Complexity and Decidability of Some Equivalence-Checking Problems

Transcription

1 Complexity nd Decidbility of Some Equivlence-Checking Problems Zdeněk Sw Ph.D. Thesis Fculty of Electricl Engineering nd Computer Science Technicl University of Ostrv 2005

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3 Acknowledgements I would like to thnk my supervisor Petr Jnčr for guidnce, comments nd fruitful discussions. I owe much to him. I would lso like to thnk ll co-uthors tht worked with me on ppers for their coopertion: Antonín Kučer, Fron Moller, nd Mrtin Kot. I would lso like to thnk Philippe Schnoebelen for drwing my ttention to the complexity of equivlence checking on finite-stte systems composed from communicting subsystems nd to Rbinovich s conjecture. Lst but not lest I would like to thnk my wife Silvie for her love nd support. iii

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5 Declrtion I declre tht this thesis ws composed by myself, nd ll presented results re my own, unless otherwise stted. Some of the mteril hs been previously published in [32], [53], [30], [52], [31], nd [35]. Zdeněk Sw v

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7 Abstrct The thesis presents results obtined by the uthor in the re of verifiction of finite-stte nd infinite-stte systems. It concentrtes on questions of complexity nd decidbility of equivlence checking, i.e., of deciding behviourl equivlences nd preorders on trnsition systems. It is shown tht deciding of ny reltion between bisimultion equivlence nd trce preorder is PTIME-hrd problem for finite-stte systems tht re given explicitly s list of sttes nd trnsitions. The problem becomes EXPTIME-hrd for ny such reltion in the cse of finite-stte systems tht re creted s composition of communicting finite-stte components. The other type of systems studied in the thesis re one-counter utomt. It is shown tht deciding simultion equivlence is n undecidble problem on one-counter utomt. A generl method for proving DP-hrdness of some problems concerning one-counter utomt is presented. Using this method is shown tht deciding ny reltion between bisimultion equivlence nd simultion preorder is DP-hrd for one-counter nets (i.e., one-counter utomt tht cnnot test for zero), nd tht deciding simultion equivlence nd simultion preorder between one-counter utomton nd finite-stte system is DP-hrd. The lst type of systems studied in the thesis re Bsic Prllel Processes (BPP). Two polynomil time lgorithms for BPP re presented. The first of these lgorithms decides bisimultion equivlence between BPP nd finite-stte system, nd the other decides distributed bisimilrity on BPP. vii

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9 Abstrkt Tto disertční práce prezentuje výsledky dosžené utorem v oblsti verifikce konečně stvových nekonečně stvových systémů. Změřuje se n otázky výpočetní složitosti rozhodnutelnosti equivlence checking, to jest problémů rozhodování behviorálních ekvivlencí kvziuspořádání n přechodových systémech. Je ukázáno, že rozhodování libovolné relce, která leží mezi bisimulční ekvivlencí trce preorder je PTIME-těžký problém pro systémy, které jsou dány explicitně jko seznm stvů přechodů. Tento problém se pro libovolnou z těchto relcí stává EXPTIME-těžkým v přípdě konečně stvových systémů, které jsou vytvořeny z komunikujících konečně stvových komponent. Dlším typem systémů zkoumným v této práci jsou utomty s jedním čítčem. Je ukázáno, že rozhodování simulční ekvivlence pro utomty s jedním čítčem je nerozhodnutelné. Dále je prezentován obecná metod pro dokzování DP-obtížnosti problémů týkjících se utomtů s jedním čítčem. Použitím této metody je ukázáno, že rozhodování libovolné relce, která leží mezi bisimulční ekvivlencí simulčním kvziuspořádáním je DP-těžké pro one-counter nets (tj. pro utomty s jedním čítčem, které nemohou testovt nulu), dále, že rozhodování simulční ekvivlence simulčního kvziuspořádání mezi utomtem s jedním čítčem konečně stvovým systémem je DP-těžké. Posledním typem systémů zkoumným v této práci jsou Bsic Prllel Processes (BPP). Jsou ukázány dv polynomiální lgoritmy pro BPP. První z těchto lgoritmů rozhoduje bisimulční ekvivlenci mezi BPP konečně stvovým systémem druhý rozhoduje distribuovnou bisimilritu n BPP. ix

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11 Contents 1 Introduction Gols of the Thesis Overview of the Results Finite-Stte Systems One-Counter Automt Bsic Prllel Processes Lyout of the Thesis Definitions Nottion Conventions Lbelled Trnsition Systems Process Rewrite Systems Process Algebr, BPA, nd BPP Petri Nets One-Counter Automt Explicit nd Composed Finite-Stte Systems Behviourl Equivlences Distributed Bisimilrity nd BPP Finite-Stte Processes Stte of the Art Own Contribution xi

12 xii Contents 3.3 Explicit Finite-Stte Systems Alternting Grphs Reduction Composed Finite-Stte Systems Liner Bounded Automt Rective Liner Bounded Automt Reduction Decomposition of Trnsitions Correctness of the Construction of the RLBA Summry of the Results One-Counter Automt Stte of the Art Undecidbility Result The OCL Frgment of Arithmetic Definition of OCL DP-hrdness of TruthOCL TruthOCL is in Π p Appliction to One-Counter Automt Problems Results for One-Counter Nets Simultion Problems for One-Counter Automt nd Finite-Stte Systems Summry of the Results Bsic Prllel Processes Stte of the Art Bisimilrity with Finite-Stte System Bsic Definitions The Algorithm Time Complexity of the Algorithm

13 Contents xiii 5.3 Distributed Bisimilrity Summry of the Results Conclusion Summry of the Results Open Problems A List of Publictions 95

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15 Chpter 1 Introduction We cn find mny exmples of complicted softwre nd hrdwre systems where bugs cn hve serious or even ctstrophic consequences. Exmples of such systems re operting systems, network communiction protocols, microprocessors, nd trffic control systems. One of the min problems in the design of such complicted systems is to ensure the correctness of this design. Correctness mens tht the system fulfills the tsk for which it ws designed. We usully hve some specifiction of the desired behvior of the system nd we wnt to ensure tht the implementtion of the system is correct with respect to this specifiction. The process of checking whether the given implementtion stisfies the given specifiction is clled verifiction. Stndrd techniques used for verifiction re testing nd simultion. In testing we run the system in different situtions nd with different inputs nd observe the behvior of the system. Simultion is similr to testing but we do not test the ctul system, but some model of it. It is used in situtions when running the ctul system is not possible or prcticl, for exmple due to enormous costs. While testing nd simultion re very useful in the erly stges of development nd llow to discover mny bugs in the system, they hve the importnt disdvntge tht one cn never be sure tht there re not some other more subtle bugs in the system. Testing nd simultion cn never gurntee the correctness of the system due to their inherent limittions becuse they cn explore only some of possible behviors of the system, but the number of ll possible behviors is usully very lrge nd often infinite. 1

16 2 Chpter 1. Introduction The limittions of testing nd simultion become even more severe in the design of systems composed of mny components running concurrently tht cn interct with ech other. The behvior of such systems is usully nondeterministic, nd it cn be difficult even to reproduce bugs in these systems since they cn occur only under some rre circumstnces. It becomes obvious tht some forml methods tht ensure correctness of ll possible behviors should be used for verifiction. The forml methods provide us with the necessry mthemticl tools tht cn be used in the construction of rigorous mthemticl proofs of the correctness of the system. The construction of such proofs cn be done either by hnd or my be utomted, t lest prtilly, by use of some sort of verifiction softwre tools. The ltter pproch is very ttrctive since the construction of proofs by hnd is usully very tedious nd error prone. The pproch using utomted tools is usully clled computer ided verifiction. Unfortuntely this tsk cn not be fully utomted in full generlity becuse mny problems concerning behvior of computer progrms re undecidble. For exmple, it is well known tht Hlting problem, i.e., the problem whether progrm hlts for given input fter some finite number of steps, is undecidble. In generl, there re two min pproches to verifiction tht llow to ensure correctness for ll possible behviors of the system theorem proving nd techniques bsed on model checking nd equivlence checking. In theorem proving, we try to construct forml proofs of correctness of the system. Computer progrms clled theorem provers cn ssist the user in this tsk nd do some simple steps utomticlly. However, the user hs to guide the progrm nd do the crucil steps of the proofs. The min disdvntge of this technique is tht it requires lot of knowledge, skill nd prctice from the user. This severely limits the prcticl pplicbility of theorem proving. Other pproches re model checking nd equivlence checking. These techniques re fully utomtic nd do not require ny interction from the user, however they cn not be pplied to rbitrry progrms due to undecidbility of Hlting problem. Insted, some properties of models tht do not hve the expressive power of Turing mchines re verified. These techniques hve their origins in the utomt nd forml lnguge theory where infinite lnguges re described finitely nd some properties of these lnguges re decidble, for exmple, it is decidble whether two finite utomt recognize

17 3 the sme lnguge. In model checking we hve given system (resp. description of the system) nd some desired property of the system expressed s formul of some temporl logic, nd the question is whether the system stisfies the given property. See [15, 17, 57, 6] for more informtion bout model checking nd temporl logic. Equivlence checking is type of problems where we hve given (descriptions of) two systems nd the question is whether these systems re equivlent with respect to some notion of equivlence. Usully one of these systems is specifiction nd the other is n implementtion nd we wnt to ensure tht their behvior is identicl. Equivlence checking problems re the min topic of this thesis. There re mny vrints of model checking nd equivlence checking problems tht differ in the wy how the systems nd their properties re expressed. Models with gret expressive power cnnot be verified utomticlly, nd models tht re too restrictive do not llow to model mny spects of rel systems. This motivtes the reserch tht concentrtes on decidbility nd complexity of verifiction problems. One ctive re of reserch concentrtes on the question which model checking nd equivlence checking problems re decidble, nd where exctly lies the dividing line between decidble nd undecidble problems. Another importnt question is wht is the exct computtionl complexity of decidble verifiction problems, becuse mny verifiction problems cn be solved by some lgorithm theoreticlly, but the lgorithm cn be used in prctice only for smll instnces due to its computtionl complexity. One of the most serious obstcles in the design of efficient verifiction lgorithms is the phenomenon known s stte explosion. This problem ppers when we hve system composed of mny components. These components cn hve resonbly smll stte spces, but the stte spce of the whole system cn be exponentilly lrger with respect to the size of descriptions of its components. Unfortuntely it shows up tht the stte explosion is unvoidble in mny cses nd tht lgorithms solving such problems require exponentil time. This thesis concentrtes on complexity nd decidbility of some equivlence checking problems, nd presents some results obtined by the uthor in this re. Some of the results presented here re joint work with other uthors Petr Jnčr, Antonín Kučer, Fron Moller, nd Mrtin Kot.

18 4 Chpter 1. Introduction It is ssumed tht the reder is fmilir with forml lnguges nd the bsics of complexity theory. 1.1 Gols of the Thesis The min gol of the thesis ws to contribute with some new results in the re of equivlence checking. The min focus ws on the decidbility nd computtionl complexity of equivlence-checking problems. The thesis presents results obtined by the uthor in this re. The solved problems re more or less independent of ech other nd cn be divided into three min groups: problems concerning finite-stte systems either given explicitly or s prllel composition of communicting components problems concerning one-counter utomt nd vrint of them clled one-counter nets problems concerning Bsic Prllel Processes (BPP) The next section gives n overview of the results presented in this theses. See Chpter 2 for forml definitions of terms used in this section. 1.2 Overview of the Results The following subsections describe shortly the min results presented in this thesis Finite-Stte Systems Finite-Stte Systems re one of the simplest type of systems. They cn be given either explicitly (s n explicit list of sttes nd trnsitions), or they cn be described s composition of explicitly given communicting finitestte systems. The systems of the former type re clled explicit finite-stte system in the thesis, nd the ltter re clled composed finite-stte systems. An exmple of composed finite-stte system is prllel composition of explicit finite-stte systems tht synchronize on common ctions nd tht use

19 1.2 Overview of the Results 5 hiding of ctions (they re clled Prllel Composition with Hiding (PCH) in this thesis), or 1-sfe Petri nets. Chpter 3 contins proofs of lower bounds of the complexity of equivlencechecking on explicit nd composed systems. These lower bounds re not specific for one type of equivlence, but pply to whole spectrum of reltions between bisimultion equivlence nd trce preorder. This spectrum includes lmost ll types of equivlences considered in the literture tht re useful in prctice. It is shown t first tht deciding ny reltion between bisimultion equivlence nd trce preorder on explicit systems is PTIME-hrd. This result ws proved in [53], however the proof presented in this thesis uses different nd simpler construction tht ws published in [52]. This construction ws used in [52] s one prt of the proof concerning composed systems. It ws shown there tht deciding ny reltion between bisimultion equivlence nd trce preorder is EXPTIME-hrd for mny types of composed systems including PCH nd 1-sfe Petri nets. This result ws conjectured for PCH by A. Rbinovich in [51], nd the result from [52] pproves his conjecture. To simplify the proof, new model of computtion clled Rective Liner Bounded Automt (RLBA) ws introduced. It ws shown tht deciding ny reltion between bisimultion equivlence nd trce preorder is EXPTIME-hrd for RLBA. Since RLBA cn be implemented (i.e., its behvior cn be esily simulted) by different types of composed systems, the EXPTIME-hrdness result extends to ll such types of systems. (However there is one notble exception, prllel composition of finite-stte systems tht synchronize on common ctions where hiding of ctions is not llowed. Such systems re not powerful enough to simulte RLBA, nd deciding for exmple trce equivlence is PSPACE-complete for them. [54]) One-Counter Automt One-Counter Automt (OCA) re like finite-stte systems extended with counter contining non-negtive integer vlue. This counter cn be incremented nd decremented by one. This kind of mchines cn test whether the vlue of the counter is zero or non-zero nd perform ctions depending on tht. There is vrint of one-counter utomt clled One-Counter Nets (OCN). One-counter nets cn not test for zero vlue of the counter, they cn only test for non-zero vlue. One-counter nets re equivlent (up to isomorphism) with Petri nets with (t most) one unbounded plce.

20 6 Chpter 1. Introduction It is shown in Chpter 4 tht deciding simultion equivlence nd simultion preorder is undecidble problem for one-counter utomt. This result ws proved in [32]. The rest of the chpter is devoted to presenttion of generl method for proving DP-hrdness of different problems concerning one-counter utomt published in [30] nd [31]. This method uses frgment of Presburger rithmetic clled One Counter Logic (OCL). This frgment is chosen in such wy tht it is possible to reduce the problem of deciding the truth of formul of OCL to some problems concerning one-counter utomt. The constructions in these reductions proceed by induction on the structure of formul. It is proved tht the problem of deciding truth of n OCL formul is DP-hrd. This implies tht the problems to which this problem cn be reduced re lso DP-hrd. This technique ws used to show DP-hrdness of deciding ny reltion between bisimultion equivlence nd simultion preorder on one-counter nets, nd to show DP-hrdness of deciding simultion preorder nd simultion equivlence of one-counter utomton nd finite-stte system. (The sme technique ws lso used in [31] by A. Kučer to show DP-hrdness of model checking the logic EF (frgment of CTL) on one-counter nets, however this result is not presented in this thesis, see [31] for more informtion.) Bsic Prllel Processes Bsic Prllel Processes (BPP) re very nturl subclss of infinite stte systems. It ws proved by P. Jnčr in [25] tht deciding bisimultion equivlence on BPP is PSPACE-complete. The technique used in this pper ws used to show two results concerning BPP presented in this thesis in Chpter 5. The first of these results is polynomil time lgorithm for deciding bisimultion equivlence between BPP nd finite-stte system with time complexity O(n 4 ) where n is the size of the instnce. This result ws presented in [35]. The second of these results is polynomil time lgorithm for deciding distributed bisimilrity on BPP with time complexity O(n 3 ). A polynomil time lgorithm for this problem ws lredy presented by Lsot in [40], however the lgorithm presented here is simpler, more efficient nd provides n explicit upper bound on the complexity of the problem. (No degree of the polynomil ws specified in the proof in [40].) The new lgorithm for this problem presented in this thesis ws not published yet.

21 1.3 Lyout of the Thesis Lyout of the Thesis Chpter 2 provides necessry bsic definitions. Problems concerning finitestte systems (both explicit nd composed) re studied in Chpter 3. Chpter 4 concentrtes on problems concerning one-counter utomt, nd Chpter 5 on problems concerning Bsic Prllel Processes. Chpter 6 contins conclusion nd n overview of results presented in this thesis. Appendix A contins list of publictions of the uthor. Becuse the types of problems solved in Chpters 3, 4, nd 5 re more or less independent, ech of these chpters contins its own section clled Stte of the Art which describes known results tht re relevnt for the given chpter.

22 8 Chpter 1. Introduction

23 Chpter 2 Definitions This chpter contins some bsic definitions tht re used in the remining chpters. Section 2.1 presents nottion conventions used in the thesis. Section 2.2 describes the notion of lbelled trnsition systems, formlism tht underlies different types of models described in the following sections process rewrite systems (Section 2.3), process lgebr, BPA nd BPP (Section 2.4), Petri nets (Section 2.5), one-counter utomt (Section 2.6), nd explicit nd composed finite stte systems (Section 2.7). The remining two sections describe different types of behviourl equivlences on lbelled trnsition systems. Section 2.8 describes equivlences from liner time brnching time spectrum, nd Section 2.9 describes distributed bisimilrity one of non-interleving equivlences. 2.1 Nottion Conventions The following nottion conventions re used in the rest of the thesis. N denotes the set of non-negtive integers, i.e., the set {0,1,2,...}. The symbol ω denotes infinity. We define N ω = N {ω}. [x,y], where x,y N, denotes the set of integers between (nd including) x nd y, i.e., the set {z x z y}. Let X be set. X denotes the crdinlity of X. X Y denotes tht X is proper subset of Y, while X Y llows equlity. P(X) denotes the power-set of X, i.e., the set {Y Y X}. 9

24 10 Chpter 2. Definitions s 2 s 4 b s 1 b,b s 3 b s 5 Figure 2.1: Exmple of n LTS X denotes the set of finite sequences of elements from X. Let x X be sequence. The length of x is the number of its elements denoted x. We use x(i) to denote i-th element of x, i.e., x(i) = x i for x = x 1 x 2 x 3. Let X be set. Prtition X of X is set X = {X 1,X 2,...,X l } of disjoint non-empty clsses whose union is X. 2.2 Lbelled Trnsition Systems There re mny possible wys how systems cn be described, for exmple different types of utomt, process rewriting systems, process lgebrs, or Petri nets. However there is one common concept underlying ll these formlisms. It is the concept of lbelled trnsition system (LTS). Formlly, lbelled trnsition system is triple (S,Act, ) where: S is set of sttes, Act is finite set of ctions, nd S Act S is trnsition reltion. Informlly, S is the set of ll possible sttes of the system, Act is set of nmes of externlly observble ctions which cn be performed by the system, nd the trnsition reltion represents behviour of the system. Insted of (s,,s ) we usully write s s, nd this cn be interpreted s tht the system in the stte s cn perform the ction nd go to the stte s.

25 2.2 Lbelled Trnsition Systems 11 See Figure 2.1 for n exmple of n LTS where S = {s 1,s 2,s 3,s 4,s 5 }, Act = {,b}, nd the trnsition reltion contins the following trnsitions: s 1 s 3 s 3 b s 1 s 2 s 4 s 2 s 5 b s 4 s 5 s 4 b s 5 s 2 s 4 s 4 s 5 b s 5 s 3 The set of sttes S in lbelled trnsition system cn be finite or infinite. Lbelled trnsition systems where S is finite re clled finite-stte lbelled trnsition systems, nd lbelled trnsition systems where S is infinite re clled infinite-stte lbelled trnsition systems. Finite-stte systems re the simplest type of systems. The sets of sttes nd trnsitions in such systems re finite nd cn be given explicitly. On the other hnd, it is not possible to work directly with infinite-stte systems. Insted we must work with some finite representtions of them. Exmples of such representtions re different kinds of utomt, Petri nets, process lgebrs, nd process rewrite systems. The nottion s s cn be extended in nturl wy to sequences of ctions. Let w Act. We write s w s iff there is sequence of sttes s 0,s 1,...,s n such tht s = s 0, s w(i) = s n, nd s i 1 s i for ech i such tht 1 i n. Recll tht w(i) denotes i-the symbol of w. A stte s is rechble from stte s, written s s, iff there is some w Act such tht s w s. A lbelled trnsition system is deterministic if for every s S nd every Act there is t most one s such tht s s. System tht is not deterministic is nondeterministic. A lbelled trnsition systems is imge-finite iff for every s S nd every Act the set {s s s } is finite. Sometimes lbelled trnsition system hs distinguished initil stte or more generlly set of initil sttes. Besides observble ctions in Act, there cn be invisible ction denoted τ with specil semntics. We define τ Act for ny set of ctions Act. In lbelled trnsition systems where τ ctions occur, the trnsition reltion is defined s subset of S (Act {τ}) S where S is the set of sttes. Agin we use the nottion s τ s insted of (s,τ,s ).

26 12 Chpter 2. Definitions 2.3 Process Rewrite Systems Process Rewrite Systems (PRS) defined by Myr in [43] provide unified view of mny formlisms presented in the following sections. Process rewrite systems re defined s follows. Let Act = {,b,c,...} be countbly infinite set of tomic ctions nd Vr = {X,Y,Z,...} be countbly infinite set of process vribles. Process terms re defined by the following bstrct grmmr P ::= ε X P 1.P 2 P 1 P 2 where ε is the empty term, X is process vrible, nd where. denotes sequentil composition nd prllel composition. Sequentil composition is ssocitive nd prllel composition is ssocitive nd commuttive. We lwys work with equivlence clsses of terms modulo ssocitivity of sequentil composition nd modulo ssocitivity nd commuttivity of prllel composition. We lso define tht ε.p = P.ε = P nd P ε = P. Process rewrite system is finite set of rules contining rules of the form t 1 t 2 where t 1 nd t 2 re process terms nd Act is n tomic ction. Let Vr( ) be the set of process vribles occurring in nd let Act( ) be the set of tomic ctions occurring in. Process rewrite system produces corresponding lbelled trnsition system (S,Act, ) where S is the set of process terms tht contin only vribles from Vr( ), Act = Act( ), nd the trnsition reltion is the smllest reltion stisfying the following inference rules where t 1,t 2,t 1,t 2 re process terms: (t 1 t 2 ) t 1 t 2 t 1 t 1 t 1.t 2 t 1.t 2 t 1 t 1 t 1 t 2 t 1 t 2 t 2 t 2 t 1 t 2 t 1 t 2 Note tht Vr( ) nd Act( ) re finite. Since is finite, the generted lbelled trnsition system is finitely brnching, which mens tht the brnching-degree is finite in every stte, however it cn be be rbitrrily high, i.e., it is possible tht there is no finite constnt depending only on bounding the brnching-degree in ll sttes. It is worth mentioning tht process rewrite systems re not Turing powerful becuse for exmple the rechbility problem is decidble for them [43].

27 2.3 Process Rewrite Systems 13 Note lso tht there is no opertor for non-deterministic choice ( + ), becuse nondeterminism cn be encoded in the set of rules which cn contin more rules with the sme term on the left side. There cn be defined different types of subclsses of process rewrite systems. At first we distinguish four clsses of process terms: 1 terms consisting of single process vrible (e.g., X), S terms consisting of ε, single vrible, or sequentil composition of process vribles (e.g., X.Y.Z), P terms consisting of ε, single vrible, or prllel composition of process vribles (e.g., X Y Z), G ny process terms without ny restriction (e.g., (X Y ).Z). Obviously 1 S, 1 P, S G, nd P G. Clsses S nd P re incomprble nd S P = 1 {ε}. Let α,β {1, S, P, G} be clsses of process terms such tht α β. We define (α,β)-prs s finite set of rules where in every rewrite rule (l r) the term l is from clss α nd l ε nd the term r is from clss β (nd r cn be ε). The hierrchy of (α, β)-prs models is depicted in Figure 2.2. Ech model in the hierrchy hs nme shown lso in the figure nd mny of these (α, β)-prs correspond to well-known clsses of infinite stte systems studied in the literture. A line from higher model to lower model mens tht the higher model is more generl thn the lower one. It is known tht the hierrchy is strict with respect to bisimilrity [43]. The clsses of process rewrite systems correspond to the following following formlisms: FS finite-stte systems, BPA Bsic Process Algebr [7], lso clled context-free processes, BPP Bsic Prllel Processes [13], PDA Pushdown Automt, lso clled pushdown processes or pushdown systems, PA Process Algebr [4],

28 14 Chpter 2. Definitions (G, G)-PRS PRS (S, G)-PRS PAD (P, G)-PRS PAN (S, S)-PRS PDA (1, G)-PRS PA (P, P)-PRS PN (1, S)-PRS BPA (1, P)-PRS BPP (1,1)-PRS FS Figure 2.2: Hierrchy of process rewrite systems PN Petri nets PRS Process Rewrite Systems Clss PAD ws introduced in [42] s the smllest common generliztion of clsses PDA nd PA. Similrly clss PAN ws introduced in [41] s the smllest common generliztion of clsses PA nd PN. 2.4 Process Algebr, BPA, nd BPP Process Algebr (PA) ws introduced in [4]. It is defined s follows. Let Act = {,b,c,...} be countbly infinite set of tomic ctions nd let Vr = {X,Y,Z,...} be countbly infinite set of process vribles. The clss of PA expressions is defined by the following bstrct syntx: P ::= 0 X.P P 1 + P 2 P 1.P 2 P 1 P 2 where 0 denotes the empty process, X is process vrible,.p is n ction prefix, + denotes non-deterministic choice,. sequentil composition, nd prllel composition.

29 2.4 Process Algebr, BPA, nd BPP 15 We work with expressions modulo ssocitivity nd commuttivity of nondeterministic choice nd prllel composition, nd modulo ssocitivity of sequentil composition. We lso define P.0 = 0.P = P nd P 0 = P. A PA-process is defined by finite fmily of recursive equtions = {X i := P i 1 i n} where ll X i re distinct nd ll P i re PA expressions contining vribles only from Vr( ), where Vr( ) denotes the set {X 1,X 2,...,X n }. The set of ctions occurring in is denoted Act( ). It ssumed tht every occurrence of vrible in the P i is gurded, i.e., tht it is within the scope of n ction prefix. The set of equtions produces lbelled trnsition system (S,Act, ) where S is the set of PA expressions, Act = Act( ), nd trnsition reltion is the lest reltion stisfying the following inference rules: P.P P P.Q P.Q P P P P P + Q P P P P Q P Q P Q X P (where (X := P) ) Q P + Q Q Q Q P Q P Q Sometimes lso the left merge opertor is included in the definition with the following semntics: P P P Q P Q It resembles prllel composition but only the left of the processes cn proceed by performing n ction. A PA-process is in norml form if ll its equtions re of the form n i X i = ij P ij j=1 where 1 i n, n i N, ij Act, nd P i re PA expression not contining ny non-deterministic choice ( + ) or ction prefix. Any PA-process cn be effectively trnsformed to the norml form s ws proved in [8].

30 16 Chpter 2. Definitions Remrk. The term Process Algebr is nowdys lso used in much wider sense denoting lso other formlisms such s for exmple CCS [44]. There re two nturl subclsses of PA-processes Bsic Process Algebr (BPA) [7] nd Bsic Prllel Processes (BPP) [13]. BPA is subclss of PA where no prllel composition ( ) is llowed. Such systems re lso clled context-free processes since the equtions of BPA in the norml form cn viewed s set of rules of context-free grmmr in Greibch norml form (GNF) where ech rule is of the form X Y 1 Y 2...Y k nd where only left derivtions re llowed. Note tht sttes of the produced lbelled trnsition system re sequences of vribles from Vr, i.e., elements of Vr. BPP is subclss of PA where no sequentil composition (. ) is llowed. Processes of the form X 1 X 2 X n where ech X i Vr nd n 0 re clled bsic processes. We identify the bsic process where n = 0 with 0. It is known tht every BPP process cn be trnsformed to norml form where if ll its equtions re of the form n i X i = ij P ij j=1 where 1 i n, ij Act, nd every P i is bsic process. The order of vribles in bsic process is not importnt due to ssocitivity nd commuttivity of, so we cn identify bsic process with multiset of vribles. We use Vr to denote the set of ll multisets of Vr. For P Vr nd X Vr we use P(X) to denote number of occurrences of X in P. The reltion on Vr is defined s P Q iff P(X) Q(X) for every X Vr. Other reltions such s, < nd > nd defined nlogously. We use P Q to denote the union of P,Q Vr, i.e., (P Q)(X) = P(X) + Q(X) for ech X Vr. We use P Q to denote the difference of P,Q Vr such tht P Q, i.e., (P Q)(X) = P(X) Q(X) for ech X Vr. Equivlently we cn represent s finite set of rules of the form X P

31 2.5 Petri Nets 17 where X Vr, Act, nd P Vr. For ech t = (X P), we define pre(t) = X, λ(t) =, nd we use F(t,X) to denote P(X). We write P t P. Remrk. Note tht the rules representing BPP in the norml form cn be viewed s set of rules of context-free grmmr in Greibch norml form (GNF) where i ny derivtion is llowed, not only the left-most. 2.5 Petri Nets A net is triple N = (P,Tr,F), where P is finite set of plces, Tr is finite set of trnsitions, nd is the flow function. F : (P Tr) (Tr P) N Let X = P Tr. For plce or trnsition x X we define sets pre(x) = {y X F(y,x) > 0} nd succ(x) = {y X F(x,y) > 0}. This nottion cn be extended to sets of plces nd trnsitions in the nturl wy, nd so for X P Tr pre(x) = pre(x) succ(x) = succ(x) x X For trnsition t Tr, the sets pre(t) nd succ(t) re clled its input plces nd output plces, respectively. A mrking is mpping M : P N. If P = {s 1,s 2,...,s k }, the mrking M cn be identified with vector (x 1,x 2,...,x k ) where x i = M(s i ) for ech i [1,k]. A (Plce/Trnsition) Petri net is pir N = (N,M 0 ) where N is net nd M 0 is the initil mrking. A trnsition t is enbled t mrking M if M(p) F(p,t) for every p pre(t). A trnsition t tht is not enbled is disbled. If t is enbled t M, then it cn fire or occur, nd its firing leds to the successor mrking M such tht M (p) = M(p) F(p,t) + F(t,p) for every p P. The expression M t M denotes tht t is enbled in M, nd M is reched from M fter firing of t. x X

32 18 Chpter 2. Definitions For sequence σ = t 1 t 2...t n of trnsitions, M σ M denotes tht there is sequence of mrkings M 0,M 1,...,M n such tht M 0 = M, M n = M, t nd M i i 1 Mi for every i [1,n]. A mrking M rechble from mrking M, written M M, iff there is some sequence of trnsitions σ such tht M σ M. A mrking M is rechble iff it is rechble from the initil mrking, i.e., when M 0 M. We use M(N) to denote the set of rechble mrkings of Petri net N. A lbelled net is fourtuple (P,Tr,F,λ), where (P,Tr,F) is net nd λ : P Act is mpping tht ssocites to ech t Tr lbel λ(t) from set of ctions Act. A lbelled Petri net is pir (N,M 0 ), where N is lbelled net nd M 0 is the initil mrking. To lbelled Petri net N we ssocite lbelled trnsition system where S = M(N), nd M λ(t) = nd M t M. (S,Act, ) M iff there is some trnsition t such tht A Petri net is 1-sfe iff M(p) 1 for every rechble mrking M nd every plce p. A Petri net is communiction-free if for ech trnsition t there is exctly one plce p such tht F(p,t) = 1 nd F(p,t) = 0 for ech p p. Lbelled trnsition systems produced by lbelled communiction-free Petri nets re isomorphic to lbelled trnsition systems produced by BPP processes. Let us hve BPP in norml form. Let V = Vr( ). We cn construct the communiction-free Petri net with the set of plces V, where for ech eqution n i X i = ij α ij j=1 from where α ij V we dd for ech j [1,n i ] new trnsition t such tht λ(t) = ij, F(X i,t) = 1, F(X,t) = 0 for ech X X i, nd F(t,X) is set to number of occurrences of X in α ij for ech X V. It is obvious tht the lbelled trnsition system produced by the constructed Petri net is isomorphic to the lbelled trnsition system produced by the BPP. The construction of the corresponding BPP for given communiction-free Petri net is similr. See [50] for more informtion bout Petri nets.

33 2.6 One-Counter Automt One-Counter Automt One-counter utomt re nondeterministic finite-stte utomt operting on single counter vrible which tkes vlues from N. Formlly this is tuple A = (Q,Act,δ =,δ >,q 0 ) where Q is finite set of control sttes, Act is finite set of ctions, δ = : Q Act P(Q {0,1}) δ > : Q Act P(Q { 1,0,1}) re trnsition functions, nd q 0 Q is distinguished initil control stte. The function δ = represents the trnsitions which re enbled when the counter vlue is zero, nd the function δ > represents the trnsitions which re enbled when the counter vlue is positive. A one-counter utomton A is one-counter net if nd only if for ll pirs (q,) Q Act we hve tht δ = (q,) δ > (q,). The set of (globl) sttes of A is the set Q N. Sttes from Q N re written s p(n) insted of (p,n). To the one-counter utomton A we ssocite the lbelled trnsition system (S,Act, ), where S = {p(n) p Q, n N}, nd is defined s follows: p(n) q(n + i) iff nd { n = 0, nd (q,i) δ = (p,); or n > 0, nd (q,i) δ > (p,). Note tht ny trnsition increments, decrements, or leves unchnged the counter vlue, nd decrementing trnsition is only possible if the counter vlue is strictly positive. Also observe tht when n > 0 the immedite trnsitions of p(n) do not depend on the ctul vlue of n. Finlly note tht one-counter net cn in sense test if its counter is nonzero (tht is, it cn perform some trnsitions only on the proviso tht its counter is nonzero), but it cnnot test in ny sense if its counter is zero. Finite-stte systems cn be viewed s one-counter nets where δ = = δ > nd where the counter is never chnged. Thus, the prts of the lbelled trnsition system produced by the utomton rechble from p(i) nd p(j) re isomorphic nd finite for ll p Q nd i,j N. Remrk. The clss of trnsition systems generted by one-counter utomt is the sme (up to isomorphism) s tht generted by the clss of reltime

34 20 Chpter 2. Definitions b b + b b,b Figure 2.3: An exmple of one-counter utomton pushdown utomt (i.e., pushdown utomt without ε-trnsitions) with single stck symbol (prt from specil bottom-of-stck mrker). The clss of trnsition systems generted by one-counter nets is the sme (up to isomorphism) s tht generted by the clss of lbelled Petri nets with (t most) one unbounded plce. One-counter utomt cn be depicted grphiclly s finite grphs with two kinds of edges (solid nd dshed ones) which re lbelled by pirs of the form (,i) Act { 1,0,1}. Insted of (, 1), (,1), nd (,0) we write simply, +, nd, respectively. A solid edge from p to q lbelled by (, i) indictes tht the represented one-counter utomton cn mke trnsition p(k) q(k + i) whenever i 0 or k > 0. A dshed edge from p to q lbelled by (,i) (where i must not be 1) represents zerotrnsition p(0) q(i). Hence, grphs representing one-counter nets do not contin ny dshed edges, nd grphs corresponding to finite-stte systems use only lbels of the form (,0) (remember tht finite-stte systems cn be viewed s specil one-counter nets). Also observe tht the grphs cnnot represent non-decrementing trnsitions which re enbled only for positive counter vlues. This does not mtter since we do not need such trnsitions in our proofs. The distinguished initil control sttes re indicted by blck circles. See Figure 2.3 for n exmple of grph representing n one-counter utomton.

35 2.7 Explicit nd Composed Finite-Stte Systems Explicit nd Composed Finite-Stte Systems We cll finite-stte trnsition system tht is given explicitly explicit trnsition system. A composed system is system given s composition of intercting explicit systems. The set of globl sttes of composed system cn be exponentilly lrger thn the sum of sizes of its prts. This phenomenon is known s stte explosion nd presents the min chllenge in the design of efficient lgorithms for verifiction of composed systems. There re severl different types of prllel composition. One of these types is the prllel composition where systems synchronize on common ctions nd where ctions cn be hidden. Synchroniztion on common ctions mens tht visible ction is executed iff every LTS tht hs in its lphbet executes it. Invisible ctions re not synchronized, tht is, when n LTS executes the invisible ction τ, other LTSs do nothing. Formlly, the prllel composition T 1 T 2 T n of LTSs T 1,..., T n where T i = (S i,act i, i ) for ech i I where I = {1, 2,..., n}, produces the LTS (S, Act, ) where: S = S 1 S 2 S n, Act = Act 1 Act 2 Act n, contins trnsition (s 1,...,s n ) (s 1,...,s n) iff either Act nd for every i I: if Act i, then s i s i, nd if Act i, then s i = s i, or τ = τ nd s i s i for some i I, nd s j = s j for ech j i. Tuples from S 1 S 2 S n re clled globl sttes. Hiding of ctions removes set of visible ctions from the lphbet of n LTS nd relbels corresponding trnsitions with the invisible ction τ. Formlly, hide B in T 1, where T 1 is n LTS (S 1,Act 1, 1 ) nd B Act 1, denotes the LTS (S,Act, ) where S = S 1, Act = Act 1 B, nd s s iff there is some (Act 1 {τ}) such tht s s nd either B nd =, or B nd = τ.

36 22 Chpter 2. Definitions A prllel composition with hiding (PCH) is n LTS T given in the form hide B in (T 1 T n ) where T 1,..., T n re explicit finite-stte systems. The size T of PCH T is T T n + B. There re lso other types of prllel composition defined in the literture, see, e.g., [59], however, most of them re more generl thn prllel composition with hiding described in this section. An exmple is prllel composition where renming of ctions is llowed. Another formlism tht cn be included in composed systems re 1-sfe Petri nets, s individul plces of 1-sfe Petri net cn be viewed s finitestte systems (with 2 sttes) tht communicte through the trnsitions of the Petri net. 2.8 Behviourl Equivlences The equivlence-checking pproch to the forml verifiction of systems is bsed on the following scheme: the specifiction S (i.e., the intended behviour) nd the ctul implementtion I of system re defined s sttes in trnsition systems, nd then it is shown tht S nd I re equivlent. There re mny possible wys how equivlence of processes cn be defined. The most prominent of equivlences defined in the literture were orgnized by vn Glbbeek into the hierrchy clled liner time brnching time spectrum [60]. The hierrchy is shown in Figure 2.4. Arrows in the digrm represent strict inclusion of equivlences, i.e., n rrow from reltion R to reltion R mens tht sttes relted by R must be relted by R, but the converse is not true in generl. As cn be seen in the digrm, bisimultion equivlence is the finest of these equivlences nd trce equivlence is the corsest. As ll equivlences except bisimilrity re defined s symmetric closure of preorder, there is lso similr hierrchy of preorders, see e.g. [21]. Bisimultion equivlence nd simultion equivlence [44, 49] re of specil importnce, s their ccompnying theory hs found its wy into mny prcticl pplictions. Another importnt equivlence is trce equivlence due to its direct correspondence to lnguge equivlence in forml lnguge theory.

37 2.8 Behviourl Equivlences 23 Bisimultion equivlence 2 nested simultion equivlence Redy simultion equivlence Possible futures equivlence Redy trce equivlence Simultion equivlence Rediness equivlence Filure trce equivlence Filures equivlence Completed trce equivlence Trce equivlence Figure 2.4: Liner time brnching time spectrum

38 24 Chpter 2. Definitions Let (S,Act, ) be lbelled trnsition system. A binry reltion R S S is bisimultion iff for every pir of sttes (s,t) R nd every ction Act the following conditions hold: If there is some s S such tht s s, then there is some t S such tht t t nd (s,t ) R. If there is some t S such tht t such tht s s nd (s,t ) R. t, then there is some s S (It is sid tht s s is mtched by t t, resp. t t is mtched by s s.) Sttes s,t re bisimilr, written s t, iff there exists some bisimultion R such tht (s,t) R. The reltion is clled bisimultion equivlence or bisimilrity. It is not difficult to show tht is reflexive, symmetric nd trnsitive. Notice tht union of fmily of bisimultion reltions is lso bisimultion reltion. This implies tht which is the union of ll bisimultions is the mximl bisimultion. A binry reltion R S S is simultion iff for every pir of sttes (s, t) R nd every ction Act the following condition holds: If there is some s S such tht s s, then there is some t S such tht t t nd (s,t ) R. Stte s is simulted by stte t, written s t, iff (s,t) R for some simultion R. Sttes s nd t re simultion equivlent, written s t, iff s t nd t s. The reltion is clled simultion preorder nd the the reltion is clled simultion equivlence. Note tht the union of fmily of simultion reltions is itself simultion reltion, hence, simultion preorder, being the union of ll simultion reltions, is in fct the mximl simultion reltion. A trce from s S is ny w Act such tht there is sequence of sttes w(i) s 0,s 1,...,s n where s 0 = s nd s i 1 s i for every 1 i n. The set of ll trces from s is denoted Trces(s). Sttes s,t re in trce preorder, written s tr t, iff Trces(s) Trces(t). Sttes s,t re trce equivlent iff s tr t nd t tr s. Let R 1, R 2 be binry reltions over S such tht R 1 R 2. We sy the reltion R is between R 1 nd R 2 iff R 1 R R 2.

39 2.9 Distributed Bisimilrity nd BPP 25 Any reltion relting sttes of lbelled trnsition system cn lso relte sttes of different trnsition systems, becuse we cn consider two trnsition systems to be single one by tking the disjoint of them. Let 1, 2 be (descriptions of) lbelled trnsition systems with distinguished initil sttes s nd t, nd let let be binry reltion relting sttes of these systems. Systems 1, 2 re relted by iff their initil sttes re relted by, formlly 1 2 iff s t. Let P nd Q be clsses of lbelled trnsition systems nd let be binry reltion relting sttes of these systems. The problem of deciding whether given systems 1 P nd 2 Q with distinguished initil sttes re relted by is denoted by P Q. For exmple the problem whether two systems from clss P re bisimilr is denoted by P P. Similrly the problem whether given process from clss P is simulted by process from clss Q is denoted by P Q. Equivlence checking problem is ny problem of the form P Q where P, Q, nd re fixed. We buse the terminology little bit, since the term equivlence checking is used even for problems where is not n equivlence reltion. See [46, 9, 55] for n overview of known results bout decidbility nd complexity of equivlence-checking problems for different types of systems nd different types of equivlences. 2.9 Distributed Bisimilrity nd BPP Distributed bisimilrity is one of non-interleving equivlences lso clled true concurrency equivlences. It ws introduced in [10]. Exmples of other non-interleving equivlences re loction equivlence [11], cusl equivlence [16], history preserving bisimilrity [61], or performnce equivlence [19]. In this thesis we concentrte on deciding distributed bisimilrity on BPP nd we use the definition from [13]. However, it is known tht distributed bisimilrity coincides on BPP with mny other non-interleving equivlences, see [40] for detils. This mens tht n lgorithm tht decides distributed bisimilrity on BPP cn be used lso for deciding ny such equivlence on BPP. When considering distributed bisimilrity we use the following definition of BPP. Let Act = {,b,c,...} be countbly infinite set of tomic ctions

40 26 Chpter 2. Definitions nd let Vr = {X,Y,Z,...} be countbly infinite set of process vribles. The clss of BPP expressions over Act nd Vr is defined by the following bstrct syntx: P ::= 0 X.P P 1 + P 2 P 1 P 2 P 1 P 2 where 0 denotes the empty process, X is process vrible,. is n ction prefix, + denotes non-deterministic choice, prllel composition, nd left merge. Remrk. The left merge opertor is similr to prllel composition, but in n ction must be performed first in the first rgument. A BPP process definition is finite fmily of recursive equtions = {X i := P i 1 i n} where ll X i re distinct nd ll P i re BPP expressions where every occurrence of vrible in P i is gurded, i.e., it is within the scope of n ction prefix. The sets of ctions nd vribles occurring in re denoted Act( ) nd Vr( ), respectively. A BPP process is pir (P, ) where is BPP process definition nd P is process expression contining only ctions nd vribles from Act( ) nd Vr( ). We usully write just P insted of (P, ) when is obvious from the context. Distributed bisimilrity is binry reltion defined over BPP expressions. Informlly, for ech BPP expression there is set of possible trnsitions going out of this expression to pir of expressions clled locl derivtive nd concurrent derivtive. The intuition behind this definition is tht processes re distributed in spce, nd locl nd concurrent derivtives re two prts of the whole process. Locl derivtive records loction t which the ction is observed, nd concurrent derivtive records the rest of the process. We write trnsitions s P [P,P ] where P is the originl process, P nd P re its locl nd concurrent derivtives, nd is the performed ction. Let us ssume we hve fixed BPP process definition. Then the possible trnsitions re defined by the following set of rules:.p [P,0] P [P,P ] P Q [P,P Q] P j [P,P ] for some j I i I P i [P,P ] Q [Q,Q ] P Q [Q,P Q ]

41 2.9 Distributed Bisimilrity nd BPP 27 P [P,P ] P Q [P,P Q] P [P,P ] def X ((X = P) ) [P,P ] A reltion R is distributed bisimultion iff for ech (P,Q) R nd ech Act two following conditions hold: if P [P,P ] then Q [Q,Q ] for some Q,Q (P,Q ) R nd (P,Q ) R, nd if Q [Q,Q ] then P [P,P ] for some P,P (P,Q ) R nd (P,Q ) R. such tht such tht Processes P nd Q re distributed bisimilr, denoted P Q, iff there is distributed bisimultion R such tht (P,Q) R. The reltion is clled distributed bisimultion equivlence or distributed bisimilrity. Remrk. Distributed bisimilrity nd (norml) bisimilrity re both denoted by the symbol in this thesis. The convention is tht represents (norml) bisimilrity unless otherwise stted, the only exception is Section 5.3 tht concentrtes on distributed bisimilrity nd where represents distributed bisimilrity. It is not difficult to show tht non-deterministic choice ( + ) nd prllel composition ( ) re ssocitive nd commuttive with respect to, so we cn work with them modulo ssocitivity nd commuttivity. Processes of the form X 1 X 2 X n, where n 0 nd ech X i Vr, re clled bsic processes. We identify the bsic process where n = 0 with 0. Every BPP process definition cn be trnsformed to equivlent norml form where ll equtions re of the form X def = i I ((.P i ) Q i ) where ech P i nd Q i re bsic processes nd where the reltion defined below is irreflexive. The reltion Vr( ) Vr( ) is defined such tht Y X holds iff X def = i I ((.P i) Q i ) nd there is some Q i such tht Y occurs in Q i. It is n esy tsk to verify if some such reltion exists nd to find it. Since is irreflexive, it cn be esily extended to some (rbitrry) liner order. In the thesis we use to denote this liner order.

42 28 Chpter 2. Definitions Remrk. Note tht some vribles in the norml form re not gurded, nd so in this sense the norml form is not correct BPP process definition. However, note tht lthough these vribles re not gurded in syntcticl sense, they re gurded in semntic sense it is not possible to rewrite vrible X to some expression without going through some ction prefix. See [40] for polynomil time lgorithm tht trnsforms BPP process definition to norml form. Due to ssocitivity nd commuttivity of prllel composition, the order of vribles in bsic process is not importnt, nd so we cn identify bsic process with multiset of vribles. We use Vr to denote the set of ll multisets of Vr( ). For P Vr nd X Vr we use P(X) to denote the number of occurrences of X in P. The reltion on Vr is defined s P Q iff P(X) Q(X) for every X Vr( ). We use P Q to denote the union of P,Q Vr, i.e., (P Q)(X) = P(X) + Q(X) for ech X Vr( ). We use P Q to denote the difference of P,Q Vr such tht P Q, i.e., (P Q)(X) = P(X) Q(X) for ech X Vr( ). For technicl convenience we use little bit different nottion for BPP process definitions in the thesis. We represent BPP process definition s finite set of rules of the form X (P,Q) where X Vr( ), Act, nd P,Q Vr. Note tht there cn be more thn one rule with the sme vrible X on the left hnd side. For ech t = (X (P,Q)), we define pre(t) = X, λ(t) =, nd we use F(t,X) nd G(t,X) to denote P(X) nd Q(X), respectively. We write t P [P,P ] iff process P goes to pir of processes [P,P ] (denoted P [P,P ]) using rule t, i.e., iff t is of the form X (P,Q ) where P = (P {X}) Q.