Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions of systems of recursion equations of the formX_1  =  P_1 X_2 + Q_1    X_n  =  P_n X_1 + Q_n  where  n  is a positive integer, and the  P_i  and the  Q_i  are process terms. The addition of multi-exit iteration to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse

Aceto, Luca

Fokkink, Willem Jan

Ingólfsdóttir, Anna

English

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BRICSRS-02-40Acetoetal.:ANoteonanExpressivenessHierarchyforMulti-exitIterationBRICSBasic Research in Computer ScienceA Note on an Expressiveness Hierarchy forMulti-exit IterationLuca AcetoWillem Jan FokkinkAnna IngólfsdóttirBRICS Report Series RS-02-40ISSN 0909-0878 September 2002Copyright c© 2002, Luca Aceto & Willem Jan Fokkink & AnnaIngólfsdóttir.BRICS, Department of Computer ScienceUniversity of Aarhus. All rights reserved.Reproduction of all or part of this workis permitted for educational or research useon condition that this copyright notice isincluded in any copy.See back inner page for a list of recent BRICS Report Series publications.Copies may be obtained by contacting:BRICSDepartment of Computer ScienceUniversity of AarhusNy Munkegade, building 540DK–8000 Aarhus CDenmarkTelephone: +45 8942 3360Telefax: +45 8942 3255Internet: BRICS@brics.dkBRICS publications are in general accessible through the World WideWeb and anonymous FTP through these URLs:http://www.brics.dkftp://ftp.brics.dkThis document in subdirectory RS/02/40/A Note on an Expressiveness Hierarchyfor Multi-exit IterationLuca Aceto∗ Wan Fokkink† Anna Ingólfsdóttir∗‡AbstractMulti-exit iteration is a generalization of the standard binary Kleenestar operation that allows for the specification of agents that, up to bisim-ulation equivalence, are solutions of systems of recursion equations of theformX1def= P1X2 + Q1...Xndef= PnX1 + Qnwhere n is a positive integer, and the Pi and the Qi are process terms. Theaddition of multi-exit iteration to Basic Process Algebra (BPA) yields amore expressive language than that obtained by augmenting BPA with thestandard binary Kleene star. This note offers an expressiveness hierarchy,modulo bisimulation equivalence, for the family of multi-exit iterationoperators proposed by Bergstra, Bethke and Ponse.AMS Subject Classification (1991): 68Q15, 68Q70.CR Subject Classification (1991): D.3.1, F.1.1, F.4.1.Keywords and Phrases: Concurrency, process algebra, Basic ProcessAlgebra (BPA), multi-exit iteration, bisimulation, expressiveness.1 BackgroundFor the sake of completeness and readability, we begin by recalling the relevantnotions from [1] that will be needed in this note. The interested reader is referredto op. cit. and [5] for motivation and further information.We assume a non-empty alphabet A of atomic actions, with typical elementsa, b. The language BPAme∗(A) of terms over Basic Process Algebra (BPA) withmulti-exit iteration is defined inductively as follows:∗BRICS (Basic Research in Computer Science), Centre of the Danish National ResearchFoundation, Department of Computer Science, Aalborg University, Fr. Bajersvej 7E, 9220Aalborg Ø, Denmark. Email: {luca,annai}@cs.auc.dk.†CWI, Department of Software Engineering, Kruislaan 413, 1098 SJ Amsterdam, TheNetherlands. Email: wan@cwi.nl.‡deCODE Genetics, Sturlugata 8, 101 Reykjavik, Iceland.1- each a ∈ A is a term;- P + Q and P · Q are terms, if so are P and Q;- (P1, ..., Pm)∗(Q1, ..., Qn) is a term, if so are P1, ..., Pm and Q1, ..., Qn forsome positive integers m and n.We shall use P, Q, R (possibly subscripted and/or superscripted) to range overBPAme∗(A). In writing terms over the above syntax, we shall always assumethat the operation · binds stronger than +. In the sequel the operation · willoften be omitted, so PQ denotes P ·Q. We shall use the symbol ≡ to stand forsyntactic equality of terms. For every natural number n, we shall write [n] inlieu of {1, . . . , n}.Apart from actions, the signature of the language BPAme∗(A) includes thebinary operations of alternative composition + and sequential composition ·familiar from the theory of Basic Process Algebra [6, 4], and a variation on theoriginal binary version of the Kleene star operation [9], that will be referredto as multi-exit iteration. For positive integers m and n, the process term(P1, ..., Pm)∗(Q1, ..., Qn) stands for an agent whose behaviour is specified by thefollowing defining equation:(P1, ..., Pm)∗(Q1, ..., Qn) = P1 · (P2, ..., Pm, P1)∗(Q2, ..., Qn, Q1) + Q1 .In order to simplify notation in the presentation of the operational semanticsfor BPAme∗(A), we shall use the notion of ‘vectors of processes’. A vector ofprocesses is a tuple (P1, ..., Pm), where m ≥ 0. We shall use ~Q, ~S to denotesuch vectors of processes. In multi-exit iteration, the expressions at the left-and right-hand sides of the star are non-empty vectors of processes. Enclosingparentheses will always be omitted from vectors of length one, i.e., (P ) will bewritten P .The operational semantics for the language BPAme∗(A) is given by the la-belled transition system(BPAme∗(A),{a→| a ∈ A},{a→X | a ∈ A}),where the transition relations a→ and the unary predicates a→X are, respec-tively, the least subsets of BPAme∗(A)×BPAme∗(A) and BPAme∗(A) satisfyingthe rules in Table 1. Intuitively, a transition P a→ Q means that the systemrepresented by the term P can perform the action a, thereby evolving into Q.The special symbol X stands for (successful) termination; therefore the inter-pretation of the statement P a→X is that the process term P can terminate byperforming a. Note that, for every term P , there is some action a for whicheither P a→ P ′ holds for some P ′, or P a→X does.Definition 1.1 The term P ′ is a derivative of P if P can evolve into P ′ byzero or more transitions. A derivative P ′ of P is proper if P can evolve intoP ′ by performing at least one transition.2aa→XPa→XP + Q a→XQa→XP + Q a→XPa→ P ′P + Q a→ P ′Qa→ Q′P + Q a→ Q′Pa→XP · Q a→ QPa→ P ′P · Q a→ P ′ · QPa→X(P, ~Q)∗(R, ~S) a→ ( ~Q, P )∗(~S, R)Pa→ P ′(P, ~Q)∗(R, ~S) a→ P ′ · ( ~Q, P )∗(~S, R)Ra→X(P, ~Q)∗(R, ~S) a→XRa→ R′(P, ~Q)∗(R, ~S) a→ R′Table 1: Transition RulesProcess terms are considered modulo bisimulation equivalence [10].Definition 1.2 Two process terms P and Q are bisimilar, denoted by P ↔ Q,if there exists a symmetric binary relation B on process terms which relates Pand Q, such that:- if R B S and R a→ R′, then there is a transition S a→ S′ such that R′ B S′,- if R B S and R a→X, then S a→X.Such a relation B will be called a bisimulation. The relation ↔ will be referredto as bisimulation equivalence.Note that if P is bisimilar to Q, then every (proper) derivative of P is bisimilarto some (proper) derivative of Q, and vice versa.The transition rules in Table 1 are in the ‘path’ format of Baeten and Verhoef[3]. Hence, bisimulation equivalence is a congruence with respect to all theoperations in the signature of BPAme∗(A).Process terms in BPAme∗(A) are normed, which means that they are ableto terminate by embarking in a finite sequence of transitions. We call such asequence a termination trace. The norm of a process term P , denoted by |P |,is the length of its shortest termination trace; this notion stems from [2]. Notethat bisimilar process terms have the same norm. The following lemma, whichis due to Caucal [8], is typical for normed processes, and will be useful in thetechnical developments to follow.Lemma 1.3 Let P, Q, R, S ∈ BPAme∗(A) be such that PQ ↔ RS. If |Q| = |S|,then P ↔ R and Q ↔ S.A technical tool we shall use below is a weight function g that associates a3natural number to each process term. This is defined thus:g(a) ∆= 0g(P + Q) ∆= max{g(P ), g(Q)} + 1g(PQ) ∆= max{g(P ), g(Q)}g((P1, ..., Pm)∗(Q1, ..., Qn)) ∆= max{g(Pi), g(Qj) + 1 | i ∈ [m], j ∈ [n]} .The basic property of this weight function that we shall need is expressed in thelemma below (cf. [1, Lemma 3.5]).Lemma 1.4 If P ′ is a derivative of P , then g(P ′) ≤ g(P ). Moreover, if- P ≡ P1 + P2 for some terms P1 and P2, and P ′ is a proper derivative ofP , or- P ≡ (P1, ..., Pm)∗(Q1, ..., Qn), for some terms Pi (i ∈ [m]) and Qj (j ∈[n]), and P ′ is a proper derivative of some Qj,then g(P ′) < g(P ).2 An Expressiveness HierarchyAs shown in [5], the addition of multi-exit iteration to BPA yields a languagethat, modulo bisimulation equivalence, is strictly more expressive than thatobtained by augmenting BPA with the standard binary Kleene star. More pre-cisely, it is proven ibidem that, in the presence of at least two actions, the process(a, a)∗(a, b) cannot be expressed, modulo bisimulation equivalence, in ACP [4],and a fortiori in BPA, enriched with the binary Kleene star (cf. Lemma 3.2.3 inop. cit.).Let us say that a term of the form (P1, . . . , Pm)∗(Q1, . . . , Qn) has n-exititeration. By analogy with the aforementioned result from [5], it was provedin [1] that, in the presence of a non-empty set of actions, the sequence of k-exit iteration operations induces a hierarchy of super-languages of BPA with astrictly increasing expressive power modulo bisimulation equivalence. To thisend, it was shown in op. cit. that, for every positive integer k, the processa∗(a, a2, . . . , ak+1)cannot be specified using h-exit iteration with h ≤ k, modulo bisimulationequivalence. (Cf. Corollary 4.5 in [1].)In light of the above result, increasing the maximum number of exits allowedin a multi-exit iteration increases the expressive power of the language modulobisimulation equivalence. Our aim in this note is to show that increasing themaximum number of processes on the left-hand side of the star in a multi-exititeration also increases the expressive power of the language modulo bisimulationequivalence.4Notation 1 For every positive integer k, we write BPAk∗ for the set of termsin the language BPAme∗(A) that may use multi-exit iteration operations whosefirst argument is a non-empty vector of processes of length at most k.For a positive integer i and action a, we write ai for the term obtained byconcatenating i copies of action a.Our aim is to prove the following theorem:Theorem 2.1 For every positive integer k, the process (a, a2, . . . , ak+1)∗a can-not be expressed in the language BPAk∗ modulo bisimulation equivalence.The remainder of this note will be devoted to a proof of the above result. Tothis end, it is sufficient to establish the following special case of the statementof our main result.Proposition 2.2 For every positive integer k, the process (a, a2, . . . , ak+1)∗acannot be expressed, modulo bisimulation equivalence, as a term in the languageBPAk∗ of the form (P1, . . . , Ph)∗(Q1, . . . , Qm) with |Qj| = 1, for every j ∈ [m].Indeed, using the above result, we can prove Theorem 2.1 thus:Proof of Theorem 2.1: Assume, towards a contradiction, that there is aterm P in the language BPAk∗ that is bisimilar to (a, a2, . . . , ak+1)∗a. Assume,furthermore, that P is a process with this property with minimum weight g(P ).We proceed with the proof by analyzing the possible forms such a P may take.It is easy to see that P can neither have the form a nor the form P1P2for some processes P1 and P2. Indeed, this follows because bisimilar processeshave equal norm, but any process of the form P1P2 has norm at least two and(a, a2, . . . , ak+1)∗a has norm one.We claim that P cannot have the form (P1, . . . , Ph)∗(Q1, . . . , Qm) either.To see this, note, first of all, that, by Proposition 2.2, P cannot have the form(P1, . . . , Ph)∗(Q1, . . . , Qm), with |Qj | = 1 for every j ∈ [m]. If there is someQj (j ∈ [m]) whose norm is greater than one, then this Qj affords a transitionQja→ Q′j for some process Q′j . It follows that, for some positive integer `,(P1, . . . , Ph)∗(Q1, . . . , Qm)a`→ Q′j .Since the terms (P1, . . . , Ph)∗(Q1, . . . , Qm) and (a, a2, . . . , ak+1)∗a are bisimilar,there is a derivative R of the latter term such that(a, a2, . . . , ak+1)∗a a`→ R and Q′j ↔ R .As (a, a2, . . . , ak+1)∗a is easily seen to be a derivative of R, we have that Q′j has aderivative that is bisimilar to (a, a2, . . . , ak+1)∗a, and thus that Qj has a properderivative Q′ that is bisimilar to (a, a2, . . . , ak+1)∗a. By Lemma 1.4, the value ofg(Q′) is strictly smaller than g(P ). This contradicts our assumption that P wasa process with minimum weight in BPAk∗ that is bisimilar to (a, a2, . . . , ak+1)∗a.5From the above reasoning, it follows that P can only have the form P1 + P2for some processes P1 and P2. Since(a, a2, . . . , ak+1)∗a an→ (a, a2, . . . , ak+1)∗a(n =(k + 1)(k + 2)2)and P ≡ P1 + P2 is bisimilar to (a, a2, . . . , ak+1)∗a, there is a process P ′ suchthatPan→ P ′ and P ′ ↔ (a, a2, . . . , ak+1)∗a .By Lemma 1.4, since P ′ is a proper derivative of P ≡ P1 + P2, the value ofg(P ′) is strictly smaller than g(P ). This contradicts our assumption that P wasa process with minimum weight in BPAk∗ that is bisimilar to (a, a2, . . . , ak+1)∗a.It follows that no term in BPAk∗ can be bisimilar to (a, a2, . . . , ak+1)∗a,which was to be shown. To complete the proof, we are therefore left to show Proposition 2.2. This resultis an immediate consequence of the second statement in the following lemma.Lemma 2.3 Assume that Q1, . . . , Qm are processes with norm one. Then thefollowing statements hold:1. For every positive integer i, if (P1, . . . , Ph)∗(Q1, . . . , Qm) is bisimilar to(ai, R1, . . . , Rn)∗a (n ≥ 0), then• P1 ↔ ai and• (P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) ↔ (R1, . . . , Rn, ai)∗a.2. For every k ≥ h ≥ 1, if (P1, . . . , Ph)∗(Q1, . . . , Qm) ↔ (a, a2, . . . , ak)∗a,then h = k, and Pi ↔ ai for every i ∈ [k].Proof: We prove the two statements separately.• Proof of Statement 1. We consider two cases, depending on whetheri = 1 or not. In both cases of the proof, we use the fact that, as(P1, . . . , Ph)∗(Q1, . . . , Qm) ↔ (ai, R1, . . . , Rn)∗a holds by assumption, P1can perform an a-labelled transition and no transition labelled with ac-tions different from a.Assume that i = 1. Then P1 has no transitions of the form P1a→ P ′1.Indeed, if P1a→ P ′1 holds, then so does(P1, . . . , Ph)∗(Q1, . . . , Qm)a→ P ′1(P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) . (1)Since (P1, . . . , Ph)∗(Q1, . . . , Qm) ↔ (a, R1, . . . , Rn)∗a holds by assump-tion, there is a transition(a, R1, . . . , Rn)∗aa→ Rfor some R such thatP ′1(P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) ↔ R .6The only candidate for this R is the term (R1, . . . , Rn, a)∗a. However, theterm (R1, . . . , Rn, a)∗a has norm one, whereas|P ′1(P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1)| ≥ 2 .It follows that P ′1(P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) cannot be bisimilar to(R1, . . . , Rn, a)∗a, and thus that P1a→X is the only transition afforded byP1. We can now conclude that– P1 ↔ a and– (P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) ↔ (R1, . . . , Rn, a)∗aboth hold, which was to be shown.Assume now that i is greater than 1. Reasoning as in the previous case,it is not hard to see that P1 only affords transitions of the form P1a→ P ′1.For every such transition, we have a transition of the form (1) out of(P1, . . . , Ph)∗(Q1, . . . , Qm). These transitions can only be matched by thetransition(ai, R1, . . . , Rn)∗aa→ ai−1(R1, . . . , Rn, ai)∗afrom (ai, R1, . . . , Rn)∗a. It follows that, for every term P ′1 such that P1a→P ′1, it holds thatP ′1(P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) ↔ ai−1(R1, . . . , Rn, ai)∗a .Since the terms (P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) and (R1, . . . , Rn, ai)∗ahave both norm one by the proviso of the lemma, Lemma 1.3 yields thatP ′1 ↔ ai−1 and(P2, . . . , Ph, P1)∗(Q2, . . . , Qm, Q1) ↔ (R1, . . . , Rn, ai)∗a .To complete the proof for this case, note that since every term that canbe reached from P1 via an a-labelled transition is bisimilar to ai−1, fromour previous observations it follows that P1 is bisimilar to ai.• Proof of Statement 2. Assume that k ≥ h ≥ 1 and(P1, . . . , Ph)∗(Q1, . . . , Qm) ↔ (a, a2, . . . , ak)∗a .Using statement 1 of the lemma repeatedly, we have that Pi ↔ ai for everyi ∈ [h], and(P1, . . . , Ph)∗(Q`+1, . . . , Qm, Q1, . . . , Q`) ↔ (ah+1, . . . , ak, a1, . . . , ah)∗a ,where ` = h mod m.If h < k, then statement 1 of the lemma would entail thata ↔ P1 ↔ ah+1 ,which is impossible because a ↔/ ah+1, as h ≥ 1. It follows that h = kholds, and we are done.7This completes the proof of the lemma. Acknowledgements: The research reported in this note originated from aquestion posed to the authors by Kim G. Larsen.References[1] L. Aceto and W. J. Fokkink. An Equational Axiomatization for Multi-exit Iteration. Information and Computation, 137(2):121–158, 15 Septem-ber 1997.[2] J. Baeten, J. Bergstra, and J. Klop, Decidability of bisimulationequivalence for processes generating context-free languages, J. Assoc. Com-put. Mach., 40 (1993), pp. 653–682.[3] J. Baeten and C. Verhoef, A congruence theorem for structured oper-ational semantics, in Best [7], pp. 477–492.[4] J. Baeten and W. Weijland, Process Algebra, Cambridge Tracts inTheoretical Computer Science 18, Cambridge University Press, 1990.[5] J. Bergstra, I. Bethke, and A. Ponse, Process algebra with iteration,Report CS-R9314, Programming Research Group, University of Amster-dam, 1993.[6] J. Bergstra and J. Klop, Fixed point semantics in process algebras,Report IW 206, Mathematisch Centrum, Amsterdam, 1982.[7] E. Best, ed., Proceedings CONCUR 93, Hildesheim, Germany, vol. 715 ofLecture Notes in Computer Science, Springer-Verlag, 1993.[8] D. Caucal, Graphes canoniques de graphes algébriques, Theoretical Infor-matics and Applications, 24 (1990), pp. 339–352.[9] S. Kleene, Representation of events in nerve nets and finite automata, inAutomata Studies, C. Shannon and J. McCarthy, eds., Princeton UniversityPress, 1956, pp. 3–41.[10] D. Park, Concurrency and automata on infinite sequences, in 5th GI Con-ference, Karlsruhe, Germany, P. Deussen, ed., vol. 104 of Lecture Notes inComputer Science, Springer-Verlag, 1981, pp. 167–183.8Recent BRICS Report Series PublicationsRS-02-40 Luca Aceto, Willem Jan Fokkink, and Anna Inǵolfsdóttir. ANote on an Expressiveness Hierarchy for Multi-exit Iteration.September 2002. 8 pp.RS-02-39 Stephen L. Bloom and Zolt́an Ésik. Some Remarks on RegularWords. September 2002.RS-02-38 Daniele Varacca. The Powerdomain of Indexed Valuations.September 2002. 54 pp. Short version appears in Plotkin, ed-itor, Seventeenth Annual IEEE Symposium on Logic in Com-puter Science, LICS ’02 Proceedings, 2002, pages 299–308.RS-02-37 Mads Sig Ager, Olivier Danvy, and Mayer Goldberg.A Sym-metric Approach to Compilation and Decompilation. August2002. To appear in Neil Jones’s Festschrift.RS-02-36 Daniel Damian and Olivier Danvy. CPS Transformation ofFlow Information, Part II: Administrative Reductions. 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A Note on an Expressiveness Hierarchy for Multi-exit Iteration

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A Note on an Expressiveness Hierarchy for Multi-exit Iteration

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