed with a number of seminal, well-known papers to the
historicaldevelopmentofthemainconceptsandresultsinthebook.Also,thepre-
sentationofcomplexresults,whichappearedoveralargespanofyears(thebibli-
ography contains about 150 citations), required a remarkable effort to make them
consistent and uniform. In addition, some important results, needed to complete
the full picture, are published in the book for the first time, as appropriate in a re-
searchmonograph.Finally,thepresentationandjustificationoftheconceptualde-
velopments and of the formal achievements is well articulated and explained, and
supportedbyplentyofexamplesasrequiredinanadvancedtextbook.
Thebookwillbepreciousforthoseinterestedinthetrulyconcurrentsemantics
ofdistributedcommunicatingsystems.Asystemdesigner,givenaspecificationin
CCS,mightwanttobetterunderstanditsbehaviorbylookingatthecorresponding
net,bycheckingtherethegivenconcurrencyrequirements,andpossiblybytaking
advantage of existing tools. More relevantly from a methodological point of view,
acomputerscientist,aimingatstudyingaparticularclassofconcurrentdistributed
systems,mightdefineaspecificationlanguagebasedonvariantsofCCSandPetri
netsbyapplyingtheapproachinthebook.
UgoMontanari Pisa,June2016
Contents
1 Introduction................................................... 1
1.1 TheAlphabetizationofDistributedSystems .................... 1
1.2 TheHierarchy ............................................. 6
1.3 StructureoftheBook ....................................... 8
1.4 InterleavingvsTrueConcurrency ............................. 11
1.5 BeyondTuring-Completeness ................................ 12
2 LabeledTransitionSystems ..................................... 15
2.1 LabeledTransitionSystems .................................. 15
2.2 BehavioralEquivalences..................................... 19
2.2.1 StrongEquivalences.................................. 19
2.2.2 WeakEquivalences................................... 26
2.3 StepTransitionSystems ..................................... 31
3 PetriNets ..................................................... 35
3.1 Introduction ............................................... 35
3.2 Place/TransitionPetriNets................................... 36
3.2.1 SomeClassesofPetriNets ............................ 41
3.2.2 DynamicallyReachableandStaticallyReachableSubnets .. 44
3.3 DecidableProperties........................................ 47
3.3.1 CoverabilityTree .................................... 48
3.3.2 Reachability,LivenessandDeadlock.................... 54
3.4 BehavioralEquivalences..................................... 57
3.4.1 NetIsomorphism .................................... 57
3.4.2 InterleavingSemantics................................ 58
3.4.3 StepSemantics ...................................... 64
3.5 NonpermissivePetriNets.................................... 67
3.5.1 BehavioralEquivalences .............................. 70
3.5.2 Turing-Completeness................................. 74
4 TheBasicCalculus:SFM ....................................... 77
4.1 Syntax.................................................... 77
4.2 OperationalLTSSemantics .................................. 80
4.2.1 Expressiveness ...................................... 81
4.2.2 Congruence......................................... 83
4.3 OperationalNetSemantics................................... 84
4.4 RepresentingAllSequentialFinite-StateMachines .............. 87
4.5 DenotationalNetSemantics.................................. 90
5 AddingAsynchronousParallelComposition:CFMandBPP ........ 95
5.1 CFM ..................................................... 95
5.1.1 InterleavingLTSSemantics............................ 96
5.1.2 StepSemantics ...................................... 99
5.1.3 OperationalNetSemantics ............................102
5.1.4 RepresentingAllConcurrentFinite-StateMachines .......106
5.1.5 Soundness ..........................................107
5.1.6 DenotationalNetSemantics ...........................108
5.2 BPP:BasicParallelProcesses ................................110
5.2.1 Expressiveness ......................................111
5.2.2 OperationalNetSemantics ............................113
5.2.3 RepresentingAllBPPNets ............................116
5.2.4 DenotationalNetSemantics ...........................118
6 AddingCommunicationandRestriction:FNC ....................121
6.1 Syntax....................................................121
6.1.1 RestrictedActionsandExtendedProcesses ..............123
6.1.2 SyntacticSubstitution ................................124
6.1.3 SequentialSubterms..................................126
6.2 OperationalLTSSemantics ..................................129
6.2.1 Expressiveness ......................................130
6.3 StepSemantics ............................................132
6.4 OperationalNetSemantics...................................134
6.4.1 PlacesandMarkings .................................134
6.4.2 NetTransitions ......................................138
6.4.3 TheReachableSubnetNet(p)..........................145
6.5 RepresentingAllFiniteCCSNets.............................147
6.6 Soundness ................................................152
6.7 DenotationalNetSemantics..................................154
6.8 RCS .....................................................166
7 AddingMulti-partyCommunication:FNM .......................169
7.1 Preliminaries ..............................................169
7.1.1 SyntaxandInformalSemantics ........................169
7.1.2 ExtendedProcessesandSequentialSubterms.............171
7.1.3 Well-FormedProcesses ...............................173
7.2 OperationalLTSSemantics ..................................175
7.2.1 Expressiveness ......................................181
7.2.2 CongruenceProblem .................................183
7.3 StepSemantics ............................................184
7.3.1 StepBisimilarityImpliesInterleavingBisimilarity ........187
7.3.2 StepBisimilarityIsaCongruence ......................191
7.4 OperationalNetSemantics...................................192
7.4.1 PlacesandMarkings .................................192
7.4.2 NetTransitions ......................................195
7.4.3 PropertiesofNetTransitions...........................196
7.4.4 TheReachableSubnetNet(p)..........................202
7.5 RepresentingAllFiniteP/TNets..............................205
7.5.1 Expressiveness ......................................211
7.6 Soundness ................................................213
7.7 DenotationalNetSemantics..................................215
7.8 RMCS....................................................223
8 AddingAtomicTestsforAbsence:NPL...........................227
8.1 Syntax....................................................227
8.2 OperationalLTSSemantics ..................................230
8.2.1 Expressiveness ......................................233
8.2.2 CongruenceProblem .................................237
8.3 StepSemantics ............................................239
8.4 OperationalNetSemantics...................................246
8.4.1 PlacesandMarkings .................................247
8.4.2 NetTransitions ......................................248
8.4.3 PropertiesofNetTransitions...........................250
8.4.4 TheReachableSubnetNet(p)..........................254
8.5 RepresentingAllFiniteNP/TNets ............................256
8.6 Soundness ................................................260
8.7 DenotationalNetSemantics..................................265
8.8 RNPL ....................................................271
9 GeneralizationsandVariantSemantics ...........................273
9.1 CommunicatingPetriNets ...................................273
9.2 VariantNetSemantics.......................................275
9.3 GeneralRestriction .........................................277
9.4 AsynchronousCommunication ...............................282
9.5 OtherLanguages? ..........................................283
9.6 FutureResearch............................................284
Glossary ..........................................................287
References.........................................................291
Index .............................................................299
Chapter 1
Introduction
Abstract This introductory chapter outlines the main problem dealt with in this
book:findingsuitablelanguagesforrepresentingclassesofPetrinets,takinginspi-
rationfromtheprocessalgebrasdevelopedinthelastfourdecades.Thestructureof
thebookisoutlinedandsomehintsonhowtoreaditarepresented.Finally,itisalso
arguedthatTuring-completenessisnotasufficientcriteriontocomparetheexpres-
sivepowerofdifferentprocessalgebras,becausethesetsofproblemsindistributed
computing that two languages can solve may be different, even if both include all
theTuring-computablefunctions.
1.1 TheAlphabetizationofDistributedSystems
A distributed system is a computer system made of several components, imple-
mentedinhardwareorsoftwareorasacombinationofboth,thatmaybelocatedat
differentsites,evenatageographicaldistance,andthatcooperatetoaccomplisha
taskorcoordinatetoofferaservicebymeansofsuitablecommunicationprotocols
basedonmessagepassing.Themostnotableexampleofadistributedsystemisthe
Internet,whosemostimportantserviceistheWorldWideWeb.
Ataveryabstractlevelofdetail,themainfeatureofadistributedsystemisdis-
tribution:theglobalstateofthesystemiscomposedofacollectionoflocalstates,
physicallylocatedatdifferentsites,andeachactivitythatthesystemperformsmay
actuallyinvolveonlyasubsetoftheselocalstates.Anotherimportantfeatureisthat
communication takes place only by message passing, so that any information ex-
changehappensbymeansofexplicitcommunicationprimitivesofsendorreceive;
in other words, there is no global memory, shared by the components. The com-
municationmechanismcanbesynchronousorasynchronous:theformerwhenthe
sendactionandthereceiveoneareperformedbytheinteractingpartnersatthesame
time;thelatterwhenthesendactionisdecoupledfromthereceiveone.
Many other features of distributed systems are relevant, such as heterogeneity
of the components, possible independent failure of components, the absence of a
© Springer International Publishing AG 2017 1
R. Gorrieri, Process Algebras for Petri Nets, Monographs in Theoretical
Computer Science. An EATCS Series, DOI 10.1007/978-3-319-55559-1_1
2 1 Introduction
global clock, scalability etc., but for the aims of this book, only distribution and
communicationareconsidered.Inparticular,communicationisassumedtobesyn-
chronous,becausethismechanismcanalsoeasilyimplementtheasynchronousone
(it is enough to put a medium, such as a buffer, between the two partners of the
communicationtogetanasynchronouscommunication),whilethereverseismore
difficulttoachieve.
Manysemanticmodelsofcomputationhavebeenproposedtomodeldistributed
systems;herewegiveashort,notexhaustive,list:
• Petrinets[Petri62,Hack76b,Pet81,Rei85,MM90,JK95,MR95,Bus02,Rei13];
• Transitionsystems[Kel76,Mil80,Plo04b,Gla01,San12,GV15];
• Eventstructures[NPW81,Win87,Win88,BMM06];
• Causaltrees[DD89,DD90,BMS15];
• Concurrenthistories[DM87];
• Statecharts[Har87,HPSS87];
• Messagesequencecharts[RGG,DH99];
• Kahnprocessnetworks[Kah74].
Eachofthesehasitsownprosandcons.However,amongthem,wechosePetri
nets,forthefollowingreasons:
1. distributionisafirst-classconcept(whichisnotthecasefor,e.g.,labeledtransi-
tionsystems);
2. Petrinetscanmodelrecursivebehaviorwithafinitestructure(whichisnotthe
casefor,e.g.,eventstructures);
3. theyareawidelystudiedmodel(see,e.g.,[RR98a,DRR04]andthereferences
therein),equippedwithasimple,precise,formalsemantics,bothfortheso-called
linear-timeandbranching-timesemantics(whichisnotthecasefor,e.g.,message
sequencecharts);
4. theyareequippedwithanalysistechniquesthataredecidableinmanycases,as
described in Section 3.3 (which is not the case for, e.g., concurrent histories),
andthataresometimessupportedbyautomaticorsemi-automaticsoftwaretools
(see,e.g.,[TA15,Tool]forsurveysonPetrinettools);and,finally,
5. there is a large literature of applications of Petri nets to the modeling of real
distributedsystems(see,e.g.,[RR98b,Rei98])andthereferencestherein).
Fromnowon,atanabstractlevel,wetakethelibertyofidentifyingadistributed
system with the Petri net which models it. Therefore, we shall consider specific
classesofPetrinetsasspecificclassesofdistributedsystems.
Many specification languages have been proposed to describe reactive, dis-
tributedsystems,startingfromtheseminalworkbyHoarewithCSP[Hoa78,Hoa85,
Ros98], and Milner with CCS [Mil80, Mil89, GV15]. These languages are usu-
ally called process algebras, to reflect the algebraic nature of their syntactic and
semantic definitions. Many process algebras have been proposed in the literature:
besides CSP and CCS, also ACP [BK84, BW90, BBR10] by Bergstra, Klop and
1.1 TheAlphabetizationofDistributedSystems 3
a.0|b.0 a.b.0+b.a.0
a b a b
0|b.0 a.0|0 b.0 a.0
b a b a
0|0 0
Fig.1.1 Interleavinglaw:twoisomorphicLTSs
a.0|b.0 a.b.0+b.a.0
{a} {b} {a} {b}
0|b.0 {a,b} a.0|0 b.0 a.0
{b} {a} {b} {a}
0|0 0
Fig.1.2 Twosteptransitionsystems
Baeten(whocoinedthetermprocessalgebra),LOTOS[BB87,BLV95]byBolog-
nesiandBrinksma(whoincludedabstractdatatypesinaprocessalgebra),CIRCAL
[Mil85]byMilne(whointroducedthestepsemanticsforaprocessalgebra),andthe
π-calculus [MPW92, Mil99, Par01, SW01] by Milner, Parrow and Walker (which
modelsmobility),justtomentionafew(see[Bae05]forahistoricaloverview,and
[BPS01]foratechnicalsurveyonthemanyfacetsofprocessalgebras).Theseman-
tics of these languages have mainly been given in an interleaving style, either in
termsofexecutiontraces(e.g.,CSP)orintermsoflabeledtransitionsystems(e.g.,
CCS),butinnowayareparallelism(orco-occurrenceofindependentactions)and
distributionmodeledinthesesemantics.
An example may help clarify the idea. The process composed of two parallel
actions a and b is denoted in CCS by the term a.0|b.0, while the sequential pro-
cessperformingthesetwoactionsineitherorderisdenotedbya.b.0+b.a.0.Inthe
labeledtransitionsystemsemanticsof[Mil89],thesetwoCCSprocessesoriginate
theisomorphiclabeledtransitionsystemsinFigure1.1,sothattheyaresemantically
equal.Thisistheessenceoftheso-calledinterleavinglaw:a(finite-state)parallel
process is semantically equivalent to a (finite-state) possibly nondeterministic, se-
quentialprocess.Thisisacleardrawbackoftheinterleavingsemantics,aswewill
arguefurtherinSection1.4.
Onemayenrichthelabelingofthetransitionsystembyusing,insteadofsingle
actions, multisets of concurrently executable actions: this is the so-called step se-
mantics(originallyintroducedin[NT84,Mil85]),whichisrefinedenoughtomodel