Axiomatizing Prefix Iteration with Silent Steps

15 Pages·1996·0.64 MB·English

InformationandComputation(cid:21)IC2577
informationandcomputation127,26(cid:21)40(1996)
articleno.0047
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LucaAceto*
BRICS,DepartmentofMathematicsandComputerScience,AalborgUniversty,Fr.Bajersvej7E,9220AalborgO3 ,Denmark
E-mail:luca(cid:20)iesd.auc.dk
RobvanGlabbeek-
ComputerScienceDepartment,StanfordUniversity,Stanford,California94305-9045
E-mail:rvg(cid:20)cs.stanford.edu
WanFokkink
DepartmentofPhilosophy,UtrechtUniversity,Heidelberglaan8,3584CSUtrecht,TheNetherlands
E-mail:fokkink(cid:20)phil.ruu.nl
and
AnnaIngo(cid:14)lfsdo(cid:14)ttir
BRICS,DepartmentofMathematicsandComputerScience,AalborgUniversity,Fr.Bajersvej7E,9220AalborgO3 ,Denmark
E-mail:annai(cid:20)iesd.auc.dk
expressivepowerofvariationsonstandardprocessdescrip-
Prefix iteration is a variation on the original binary version of the tion languages in which infinite behaviours are defined by
KleenestaroperationP*Q,obtainedbyrestrictingthefirstargumentto means of Kleene’s star operation [28,11] rather than by
beanatomicaction.Theinteractionofprefixiterationwithsilentsteps means ofsystems ofrecursionequations.Someothers (see,
is studied in the setting of Milner’s basic CCS. Complete equational
e.g.,[18,34,15,2,17])havestudiedthepossibilityofgiving
axiomatizationsare given forfournotionsofbehaviouralcongruence
finite equational axiomatizations of strong bisimulation
over basic CCS with prefix iteration, viz., branching congruence,
’-congruence, delay congruence, and weak congruence. The com- equivalence [32,30] over simple process algebras that
pletenessproofsfor’-, delay, andweakcongruenceareobtainedby include variations on Kleene’s star operation. De Nicola
reductiontothecompletenesstheoremforbranchingcongruence.Itis andco-workers[13,12] haveinsteadfocusedonthestudy
alsoarguedthattheuseofthecompletenessresultforbranchingcon-
of tree-based models for what they call ‘‘nondeterministic
gruence in obtaining the completeness result for weak congruence
Kleene algebras’’ and on the proof systems these models
leads to a considerable simplification with respect to the only direct
proofpresentedintheliterature.Thepreliminaries andthe complete- support to reason about regular expressions and more
nessproofsfocusonopenterms,i.e.,termsthatmaycontainprocess expressivelanguagesbuiltontopofthose.
variables.Asaby-product,the|-completenessoftheaxiomatizations This paper aims at giving a contribution to the study of
is obtained, as well as their completeness for closed terms. ]1996 complete equational axiomatizations for Kleene star-like
AcademicPress,Inc.
operations from the point of view of process theory. Our
starting point is the work presented in [15]. In that
reference, a finite, complete equational axiomatization
1. INTRODUCTION of strong bisimulation equivalence has been given for
T(BCCS)p*(A ),i.e.,thelanguageofclosedtermsobtained
{
Theresearchliteratureonprocesstheoryhasrecentlywit- byextendingthefragmentofMilner’sCCS[30]containing
nessed a resurgence of interest in the study of Kleene star- the basic operations needed to express finite synchroniza-
like operations (cf., e.g., [8, 18, 15, 13, 34, 12, 16, 3, 2]). tiontreeswithprefixiteration.Prefixiterationisavariation
Some of these studies, notably [8], have investigated the on the original binary version of the Kleenestar operation
P*Q [28] obtained by restricting the first argument to be
*On leave from the School of Cognitive and Computing Sciences,
an atomic action. Intuitively, at any time the process term
UniversityofSussex,BrightonBN19QH,UnitedKingdom.Partiallysup-
a*P can decide to perform action a and evolve to itself,
portedbyHCMprojectEXPRESS.
-PartiallysupportedbyONRGrantN00014-92-J-1974. or an action from P, by which it exits the a-loop. The
0890-5401(cid:18)96(cid:30)18.00 26
Copyright(cid:23)1996byAcademicPress,Inc.
Allrightsofreproductioninanyformreserved.
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AXIOMATIZING PREFIX ITERATION WITH SILENT STEPS 27
behaviour of a*P is captured very clearly by the rules that [3], and describe a rather subtle interplay between prefix
giveitsPlotkin-stylestructuraloperationalsemantics: iterationand thesilentaction{.Alltheaxiomatizationswe
present are finite, if so is the set of abservable actions, and
Pw(cid:20)b P$ irredundant.
a*Pw(cid:20)a a*P a*Pw(cid:20)b P$ The strategy we adopt in establishing the completeness
resultsisbasedupontheuseofbranchingequivalenceinthe
analysis of weak, delay and ’-equivalence advocated in
In [15], itisshownthat, instrongbisimulationsemantics,
[20]. Following [20], complete axiomatizationsfor weak,
such an operation can be characterized by the standard
delay and ’-congruence can be obtained from one for
equationsforCCSsummantion (cf.[30]andTable1) and
branchingcongruenceby:
thefollowingtwonaturallaws:
1. identifying a collection of process terms on which
a.(a*x)+x=a*x
branching congruence coincides with the congruence one
a*(a*x)=a*x. aimsataxiomatizing, and
2. findinganaxiomsystemthatallowsforthereduction
The reader familiar with Hennessy’s work on complete ofeveryprocesstermtooneoftherequiredform.
axiomatizations for the delay operation of Milner’s SCCS
[24,25] willhavenoticedthesimilaritybetweentheabove For example, the completeness result for weak congruence
lawsandthosepresentedin[24](seealso[1, p.40]).This isobtainedbyprovingthatbranchingandweakcongruence
isnotsurprising, assuchadelayoperationisaninstanceof coincideoverthecollectionofw-saturatedprocessterms(cf.
theprefixiterationconstruct. Definition4.5), and that, using the axiom system for weak
congruence, every term is provably equal to a w-saturated
1.1. Results one.
Thecompletenessresultsfor’-anddelaycongruenceare
In this paper, we extend the results in [15] to a setting
new, while those for weak and branching congruence were
withtheunobservableaction{.Moreprecisely,weconsider
firstprovenin[3,16],respectively.However,theproofsfor
four versions of bisimulation equivalence that, to different
these last two results that are presented in this paper are
degrees, abstract away from the internal evolution of
new, and we consider them to be an improvement on the
processes (viz., delay equivalence [29], weak equivalence
original ones. In particular, unlike the one given in [16],
[30], ’-equivalence [5] and branching equivalence [21]),
the proof for branching congruence does not rely on the
and provide complete equational axiomatizations for each
completeness result for strong bisimulation presented in
of the congruences they induce over the language
[15]. Perhaps surprisingly, the proof for weak congruence
T(BCCS)p*(A ) of open terms over the signature of
{ presented here is simpler than the one given in [3] which
T(BCCS)p*(A ). The axiomatizations we present are
{ only uses properties of weak congruence. The direct proof
obtainedbyextendingtheaxiomsystemfrom[15]withthe
methodemployedin[3]yieldsalongproofwithmanycase
relevant {-laws known from the literature for each of the
distinctions, while the indirect proof via branching con-
congruences we consider (cf. [22] for a discussion ofthese
gruence,whichwepresenthere,isconsiderablyshorter,and
laws)andwithcollectionsoflawsthatdescribetheinterplay
reliesonageneralrelationshipbetweenthetwocongruences.
between the silent nature of { and prefix iteration. For
All the authors’ attempts to obtain a direct proof of the
instance, the axiomatization of weak congruence uses
completenesstheoremforweakcongruencewhichissimpler
Milner’swell-known{-laws [30] andthefollowingaxioms
than the one presented in [3] have been to no avail. It
describing the interaction of prefix iteration with the silent
shouldbenoted,however,thatdelicatecaseanalysesappear
action{:
to be inescapable components of completeness proofs for
equationalaxiomatizationsofbehaviouralcongruencesover
{*x={.x variationsonKleenealgebras(cf.,e.g.,theproofsin[18,15,
3, 16, 2, 17]), and they are present in our completeness
{.(a*x)=a*({.a*x)
proofsjustaswell.
a*(x+{.y)=a*(x+{.y+a.y). Anothernotablefeatureoftheproofsofthecompleteness
theoremsweofferisthat,unlikethosein[3,16],theyapply
Thefirstoftheseequationswasintroducedin[8]underthe toopentermsdirectly,andthusyieldthe|-completenessof
name of Fair Iteration Rule, and expresses a fundamental theaxiomatizationsaswellastheircompletenessforclosed
propertyofweakcongruence,namely,theabstractionfrom terms. Following [31,19], this is achieved by defining a
{-loops, that underlies the soundness of Koomen’s Fair structural operational semantics and notions of bisimula-
Abstraction Rule [4]. The other two equations are from tiondirectlyonopenterms.Forallthenotionsofbisimula-
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28 ACETO ET AL.
tion equivalence so defined for open terms in the language R, S, T to range over T(BCCS)p*(A ). In writing terms
{
T(BCCS)p*(A ),weprovethattwotermsareequivalentiff over the above syntax, we shall always assume that the
{
all their closed instantiations are. This ensures that our operations:*and :. bindstrongerthan+.Weshalluse
&
definitions are in agreement with the standard ones in the the symbol # to stand for syntactic equality of terms. The
literatureonprocesstheory. setofprocessvariablesoccurringinatermPwillbewritten
The|-completenessoftheaxiomatizationforbranching, Var(P).
’-anddelaycongruenceareallnew.Theaxiomatizationfor A (closed) substitution is a mapping from process
weak congruence was first shown to be |-complete in [3] variables to (closed) terms over BCCSp*(A ). For every
{
in the presence of a countably infinite set of observable term P and (closed) substitution _, the (closed) term
actions, usingatechniquefromGroote [23].Ourresultin obtainedbyreplacingeveryoccurrenceofavariablexinP
thispapersharpenstheoneintheaforementionedreference withthe(closed)term_(x)willbewrittenP_.Weshalluse
in that, like the ones for branching, ’- and delay con- [x[P] tostandforthesubstitutionmappingxtoP, and
gruence,itonlyrequiresthatthesetofobservableactionsbe actingliketheidentityonalltheothervariables.
non-empty. TheoperationalsemanticsforthelanguageBCCSp*(A )
{
isgivenbythelabelledtransitionsystem[27,33]
1.2. OutlineofthePaper
(T(BCCSp*(A ),[w(cid:20)! |!#A _Var])
{ {
The paper is organized as follows. Section2 introduces
the language of basic CCS with prefix iteration, where the transition relations w(cid:20)! are the least subsets of
T(BCCS)p*(A ),anditsoperationalsemantics.Inthatsec-
{ T(BCCS)p*(A )_T(BCCS)p*(A ) satisfying the rules in
tion, wealsogivethedefinitionofbranching,’-,delay,and Fig.1. Intuitive{ly, a transition Pw(cid:20):{ Q (:#A ) means that
weakcongruenceoveropenterms,andshowthattwoopen {
thesystemrepresentedbythetermPcanperformtheaction
terms are related by any of those congruences iff all their :,therebyevolvingintoQ,whereasPw(cid:20)x P$meansthatthe
closed instantiations are. Section2 concludes with a study
initial behaviour of P may depend on the term that is sub-
ofseveralpropertiesofthecongruencerelationsweconsider
stitutedfortheprocessvariablex.Itisnothardtoseethat
that will be used in the remainder ofthe paper. The axiom ifPw(cid:20)x P$thenP$#x.
systems that will be shown to completely characterize the The derived transition relations =O= and =O! (!#A _
aforementioned congruences over T(BCCS)p*(A ) are {
{ Var)aredefinedin thestandardwayasfollows:
analyzed in Section3. Detailed proofs of the completeness
of our axiom systems with respect to the relevant congru-
encesoverT(BCCS)p*(A )arepresentedinSection4. {=O= is the reflexive , transitive closure of w(cid:20){
{
P =O! Q iff _P ,P :P =O= P w(cid:20)! P =O= Q.
1 2 1 2
2. BASICCCSWITHPREFIXITERATION
Definition 2.1. Thesetder(P)ofderivativesofPisthe
leastsetcontainingPthatisclosedunderaction-transitions.
We assume a non-empty, countable set A of observable Formally, der(P)istheleastsetsatisfying:
actions not containing the distinguished symbol{. Follow-
ing Milner [30], the symbol { will be used to denote an 1. P#der(P);
internal, unobservable action of a system. We define 2. if Q#der(P) and Qw(cid:20): Q$ for some :#A , then
{
A{]A_[{], and use a,b to range over A and :,;,# to Q$#der(P).
range over A . We also assume a countably infinite set of
{
processvariablesVar,rangedoverbyx,y,z,thatisdisjoint Thefollowingbasicfactcanbeeasilyshownbystructural
from A . The meta-variable ! will stand for a typical inductiononterms:
{
memberofthesetA _Var.
{ Fact 2.2. For every P#T(BCCS)p*(A ), the set of
ThelanguageofbasicCCSwithprefixiteration, denoted {
derivativesofPisfinite.
byBCCSp*(A ), isgivenbythefollowingBNFgrammar:
{
P::=x |0| :.P|P+P| :*P.
Here x#Var and :#A . The set of (open) terms over
{
BCCSp*(A )isdenotedbyT(BCCS)p*(A ),andthesetof
{ {
closed terms, i.e., terms that do not contain occurrences of
process variables, by T(BCCS)p*(A ). We shall use P, Q, FIG. 1. Transitionrules.
{
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AXIOMATIZING PREFIX ITERATION WITH SILENT STEPS 29
Afundamentalsemanticequivalenceinthestudyofreactive 1. IfPW Q, thenPW QandPW Q;
b ’ d
systems is bisimulation equivalence [32,30]. In this study, 2. IfPW QorPW Q, thenPW Q.
n d w
weshallconsider four versionsof thisnotionwhich, todif-
Proof. For +#[’,d,w], the identity relation, the con-
ferent degrees, abstract away from invisible actions, viz.
branching equivalence [21], ’-equivalence [5], delay verse of a +-bisimulation and the symmetric closure of the
compositionoftwo+-bisimulationsareall+-bisimulations.
equivalence [29] and weak equivalence [30]. These we
Hence W is an equivalence relation. This argument does
now proceed to define for the sake of completeness. The +
not apply for +=b because the symmetric closure of
interested reader is referred to the aforementioned refer-
encesandto[22, 31, 19]fordiscussionandmotivation. the composition of two b-bisimulations need not be a
b-bisimulation, but in [7] it is shown that also W is an
b
Definition 2.3 (Branching Equivalence). A binary equivalencerelation.
relation B over T(BCCS)p*(A ) is a branching bisimula- That W is the largest +-bisimulation (for +#
{ +
tion, or b-bisimulation for short, iff it is symmetric and, [b,’,d,w])followsimmediatelyfromtheobservationthat
wheneverPBQ, forall!#A _Var, the set of+-bisimulations is closed under arbitraryunions.
{
Theimplicationsholdbydefinition. K
ifPw(cid:20)! P$then
Thereader familiar with the literatureon processtheory
v !={andP$BQ, or
might have noticed that, in the above definitions, we have
v Q =O= Q w(cid:20)! Q =O= Q$forsomeQ ,Q ,Q$suchthat departedfromthestandardapproachfollowedin,e.g.,[30]
1 2 1 2
PBQ , P$BQ , andP$BQ$. inthatwehavedefinednotionsofbisimulationequivalence
1 2
thatapplytoopentermsdirectly.Indeed,withtheexception
Two process terms P,Q are branching equivalent, denoted
ofstudieslike[31,19],bisimulationequivalenceslikethose
byPW Q, iffthereexistsabranchingbisimulationBsuch
b presented in Definitions2.3(cid:21)2.4 are usually defined for
thatPBQ.
closed process expressions only, and are extended to open
The notions of ’-, delay and weak bisimulation are processexpressionthus(+#[b,’,d,w]):
obtained by relaxing (some of) the constraints imposed by
branching bisimulation on the way that two processes can PW QiffP_W Q_ for every closed substitution _.
+ +
match each other’s behaviours. Compare the following
definitions: By the following result, first shown in [19] for branching
bisimulation over basic CCS with recursion, both
Definition 2.4 (’-, Delay, and Weak Equivalence). approaches yield the same equivalence relation over open
The notion of ’-bisimulation is defined just as a branching termsinthelanguageBCCSp*(A ).
{
bisimulation above, but without the requirement P$BQ .
2 Proposition 2.6. For all P,Q#T(BCCS)p*(A ) and
Two process terms P,Q are ’-equivalent, denoted by {
PW Q,iffthereexistsan’-bisimulationBsuchthatPBQ. +#[b,’,d,w],
’
Likewise,adelaybisimulation,ord-bisimulationforshort,
isdefinedjustasabranchingbisimulation,butomittingthe PW Q iff P_W Q_ for every closed substitution
+ +
requirement PBQ1. Two process terms P,Q are delay _:Var(cid:20)T(BCCS)p*(A ).
equivalent, denoted by PW Q, iff there exists a delay {
d
bisimulationBsuchthatPBQ.
Proof. In the proof of this result, we shall make use of
Finally, a weak bisimulation, or w-bisimulation, lacks
the following, easily established, facts, which relate the
both therequirements PBQ and P$BQ , andtwo process
1 2 transitionsofatermP_tothoseofPandthoseoftheterms
terms P,Q are weakly equivalent, denoted by PW Q, iff
w _(x):
thereexistsaweakbisimulationBsuchthatPBQ.
1. IfPw(cid:20): P$, thenP_w(cid:20): P$_.
Remark. It is easy to see that inthe definitions ofboth
2. IfPw(cid:20)x xand_(x)w(cid:20)! Q, thenP_w(cid:20)! Q.
branchinganddelaybisimulationtheexistencerequirement
ofatermQ$suchthatQ =O= Q$andP$BQ$isredundant. 3. IfP_w(cid:20)! Q, theneither
2
The notions of delay and weak equivalence were (a) !#A andthereexistsaP$suchthatPw(cid:20)! P$and
{
originally both introduced by Milner under the name of Q#P$_, or
observation(al)equivalence. (b) there exists an x#Var such that Pw(cid:20)x x and
_(x)w(cid:20)! Q.
Proposition 2.5. Each of the relations W (+#
+
[b,’,d,w]) is an equivalence relation and the largest We prove the two implications in the statement of the
+-bisimulation.Furthermore, forallP,Q, propositionseparately.
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30 ACETO ET AL.
v Only If Implication. Assume that PW Q is a +-bisimulation. The details of this verification are
+
(+#[b,’,d,w]). We shall show that P_W Q_ for every straightforward, using facts 1(cid:21)3 above and (2). In par-
+
closedsubstitution _:Var(cid:20)T(BCCS)p*(A ).To thisend, ticular, condition (2) ensures that whenever SB T and
{ +
itissufficienttoprovethattherelation: Sw(cid:20)x x, thenT =Ox x.
B ][(S_,T_)|SW T, _ a closed substitution] This completes the proof of the inductive step, and
+ +
therebyofthe‘‘if’’implication.
isan+-bisimulation.Thisisstraightforwardusingfacts1(cid:21)3 Theproofofthe propositionisnowcomplete. K
above.
Remark. The reader may have noticed that the ‘‘if’’
v If Implication. Let +#[b,’,d,w]. Assume that
implicationintheabovestatementwouldnotholdiftheset
P_W Q_ for every closed substitution _. We shall show
+ ofobservableactionsAwereempty.Infact, inthat, admit-
thatPW Qholds.Thisweprovebyinductiononthenum-
+ tedly uninteresting, case, the universal relation over
berofvariables occurring in PorQ, i.e., on thecardinality
T(BCCS)p*(A ) would be a branching bisimulation. This
ofVar(P)_Var(Q). {
wouldimply,forinstance,that,foreveryclosedsubstitution
(cid:21)(cid:21) Basis. Var(P)_Var(Q)=<. In this case, P and Q _andvariablesx, y,
areclosedterms, andtheclaimfollowsimmediately.
(cid:21)(cid:21) Inductive Step. Var(P)_Var(Q){<. Choose a x_W y_.
b
variable x in Var(P)_Var(Q). As the set of observable
actions A is non-empty, we can pick a#A. It is easy tosee
Onthe otherhand, xisnotbranchingequivalenttoy.
that, forpositiveintegersn,m,
For the standard reasons explained at length in, e.g.,
an.0W am.0(cid:15)n=m.
+ Milner’s textbook [30], none of the aforementioned
equivalences is a congruence with respect to the summa-
ByFact2.2,der(P)_der(Q)isafinitesetofprocessterms.
tion operation. In fact, it is also the case that none of
Therefore it is possible to choose a positive integer n such
the aforementioned equivalences is preserved by the
that, foreveryR#der(P)_der(Q),
prefix iteration operation. As a simple example of this
an.0W3 R. (1) phenomenon,considerthetermsb.0and{.b.0.Asitiswell-
+ known, b.0W {.b.0 (+#[b,’,d,w]); however, it is not
+
Note that the above inequality implies that, for every difficult to check that a*(b.0)W3 a*({.b.0). Following
+
R#der(P)_der(Q), Milner [30], the solution to these congruence problems is
bynowstandard;itissufficienttoconsider,foreachequiv-
an.0W3 +R[x[an+1.0]. (2) alence W+, the largest congruence over T(BCCS)p*(A{)
contained init.Wenow proceedtocharacterize the result-
Thisisimmediateby(1)ifxdoesnotoccurinR.Otherwise,
ingcongruencesexplicitly.
xoccursinR, anditisnothardtoseethatR[x[an+1.0]
can perform a sequenceof transitionsleading to0 that has Definition 2.7. Wesaythat
a suffix consisting of at least n+1 a-transitions, whereas
an.0cannot. v P and Q are branching congruent, written PWcbQ, iff
forall!#A _Var,
Now, notethat, foreveryclosedsubstitution_, {
1. if Pw(cid:20)! P$, then Qw(cid:20)! Q$ for some Q$ such that
(P[x[an+1.0])_W (Q[x[an+1.0])_. (3)
+ P$W Q$;
b
As the set of variables occuring in P[x[an+1.0] or 2. if Qw(cid:20)! Q$, then Pw(cid:20)! P$ for some P$ such that
Q[x[an+1.0] is strictly contained in Var(P)_Var(Q), P$W Q$.
b
wemayapplytheinductivehypothesisto(3) toinferthat
v P and Q are ’-congruent, written PWcQ, iff for all
’
P[x[an+1.0]W Q[x[an+1.0]. (4) !#A{_Var,
+
1. if Pw(cid:20)! P$, then Qw(cid:20)! Q =O= Q$ for some Q ,Q$
We prove that this implies PW+Q, as required. To this suchthatP$W Q$; 1 1
end,inviewof(4),itissufficienttoshowthatthesymmetric ’
2. if Qw(cid:20)! Q$, then Pw(cid:20)! P =O= P$ for some P ,P$
closureoftherelation 1 1
suchthatP$W Q$.
’
B+][(S,T)|(S,T)#der(P)_der(Q) v PandQaredelaycongruent,writtenPWcQ,iffforall
d
and S[x[an+1.0]W T[x[an+1.0]] !#A _Var,
+ {
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AXIOMATIZING PREFIX ITERATION WITH SILENT STEPS 31
1. if Pw(cid:20)! P$, then Q =O= Q w(cid:20)! Q$ for some Q , Q$ Remark. Bloom [10] has formulated the ‘‘RWB cool’’
1 1
suchthatP$W Q$; and ‘‘RBB cool’’ formats for transition rules, which ensure
d
2. if Qw(cid:20)! Q$, then P =O= P w(cid:20)! P$ for some P , P$ that the relations Wc and Wc, respectively, are con-
1 1 w b
suchthatP$W Q$. gruences.
d Although both Wc and Wc are congruences for
v PandQareweaklycongruent, writtenPWc Q,ifffor w b
w T(BCCS)p*(A ), the transition rules for BCCSp*(A ) do
all!#A _Var, { {
{ not fit the RWB and RBB cool formats. In particular,
1. if Pw(cid:20)! P$, then Q =O! Q$ for some Q$ such that Bloom’s formats require that operators for which weak or
P$W Q$; branchingequivalenceisnotacongruencearenottooccur
w
2. if Qw(cid:20) Q$, then P =O! P$ for some P$ such that in the right-hand sides of conclusions of transition rules.
!
P$W Q$. However, we already remarked that weak and branching
w
equivalencearenotcongruencesforprefixiteration,butthis
Proposition 2.8. Forevery+#[b,’,d,w],therelation
operatordoesoccurattheright-handsideofthetransition
Wc+ is the largest congruence over T(BCCS)p*(A{) con- rulea*Pw(cid:20)a a*P.
tainedinW .
+ Hence, we obtain a positive answer to the fourth open
Proof. It is straightforward to check that Wc is an question at the end of [10], namely, whether there exist
+
equivalence relation for +#[b,’,d,w], using that this is transition rules outside the RWB and RBB cool formats
the case for W . Moreover, it is trivial to see that Wc is whichdefine‘‘interesting’’operatorsforwhichWc andWc
+ + w b
includedinW . arecongruences.
+
That Wc is a congruencerelation overT(BCCS)p*(A )
+ { Thefollowingresultisthecounter-partofProposition2.6
followseasilyfromDefinition2.7, usingthattherelation
fortheaforementionedcongruencerelations.
[(:*P,:*Q)|:#A{,PWc+Q]_W+ Proposition 2.9. For P,Q#T(BCCS)p*(A ) and
{
+#[b,’,d,w],
is an +-bisimulation. Here it is essential that, unlike W ,
+
the relations Wc require that an initial {-transition in a
+ PWc Q iff P_Wc Q_ for every closed substitution _.
processcannotbematchedbytheotherstayingidle. + +
Tosee thatWc is indeed thelargest congruencerelation
+
overT(BCCS)p*(A )containedinW ,assumethat= is Proof. A straightforward modification of the proof of
{ + +
anotherrelationwiththesepropertiesandthatP= Q.We Proposition2.6. K
+
showthatPWc Qholds.
+ We end this section with two lemmas that will be of use
As A is non-empty, we can pick an action a#A. By
inthecompletenessproofforbranchingcongruence(cf.the
Fact2.2, der(P)_der(Q) is a finite set of process terms.
proof of Proposition4.3). The first of these lemmas is a
Therefore it is possible to choose a positive integer n such
standard result for branching bisimulation equivalence,
that, foreveryR#der(P)_der(Q),
whoseproofmaybefoundin [22,14].
an.0W3 +R. Lemma 2.10 (Stuttering Lemma). If P w(cid:20){ }}} w(cid:20){ P
0 n
andP W P , thenP W P fori=1,...,n&1.
As P= Q and = is a congruence relation contained in n b 0 i b 0
+ +
W , it follows that P+an+1.0W Q+an+1.0. For every The following result about the expressiveness of the
+ +
+#[b,’,d,w], this implies that PWc Q. Consider, for languageT(BCCS)p*(A )stemsfrom[3].
instance, thecase+=b.LetPw(cid:20)! P$for+some!#A _Var. {
As P$W3 Q+an+1.0, it must be that Q+an+{1.0 =O= Lemma 2.11.
b
Q w(cid:20)! Q$ with P+an+1.0W Q and P$W Q$.
1 b 1 b 1. If P =aOn P for n=0,1,2,..., then there is an N
Moreover,asP+an+1.0cannotbebranchingequivalentto n n+1
suchthata =a forn>N.
a derivative of Q, it follows that Q #Q+an+1.0. Finally n N
PW3 an.0, so Qw(cid:20)! Q$, even when1!=a. By symmetry, it 2. Let a,b#A. If a*PW b*Q (+#[b,’,d,w]), then
b +
followsthatPWcQ, whichwastobeshown. K a=b.
b
Remark. Again, note that, if the set of observable Proof. Theproofofthefirststatementisaneasyexercise
actions A were empty, then the relations Wc (+# bystructuralinductiononterms,whichislefttothereader.
+
[b,’,d,w])wouldnotbethelargestcongruencescontained Toshow statement2, notethat, inlightofProposition2.5,
inW overT(BCCS)p*(A ).Infact,inthatcase,W itself itissufficienttodealwiththecase+=w.Assume, towards
+ { +
would be a congruence, and it is easy to see that, e.g., a contradiction, that a*PW b*Q and a{b. Then there
w
{.0W 0, but{.0W3 c 0. existtermsP$,Q$suchthat:
+ +
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32 ACETO ET AL.
v a*P =Ob P$W b*Q, and TABLE 1
w
v b*Q =Oa Q$W a*P. TheAxiomSystemF
w
This implies that a*P and b*Q both exhibit, for example, A1 x+y=y+x
an infinite sequence where a and b alternate, i.e., A2 (x+y)+z=x+(y+z)
=Oa =Ob =Oa =Ob }}}.Thiswouldcontradictstatement1ofthe A3 x+x=x
lemma. K A4 x+0=x
PA1 a.(a*x)+x=a*x
PA2 a*(a*x)=a*x
3. AXIOMSYSTEMS
TABLE 2
Themain aim ofthis study is to provide complete equa- AxiomsforE andforE
b ’
tional axiomatizations for branching, ’-, delay, and weak
congruence over the language T(BCCS)p*(A ). In this B1 :.({.(x+y)+x)=:.(x+y)
{ PB1 {*x={.x+x
section, wepresenttheaxiomsystemsthatwillbeshownto
PB2 {.a*({.a*(x+y)+x)={.a*(x+y)
completely characterize these congruence relations over
T(BCCS)p*(A ), and prove their soundness. We also pre-
{
TABLE 3
sent a proposition on the inter-derivability of these axiom
systems that will be useful in the proofs of the promised AxiomsforE andforE
d w
completeness theorems, and address the issue of the irre-
dundancyoftheaxiomsystems. T1 :.{.x=:.x
T2 {.x={.x+x
PT1 {*x={.x
3.1. TheAxioms PT2 {.(a*x)=a*({.(a*x))
Table1 presents the axiom system F, which was shown
TABLE 4
in [15] to characterize strong bisimulation over
T(BCCS)p*(A).InadditiontotheaxiomsinF,theaxiom ExtraAxiomsforE’
systems E (+#[b,’,d,w]) include equations which
+ T3 :.(x+{.y)=:.(x+{.y)+:.y
expresstheunobservablenatureofthe{action.Theseequa-
PT3 a*(x+{.y)=a*(x+{.y+a.y)
tions may be found in Tables2(cid:21)5; they reflect the different
ways in which the congruences we consider abstract away
TABLE 5
from internal computations in process behaviours. The
axiomsystemEbisobtainedbyaddingtheaxiomspresented ExtraAxiomsforEw
in Table2 to F, and E extends E with the equations in
’ b
AT3 a.(x+{.y)=a.(x+{.y)+a.y
Table4. The set of axioms E includes the equations in F
d PT3 a*(x+{.y)=a*(x+{.y+a.y)
andthoseinTable3.Finally,E extendsE withthelawsin
w d
Table5.
The two versions of equation PB2 are easily shown to be
The law B1 and the equations T1(cid:21)3, AT3 are standard inter-derivable;eachofthemprovestheircommongeneral-
characterizations of the silent action { in branching and ization
weak congruence, respectively. (Note that AT3 is the
instance of T3 with :#A. We distinguish the laws T3 and
PB2$ #.a*({.a*(x+y)+x)=#.a*(x+y)
AT3 in order to obtain an irredundant axiom system for
weak congruence. Cf. Proposition3.4 and the subsequent
for##A , usinglawsA1, A2, A4, PA1, andB1.)Theequa-
remark for more details.) The origins of the five remaining {
tion PT1 was introduced in [8] under the name of FIR
axioms, which describe the interplay between { and prefix 1
(FairIterationRule).In[8], itwasalsonotedthatthislaw
iteration, are as follows. The equations PB1 and PB2 stem
is an equational formulation of Koomen’s Fair Abstraction
from[16], whereacompleteaxiomatizationforbranching
Rule [4]. (To be precise, Koomen’s Fair Abstraction Rule
congruenceoverclosedtermsinthelanguageBPA[9]with
isageneralnameforafamilyofproofrulesKFAR , n(cid:30)1.
prefixiterationwaspresented.(Forthesakeofprecision,we n
PT1 corresponds to KFAR .) The laws PT2 and PT3
remark here that equation PB2 was formulated in [16] 1
originate from [3], where the axiom system E wasshown
thus: w
tobecompleteforweakcongruenceoverT(BCCS)p*(A ),
{
and |-complete in the presence of a denumerable set of
a.a*({.a*(x+y)+x)=a.a*(x+y). observableactionsA.
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AXIOMATIZING PREFIX ITERATION WITH SILENT STEPS 33
Note that each of the axiom system E+ (+#[b,’,d,w]) B1 :.({.(x+y)+x)=(5) :.{.(x+y)=T1 :.(x+y).
isfiniteifsoisthesetofactionsA.
Notation 3.1. For an axiom system T, we write PB1 {*xP=T1{.x=T2 {.x+x.
T&| P=Q iff the equation P=Q is provable from the
PB2 {.a*({.a*(x+y)+x)
axiom system T using the rules of equational logic. For
axiom systems T, T$, we write T&| T$ iff T&| P=Q for
every equation (P=Q)#T$. For a collection of equations =(6) {.a*({.(a*(x+y)+x+y)+x)
X over the signature ofBCCSp*(A ), we write P=X Q as a
{
short-handforA1, A2, X&| P=Q. =(5) {.a*({.(a*(x+y)+x+y))
For I=[i ,...,i ] a finite index set, we write (cid:29) P or
1 n i#I i
(cid:29)[Pi|i#I] for Pi1+}}}+Pin. By convention, (cid:29)i#<Pi =(6) {.a*({.a*(x+y))
standsfor0.
P=T2{.({.a*(x+y))
Weestablishthesoundnessoftheaxiomsystems.
Proposition 3.2. Let +#[b,’,d,w]. If E &| P=Q, =T1 {.a*(x+y). K
+
thenPWc Q.
+
Proof. As Wc (+#[b,’,d,w]) is a congruence, it is
+ 3.3. Irredundancy oftheAxiomSystems
sufficient to show that each equation in E is sound with
+
respect to it. The equations in the axiom system F are AcollectionTofequationsissaidtobeirredundant[35,
known to be sound with respect to strong bisimulation
p.389] iff for every proper subset T$ ofT there exists an
equivalence over T(BCCS)p*(A ); therefore they are,
{ equationwhichisderivablefromT, butnotfromT$.
afortiori, soundwithrespecttoeachofthecongruenceswe
Experience has shown that axiom systems can contain
consider.ThesoundnessoftheaxiomsB1,T1(cid:21)3andAT3is
redundancies; in the field of equational axiomatizations of
well-known, and that of PB1(cid:21)2 and PT1(cid:21)3 is easy to
behaviouralcongruencesthishappens,forinstance,in[19].
check. K
Therefore, wefind itinterestingtoconclude this sectionby
addressing the issue of the irredundancy of the axiom
3.2. ExpressivenessoftheAxiomSystems systemsE (+#[b,’,d,w]).
+
Foruse inthe promisedcompletenesstheorems, wenow Proposition 3.4. For each +#[b,’,d,w], the axiom
studytherelativeexpressivepoweroftheaxiomsystems.
systemE isirredundant.
+
Proposition 3.3. E &| E &| E andE &| E &| E .
w d b w n b Proof. To show the irredundancy of the axiom system
Proof. SinceE incorporatesE ,andE incorporatesE , E (+#[b,’,d,w]), it is sufficient to prove that, for every
w d ’ b +
the statements E &| E and E &| E are trivially true. In axiom(P=Q)#E ,
w d ’ b +
order to prove the remaining two statements, E &| E and
d b
E &| E, itsufficestoshowthatthethreeaxiomsinTable2
w ’
andtheinstanceofT3for:={arederivablefromE.First E "[P=Q]&|3 P=Q. (7)
d +
ofall, notethat
The standard proof strategy to establish this kind of result
{.(x+y)A=3,T2{.(x+y)+x. (5) istofindamodelfortheaxiomsystemE "[P=Q]inwhich
+
the equation P=Q is not valid. As the axiom systems E
b
The derivability of the instance of T3 with :={ from E and E are contained in E and E , respectively, it is
d d ’ w
follows immediately by observing that, modulo com- sufficienttoshow(7)forE andE .Inwhatfollows,welimit
’ w
mutativity of +, that equality is a substitution instance of ourselvestotheproofsfortheaxiomsPTn(n=1,2,3)and
(5). In deriving the laws in Table2 from E , weshallmake PBn (n=1,2). We present the model explicitly only for
d
useofthefollowingderivedequation: axioms PT2, PB2, and PT3. For axioms PT1 and PB1 we
merely give the intuition underlying the construction ofan
appropriate model. The reader will not have too much
a*x+xA3=,PA1a*x. (6)
troubleinfinding modelswhichcapturethisintuition.
The derivation of the three axioms in Table2 from E now AxiomsPT1andPB1. Intuitively,thereasonwhyequa-
d
proceedsasfollows: tions PT1 and PB1 are not derivable from the axiom
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34 ACETO ET AL.
systemsE "[PT1]andE"[PB1], respectively, isthatPT1 Formally, define a denotational semantics for
w ’
and PB1 are the only equations that can be used to com- T(BCCS)p*(A ) inthe domain2[0,1]by
{
pletely eliminate occurrences of the operation {* from
terms. (cid:26)x(cid:25)\=\(x)
Axioms PT2 and PB2. These axioms can actually be
(cid:26)0(cid:25)\=<
regarded as axiom schemes, in the sense that there is one
axiomfor eachchoice ofan actiona#A. Call these instan- (cid:26){.P(cid:25)\=(cid:26)P(cid:25) \
tiationsPT2(a)andPT3(a).Wenowshowthatforalla#A
(cid:26)a.P(cid:25)\=(cid:26)P(cid:25) \_[1]
E"[PT2(a)]&|3 PT2(a) and E"[PB2(a)]&|3 PB2(a). (cid:26)b.P(cid:25)\=(cid:26)P(cid:25) \"[1] for b{a
w b
Let a#A. We say that a termP is stableiff Pw(cid:20){ P$for no (cid:26)P+Q(cid:25)\=(cid:26)P(cid:25) \_(cid:26)Q(cid:25)\
P$. A term whose sub-terms of the form a*P$ are stable is
(cid:26):*P(cid:25)\=(cid:26)P(cid:25) \ for :{a
saidtobea*-stable.Intuitively,thereasonwhyPT2(a)and
PB2(a) cannot be derived from the other equations is that {[0,1], if 1(cid:18)(cid:26)P(cid:25)\,
(cid:26)a*P(cid:25)\=
PT2(a) and PB2(a) are the only axioms in E and E , (cid:26)P(cid:25) \, otherwise,
w ’
respectively,thatcanbeusedtoequateana*-stabletermto
onethatisnot. where \:Var(cid:20)2[0,1]. Here 1(cid:18)(cid:26)P(cid:25) \ denotes a-stability
Formally, define a denotational semantics for and 0#(cid:26)P(cid:25) \ denotes the property of having a subterm
T(BCCS)p*(A{)inthedomain2[0,1]by a*P$ with P$ a-stable. It is now simple tocheck thatthis is
a model for the axiom system E "[PT3(a)]. However,
(cid:26)x(cid:25)\=\(x) w
letting\ mapeachvariableinVarto<,
<
(cid:26)0(cid:25)\=<
(cid:26){.P(cid:25)\=(cid:26)P(cid:25) \_[1] (cid:26)a*(x+{.y)(cid:25)\<=[0,1]{[1]=(cid:26)a*(x+{.y+a.y)(cid:25)\<
(cid:26)b.P(cid:25)\=(cid:26)P(cid:25) \"[1] for b#A
andsotheaboveisnotamodelofE . K
w
(cid:26)P+Q(cid:25)\=(cid:26)P(cid:25) \_(cid:26)Q(cid:25) \
Remark. In light of (5), the instance of axiom T3 with
(cid:26){*P(cid:25)\=(cid:26)P(cid:25) \_[1] :={isderivablefromtheaxiomsystemE , and, afortiori,
d
{(cid:26)P(cid:25) \_[0], if b=a71#(cid:26)P(cid:25)\, from Ew. Thus defining the axioms for weak congruence to
(cid:26)b*P(cid:25)\= include T3 in lieu of AT3 would lead to a redundant
(cid:26)P(cid:25)\, otherwise,
axiomatization, likethosepresentedin, e.g., [26,31].
where\:Var(cid:20)2[0,1]. Here 1(cid:18)(cid:26)P(cid:25)\ denotes stabilityand
0(cid:18)(cid:26)P(cid:25) \denotesa*-stability.Itisnowsimpletocheckthat 4. COMPLETENESS
thisisamodelforboththeaxiomsystemsE "[PT2(a)]and
w
E"[PB2(a)].However,letting\ mapeachvariableinVar This section is entirely devoted to detailed proofs of the
’ <
to<, completenessoftheaxiomsystemsE (+#[b,’,d,w])with
+
respect to Wc over the language of open terms
+
(cid:26){.(a*x)(cid:25)\<=[1]{[0,1]=(cid:26)a*({.(a*x))(cid:25) \< T(BCCS)p*(A{). A common and, webelieve, aesthetically
pleasing featuring of our completeness proofs for the
and
behaviouralcongruencesWc (+#[’,d,w])isthattheyare
+
derived in uniform fashion from the corresponding result
(cid:26){.a*({.a*(x+y)+x)(cid:25) \ =[0,1]
< for branching congruence. Moreover, we shall also argue
{[1]=(cid:26){.a*(x+y)(cid:25)\ . that the proof of completeness for weak congruence via
<
reduction to the completeness result for branching con-
ThereforetheaboveisneitheramodelofEwnoroneofE’. gruenceisconsiderablyshorterthantheonlydirectproofof
Axiom PT3. Again we consider the instantiations thisresultpresentedintheliterature (cf.[3]).
PT3(a) and show E "[PT3(a)]&|3 PT3(a). We say that a Becauseoftheprominentro^leplayedbythecompleteness
w
term P is a-stable iff P=Oa P$ for no P$. Intuitively, the theorem for branching congruence in the developments to
reasonwhyPT3(a)cannotbederivedfromtheotherequa- follow, we begin by presenting our proof of this result.
tionsisthatPT3(a)istheonlyaxiominE thatcanbeused We remark here that the completeness of the theory E
w b
to equate a term P with a sub-term of the form a*P$ such with respect to Wc over the language of closed terms
b
that P$ is a a-stable to a term Q that does not have this T(BCCS)p*(A ) was first shown in [16]. The proof pre-
{
property. sentedbelowis, however, new, andyieldsthecompleteness
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AXIOMATIZING PREFIX ITERATION WITH SILENT STEPS 35
of the axiom system E for the whole of the language by complete induction on the sum of the sizes of P and Q.
b
T(BCCS)p*(A ). Moreover it may be argued that, even Recallthatnormalformscantakethefollowingtwoforms:
{
when restricted to the language of closed terms, our proof
improvesontheoneofferedintheaforementionedreference \ +
:: .P +:x or a* :: .P+:x ,
inthat, unlike thatproof, it does notrely onthecomplete- i i j i i j
i j i j
nessresultforstrongbisimulationfrom[15].
wheretheP ’sarethemselvesnormalforms.So,inparticular,
i
4.1. CompletenessforBranchingCongruence P and Q have one of these forms. By symmetry, it is suf-
ficienttodealwiththefollowingthreecases:
We aim at identifying a subset of process terms of a
1. P= (cid:29) : .P+(cid:29) x andQ= (cid:29) ; .Q +(cid:29) y ;
special form, which will be convenient in the proof of the AC i i i k k AC j j j l l
completeness result for branching congruence. Following 2. P= a*((cid:29) : .P +(cid:29) x )and
AC i i i k k
a long-established tradition in the literature on process Q= b*((cid:29) ; .Q +(cid:29) y);and
AC j j j l l
theory,weshallrefertothesetermsasnormalforms.Theset 3. P= (cid:29) : .P+(cid:29) x and
AC i i i k k
of normal forms we are after is the smallest subset of Q= a*((cid:29) ; .Q +(cid:29) y).
AC j j j l l
T(BCCS)p*(A )includingprocesstermshavingoneofthe
{
Wetreatthesethreecasesseparately.
followingtwoforms:
1. Case. P= (cid:29) : .P +(cid:29) x and Q=
AC i i i k k AC
(cid:29) ; .Q+(cid:29) y .Considerthe followingtwoconditions:
: : .P+: x or a*\: : .P +: x+. j j j l l
i i j i i j A. : ={andP W Qforsomei;
i#I j#J i#I j#J i i b
B. ; ={andQ W Pforsomej.
j j b
HerethetermsP arethemselvesnormalforms,andI,Jare
i
We distinguish three sub-cases in the proof, depending on
finiteindexsets. (Recallthattheemptysumrepresents0.)
whichoftheaboveconditionshold.
Lemma 4.1. Each term in T(BCCS)p*(A ) can be
{ I. Suppose that neither A nor B holds. Then, as
provenequaltoanormalformusingequationsA4, PA1, and
PW Q, each transition Pw(cid:20)! P$ must be matched by a
PB1. b
transition Qw(cid:20)! Q$ with P$W Q$. Hence, each summand
b
Proof. A straightforward induction on the structure of : .P ofPmatcheswithasummand; .Q ofQ,inthesense
i i j j
process terms. For example, the term {*(a*x) can be that : =; and P W Q . For each such pair of related
i j i b j
reducedtoanormalformthus: summands, inductionyields
{*(a*x)P=B1{.(a*x)+a*x Eb&| :i.Pi=:i.Qj=;j.Qj.
P=A1{.(a*x)+a.(a*x)+x Moreover,eachsummandx ofPmustbeasummandofQ.
k
=A4 {.(a*(0+x))+a.(a*(0+x))+x. K Hence, possiblyusingaxiomA3, itfollowsthatEb&| P+Q
=Q. By symmetry, we infer that E &| P=P+Q=Q. The
b
factthatE &| #.P=#.Qforall##A isnowimmediate.
b {
Notation 4.2. P= Q denotes that P and Q are equal
AC II. Suppose that both of A and B hold. In this
moduloassociativityandcommutativityof+,i.e.,thatA1,
case, there exist i and j such that :=; ={ and
A2&| P=Q. i j
P W QW PW Q . Applying the inductive hypothesis
i b b b j
The following result is the key to the completeness to the equivalences PW Q , P W Q and P W Q, we
b j j b j i b
theoremforbranchingcongruence. inferthat, forall##A ,
{
PWProQpo,stihteionn, fo4r.3a.ll#F#orA a,lEl &|P,#Q.P#=T#(B.QC.CS)p*(A{), if Eb&| #.P=#.Qj=#.Pi=#.Q
b { b
andtheinductivestepfollows.
Proof. First of all, note that, as the equations in E are
b
sound with respect to Wc, and, a fortiori, for W , by III. SupposethatonlyoneofAandBholds.Inthe
b b
Lemma4.1itissufficienttoprovethatthestatementofthe remainder of the proof for this case, we shall assume,
propositionholdsforbranchingequivalentnormalformsP withoutlossofgenerality,thatonlyAholds.Foreverysum-
andQ. mand{.P ofPwithP W Qweobtain,byinduction,that
i i b
So let us assume that P and Q be branching equivalent
normalforms. Weprove thatE &| #.P=#.Q forall ##A , E &| {.P={.Q.
b { b i
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