Complete Axiomatization for Divergent-Sensitive Bisimulations in Basic Process Algebra with Prefix Iteration 1

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1 Electronc Notes n Theoretcal Computer Scence 212 (2008) Complete Axomatzaton for Dvergent-Senstve Bsmulatons n Basc Process Algebra wth Prefx Iteraton 1 Taolue Chen 2 CWI, Department of Software Engneerng, PO Box 94079, 1090 GB Amsterdam, The Netherlands Jan Lu 3 State Key Laboratory of Novel Software Technology Nanjng Unversty, Nanjng, P.R.Chna, Abstract We study the dvergent-senstve spectrum of weak bsmulaton equvalences n the settng of process algebra. To represent the nfnte behavor, we consder the prefx teraton extenson of a fragment of Mlner s CCS. The prefx teraton operator s a varant on the bnary verson of the Kleene star operator obtaned by restrctng the frst argument to be an atomc acton and allows us to capture the noton of recurson n a pure algebrac way. We nvestgate four typcal dvergent-senstve weak bsmulaton equvalences, namely dvergent, stable, completed and dvergent stable weak bsmulaton equvalences from an axomatc perspectve. A lattce of dstngushng axoms s developed and thus pure equatonal axomatzatons for these congruences are obtaned. A large part of the current paper s devoted to a consderable complcated proof for completeness. Ths work, to some extent, sheds lght on dstnct semantcs of dvergence. Keywords: Concurrency, Process Algebra, Axomtzaton, Prefx Iteraton, Dvergence, Bsmulaton 1 Introducton Labeled transton systems consttute a wdely used model of concurrent computaton. They model processes by explctly descrbng ther states and ther transtons from state to state, together wth the actons that produce these transtons. Several notons of behavoral semantcs have been proposed, wth the am to dentfy 1 Ths work s partally supported by the Dutch Bsk project BRICKS, the Chnese natonal 863 program (2007AA01Z178), NSFC ( ) and JSNSF (BK ). 2 Emal: chen@cw.nl 3 Emal: lj@nju.edu.cn /$ see front matter 2008 Elsever B.V. All rghts reserved. do: /j.entcs

2 56 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) those states that afford the same observatons. The lack of consensus on what consttutes an approprate noton of observable behavor for reactve systems has led to a large number of proposals for behavoral semantcs for concurrent processes. In lght of ths, one of the man tasks of concurrency theory s to provde a unform classfcaton of system (or process) behavor. Ths forms the study of comparatve concurrency semantcs, whch has, n a large sense, cumulated n van Glabbeek s celebrated lnear tme-branchng tme spectrum [6][7]. Techncally speakng, hs spectrum has two parts, one deals wth concrete processes (wthout slent moves) [6] whle the other explores processes wth abstracton (slent moves) [7]. The current paper focuses on the latter. Bascally, the spectrum n [7] enjoys two-dmenson structure: the vertcal dmenson spans between trace equvalence and branchng bsmulaton, whch dentfes dfferent ways to dscrmnate processes accordng to ther lnear or branchng structure. Due to the presence of slent moves, ths spectrum s spread n horzontal dmenson as well, determned by the semantcs of dvergence. In a nutshell, a process s dvergent f the assocated labeled transton system exhbts nfnte slent move loops. A dstnct characterstc of tradtonal weak bsmulaton [12] les n that t s possble for two processes, exactly one of whch s dvergent, to be bsmulaton equvalent. However, ths ncurs some crtcsm snce t may be useful and approprate n crcumstances not to dsregard the presence of such slent move loops. In lght of ths, n the fragment spannng from weak bsmulaton to branchng bsmulaton, several dfferent crtera to dstngush dvergence are proposed, whch nduces a horzontal lattce, and ths lattce appears for all the bsmulaton relatons. To gve further nsght nto the dentfcatons made by the respectve behavoral equvalences n the spectrum, one of the classcal research topcs n concurrency theory s to study them n the settng of the process algebra, whch often provdes an elegant way to compare semantcs, by presentng dstngushng axoms that capture the essence of the semantcs. For nstance, van Glabbeek [6] studed hs spectrum axomatcally n the settng of process algebra BCCSP, whch contans only the basc process algebrac operators from CCS and CSP. For the vertcal dmenson of the spectrum, the dstngushng axoms are well-known (see, e.g. [12][8]). However, t s far to say that the horzontal dmenson receves relatvely less attenton. We beleve that ths s manly due to the fact that dvergence only makes sense n the presence of recurson, whle recurson s often hard to tackle axomatcally. Wth ths n mnd, the current paper ams at nvestgatng the dvergent-senstve spectrum of weak bsmulaton equvalences from an axomatc perspectve and thus enrches the research on ths aspect to some extent. Note that for ths purpose, BCCSP does not suffce any more, snce t lacks the slent moves, whch prevents us from studyng weak semantcs and lacks operators expressng nfnte behavor, whch mpedes the characterzaton of dvergence. In lght of ths, we consder the teraton-lke extenson of some basc process languages whch paves the way to brng an axomatc treatment nto the vertcal dmenson of the spectrum. Generally, Kleene [10] defned a bnary operator -*- n the context of fnte automata, called Kleene star or teraton. Intutvely, the expresson P Q

3 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) can choose to execute ether P, after whch t evolves nto P Q agan, or Q, after whch t termnates. An advantage of the Kleene star s that on the one hand t can express recurson n a very natural way, but that on the other hand one can capture ths operator by equatonal laws, thus can take an pure algebrac theory nto account (see [2] for thorough dscussons). Hence, one does not need meta-prncples, such as Unque Fxpont Inducton Prncple [11] (a.k.a. Recursve Specfcaton Prncple). Unfortunately general Kleene star operator s so powerful that t often obstructs a complete fnte equatonal axomatzaton even under very mld assumptons, c.f. [13]. In lght of ths, one way to obtan an equatonal axomatzaton s to restrct the range of the terms that mght occur at the left-hand sde of the bnary Kleene star. We consder the prefx teraton, ntroduced by Fokknk [4], a varant on the bnary verson of the Kleene star operator obtaned by restrctng the frst argument of P Q to be an atomc acton. We thnk the prefx teraton provdes us a sutable context to explore the dvergent-senstve bsmulaton equvalences snce t s a tradeoff between the expressve power of the underlyng language and the axomatzablty of nterestng bsmulaton congruences. For more works on prefx teraton and Kleene star n general, we refer the readers to [4][1][2] and the references theren. Ths paper develops a lattce of dstngushng axoms to characterze four typcal semantcs of dvergence n the horzontal dmenson of weak bsmulaton equvalences, namely, dvergent, stable, completed and dvergent stable weak bsmulatons. A large part of the paper s devoted to a consderable complcated proof for completeness w.r.t. a fragment of CCS wth prefx teraton. Ths work helps us wth n-depth understandng on dstnct semantcs of dvergence. The rest of paper s set up as follows: Secton 2 presents basc notons regardng the language of basc CCS wth prefx teraton and ts operatonal semantcs. Secton 3 provdes the defnton of four dvergent-senstve bsmulaton equvalences. The axomatzaton and the completeness proof are gven n Secton 4. The paper s concluded n Secton 5, where related works are also dscussed. 2 Prelmnares We assume a non-empty, countable set A of observable actons not contanng the dstngushed symbol. Followng Mlner, the symbol wll be used to denote an nternal, unobservable acton (slent move) of a system. We wrte A for A {}, and use a, b to range over A and, β to range over A. Note that we follow ths conventon strctly, so for any a A, a. We also assume a countably nfnte set of process varables V, ranged over by x, y, z, that s dsjont from A. The language of basc CCS wth prefx teraton, denoted by BCCS p (A ), s gven by the followng BNF grammar: P ::= x 0.P P + P P where x V and A. The set of closed terms,.e. terms that do not contan occurrences of process varables, generated by the above grammar wll be denoted by T(BCCS p (A )). We shall use P, Q (possbly subscrpted and/or superscrpted)

4 58 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) P P P P P + Q P Q Q P + Q Q P P P P β P P Fg. 1. Operatonal Semantcs. to range over T(BCCS p (A )). The operatonal semantcs for the language BCCS p (A ) s gven by the labeled transton system < T(BCCS p (A )), { A } >, where satsfes the rules n Fg. 1. Followng Mlner [12], the derved transton relaton s defned as the reflexve, transtve closure of as follows: def ˆ def = =, and f = o.w. ( ), and, ˆ are defned n the standard way We also wrte P f P can not ssue a transton; and P f P does not have any transton (a.k.a. deadlock). Defnton 2.1 [Weak Bsmulaton] A bnary relaton R over T(BCCS p (A )) s a weak bsmulaton f t s symmetrc and whenever P RQ and P P, then Q exsts s.t. Q ˆ Q and P RQ. Two process terms P, Q are observaton equvalent, denoted by P Q, f there exsts a bsmulaton R s.t. P RQ. 3 Dvergent-Senstve Weak Bsmulatons To lft the standard defnton of weak bsmulaton (Defnton 2.1) to dvergentsenstve varants, we frst defne a few propertes that a term may or may not have. We shall use a suffx notaton for these propertes. Namely, for any term P and a property of P, we wrte P. We shall defne the propertes, σ, and respectvely, as follows. Note that n the remander of ths paper, ranges over {,σ,, } unless stated otherwse. Defnton 3.1 For any term P, P (P s dvergent) f there exst P for N such that P P 1 P2 ; Pσ (P s stable) ff P ; P (P s complete) ff P ; and P (P s dvergent stable) f P or there exsts some P such that P P wth P. We are now n a poston to ntroduce the dvergent-senstve bsmulaton equvalences studed n ths paper.

5 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) Defnton 3.2 [Dvergent-Senstve Weak Bsmulaton] We say that a symmetrc relaton R preserves a property, f f (P, Q) Rand P, then Q.Q Q A weak bsmulaton R s a -weak bsmulaton f t preserves. As usually, two process terms P, Q are -observaton equvalent, denoted by P Q, f there exsts a -bsmulaton R s.t. P R Q, where {,σ,, }. (resp. σ,, ) s called a dvergent (resp. stable, completed, dvergent stable)-observaton equvalent. The followng theorem states the relatonshp among the four dvergent-senstve bsmulaton equvalences. Theorem 3.3 σ and. As s well-known [12], s an equvalence relaton. However, t s not a congruence w.r.t. summaton operator or prefx teraton operator. The same apples to. A standard soluton s to ntroduce the so called rooted condtons, thusthe coarsest congruence whch s contaned n ( ) can be obtaned, respectvely. Defnton 3.4 [Observaton Congruence] For all process terms P, Q T(BCCS p (A )), P Q ff for any A, If P P, then Q exsts s.t. Q Q, s.t. P Q. If Q Q, then P exsts s.t. P P, s.t. P Q. In the same ven, we defne P Q ff for any A, If P P, then Q exsts s.t. Q Q, s.t. P Q. If Q Q, then P exsts s.t. P P, s.t. P Q. 4 Axomatzaton and Completeness One of man ams of the current paper s to gve complete equatonal axomatzatons for dvergent-senstve observaton congruences ntroduced n prevous secton. We frst present a lattce of axomatc systems. The axomatc system F s the one that s shown n [4] to characterze strong bsmulaton, whch s gven n Fg. 2. In addton, [1] extends t wth two of Mlner s standard -laws and three auxlary equatons that descrbe the nterplay between the slent nature of and prefx teratons, part of whch s reported n Fg. 3. Moreover, we provde new axoms for dfferent dvergent-senstve observaton congruences, whch are reported n Fg. 4. For {,σ, }, we let E denote the system F and those axoms n Fg. 3 plus axom from Fg. 4. For =, E conssts of E plus axom from Fg. 4. For an axomatc system T, we wrte T P = Q ff the equaton P = Q s provable from the axomatc system T usng the rules of standard equatonal logc. Often for convenence, we omt T and abbrevate t as P = Q. We wrte P = X Q

6 60 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) A1 x + y = y + x A2 (x + y)+z = x +(y + z) A3 x + x = x A4 x + 0 = x PA1.( x)+x = x PA2 ( x) = x Fg. 2. The Axomatc System F. T1..x =.x T2.x =.x + x T3.(x +.y) =.(x +.y)+.y PT1.( x) = (.( x)) PT2 (x +.y) = (x +.y +.y) Fg. 3. Laws for Observaton Congruence. ( x + y) =.( x + y) σ (.x + y) =.(.x + y) (.x + y) =.(x + y) (0) =.0 Fg. 4. Dstngushng laws. as a short-hand for A1, A2,X P = Q. We use P = AC Q to denote that P and Q are equal modulo assocatvty and mutatvty of +,.e. A1, A2 P = Q. Some remarks are n order: In [1], among others, the tradtonal weak bsmulaton [12] s consdered where along wth the axoms n Fg. 3, an mportant axom x =.x s also ntroduced. Ths s an equatonal formulaton of Koomen s Far Abstracton Rule, representng Mlner s far settng, where dvergence s never dstngushed. However, t does not hold n our settng, whch s the tenet of dvergent-senstve bsmulatons. Ths has a far-reachng effect on the axomatzaton, snce we have to provde dstngushng axoms, σ, and to make up the lose of t. And as we wll see later, ths complcates the completeness proof n a large extent. The soundness of all these axomatc systems s obvous. The remander of the secton s devoted to the proof of completeness.

7 4.1 Completeness T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) Lemma 4.1 If P n n Pn+1 for n =0, 1,, then there exsts some N such that n = N for n>n. Proof. By structure nducton on terms. Remark 4.2 Ths lemma crucally depends on the smplcty of prefx teraton. It fals n presence of some more complcated constructon, say, flat teraton, where (a + ) P s allowed. The followng lemma s essentally [1, Lemma 2.11], whch s for tradtonal weak bsmulaton, snce for any {,σ,, }. Lemma 4.3 Let a, b A. Ifa P b Q, where {,σ,, }, then a = b. Lemma 4.4 The followng propertes hold: () If P a Q, then P. () If P σ a Q, then P. () If P a Q, then P. Proof. [Sketch] Frst, we note that accordng to Lemma 4.1, for any term P and a A, t can not be the case that P a a Namely, P can not exhbts an nfnte sequence where a and alternate. () Snce P and P a Q, a Q and thus Q. Namely, there exsts Q = Q 0 Q1. Snce P a Q, there exsts P P 1 such that for each, Q P. Assume, towards a contradcton, that P. Then t must be the case that for any 1, P = P. Namely, Q P a Q for each. It follows easly that there exsts some N, such that for any n>n, R = Q n = Q n+1 = for some R. Namely, Q R wth R P a Q. Furthermore, snce a Q a a Q a a Q and R a Q, t must be the case that R a a R 1 R2 such that for any 1, a Q R. Then there exsts some N such that for any n>n, a S = R n = R n+1 =. Hence a S a Q P, and Q R a a S. Applyng the aforementoned argument on P a S, we can construct a sequence of transtons from Q such that Q a a, whch leads to a contradcton. () Snce P, a Q, and thus Q. Namely, there exsts Q = Q 0 Q1. Snce P σ a Q, there exsts P P 1 such that Q 1 σ P 1. Assume, towards a contradcton, that P. Then t must be the case that P 1 = P. Namely, Q 1 σ P σ a Q. Repeat the same argument, we can buld a sequence of transtons such that Q 0 Q1 Q2 and for each, Q σ P σ a Q. Now we can reuse the smlar argument as n the prevous case and the concluson follows.

8 62 () Smlar to Case (). T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) Lemma 4.5 (Absorpton Lemma) For any P, Q T(BCCS p (A )), fp P, then E P = P +.P for {,σ,, }. Proof. By nducton on the length of the dervaton P P. The detals are standard. Followng a long-establshed tradton of the lterature on process theory, we frst dentfy a subset of process terms of a specal form, namely the normal forms (nfs n short). Defnton 4.6 The defntons of -nf, σ-nf, -nf and -nf are n order: () The set of -nfs s the smallest subset ncludng process terms havng one of the followng two forms: I.P, where the term P are themselves -nfs; or ( I.P ), where the term P are themselves -nfs and f =, P for each I. () The set of σ-nfs s the smallest subset ncludng process terms havng one of the followng two forms: I.P, where the term P are themselves σ-nfs; or ( I.P ), where the term P are themselves σ-nfs and =, for each I,. () The set of -nfs s the smallest subset ncludng process terms havng one of the followng three forms: I.P, where the term P are themselves -nfs; or a ( I.P ) for some a A where the term P are themselves -nfs; or 0. (v) The set of -nfs s the smallest subset ncludng process terms havng one of the followng two forms: I.P, where the term P are themselves -nfs; or ( I.P ), where the term P are themselves -nfs and f =, P 0 or P for each I. Note that below I,J are often used to denote fnte ndex sets and recall that the empty sum represents 0. Lemma 4.7 Each term can be proven equal to a -nf usng axoms n E. Proof. [Sketch] By nducton on the structure of the term. We only present some nterestng case. ( ) The only nterestng case s P = P. Then by nducton we can rewrte P as a -nf, whch s, stll, denoted by P. If P, then we are done; Otherwse P, then there exsts some P P1 P2. Clearly, t must be the case

9 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) that for some n, P n s of the form P n = S, wth P S. By Lemma 4.5, E P = P +. S. It follows that E P = (P +. S) = (P +. S + S) =.(P +. S + S) =.(P +. S) =.P Note that.p s a -nf, snce P s a -nf. We are done. (σ) The only nterestng case s P = P. Then by nducton we can rewrte P as a -nf, whch s, stll, denoted by P.IfP, then we are done; Otherwse P, then t must be the case that E σ P = P +.S. E σ P = (P +.S) σ =.(P +.S)=.P ( ) The only nterestng case s P = P. Then by nducton we can rewrte P as a -nf, whch s, stll, denoted by P.IfP a for any a A, then we are done; Or P a S for some a, then by Lemma 4.5, E P = P + a.s. It follows that E P = (P + a.s) =.(P + a.s) =.P ( ) The only nterestng case s P = P. Then by nducton we can rewrte P as a -nf, whch s, stll, denoted by P. If P, then we are done; Otherwse, ether P P wth P or P. We dstngush these two cases: P. Then followng the argument n case ( ), we can obtan that E P =.P, whch s what we desre. P P wth P. Clearly, E P = 0. By Lemma 4.5, E P = P +.P = P +.0. It follows that E P = (P +.0) =.(P + 0) =.(P +.0) =.(P +.0) =.P In the proof of the completeness result to come, we shall make use of a weght functon w : T(BCCS p (A )) N. Ths s defned by structure nducton on terms as follows: w(0) = 0 w(.p ) = w(p )+1 w(p + Q) = w(p )+w(q)+1 w( P ) = w(p )+1 Proposton 4.8 For all P, Q T(BCCS p (A )), fp Q, then E.P =.Q. Proof. Frst note we shall separate the proof for each when t s necessary. And P = Q denotes genercally E P = Q. By Lemma 4.7, t s suffcent to assume that P and Q are -observaton equvalent normal forms and we show that

10 64 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) P =.Q by nducton on w(p )+w(q). Recall that -nfs can take the followng two forms:.p or (.P ) I I where the P are themselves -nfs. So, n partcular, P and Q have one of these forms. By symmetry, t s suffcent to deal wth the followng three cases: () P = I.P and Q = j J β j.q j. () P = ( I.P ) and Q = β ( j J β j.q j ). () P = I.P and Q = ( j J β j.q j ). We treat these three cases separately. () CASE: P = I.P and Q = j J β j.q j. Consder a summand.p of P, t gves rse to a transton P P, and hence snce P Q, we can dstngush two subcases n the proof:. Then we have Q Q and P Q. Snce both P and Q are n normal form and w(p )+w(q ) <w(p )+w(q), by nducton,.p =.Q. It follows that Q = Q +.Q (Lemma 4.5) T1 = Q +..Q = Q +..P T1 = Q +.P Thus we can obtan that Q = Q +.P. =. Then ether Q Q or Q = Q, n both cases, P Q. By nducton,.p =.Q. For the frst case, t can be easly shown that Q = Q +.P. In the second case one has.p =.Q. In summary, for I, we have ether.q =.P or Q = Q+.P. It follows that Q+P = Q+ { }.P + { =}.P = Q+ k K.Q for some ndex set K = {k k = and P Q and k I}. Consequently we have.(p + Q) =.(Q + T2,A3 k K.Q) =.Q. Symmetrcally.(P + Q) =.P, therefore.p =.Q. () CASE:P = AC ( I.P ) and Q = AC β ( j J β j.q j ). We have the followng subcases:, β A. Then by Lemma 4.3, t must be the case that a = b. For convenence, we wrte P 1 = I.P and Q 1 = j J β j.q j, then P = a P 1 and Q = a Q 1. For each transton P P, snce P Q, we can dstngush three cases n the proof: P 1 P and a,. Then t must be Q 1 Q and P Q. By nducton,.p =.Q. Followng the same lnes as n CASE (1), we have Q 1 = Q 1 +.P. Thus we can obtan that Q 1 = Q 1 + {,a}.p. P 1 a P. Then there are three subcases:

11 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) a () Q 1 Q and P Q. Then by nducton,.p =.Q, t follows that Q 1 = Q 1 + a.p ; () Q a Q, Q 1 Q and P Q. Then by nducton,.p =.Q. Moreover, we have Q 1 = Q 1 +.P. Thus we can show that Q 1 = Q 1 + k K 1.P k for some ndex set K 1 {k k = a and k I}; () Q a Q and P Q. Then by nducton,.p =.Q, thus a.p = a..p = a..q = a.q. Therefore, from ()()(), we can conclude that Q 1 + k K 1 K 2 a.p k = Q 1 + { =a} a.p for the ndex set K 1 and some ndex set K 2, where K 1,K 2 {k k = a and k I} and K 1 K 2 =. Moreover, Q 1 = Q 1 + k K 1.P k and.p k =.Q for k K 2. Note that n the sequel, we wrte K = K 1 K 2. P 1 P. Then ether Q 1 Q or Q = Q (note that a ). In both cases, P Q. By nducton,.p =.Q. For the frst case, t can be easly shown that Q 1 = Q 1 +.P. In the second case one has.p =.Q. Thus we have Q 1 + l L.Q = Q 1 + { =}.P for some ndex set L = {l l = and P l.q and l I}. Combnng the above three cases, actually we have Q 1 + P 1 = Q 1 + l L.Q + k K 1 K 2 a.p k. Moreover Q 1 = Q 1 +.P k (1) k K 1.P k =.Q for k K 2 (2) It follows that a (P 1 + Q 1 ) = a (Q 1 +.Q + a.p k ) l L k K 1 K 2 (1) = a (Q 1 +.Q + a.p k + a.p k +.P k ) l L k K 1 k K 2 k K 1 PT3 = a (Q 1 +.Q +.P k + a.p k ) l L k K 1 k K 2 (1) = a (Q 1 +.Q + a.p k ) l L k K 2 Now, we have to dstngush two cases: If K 2, then we have a (P 1 + Q 1 ) = a (Q 1 + l L.Q + (2) = a (Q 1 +.Q + a.q) l L k K 2 a.p k ) PA1 = a (Q +.Q) l L We proceed by consderng the followng two subcases:

12 66 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) L =. Then a (P 1 + Q 1 )=a Q PB1 = Q. L. Then t follows that a (P 1 + Q 1 ) = a (Q + l L.Q) T2 = a (.Q) = a (.a Q 1 ) PT2 =.Q If K 2, then actually a (P 1 + Q 1 )=a (Q 1 + l L.Q). Smlarly, we also proceed by consderng the followng two subcases: L =. Then a (P 1 + Q 1 )=a Q 1 = Q. L. Then t follows that a (P 1 + Q 1 ) = a (Q 1 + l L.Q) A3,T2 = a (Q 1 +.Q + a Q 1 ) PA1 = a (Q 1 +.Q + a.a Q 1 + Q 1 ) A3,PA1 = a (.Q + Q) T2 = a (.a Q 1 ) PT2 =.Q From above, we can conclude that ether.q = a (P 1 + Q 1 )or Q = a (P 1 + Q 1 ). Symmetrcally.P = a (P 1 + Q 1 )orp = a (P 1 + Q 1 ). For each of four combnatons, clearly, we have.p =.Q, possbly usng T1. = and β A. The proof s parametrc by, σ, and. ( ) Snce P and Q are n -nfs, for each I,.P. Accordng to Lemma 4.4(), ths case s actually vacuous. (σ) Snce P and Q are n σ-nfs, for each I,. Accordng to Lemma 4.4(), ths case s actually vacuous. ( ) Snce P and Q are n -nfs, for each a A, P. However, snce β Q β β Q where β A, P β P for some P wth P β Q. Ths s clearly a contracton, and thus ths case s actually vacuous. ( ) Snce P and Q are n -nfs, ( I.P ). Accordng to Lemma 4.4(), ths case s actually vacuous. = β =. The proof s parametrc by, σ, and. For brevty, let P 1 denote I.P and Q 1 denote j J β j.q j. ( ) Snce P and Q are n -nfs, P 1 and Q 1. For each I, P 1 P, clearly, t must be the case that Q 1 Q 1 wth P Q 1. Snce w(p + Q 1 ) < w(p )+w(q). By nducton, E.P =.Q 1. By Lemma 4.5, E Q 1 = Q 1 +.Q 1. It follows that E Q 1 = Q 1 +..Q 1 = Q 1 +..P = a

13 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) Q 1 +.P. Hence E Q 1 = Q 1 + P 1. By symmetry, E P 1 = P 1 + Q 1, and thus we obtan E P 1 = Q 1. It follows that E.P =.Q. (σ) Snce P and Q are n σ-nfs, P 1 and Q 1. For each I, P 1 P, clearly, t must be the case that Q 1 Q 1 wth P Q 1. By the same argument as n the prevous case, we can obtan that E σ.p =.Q. ( ) Snce P and Q are n -nfs, t must be the case that P 1 = Q 1 = 0. The concluson follows trvally. ( ) Ths case s smlar to the case ( ) and we can obtan, agan that E.P =.Q. () CASE: P = AC I.P and Q = AC β ( j J β j.q j ). For convenence, we denote Q 1 = j j β j.q j. We have the followng two cases: β. We dscuss accordng to the two drectons of bsmulaton. For each transton from P, we consder three subcases n the proof: P P and,a. Snce P Q, ths transton from P must be matched by a transton Q 1 Q and P Q. P a P. Snce P Q, ths transton from P must be matched by transtons Q a Q, Q 1 Q and P Q,orQ a Q and P Q,or a Q 1 Q and P Q. P P. Snce P Q, ths transton from P must be matched by transtons Q 1 Q or just Q = Q, n both cases, P Q. Followng the same lnes of CASE (2), we can conclude that ether.q = a (P + Q 1 )or Q = a (P + Q 1 ). For the transton from Q, we only need consder the followng two stuatons: β j Q 1 Q j, snce P Q, t must be matched by a transton P ˆβ j P and P Q j. Followng the same lne of CASE (1), we obtan that.(p + Q 1 )=.P. Q a Q, snce P Q, t must be matched by a transton P a P and P Q. By nducton, we have.p =.Q. Clearly, t follows that P = P + a.q. Then because P = P + a.q = P + a.a (P + Q 1 )or P = P + a.q = P +a..q = P +a.a (P +Q 1 ), we have P +Q 1 = P +Q 1 +a.a (P +Q 1 ) PA1 = a (P + Q 1 ). It follows that.q =.(P + Q 1 )=.P. β =. The proof s parametrc by, σ, and. ( ) Then for each j J, β j.q j. And snce Q, there exsts some I, such that.p. Let K = { I.P }. Then K. For each K, clearly =, and snce P P, P and for each j J, β j.q j, t must be the case that Q Q such that P Q. By nducton,.p =.Q. It follows that E P = / K.P +.Q

14 68 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) Moreover, for / K, P P wth P. Snce P Q,tmustbethe case that Q 1 Q 4 1 wth P Q 1. By nducton, E.P =.Q 1. By Lemma 4.5, E Q 1 = Q 1 +.Q 1 = Q 1 +..Q 1 = Q 1 +..P = Q 1 +.P. It follows that E Q 1 = Q 1 + / K.P Hence we obtan E.Q =.Q + Q 1 = Q + Q 1 + / K.P =.Q + / K.P = P (σ) Snce Q s n σ-nf, for each j J, β j. Snce Q and P σ Q, there exsts some I, =, for whch, t must be the case that Q Q wth P σ Q. By nducton E σ.p =.Q. It follows that E σ P =.P +.Q For I wth, t must be the case that Q 1 Q 1 wth P σ Q 1. By Lemma 4.5, Q 1 = Q 1 +.Q 1 = Q 1 +..Q 1 = Q 1 +..P = Q 1 +.P. It follows that E σ Q 1 = Q 1 +.P Hence we obtan E σ.q =.Q + Q 1 =.Q + Q 1 +.P =.Q + / K.P = P ( ) Snce Q s n -nf, t must be the case that Q 1 = 0 and thus for each I, =. Moreover, P 0 for each I. Clearly E.P =.Q. It follows that E P =.Q and thus E.P =.Q. ( ) The proof s smlar to the case ( ). The proof s complete. Theorem 4.9 (Completeness) If P Q, then P = Q. Proof. Consder two process terms P and Q that are observaton congruent. We shall show that P + Q s provably equal to Q. Of course, by symmetry, P + Q s also provably equal to P. Hence we obtan completeness. To ths end, note that by Lemma 4.7, P and Q may be proven equal to some normal forms usng A4, PA1 and PA2. Possbly usng equaton PA1 agan, we may 4 Note that here Q 1 can not be Q 1 tself, so the transton can not be empty.

15 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) therefore derve that P = I.P and Q = j J β j.q j for some fnte ndex sets I,J. Consder a summand.p of P. It gves rse to a transton P P and hence, snce P Q, there exsts some Q such that Q Q and P = Q. By Proposton 4.8, we have.p =.Q. It follows that Q = Q +.Q (Lemma 4.5) T1 = Q +..Q = Q +..P T1 = Q +.P Consequently, we have Q = Q+ I.P = Q+P. By symmetry, P = P +Q. Thus P = P + Q = Q. The proof s complete. 5 Concluson In ths paper, we have studed the equatonal theory of dvergent-senstve spectrum of weak bsmulaton equvalences. We work n the settng of a fragment of Mlner s CCS extended by prefx teraton, whch allows us to express nfnte behavor n a pure algebrac way. We developed pure equatonal axomatzaton for four typcal dvergent-senstve observaton congruences, namely, the dvergent, stable, completed and dvergent stable observaton congruences. A lattce of dstngushng axoms are presented and the completeness of these axomatzatons s shown. Ths work deepens our understandng on dstnct semantcs of dvergence largely. Related Works. [9] carred out smlar research as n the current paper. Instead, the authors consdered the general recursve (fxpont) operator n [11] to express the nfnte behavor. Admttedly, ths mechansm s more general than the prefx teraton operator. However, the prce to be pad s that the axomatzaton can not be gven n an algebrac way, snce the unque fxpont nducton has to be exploted. In our work, t suffces only to resort to pure equatonal reasonng. In our pont of vew, our work s complementary to the work presented n [9]. As for the proof technques, [9] manly adapted the standard proof by Mlner [11], whch s completely dfferent from ours. Another closely related work s [1], whch deals at once wth weak, branchng, delay and η-bsmulaton. In a nutshell, the authors worked wthn the scope of vertcal dmenson whle we deal wth horzontal dmenson. Note that n the current paper, although we restrct ourselves to weak bsmulaton and ground-completeness, these are not real shortcomngs: We clam that the results reported n the current paper can be adapted to branchng (also to delay and η) bsmulatons smoothly and by the same way as [1], the ω-completeness can also be obtaned. Moreover, the proof technque exploted n the current paper, actually provdes an alternatve proof for the completeness of axomatzaton n [1] for tradtonal weak bsmulaton. It has ndependently nterests snce the prevous proof n [1] s only acheved by reducton to branchng bsmulaton whch s ndrect. For more detals, see [3]. An mmedate future work ncludes extendng the current work to Kleene star

16 70 T. Chen, J. Lu / Electronc Notes n Theoretcal Computer Scence 212 (2008) operator (note that some care s necessary to crcumvent the mpossblty result n [13]). Ths s very challengng though, snce we even do not know whether such an axomatzaton exsts, let along that even for strong case (wthout dvergence), the completeness proof s staggerng enough. Acknowledgement We are grateful to Tngtng Han who read a draft of ths paper carefully and provded valuable comments. References [1] L. Aceto, R. van Glabbeek, W. Fokknk and A. Ingólfsdóttr. Axomatzng prefx teraton wth slent steps. Informaton and Computaton, 127(1):26 40, [2] J. Bergstra, W. Fokknk and A. Ponse. Process algebra wth recursve operatons, n Handbook of Process Algebra, pp , Elsever, [3] T. Chen, T. Han, J. Lu. On the complete axomatzaton for prefx teraton modulo observaton congruence. Acta Cybernetca, 17(3): , [4] W. Fokknk. A complete equatonal axomatzaton for prefx teraton. Informaton Processng Letter, 52(6): , [5] W. Fokknk. On the completeness of the equatons for the Kleene star n bsmulaton. In Proc. AMAST 96, LNCS 1101, pp , Sprnger, [6] R. van Glabbeek. The lnear tme branchng tme spectrum I. The semantcs of concrete, sequental processes. In Handbook of Process Algebra, pp. 3 99, Elsever, [7] R. van Glabbeek. The lnear tme branchng tme spectrum II. The semantcs of sequental systems wth slent moves. In Proc. CONCUR 93, LNCS 715, pp , Sprnger, [8] R. van Glabbeek and W. P. Wejland. Branchng tme and abstracton n bsmulaton semantcs. Journal of ACM, 43(3): , [9] M. Lohrey, P. D Argeno and H. Hermanns. Axomatsng dvergence. Informaton and Computaton, 203(2): , [10] S. Kleene. Representaton of events n nerve nets and fnte automata. In Automata studes, pages Prnceton Unversty Press, [11] R. Mlner. A complete nference system for a class of regular behavours. Journal of Computer System and Scence. 28(3): , [12] R. Mlner. Communcaton and Concurrency. Prentce Hall, [13] P. Sewell. Nonaxomatsablty of equvalences over fnte state processes. Annual of Pure and Appled Logc, 90(1-3): , 1997.