Bisimilarity is not Finitely Based over BPA with Interrupt - brics

36 Pages·2005·0.21 MB·English

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BRICSReport Series RS-05-33
ISSN 0909-0878 October2005
Copyright(cid:13)c 2005, Luca Aceto & WillemJanFokkink& Anna
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This document insubdirectory RS/05/33/
Bisimilarity is not Finitely Based over BPA with
Interrupt
∗
c,a, e,b c,a
Luca Aceto , Wan Fokkink , Anna Ingolfsdottir
d
Sumit Nain
a
BRICS(BasicResearchinComputerScience), CentreoftheDanishNationalResearch
Foundation, DepartmentofComputerScience, AalborgUniversity, Fr.Bajersvej7B,9220
AalborgØ,Denmark.
b
CWI,DepartmentofSoftwareEngineering, Kruislaan 413,1098SJAmsterdam,The
Netherlands.
c
DepartmentofComputerScience, SchoolofScience andEngineering, Reykjav´ık
University, Ofanleiti2,103Reykjav´ık, Iceland.
d
DepartmentofComputerScience, MailStop132,RiceUniversity, 6100S.MainStreet,
Houston,TX77005-1892, USA.
e
VrijeUniversiteit Amsterdam,DepartmentofComputerScience, SectionTheoretical
ComputerScience, DeBoelelaan 1081a,1081HVAmsterdam,TheNetherlands.
Abstract
Thispapershowsthatbisimulationequivalencedoesnotaffordafiniteequationalaxiomati-
zationoverthelanguageobtained byenriching BergstraandKlop’sBasicProcessAlgebra
with the interrupt operator. Moreover, it is shown that the collection of closed equations
overthislanguageisalsonotfinitelybased.Insharpcontrasttotheseresults,thecollection
ofclosed equations overthe language BPAenriched withthedisrupt operator isproven to
befinitelybased.
Keywords: Concurrency, process algebra, BasicProcessAlgebra(BPA),interrupt,
disrupt, bisimulation, equational logic,completeaxiomatizations, non-finitely based
algebras, expressiveness
1991MSC:08A70,03B45,03C05,68Q10,68Q45,68Q55,68Q70
∗
Corresponding author.
Emailaddresses: [email protected],[email protected](LucaAceto),
[email protected](WanFokkink),[email protected],[email protected](Anna
Ingolfsdottir), [email protected](SumitNain).
URLs:http://www.cs.aau.dk/˜luca(LucaAceto),
http://www.cs.vu.nl/˜wanf(WanFokkink),
http://www.cs.aau.dk/˜annai (AnnaIngolfsdottir).
PreprintsubmittedtoElsevierScience
1 Introduction
Programmingandspecificationlanguagesoftenincludeconstructstospecifymode
switches (see, e.g., [8,11,23,24,26]). Indeed, some form of mode transfer in com-
putation appears in the time-honoured theory of operating systems in the guise of,
e.g., interrupts, in programming languages as exceptions, and in the behaviour of
control programs and embedded systemsas discrete“modeswitches”triggered by
changes inthestateoftheirenvironment.
In light of the ubiquitousnature of mode changes in computation,it is not surpris-
ing that classic process description languages either include primitiveoperators to
describe mode changes—for example, LOTOS [15,23] offers the so-calleddisrup-
tionoperator—or have been extended with variations on mode transfer operators.
For instance, examples of such operators that may be added to CCS are discussed
byMilnerin[25,pp.192–193],andthereference[17]offerssomediscussionofthe
benefits ofaddingoneofthose,viz.thecheckpointingoperator, tothat language.
In thesettingofBasic Process Algebra(BPA), as introducedby Bergstra andKlop
in [12], some of these extensions, and their relativeexpressiveness, have been dis-
cussed in the early paper [11]. That preprint of Bergstra’s has later been revised
and extended in [7]. There, Baeten and Bergstra study the equational theory and
expressivenessofBPA (theextensionofBPA withaconstantδ todescribe“dead-
δ
lock”) enriched with two mode transfer operators, viz. the disruptand interrupt
operators. In particular, they offer an equational axiomatization of bisimulation
equivalence [25,29] over the resulting extension of the language BPA . This ax-
δ
iomatizationisfinite, ifso istheunderlyingsetofactions—astateofaffairs thatis
mostpleasingforprocessalgebraists.
However, the axiomatization of bisimulation equivalence offered by Baeten and
Bergstrain[7]reliesontheuseoffourauxiliaryoperators—twopermodetransfer
operator. (Two of those auxiliary operators are, however, redundant since they are
derivedBPA operators.)Althoughtheuseofauxiliaryoperators intheaxiomatiza-
tion of behavioral equivalences over process description languages has been well
established since Bergstra and Klop’s axiomatization of parallel composition us-
ing theleft and communicationmerge operators [13], to ourmind,a result likethe
aforementionedonealwaysbegsthequestionwhethertheuseofauxiliaryoperators
isnecessary toobtainafiniteaxiomatizationofbisimulationequivalence.
For the case of parallel composition, Moller showed in [27,28] that strong bisim-
ulation equivalence is not finitely based over CCS [25] and PA [13] without the
left merge operator. (The process algebra PA [13] contains a parallel composition
operatorbasedonpureinterleavingwithoutcommunication,andtheleftmergeop-
erator.) Thus auxiliary operators are necessary to obtain a finite axiomatization of
parallelcomposition.But,istheuseofauxiliaryoperatorsnecessarytogiveafinite
2
axiomatization of bisimulation equivalence over the language BPA enriched with
themodetransferoperators studiedby Baeten andBergstra in[7]?
We address the above natural question in this paper. In particular, we mostly fo-
cus on BPA enriched with the interrupt operator. Intuitively, “p interrupted by q”
describes a process that normally behaves like p. However, at each point of the
computation before p terminates, q can interrupt it, and begin its execution. If this
happens,p resumes itscomputationuponterminationofq.
We show that, in the presence of a single action, bisimulation equivalence is not
finitely based over BPA with the interrupt operator. Moreover, we prove that the
collectionofclosed equationsoverthislanguageisalso notfinitelybased.Thisre-
sult provides evidence that the use of auxiliary operators in the technical develop-
mentspresentedin[7]isindeednecessaryinordertoobtainafiniteaxiomatization
ofbisimulationequivalence.
Our main result adds the interrupt operator to the list of operators whose addition
toaprocessalgebraspoilsfiniteaxiomatizabilitymodulobisimulationequivalence;
see, e.g., [3,4,14,16,20,30,31] forother examplesofnon-finiteaxiomatizabilityre-
sults over process algebras, and some of their precursors in the setting of formal
language theory. Of special relevance for concurrency theory are the aforemen-
tionedresultsofMoller’stotheeffectthattheprocessalgebrasCCSandPAwithout
the auxiliary left merge operator from [12] do not have a finite equational axiom-
atizationmodulobisimulationequivalence[27,28]. Recently, in collaborationwith
Luttik,thefirstthreeauthorshaveshownin[5]thattheprocessalgebraobtainedby
adding Hennessy’s merge operator from [22] to CCS does not have a finite equa-
tionalaxiomatizationmodulobisimulationequivalence.Thisresultisinsharpcon-
trastwithatheoremestablishedby FokkinkandLuttikin[18]to theeffect thatthe
process algebra PA [13] affords an ω-completeaxiomatization that is finite if so is
the underlyingset ofactions. Aceto, E´sik and Ingolfsdottirprovedin [2] that there
is no finite equational axiomatization that is ω-complete for the max-plus algebra
ofthenaturalnumbers,aresultwhoseprocessalgebraicimplicationsarediscussed
in [1]. Fokkinkand Nainhaveshownin[19] thatno congruenceoverthelanguage
BCCSP, a basic formalism to express finite process behaviour, that is included in
possible worlds equivalence, and includes ready trace equivalence, affords a finite
ω-completeequationalaxiomatization.
Having established that the addition of the interrupt operator to BPA spoils finite
axiomatizability modulo bisimulation equivalence, it is natural to ask ourselves
whether the same holds true for the disrupt operator from [7]. Intuitively, “p dis-
rupted by q” describes a process that normally behaves like p. However, at each
point of the computation before p terminates, q can pre-empt it, and begin its exe-
cution.Ifthishappens,p neverresumes itscomputation.
Weshowthat,perhapssurprisingly,insharpcontrasttothemainresultofthepaper,
3
the use of auxiliary operators isnotnecessary in order to obtain a finite axiomati-
zation of bisimulation equivalence over closed terms in the language obtained by
enriching BPA with the disrupt operator. The key to this positive result is the dis-
tributivity of the disrupt operator with respect to the non-deterministic choice op-
erator of BPA in its first argument—aproperty that is not afforded by the interrupt
operator.
The paper is organized as follows.We begin by presenting the language BPA with
the interrupt operator, its operational semantics and preliminaries on equational
logic in Section 2. There we also show that the interrupt operator is not definable
in BPA modulo bisimilarity. The general structure of the proof of our main result,
to the effect that bisimilarity is not finitely based over the language BPA with the
interrupt operator, is presented in Section 3. In that section, we also show how to
reducetheproofofourmainresulttothatofatechnicalstatementdescribingakey
property of closed instantiations of sound equations that is preserved under equa-
tionalderivations(Proposition13).WeofferaproofofProposition13inSection4.
WeconcludethepaperbypresentinginSection5anaxiomatizationofbisimulation
equivalenceoverclosed terms in thelanguageobtainedby enrichingBPA with the
disruptoperatorfrom[7].Suchanaxiomatizationisfiniteinthepresenceofafinite
set ofactions,anddoes notemployauxiliaryoperators.
An extended abstract of this paper appeared as [6]. There we announced without
proof our main result (namely, Theorem 9) under the assumption that the set of
actions contains two distinct actions. The present version of the paper sharpens
Theorem9inthatitnowappliestoanynon-emptysetofactions,andoffersthefull
proofofourmainresult(allofthematerialinSection4).Moreover,Proposition20
inthecurrent paperis new.
2 Preliminaries
We begin by introducing the basic definitions and results on which the technical
developments to follow are based. The interested reader is referred to [7,12] for
moreinformation.
2.1 The LanguageBPA
int
Weassumeanon-emptyalphabet Aofatomicactions,withtypicalelementa. The
language for processes we shall consider in the main body of this paper, hence-
forth referred to as BPA , is obtained by adding the interrupt operator from [7] to
int
Bergstra andKlop’sBPA [12]. Thislanguageisgivenbythefollowinggrammar:
t ::= x |a| t·t| t+t| t (cid:3) t ,
4
where x is a variable drawn from a countably infinite set V and a is an action. In
theabovegrammar,weusethesymbol (cid:3) for theinterruptoperator.
Intuitively,atermoftheformp (cid:3) q describesaprocessthatnormallybehaveslike
p.However,ateachpointofthecomputationbeforepterminates,q caninterruptit,
andbeginitsexecution.Ifthishappens,presumesitscomputationupontermination
ofq.Analternativecompositionp+q nondeterministicallybehavesaseitherporq.
Asequentialcompositionp·q firstbehavesasp,anduponterminationofpbehaves
as q.
We shall use the meta-variables t,u,v,w to range over process terms, and write
var(t) for the collection of variables occurring in the term t. Thesizeof a term is
thenumberofsymbolsinit. Formally,
• thesizeofvariablesand actionsis1, and
• thatoft·u,t+uand t (cid:3) uis oneplusthesumofthesizesoftand u.
A process term isclosedif it does not contain any variables. Closed terms will be
typically denoted by p,q,r,s. As usual, we shall often writetu in lieu of t·u, and
weassumethat ·bindsstrongerthan + and(cid:3).
A (closed) substitution is a mapping from process variables to (closed) BPA
int
terms. For every term t and substitution σ, the term obtained by replacing every
occurrence of a variable x in t with the term σ(x) will be written σ(t). Note that
σ(t)isclosed,ifsoisσ.Inwhatfollows,weshallusethenotationσ[x 7→ p],where
σ isaclosedsubstitutionandpisaclosedBPA term,tostandforthesubstitution
int
mappingxtop, and actinglikeσ onall oftheothervariablesinV.
In the remainder of this paper, we let a1 denote a, and am+1 denote a(am). More-
over, we consider terms modulo associativity and commutativity of +. In other
words,wedonotdistinguisht+uandu+t,nor(t+u)+v andt+(u+v).This
is justified because + is associative and commutative with respect to the notion
of equivalence we shall consider over BPA . (See axioms A1, A2 in Table 3 on
int
page 12.) In what follows,the symbol= will denote equality moduloassociativity
and commutativityof+.
Wesaythatatermthas+asheadoperatorift = t +t forsometermst andt .
1 2 1 2
Forexample,a+bhas + as head operator, but(a+b)a doesnot.
P
For k ≥ 1, we use a summation t to denote t + ···+ t . It is easy to
i∈{1,...,Pk} i 1 k
see that every BPA term t has the form t , for somefinite, non-emptyindex
int i∈I i
set I, and termst (i ∈ I)that donot have+ as head operator. Thetermst (i ∈ I)
i i
willbereferred toasthe(syntactic)summandsoft.Forexample,theterm(a+b)a
has onlyitselfas (syntactic)summand.
TheoperationalsemanticsforthelanguageBPA isgivenbythelabelledtransition
int
5
a
a →X
t →aX u →aX t →a t0 u →a u0
t+u →aX t+u →aX t+u →a t0 t+u →a u0
t →aX t →a t0
t·u →a u t·u →a t0·u
t →aX t →a t0 u →aX u →a u0
t (cid:3) u →aX t (cid:3) u →a t0 (cid:3) u t (cid:3) u →a t t (cid:3) u →a u0·t
Table1
TransitionRulesforBPA
int
system
(cid:16) n o n o(cid:17)
BPA , →a | a ∈ A , →a X | a ∈ A ,
int
wherethetransitionrelations→a andtheunarypredicates→a Xare,respectively,the
leastsubsetsofBPA ×BPA andBPA satisfyingtherulesinTable1.Intuitively,
int int int
atransitiont →a umeansthatthesystemrepresentedbytheterm tcan performthe
action a, thereby evolving into u. The special symbol X stands for (successful)
termination; therefore the interpretation of the statement t →a X is that the process
term t can terminate by performing a. Note that, for every closed term p, there is
someactionaforwhicheither p →a p0 holdsforsomep0, orp →a X does.
Fortermst,u,and action a, wesaythat uisan a-derivativeoftift →a u.
Thetransitionrelations→a naturallycomposetodeterminethepossibleeffectsthat
performingasequenceofactionsmay haveon aBPA term.
int
Definition 1 Forasequenceofactionsa ···a (k ≥ 0),andBPA termst,t0,we
1 k int
writet a1→···ak t0 iffthereexists asequenceof transitions
t = t →a1 t →a2 ··· →ak t = t0 .
0 1 k
Similarly,we say that a ···a (k ≥ 1) is a termination traceof a BPA term t iff
1 k int
thereexistsa term t0 suchthat
t a1−···→ak−1 t0 →akX .
If t a−1·→··ak t0 holds for some BPA term t0, or a ···a is a termination trace of t,
int 1 k
then a ···a is atraceoft.
1 k
Thedepthofa termt, written depth(t), isthelengthofthelongesttraceitaffords.
6
Thenormof a term t, denoted by norm(t), is thelength of its shortest termination
trace;thisnotionstemsfrom[9].
Thedepthandthenormofclosedtermscanalsobecharacterized inductivelythus:
depth(a) = 1
depth(p+q) = max{depth(p),depth(q)}
depth(pq) = depth(p)+depth(q)
depth(p (cid:3) q) = depth(p)+depth(q)
norm(a) = 1
norm(p+q) = min{norm(p),norm(q)}
norm(pq) = norm(p)+norm(q)
norm(p (cid:3) q) = norm(p) .
NotethatthedepthandthenormofeachclosedBPA termarepositive,andthere-
int
fore that thenorm of each closed term of theform pq is at least 2. Thissimple,but
useful,observationwillbeused repeatedly intheremainderofthisstudy.
Inwhatfollows,weshallsometimesneedtoconsiderthepossibleoriginsofatran-
sition of the form σ(t) →a p, for some action a, closed substitution σ, BPA term
int
t and closed term p. Naturally enough, we expect that σ(t) affords that transition
if t →a t0, for some t0 such that p = σ(t0). However, the above transition may also
derive from the initial behaviour of some closed term σ(x), provided that the col-
lectionofinitialmovesofσ(t)depends,insomeformalsense,onthatoftheclosed
term substitutedfor the variable x. Similarly,we shall sometimesneed to consider
thepossibleorigins ofa transitionof theform σ(t) →a X, for someaction a, closed
substitutionσ and BPA term t.
int
Tofullydescribethesesituations,weintroducetheauxiliarynotionofconfiguration
ofaBPA term.To thisend,we assumeasetofsymbols
int
V = {x | x ∈ V}
d d
disjointfromV.Intuitively,thesymbolx (read“duringx”)willbeusedtodenote
d
that the closed term substitutedfor variable x has begun executing,but has not yet
terminated.
Definition 2 The collectionofBPA configurationsisgiven bythegrammar:
int
c ::= t|x | c·t|c (cid:3) t ,
d
where tisa BPA term,and x ∈ V .
int d d
7
x →xs x x →xX
d
t →x t0 t →xs c t →xX
t+u →x t0 t+u →xs c t+u →xX
u→x u0 u →xs c u →xX
t+u →x u0 t+u →xs c t+u →xX
t →x t0 t →xs c t →xX
tu →x t0u tu →xs cu tu →x u
t →x t0 t →xs c t →xX
t (cid:3) u →x t0 (cid:3) u t (cid:3) u →xs c (cid:3) u t (cid:3) u →xX
u →x u0 u →xs c u →xX
t (cid:3) u →x u0t t (cid:3) u →xs ct t (cid:3) u →x t
Table2
SOSRulesfortheAuxiliaryTransitions →x ,→xs and→xX(x∈ V)
For example, the configuration x · (a (cid:3) x) is meant to describe a state of the
d
computationofsometerminwhichthe(closedtermsubstitutedforthe)occurrence
of variable x on the left-hand side of the · operator has begun its execution (and
has not terminated), but the one on the right-hand side has not. Note that each
configurationcontainsat mostoneoccurrence ofan x ∈ V .
d d
We shall consider the symbols x as variables, and use the notation σ[x 7→ p],
d d
where σ is a closed substitution and p is a closed BPA term, to stand for the
int
substitutionmappingx to p, andacting likeσ on alloftheothervariables.
d
Thewayinwhichtheinitialbehaviourofatermmaydependonthatofthevariables
that occur in it is formally described by three auxiliary transition relations whose
elementshavethefollowingforms:
• t →xs c (read “t can start executing x and become c in doing so”), where t is a
term,xis avariable,andc isaconfiguration,
• t →x t0, wheret andt0 areterms andx isavariable,or
• t →xX,where tisaterm.
The first of these types of transitions will be used to account for those transitions
of the form σ(t) →a p that are due to a-labelled transitions of the closed term σ(x)
thatdonotleadtoitstermination.Thesecondwilldescribetheoriginoftransitions
of the form σ(t) →a σ(t0) that are due to a-labelled transitions of the closed term
8