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BRICSBasic Research in Computer Science

A Cook’s Tour ofEquational Axiomatizations forPrefix Iteration

Luca AcetoWillem Jan FokkinkAnna Ingolfsdottir

BRICS Report Series RS-98-49

ISSN 0909-0878 December 1998

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A Cook's Tour of Equational Axiomatizationsfor Pre�x IterationLu a A eto1?, Wan Fokkink2?? and Anna Ing�olfsd�ottir3? ? ?1 BRICS (Basi Resear h in Computer S ien e),Department of Computer S ien e, Aalborg University,Fredrik Bajers Vej 7-E, DK-9220 Aalborg �, Denmark.2 Department of Computer S ien e, University of Wales Swansea,Singleton Park, Swansea SA2 8PP, Wales.3 Dipartimento di Sistemi ed Informati a, Universit�a di Firenze,Via Lombroso 6/17, 50134 Firenze, Italy.Abstra t. Pre�x iteration is a variation on the original binary version of theKleene star operation P �Q, obtained by restri ting the �rst argument to bean atomi a tion, and yields simple iterative behaviours that an be equation-ally hara terized by means of �nite olle tions of axioms. In this paper, wepresent axiomati hara terizations for a signi� ant fragment of the notionsof equivalen e and preorder in van Glabbeek's linear-time/bran hing-time spe -trum over Milner's basi CCS extended with pre�x iteration. More pre isely, we onsider ready simulation, simulation, readiness, tra e and language semanti s,and provide omplete (in)equational axiomatizations for ea h of these notionsover BCCS with pre�x iteration. All of the axiom systems we present are �nite,if so is the set of atomi a tions under onsideration.1 Introdu tionEquationally based proof systems play an important role in both the pra ti eand the theory of pro ess algebras. From the point of view of pra ti e, theseproof systems an be used to perform system veri� ations in a purely synta ti way, and form the basis of axiomati veri� ation tools like, e.g., PAM [10℄. Fromthe theoreti al point of view, omplete axiomatizations of behavioural equiva-len es apture the essen e of di�erent notions of semanti s for pro esses in termsof a basi olle tion of identities, and this often allows one to ompare seman-ti s whi h may have been de�ned in very di�erent styles and frameworks. Someresear hers also measure the naturalness of a pro ess semanti s by using theexisten e of a �nite omplete axiomatization for it over, say, �nite behavioursas an a id test.An ex ellent example of the unifying role played by equational axiomatiza-tions of pro ess semanti s may be found in [7℄. Ibidem van Glabbeek presentsthe so- alled linear time/bran hing time spe trum, i.e., the latti e of all theknown behavioural equivalen es over labelled transition systems ordered byin lusion. The di�erent identi� ations made by these semanti equivalen es? Partially supported by the Human Capital and Mobility proje t Express. Email:lu a� s.au .dk. Fax: +45 9815 9889.?? Email: W.J.Fokkink�swansea.a .uk. Fax: +44 1792 295708.? ? ? Supported by a grant from the Danish National Resear h Foundation. Email:annai�dsi2.ing.unifi.it. Fax: +39 55 4796730.

over �nite syn hronization trees are beautifully hara terized by the author ofop. it. in terms of a few simple axioms. This permits an illuminating om-parison of these semanti s within a uniform axiomati framework. However,despite the omplete inferen e systems for bisimulation-based equivalen es overregular pro esses presented in, e.g., [11, 8℄ and years of intense resear h, littleis still known on the topi of e�e tive omplete axiomatizations of the notionsof semanti s studied in [7℄ over iterative pro esses.In this study, we shall present a ontribution to this line of resear h byinvestigating a signi� ant fragment of the notions of equivalen e and preorderfrom [7℄ over Milner's basi CCS (hen eforth referred to as BCCS) [12℄ extendedwith pre�x iteration. Pre�x iteration [6℄ is a variation on the original binaryversion of the Kleene star operation P �Q [9℄, obtained by restri ting the �rstargument to be an atomi a tion, and yields simple iterative behaviours that an be equationally hara terized by means of �nite olle tions of axioms. Fur-thermore, pre�x iteration ombines better with the a tion pre�xing operatorof CCS than the more general binary Kleene star. A signi� ant advantage ofiteration over re ursion, as a means to express in�nite pro esses, is that it doesnot involve a parametri pro ess de�nition, be ause the development of pro esstheory is easier if parameterization does not have to be taken as primitive (see,e.g., Milner [13, page 212℄).Our study of equational axiomatizations for BCCS with pre�x iterationhas so far yielded omplete equational axiomatizations for all the main no-tions of bisimulation equivalen e [6, 1℄. In this paper, we ontinue this resear hprogramme by studying axiomati hara terizations for more abstra t seman-ti s over this language than those based on variations of bisimulation. Morepre isely, we onsider ready simulation, simulation, readiness, tra e and lan-guage semanti s, and provide omplete (in)equational axiomatizations for ea hof these notions over BCCS with pre�x iteration. All of the axiom systemswe present are �nite, if so is the set of atomi a tions under onsideration. Al-though the high level stru ture of the proofs of our main results follows standardlines in the literature on pro ess theory, the a tual details of the arguments are,however, rather subtle ( f., e.g., the proof of Thm. 4.6). To our mind, this showshow the analysis of the olle tion of valid identities for the semanti s onsid-ered in this paper already be omes diÆ ult even in the presen e of very simpleiterative behaviours, like those that an be expressed using pre�x iteration.The paper is organized as follows. After a brief review of the basi notionsfrom pro ess theory needed in the remainder of the paper (Se t. 2), we presentthe language BCCS with pre�x iteration and its labelled transition systemsemanti s (Se t. 3). Se t. 4 is devoted to a guided tour of our ompletenessresults. The paper on ludes with a mention of further results that will bepresented in a full a ount of this work, and a dis ussion of ongoing resear h(Se t. 5).2 PreliminariesIn this se tion we present the basi notions from pro ess theory that will beneeded in the remainder of this study. 2

2.1 Labelled Transitions SystemsA labelled transition system is a triple (Pro ; Lab;n!j ` 2 Labo), where:{ Pro is a set of states, ranged over by s, possibly subs ripted or super-s ripted;{ Lab is a set of labels, ranged over by `, possibly subs ripted;{ !� Pro � Pro is a transition relation, for every ` 2 Lab. As usual, weshall use the more suggestive notation s ! s0 in lieu of (s; s0) 2!, and writes 9 i� s ! s0 for no state s0.All the labelled transition systems we shall onsider in this paper will have aspe ial label X in their label set|used to represent su essful termination|,and will enjoy the following property: if s X! s0, then s0 9 for every label `.For n � 0 and & = `1 : : : `n 2 Lab�, we write s &! s0 i� there exist statess0; : : : ; sn su h that s = s0 `1! s1 `2! � � �sn�1 `n! sn = s0. In that ase, we saythat & is a tra e (of length n) of the state s. For a state s 2 Pro we de�ne:initials(s) �= n` 2 Lab j 9s0 : s ! s0o :2.2 From Ready Simulation to Language Equivalen eLabelled transition systems des ribe the operational behaviour of pro esses ingreat detail. In order to abstra t from irrelevant information on the way pro- esses ompute, a wealth of notions of behavioural equivalen e or approximationhave been studied in the literature on pro ess theory. A systemati investiga-tion of these notions is presented in [7℄, where van Glabbeek studies the so- alled linear time/bran hing time spe trum, i.e., the latti e of all the knownbehavioural equivalen es over labelled transition systems ordered by in lusion.In this study, we shall investigate a signi� ant fragment of the notions of equiv-alen e and preorder from [7℄. These we now pro eed to present for the sake of ompleteness.De�nition 2.1 (Simulation, Ready Simulation and Bisimulation).{ A binary relation R on states is a simulation i� whenever s1 R s2 and ` isa label:- if s1 ! s01, then there is a transition s2 ! s02 su h that s01 R s02.{ A binary relation R on states is a ready simulation i� it is a simulation withthe property that, whenever s1 R s2 and ` is a label:- if s1 9, then s2 9.{ A bisimulation is a symmetri simulation.Two states s and s0 are bisimilar, written s$ s0, i� there is a bisimulation thatrelates them. Hen eforth the relation $ will be referred to as bisimulationequivalen e. We write s ��S s0 (resp. s ��RS s0) i� there is a simulation (resp. aready simulation) R with s R s0. 3

Bisimulation equivalen e [14℄ relates two states in a labelled transition systempre isely when they have the same bran hing stru ture. Simulation (see, e.g.,[14℄) and ready simulation [3℄ relax this requirement to di�erent degrees. Thefollowing notion, whi h is based on a version of de orated tra es, is indu ed byyet another way of abstra ting from the full bran hing stru ture of pro esses.De�nition 2.2 (Readiness Semanti s). For a state s we de�ne:readies(s) �= n(&; X) j & 2 Lab�; X � Lab and 9s0 : s &! s0 and initials�s0� = XoFor states s; s0 we write s ��R s0 i� readies(s) is in luded in readies�s0�.The lassi al notion of language equivalen e for �nite state automata may bereadily de�ned over labelled transition systems. To this end, it is suÆ ientto onsider the states from whi h a X-labelled transition is possible as a eptstates.De�nition 2.3 (Language and Tra e Semanti s).{ We say that a sequen e of labels & is a epted by a state s i� s &X! s0 forsome state s0. For states s; s0 we write s ��L s0 i� every sequen e a eptedby s is also a epted by s0.{ For states s; s0 we write s ��T s0 i� the set of tra es of s is in luded in thatof s0.For � 2 fS;RS; L;R;Tg, the relation ��� is a preorder over states of an arbi-trary labelled transition system; its kernel will be denoted by '�.3 BCCS with Pre�x IterationWe begin by presenting the language of Basi CCS (hen eforth often abbrevi-ated to BCCS) with pre�x iteration [6℄, together with its operational semanti s.3.1 The SyntaxWe assume a non-empty alphabet A t of atomi a tions, with typi al elementsa; b; . The language BCCSp� of Basi CCS with pre�x iteration is given by thefollowing BNF grammar:P ::= 0 j 1 j a:P j P + P j a�P :We shall use P;Q;R; S; T to range over BCCSp�. In writing terms over theabove syntax, we shall always assume that the operator a: binds stronger than+. We shall use the symbol � to stand for synta ti equality of terms. Theexpression P [+Q℄ will be used to denote the fa t thatQ is an optional summand.The size of a term is the number of operators o urring in it.Remark 3.1. The reader might have noti ed that the syntax for the languageBCCSp� presented above in ludes two distinguished onstants, viz. 0 and 1.Intuitively, the term 0 will stand for a deadlo ked pro ess, whereas 1 will standfor a pro ess that an only terminate immediately with su ess. Our hoi e ofnotation is in keeping with a standard one for regular expressions, f., e.g., [5℄.4

3.2 Operational Semanti sLet X be a distinguished symbol not ontained in A t. We shall use X to standfor the a tion performed by a pro ess as it reports its su essful termination.The meta-variable � will range over the set A t[fXg. The operational semanti sfor the language BCCSp� is given by the labelled transition system�BCCSp�;A t [ fXg;� �!j � 2 A t [ fXg��where the transition relations �! are the least binary relations over BCCSp�satisfying the rules in Table 1. Intuitively, a transition P a! Q means that thesystem represented by the term P an perform the a tion a, thereby evolvinginto Q. On the other hand, P X! Q means that P an terminate immediatelywith su ess; the reader will immediately realize that, in that ase, Q � 0.a:P a! P 1 X! 0P �! P 0P + Q �! P 0 Q �! Q0P +Q �! Q0a�P a! a�P P �! P 0a�P �! P 0Table 1. Transition RulesWith the above de�nitions, the language BCCSp� inherits all the notionsof equivalen e and preorder over pro esses de�ned in Se t. 2.2. The followingresult is standard.Proposition 3.2. For � 2 fRS; S; L;R;Tg, the relations ��� and '� are pre-served by the operators in the signature of BCCSp�. The same holds for bisim-ulation equivalen e.4 Equational AxiomatizationsThe study of equational axiomatizations of behavioural equivalen es and pre-orders over BCCSp� was initiated in the paper [6℄. In op. it. it is shown that theaxiom system in Table 2 ompletely axiomatizes bisimulation equivalen e overthe language of 1-free BCCSp� terms. Our aim in the remainder of this studywill be to extend this result to the semanti s in the linear-time/bran hing-timespe trum dis ussed in Se t. 2.2. 5

A1 x+ y = y + xA2 (x+ y) + z = x+ (y + z)A3 x+ x = xA4 x+ 0 = xPA1 a:(a�x) + x = a�xPA2 a�(a�x) = a�xTable 2. The axiom system FFor an axiom system T , we write T ` P � Q i� the inequation P � Qis provable from the axioms in T using the rules of inequational logi . Anequation P = Q will be used as a short-hand for the pair of inequations P � Qand Q � P . Whenever we write an inequation of the form P [+1℄ � Q[+1℄, wemean that if the 1 summand appears on the left-hand side of the inequation,then it also appears on the right-hand side. P =AC Q denotes that P and Q areequal modulo asso iativity and ommutativity of +, i.e., that A1,A2 ` P = Q.For a olle tion of (in)equations X over the signature of BCCSp�, we writeP (X)� Q as a short-hand for A1,A2,X ` P � Q. For I = fi1; : : : ; ing a �niteindex set, we write Pi2I Pi for Pi1 + � � �+ Pin . By onvention, Pi2? Pi standsfor 0.Hen eforth pro ess terms will be onsidered modulo asso iativity and om-mutativity of the +-operation, i.e., modulo axioms A1{2.We begin the te hni al developments by noting that the proof of the om-pleteness of the axiom system F with respe t to bisimulation equivalen e overthe language of 1-free BCCSp� terms applies mutatis mutandis to the whole ofthe language BCCSp�.Proposition 4.1. For every P;Q 2 BCCSp�, P $ Q i� F ` P = Q.The olle tion of possible transitions of ea h pro ess term P is �nite, say fP ai!Pi j i = 1; :::; mg[ nP X! 0 j j = 1; : : : ; no. We all the termexp(P ) �= mXi=1 ai:Pi + nXj=1 1the expansion of P . The terms aiPi and 1 will be referred to as the summandsof P . A straightforward stru tural indu tion on terms, using axiom PA1, yields:Lemma 4.2. Ea h pro ess term is provably equal to its expansion.We aim at identifying a subset of pro ess terms of a spe ial form, whi h willbe onvenient in the proof of the ompleteness results to follow. Followinga long-established tradition in the literature on pro ess theory, we shall referto these terms as normal forms. The set of normal forms we are after is the6

smallest subset of BCCSp� in luding pro ess terms having one of the followingtwo forms: Xi2I ai:Pi[+1℄ or a�(Xi2I ai:Pi[+1℄);where the terms Pi are themselves normal forms, and I is a �nite index set.(Re all that the empty sum represents 0, and the notation [+1℄ stands foroptional in lusion of 1 as a summand.)Lemma 4.3. Ea h term in BCCSp� an be proven equal to a normal form usingequations A3, A4 and PA1.4.1 Ready SimulationWe begin our tour of equational axiomatizations for pre�x iteration by present-ing a omplete axiom system for the ready simulation preorder ( f. Defn. 2.1for the de�nition of this relation). The axiom system ERS onsists of the lawsfor bisimulation equivalen e ( f. Table 2) and of the inequations RS1{2 below:RS1 a:x � a:x+ a:yRS2 a�x � a�(x+ a:y) :Theorem 4.4. For every P;Q 2 BCCSp�, P ��RS Q i� ERS ` P � Q.Proof. We leave it to the reader to he k the soundness of the axiom systemERS , and on entrate on its ompleteness. In view of Lem. 4.3, it is suÆ ientto show that if P ��RS Q holds for normal forms P and Q, then ERS ` P � Q.This we now pro eed to prove by indu tion on the sum of the sizes of P and Q.We pro eed by a ase analysis on the form the normal forms P and Q maytake.{ Case: P =AC Pi2I ai:Pi[+1℄ and Q =AC Pj2J bj :Qj [+1℄.As P ��RS Q, we infer that:1. for every i there exists an index ji su h that ai = bji and Pi ��RS Qji ,2. 1 is a summand of P i� it is a summand of Q, and3. the olle tions of a tions fai j i 2 Ig and fbj j j 2 Jg are equal.The indu tion hypothesis and substitutivity yield that, for every i 2 I ,ERS ` ai:Pi � bji :Qji :Again using substitutivity, we obtain thatERS ` P �Xi bji :Qji [+1℄ :Note now that, for every index j that is not ontained in the set fji j i 2 Ig,there is an index jl (l 2 I) su h that bj = bjl . We an therefore apply axiomRS1 as ne essary to infer thatERS `Xi bji :Qji [+1℄ � Q :The provability of the inequation P � Q from the axiom system ERS nowfollows immediately by transitivity.7

{ Case: P =AC Pi2I ai:Pi[+1℄ and Q =AC b�(Pj2J bj :Qj [+1℄).To deal with this ase, begin by applying PA1 to Q to obtain the equalityQ = b:Q+Xj2J bj:Qj [+1℄ :We an now reason as in the �rst ase of the proof to derive thatP � b:Q+Xj2J bj :Qj [+1℄ :Transitivity now yields the inequation P � Q.{ Case: P =AC a�(Pi ai:Pi[+1℄) and Q =AC Pj bj :Qj [+1℄.Apply PA1 to P , and reason as in the previous ase.{ Case: P =AC a�(Pi ai:Pi[+1℄) and Q =AC b�(Pj bj :Qj [+1℄).As P ��RS Q, we infer that:1. there exists a Q0 su h that Q a! Q0 and P ��RS Q0,2. for every i there exists a Q(i) su h that Q ai! Q(i) and Pi ��RS Q(i),3. 1 is a summand of P i� it is a summand of Q, and4. the olle tions of a tions fai j i 2 Ig [ fag and fbj j j 2 Jg [ fbg areequal.Be ause of the form Q takes, Q0 and every Q(i) is either Q itself or one ofthe Qj 's. Therefore we may apply the indu tive hypothesis to ea h of theinequivalen es Pi ��RS Q(i) and substitutivity to infer thatERS `Xi ai:Pi �Xi ai:Q(i) : (1)We pro eed with the proof by onsidering the following two sub- ases:A. There is an index j su h that a = bj and P ��RS Qj ;B. For no index j with a = bj it holds that P ��RS Qj .We onsider these two ases in turn.A. Assume that there is an index j su h that a = bj and P ��RS Qj . In this ase, we may apply the indu tive hypothesis to derive thatERS ` P � Qj : (2)We an now �nish the proof of the inequation P � Q from the axiomsystem ERS as follows:P (PA1)= a:P +Xi ai:Pi[+1℄(1),(2)� bj :Qj +Xi ai:Q(i)[+1℄(RS1)� bj :Qj +Xi ai:Q(i) + exp(Q)[+1℄(A3),(PA1)= Q : 8

B. Assume that for no index j with a = bj it holds that P ��RS Qj . In this ase, we infer that a = b. We an now reason as follows:P � a�(Xi ai:Pi[+1℄) (1)� a�(Xi ai:Q(i)[+1℄)(RS1),(RS2)� a�(Xi ai:Q(i) + a:Q+Xj bj :Qj [+1℄)(A3),(PA1)� a�Q(PA2)= Q :This ompletes the proof of the theorem.4.2 SimulationThe axiom system ES onsists of the laws for bisimulation equivalen e in Table 2and of the axiom S x � x+ y :Inequation S is well-known to hara terize the simulation preorder over �nitesyn hronization trees. Unlike in the ase of ready simulation, no extra law isneeded to deal with pre�x iteration expli itly.Theorem 4.5. For every P;Q 2 BCCSp�, P ��S Q i� ES ` P � Q.4.3 ReadinessIn this se tion we present a omplete axiom system for pre�x iteration withrespe t to the readiness preorder. The axiom system ER onsists of the olle tionof laws for ready simulation and of those listed below:R1 a:(b:x+ b:y + v) � a:(b:x+ v) + a:(b:y+ w)R2 a:a�(b:x+ b:y + v) � a:a�(b:x+ v) + a:a�(b:y + w)R3 a�(b:x+ b:y + v + a:(b:y + w)) = a�(b:x+ v + a:(b:y + w)) + b:yTheorem 4.6. For every P;Q 2 BCCSp�, P ��R Q i� ER ` P � Q.We fo us on the ompleteness of ER, and leave soundness to the reader. Beforeproving this ompleteness theorem, we introdu e some auxiliary de�nitions andresults.De�nition 4.7. A term P is saturated if for ea h pair of derivations P a! Q b!Q0 and P a! R with b 2 initials(R) we have R b! R0 with Q0 ��R R0.The following lemma stems from [2℄.Lemma 4.8. If P ��R Q and P a! P 0 and Q is saturated, then Q a! Q0 withP 0 ��R Q0. 9

De�nition 4.9. A normal form P is strongly saturated if:1. P is saturated;2. if P =AC Pi2I ai:Pi[+1℄, then the term Pi is strongly saturated, for everyi 2 I .Axioms R1{R3 play a ru ial role in the proof of the following key result.Lemma 4.10. Ea h term is provably equal, by the axioms in ER, to a stronglysaturated normal form, in whi h ea h subterm of the form a�R o urs in the ontext a: .Finally we are in a position to prove Thm. 4.6.Proof. Suppose that P ��R Q; we prove that ER ` P � Q. By Lem. 4.10 itis not hard to see that it suÆ es to establish the laim under the followingassumptions:1. P and Q are normal forms;2. Q is strongly saturated;3. proper subterms of P and Q of the form a�R o ur in the ontext a: ;4. if P =AC a�R and Q =AC b�S, then a = b.(In fa t, a ording to Lem. 4.10, the last two onditions ould be repla ed bythe stronger ondition that all subterms of P and Q of the form a�R o ur inthe ontext a: . However, we shall need the weaker formulation above to be ableto satisfy the indu tion hypothesis.) We derive the desired inequality P � Qfrom ER by indu tion with respe t to the following lexi ographi ordering onpairs of pro ess terms: (P;Q) < (R; S) if- either size(P ) < size(R);- or size(P ) = size(R) and size(Q) < size(S).The next two ases distinguish the possible synta ti forms of P .{ Case 1: P =AC Pi2I ai:Pi[+1℄.Sin e P ��R Q, P ai! Pi and Q is saturated, Lem. 4.8 implies that for ea hi 2 I we have Q ai! Qi for some Qi su h that Pi ��R Qi. A ording toLem. 4.10, ER ` Qi = Ri, with Ri a strongly saturated normal form, inwhi h ea h subterm of the form �S o urs in the ontext : . Moreover,ea h Pi is a normal form, in whi h all proper subterms of the form �S o urin the ontext : , with size(Pi) < size(P ). Hen e, we an apply indu tionto Pi ��R Ri to derive ER ` Pi � Ri. Therefore, for ea h i 2 I ,ER ` ai:Pi � ai:Ri = ai:Qi : (3)By substitutivity, we have thatP =AC Xi2I ai:Pi[+1℄ (3)�Xi2I ai:Qi[+1℄ : (4)10

Sin e P ��R Q implies initials(P ) = initials(Q), it follows that initials(Q)nfXgis equal to fai j i 2 Ig. Furthermore, P ��R Q implies that P has a summand1 if and only if Q X! 0. Hen e,Xi2I ai:Qi[+1℄ (RS1)� exp(Q) (Lem.4.2)= Qwhi h together with equation (4) yields ER ` P � Q.{ Case 2: P =AC a�(Pi2I ai:Pi[+1℄).The next two ases distinguish the possible synta ti forms of Q.{ Case 2.1: Q =AC Pj2J bj:Qj [+1℄.Suppose that P ! P 0. Sin e P ��R Q and Q is saturated, Lem. 4.8 impliesthat there is a j 2 J su h that = bj and P 0 ��R Qj . Both P 0 and Qjare normal forms, and sin e Q is strongly saturated, by Defn. 4.9(2) Qj isstrongly saturated too. Furthermore, if P 0 =AC d�R and Qj =AC e�S, then = d and bj = e, owing to property 3 of P and Q, and so d = = bj = e.Moreover, it is easy to see that property 3 of P and Q implies that the sameproperty holds for P 0 and Qj . Finally, size(P 0) � size(P ) and size(Qj) <size(Q). Hen e, we an apply indu tion to P 0 ��R Qj to derive ER ` P 0 � Qj .Substitutivity now yields ER ` :P 0 � bj:Qj : (5)Hen e, P (Lem.4.2)= exp(P ) (5)� Xj2J0 bj :Qj [+1℄ (6)for some J0 � J . It is easy to see that P ��R Q implies initials(Q) n fXg =initials(P ) n fXg = fbj j j 2 J0g. Moreover, P X! 0 if and only if Q has asummand 1. Hen e,Xj2J0 bj :Qj [+1℄ (RS1)� Xj2J bj :Qj[+1℄ =AC Q :Together with equation (6) this yields ER ` P � Q.{ Case 2.2: Q =AC a�(Pj2J bj :Qj [+1℄).Sin e P ��R Q and P ai! Pi and Q is saturated, Lem. 4.8 implies that forea h i 2 I1. either ai = a and Pi ��R Q,2. or there is a j su h that ai = bj and Pi ��R Qj .Clearly, ea h Pi is a normal form in whi h all proper subterms of the form �S o ur in the ontext : , and with size(Pi) < size(P ).In the �rst ase, applying indu tion to Pi ��R Q, we infer that ER ` Pi � Q.Therefore, by substitutivity,ER ` ai:Pi � a:Q : (7)11

In the se ond ase, Lem. 4.10 implies ER ` Qj = Rj, with Rj a stronglysaturated normal form, in whi h ea h subterm of the form �S o urs in the ontext : . Then by indu tion Pi ��R Rj implies ER ` Pi � Rj . It follows,by substitutivity, that ER ` ai:Pi � ai:Rj = bj :Qj : (8)Hen e, for some J0 � J :P (RS2)� a�(a:Q+Xi2I ai:Pi[+1℄) (8),(7)� a�(a:Q+ Xj2J0 bj :Qj [+1℄) : (9)It is easy to see that P ��R Q implies that initials(Q) n fXg = fbj j j 2J0g [ fag, and that P X! 0 if and only if Q X! 0. Hen ea�(a:Q+ Xj2J0 bj :Qj[+1℄) (RS1)� a�(a:Q+Xj2J bj :Qj[+1℄) (PA1),(PA2)= Q :Together with equation (9) this yields ER ` P � Q.The proof is now omplete.4.4 Tra esThe axiom system ET onsists of the laws for bisimulation equivalen e in Table 2and of T1 a:(x+ y) = a:x+ a:yT2 a�(x+ y) = a�x+ a�yT3 a�(a:x) = a:(a�x) :Axiom T1 is a well-known equation used to hara terize tra e equivalen e over�nite syn hronization trees, and axiom T2 is the adaptation of this equation tothe ase of pre�x iteration. Finally, T3 is, to the best of our knowledge, a newaxiom.Theorem 4.11. For every P;Q 2 BCCSp�,1. P 'T Q i� ET ` P = Q;2. P ��T Q i� ET [ f(S)g ` P � Q.4.5 Language Semanti sThe axiom system EL onsists of the laws for bisimulation equivalen e in Table 2,T1{3 and the equations L1 a:0 = 0L2 a�0 = 0 :Axiom L1 is an adaptation to a tion pre�xing of a well-known equation fromregular algebra, and axiom L2 is the generalization of this equation to the aseof pre�x iteration. 12

Theorem 4.12. For every P;Q 2 BCCSp�,1. P 'L Q i� EL ` P = Q;2. P ��L Q i� EL [ (S) ` P � Q.Proof. We leave it to the reader to he k the soundness of the axiom systemEL [ (S), and on entrate on the ompleteness results.1. Assume that P 'L Q. We shall prove that EL ` P = Q. A simple termrewriting analysis (whi h is omitted here) shows that ea h pro ess term isprovably equal to a term whi h is either 0-free, or of the form 0.Suppose that two terms P and Q are language equivalent. We distinguishtwo ases.- Case 1: P � 0. Then learly also Q � 0, so P � 0 � Q.- Case 2: P is 0-free. Then learly Q is also 0-free. Sin e P and Q are0-free and language equivalent, it is not hard to see that they are alsotra e equivalent. So, a ording to Thm. 4.11, the equation P = Q anbe derived from ET , whi h is in luded in EL.2. Note that, for every P;Q 2 BCCSp�, the following holds:P ��L Q i� P + Q 'L Q :Thus the ompleteness of the axiom system EL [ f(S)g with respe t to ��Tis an immediate onsequen e of the �rst statement of the theorem.5 Further WorkThe ompleteness results presented in this paper deal with a signi� ant fragmentof the notions of semanti s dis ussed in [7℄. To our mind, the most importantomission is a omplete proof system for failures semanti s [4℄ over BCCS withpre�x iteration. We onje ture that a omplete axiomatization for the failurepreorder an be obtained by adding the lawsa:(x+ y) � a:x+ a:(y + z)a:a�(x+ y) � a:a�x+ a:a�(y + z)a:a�x � a�a:(x+ y)a�(x+ y + a:(y + z)) � a�(x+ a:(y + z)) + ya�x � a�(x+ a:y)to those for bisimulation equivalen e ( f. Table 2), and we are urrently workingon the details of su h a proof. The rux of the argument is a proof to thee�e t that the suggested inequations are suÆ ient to onvexly saturate ea hpro ess term, in the sense of [2℄. We have also obtained irredundan y resultsfor the axioms systems for ready simulation, simulation, tra e and languageequivalen e. These will be presented in the full version of this paper, togetherwith a hara terization of the expressive power of BCCS with pre�x iteration.A knowledgements: The resear h reported in this paper originates from aquestion posed by Ro o De Ni ola. We thank the anonymous referees for their omments. 13

Referen es1. L. A eto, W. J. Fokkink, R. J. van Glabbeek, and A. Ing�olfsd�ottir, Axiomatizingpre�x iteration with silent steps, Information and Computation, 127 (1996), pp. 26{40.2. J. Bergstra, J. W. Klop, and E.-R. Olderog, Readies and failures in the algebra of ommuni ating pro esses, SIAM J. Comput., 17 (1988), pp. 1134{1177.3. B. Bloom, S. Istrail, and A. R. Meyer, Bisimulation an't be tra ed, J. Asso . Com-put. Ma h., 42 (1995), pp. 232{268.4. S. Brookes, C. Hoare, and A. Ros oe, A theory of ommuni ating sequential pro- esses, J. Asso . Comput. Ma h., 31 (1984), pp. 560{599.5. J. H. Conway, Regular Algebra and Finite Ma hines, Mathemati s Series (R. Brown andJ. De Wet eds.), Chapman and Hall, London, United Kingdom, 1971.6. W. J. Fokkink, A omplete equational axiomatization for pre�x iteration, Inf. Pro ess.Lett., 52 (1994), pp. 333{337.7. R. J. v. Glabbeek, The linear time { bran hing time spe trum, in Pro eedings CONCUR90, Amsterdam, J. Baeten and J. Klop, eds., vol. 458 of Le ture Notes in ComputerS ien e, Springer-Verlag, 1990, pp. 278{297.8. , A omplete axiomatization for bran hing bisimulation ongruen e of �nite-state behaviours, in Mathemati al Foundations of Computer S ien e 1993, Gdansk,Poland, A. Borzyszkowski and S. Soko lowski, eds., vol. 711 of Le ture Notes in Com-puter S ien e, Springer-Verlag, 1993, pp. 473{484. Available by anonymous ftp fromBoole.stanford.edu.9. S. Kleene, Representation of events in nerve nets and �nite automata, in AutomataStudies, C. Shannon and J. M Carthy, eds., Prin eton University Press, 1956, pp. 3{41.10. H. Lin, An intera tive proof tool for pro ess algebras, in 9th Annual Symposium on The-oreti al Aspe ts of Computer S ien e, vol. 577 of Le ture Notes in Computer S ien e,Ca han, Fran e, 13{15 Feb. 1992, Springer, pp. 617{618.11. R. Milner, A omplete inferen e system for a lass of regular behaviours, J. Comput.System S i., 28 (1984), pp. 439{466.12. , Communi ation and Con urren y, Prenti e-Hall International, Englewood Cli�s,1989.13. , The polyadi �- al ulus: a tutorial, in Pro eedings Marktoberdorf SummerS hool '91, Logi and Algebra of Spe i� ation, NATO ASI Series F94, Springer-Verlag,1993, pp. 203{246.14. D. Park, Con urren y and automata on in�nite sequen es, in 5th GI Conferen e, Karl-sruhe, Germany, P. Deussen, ed., vol. 104 of Le ture Notes in Computer S ien e, Springer-Verlag, 1981, pp. 167{183.14

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