Bisimulation. R.J. van Glabbeek

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1 Bisimultion R.J. vn Glbbeek NICTA, Sydney, Austrli. School of Computer Science nd Engineering, The University of New South Wles, Sydney, Austrli. Computer Science Deprtment, Stnford University, CA , USA Bisimultion equivlence is semntic equivlence reltion on lbelled trnsition systems, which re used to represent distributed systems. It identifies systems with the sme brnching structure. Lbelled trnsition systems A lbelled trnsition system consists of collection of sttes nd collection of trnsitions between them. The trnsitions re lbelled by ctions from given set A tht hppen when the trnsition is tken, nd the sttes my be lbelled by predictes from given set P tht hold in tht stte. Definition 1 Let A nd P be sets (of ctions nd predictes, respectively). A lbelled trnsition system (LTS) over A nd P is triple (S,, =) with S clss (of sttes), collection of binry reltions S S one for every A (the trnsitions), such tht for ll s S the clss {t S s t} is set, nd = S P. s = p sys tht predicte p P holds in stte s S. LTSs with A singleton (i.e. with single binry reltion on S) re known s Kripke structures, the models of modl logic. Generl LTSs (with A rbitrry) re the Kripke models for polymodl logic. The nme lbelled trnsition system is employed in concurrency theory. There, the elements of S represent the systems one is interested in, nd s t mens tht system s cn evolve into system t while performing the ction. This pproch identifies sttes nd systems: the sttes of system s re the systems rechble from s by following the trnsitions. In this relm P is often tken to be empty, or it contins single predicte indicting successful termintion. Definition 2 A process grph over A nd P is tuple g = (S,I,, =) with (S,, =) n LTS over A nd P in which S is set, nd I S. Process grphs re used in concurrency theory to dismbigute between sttes nd systems. A process grph (S,I,, =) represents single system, with S the set of its sttes nd I its initil stte. In the context of n LTS (S,, =) two concurrent systems re modelled by two members of S; in the context of process grphs, they re two different grphs. The nondeterministic finite utomt used in utomt theory re process grphs with finite set of sttes over finite lphbet A nd set P consisting of single predicte denoting cceptnce. Written August 2000 for the forgotten Encyclopedi of Distributed Computing (J.E. Urbn & P. Dsgupt, eds.). Further reding dded November To pper in the Encyclopedi of Prllel Computing (D. Pdu, ed.), Springer,

2 Bisimultion equivlence Bisimultion equivlence is defined on the sttes of given LTS, or between different process grphs. Definition 3 Let (S,, =) be n LTS over A nd P. A bisimultion is binry reltion R S S, stisfying: if srt then s = p t = p for ll p P, if srt nd s s with A, then there exists t with t t nd s Rt, if srt nd t t with A, then there exists n s with s s nd s Rt. Two sttes s,t S re bisimilr, denoted s t, if there exists bisimultion R with srt. Bisimilrity turns out to be n equivlence reltion on S, nd is lso clled bisimultion equivlence. Definition 4 Let g = (S,I,, =) nd h = (S,I,, = ) be process grphs over A nd P. A bisimultion between g nd h is binry reltion R S S, stisfying IRI nd the sme three cluses s bove. g nd h re bisimilr, denoted g h, if there exists bisimultion between them. b c b c Exmple The two process grphs bove (over A = {,b,c} nd P = { }), in which the initil sttes re indicted by short incoming rrows nd the finl sttes (the ones lbelled with ) by double circles, re not bisimultion equivlent, even though in utomt theory they ccept the sme lnguge. The choice between b nd c is mde t different moment (nmely before vs. fter the -ction); i.e. the two systems hve different brnching structure. Bisimultion semntics distinguishes systems tht differ in this mnner. Modl logic (Poly)modl logic is n extension of propositionl logic with formuls / \ \ /ϕ, sying tht it is possible to follow n -trnsition fter which the formul ϕ holds. Modl formuls re interpreted on the sttes of lbelled trnsition systems. Two systems re bisimilr iff they stisfy the sme infinitry modl formuls. Definition 5 The lnguge L of polymodl logic over A nd P is given by: L, p L for ll p P, if ϕ,ψ L for then ϕ ψ L, if ϕ L then ϕ L, if ϕ L nd A then / \ \ /ϕ L. 2

3 Bsic (s opposed to poly-) modl logic is the specil cse where A = 1; there / \ \ /ϕ is simply denoted ϕ. The Hennessy-Milner logic is polymodl logic with P =. The lnguge L of infinitry polymodl logic over A nd P is obtined from L by dditionlly llowing i I ϕ i to be in L for rbitrry index sets I nd ϕ i L for i I. The connectives nd re then the specil cses I = nd I = 2. Definition 6 Let (S,, =) be n LTS over A nd P. The reltion = S P cn be extended to the stisfction reltion = S L, by defining s = i I ϕ i if s = ϕ i for ll i I in prticulr, s = for ny stte s S, s = ϕ if s = ϕ, s = / \ \ /ϕ if there is stte t with s t nd t = ϕ. Write L(s) for {ϕ L s = ϕ}. Theorem 1 [5] Let (S,, =) be n LTS nd s,t S. Then s t L (s) = L (t). In cse the systems s nd t re imge finite, it suffices to consider finitry polymodl formuls only [3]. In fct, for this purpose it is enough to require tht one of s nd t is imge finite. Definition 7 Let (S,, =) be n LTS. A stte t S is rechble from s S if there re s i S nd i A for i = 0,...,n with s = s 0, s i i 1 si for i = 1,...,n, nd s n = t. A stte s S is imge finite if for every stte t S rechble from s nd for every A, the set {u S t u} is finite. Theorem 2 [4] Let (S,, =) be n LTS nd s,t S with s imge finite. Then s t L(s) = L(t). Non-well-founded sets Another chrcteriztion of bisimultion semntics cn be given by mens of Aczel s universe V of non-well-founded sets [1]. This universe is n extension of the Von Neumnn universe of wellfounded sets, where the xiom of foundtion (every chin x 0 x 1 termintes) is replced by n nti-foundtion xiom. Definition 8 Let (S,, =) be n LTS, nd let B denote the unique function M : S V stisfying, for ll s S, M(s) = { / \, M(t) \ / s t}. It follows from Aczel s nti-foundtion xiom tht such function exists. In fct, the xiom mounts to sying tht systems of equtions like the one bove hve unique solutions. B(s) could be tken to be the brnching structure of s. The following theorem then sys tht two systems re bisimilr iff they hve the sme brnching structure. Theorem 3 [2] Let (S,, =) be n LTS nd s,t S. Then s t B(s) = B(t). 3

4 Abstrction In concurrency theory it is often useful to distinguish between internl ctions, tht do not dmit interctions with the outside world, nd externl ones. As normlly there is no need to distinguish the internl ctions from ech other, they ll hve the sme nme, nmely. τ. If A is the set of externl ctions certin clss of systems my perform, then A τ := A {τ}. Systems in tht clss re then represented by lbelled trnsition systems over A τ nd set of predictes P. The vrint of bisimultion equivlence tht trets τ just like ny ction of A is clled strong bisimultion equivlence. Often, however, one wnts to bstrct from internl ctions to vrious degrees. A system doing two τ ctions in succession is then considered equivlent to system doing just one. However, system tht cn do either or b is considered different from system tht cn do either or first τ nd then b, becuse if the former system is plced in n environment where b cnnot hppen, it cn still do insted, wheres the ltter system my rech stte (by executing the τ ction) in which is no longer possible. Severl versions of bisimultion equivlence tht formlize these desidert occur in the literture. Brnching bisimultion equivlence [2], like strong bisimultion, fithfully preserves the brnching structure of relted systems. The notions of wek nd dely bisimultion equivlence, which were both introduced by Milner under the nme observtionl equivlence, mke more identifictions, motivted by observble mchine-behvior ccording to certin testing scenrios. τ τ τ Write s = t for n 0 : s 0,...,s n : s = s 0 s 1 s n = t, i.e. (possibly empty) pth of τ-steps from s to t. Furthermore, for A τ, write s () t for s t ( = τ s = t). Thus () is the sme s for A, nd (τ) denotes zero or one τ-steps. Definition 9 Let (S,, =) be n LTS over A τ nd P. Two sttes s,t S re brnching bisimultion equivlent, denoted s b t, if they re relted by binry reltion R S S ( brnching bisimultion), stisfying: if srt nd s = p with p P, then there is t 1 with t = t 1 = p nd srt 1, if srt nd t = p with p P, then there is s 1 with s = s 1 = p nd s 1 Rt, if srt nd s s with A τ, then there re t 1,t 2,t () with t = t 1 t 2 = t, srt 1 nd s Rt, if srt nd t t with A τ, then there re s 1,s 2,s () with s= s 1 s 2 = s, s 1 Rt nd s Rt. Dely bisimultion equivlence, d, is obtined by dropping the requirements srt 1 nd s 1 Rt. Wek bisimultion equivlence [5], w, is obtined by furthermore relxing the requirements t 2 = t nd s 2 = s to t 2 = t nd s 2 = s. These definition stem from concurrency theory. On Kripke structures, when studying modl or temporl logics, normlly stronger version of the first two conditions is imposed: if srt nd p P, then s = p t = p. For systems without τ s ll these notions coincide with strong bisimultion equivlence. Concurrency When pplied to prllel systems, cpble of performing different ctions t the sme time, the versions of bisimultion discussed here employ interleving semntics: no distinction is mde between true prllelism nd its nondeterministic sequentil simultion. Versions of bisimultion tht do mke such distinction hve been developed s well, most notbly the ST-bisimultion [2], tht 4

5 tkes temporl overlp of ctions into ccount, nd the history preserving bisimultion [2] tht even keeps trck of cusl reltions between ctions. For this purpose, system representtions such s Petri nets or event structures re often used insted of lbelled trnsition systems. References [1] P. Aczel (1988): Non-well-founded Sets, CSLI Lecture Notes 14. Stnford University. [2] R.J. vn Glbbeek (1990): Comprtive Concurrency Semntics nd Refinement of Actions. PhD thesis, Free University, Amsterdm. Second edition vilble s CWI trct 109, CWI, Amsterdm [3] M. Hennessy & R. Milner (1985): Algebric lws for nondeterminism nd concurrency. Journl of the ACM 32(1), pp [4] M.J. Hollenberg (1995): Hennessy-Milner clsses nd process lgebr. In A. Ponse, M. de Rijke & Y. Venem, editors: Modl Logic nd Process Algebr: Bisimultion Perspective, CSLI Lecture Notes 53, CSLI Publictions, Stnford, Cliforni, pp [5] R. Milner (1990): Opertionl nd lgebric semntics of concurrent processes. In J. vn Leeuwen, editor: Hndbook of Theoreticl Computer Science, chpter 19, Elsevier Science Publishers B.V. (North-Hollnd), pp Further reding Gentle introductions to bisimultion semntics, with mny exmples of pplictions, cn be found in the textbooks: J.C.M. Beten & W.P. Weijlnd (1990): Process Algebr, Cmbridge University Press. R. Milner (1989): Communiction nd Concurrency, Prentice Hll. An historicl perspective on bisimultion ppers in D. Sngiorgi (2009): On the origins of bisimultion nd coinduction, ACM Trnsctions on Progrmming Lnguges nd Systems 31(4). 5