Expressiveness modulo Bisimilrity of Regulr Expressions with Prllel Composition (Extended Abstrct) Jos C. M. Beten Eindhoven University of Technology, The Netherlnds j.c.m.beten@tue.nl Tim Muller University of Luxembourg, Luxembourg tim.muller@uni.lu Bs Luttik Eindhoven University of Technology, The Netherlnds Vrije Universiteit Amsterdm, The Netherlnds s.p.luttik@tue.nl Pul vn Tilburg Eindhoven University of Technology, The Netherlnds p.j..v.tilburg@tue.nl The lnguges ccepted by finite utomt re precisely the lnguges denoted by regulr expressions. In contrst, finite utomt my exhibit behviours tht cnnot be described by regulr expressions up to bisimilrity. In this pper, we consider extensions of the theory of regulr expressions with vrious forms of prllel composition nd study the effect on expressiveness. First we prove tht dding pure interleving to the theory of regulr expressions strictly increses its expressiveness modulo bisimilrity. Then, we prove tht replcing the opertion for pure interleving by ACP-style prllel composition gives further increse in expressiveness. Finlly, we prove tht the theory of regulr expressions with ACP-style prllel composition nd encpsultion is expressive enough to express ll finite utomt modulo bisimilrity. Our results extend the expressiveness results obtined by Bergstr, Bethke nd Ponse for process lgebrs with (the binry vrint of) Kleene s str opertion. 1 Introduction A well-known theorem by Kleene sttes tht the lnguges ccepted by finite utomt re precisely the lnguges denoted by regulr expression (see, e.g., [8]). Milner, in [10], showed how regulr expressions cn be used to describe behviour by defining n interprettion of regulr expressions directly s finite utomt. He then observed tht the process-theoretic counterprt of Kleene s theorem stting tht every finite utomton is described by regulr expression fils: there exist finite utomt whose behviours cnnot fithfully, i.e., up to bisimilrity, be described by regulr expressions. Beten, Corrdini nd Grbmyer [1] recently found structurl property on finite utomt tht chrcterises those tht re denoted with regulr expression modulo bisimilrity. In this pper, we study to wht extent the expressiveness of regulr expressions increses when vrious forms of prllel composition re dded. Our first contribution, in Section 3, is to show tht dding n opertion for pure interleving to regulr expressions strictly increses their expressiveness modulo bisimilrity. A crucil step in our proof consists of chrcterising the strongly connected components in finite utomt denoted by regulr expressions. The chrcteristion llows us to prove property pertining to the exit trnsitions from such strongly connected components. If interleving is dded, then it is possible to denote finite utomt violting this property. Our second contribution, in Section 4, is to show tht replcing the opertion for pure interleving by ACP-style prllel composition [5], which implements form of synchronistion by communiction S. Fröschle, F.D. Vlenci (Eds.): Workshop on Expressiveness in Concurrency 2010 (EXPRESS 10). EPTCS 41, 2010, pp. 1 15, doi:10.4204/eptcs.41.1
2 Regulr Expressions with Prllel Composition modulo Bisimilrity between components, leds to further increse in expressiveness. To this end, we first chrcterise the strongly connected components in finite utomt denoted by regulr expressions with interleving, nd deduce property on the exit trnsitions from such strongly connected components. Then, we present n expression in the theory of regulr expressions with ACP-style prllel composition tht denotes finite utomton violting this property. Our third contribution, in Section 5, is to estblish tht dding ACP-style prllel composition nd encpsultion to the theory of regulr expressions ctully yields theory in which every finite utomton cn be expressed up to isomorphism, nd hence, since bisimilrity is corser thn isomorphism, lso up to bisimilrity. Every expression in the resulting theory, in turn, denotes finite utomton, so this result cn be thought of s n lterntive process-theoretic counterprt of Kleene s theorem. The results in this pper re inspired by the results of Bergstr, Bethke nd Ponse pertining to the reltive expressiveness of process lgebrs with binry vrint of Kleene s str opertion. In [3] they estblish n expressiveness hierrchy on the extensions of the process theories BPA(A), BPA δ (A), PA(A), PA δ (A), ACP(A,γ), nd ACP τ (A,γ) with binry Kleene str. The reson tht their results re bsed on extensions with the binry version of the Kleene str is tht they wnt to void the processtheoretic complictions rising from the notion of intermedite termintion (we sy tht stte in finite utomton is intermeditely terminting if it is terminting but lso dmits trnsition). Most of the expressiveness results in [3] re included in [4], with more elborte proofs. Csting our contributions mentioned bove in process-theoretic terminology, we estblish strict expressiveness hierrchy on the process theories BPA 0,1 (A) (regulr expressions) modulo bisimilrity, PA 0,1 (A) (regulr expressions with interleving) modulo bisimilrity nd ACP 0,1 (A,γ) (regulr expressions with ACP-style prllel composition nd encpsultion) modulo bisimilrity. The differences between the process theories BPA δ (A), PA δ (A) nd ACP(A,γ) considered [3, 4] nd the process theories BPA 0,1 (A), PA 0,1 (A) nd ACP 0,1 (A,γ) considered in this pper re s follows: we write 0 for the constnt dedlock which is denoted by δ in [3, 4], we include the unry Kleene str insted of its binry vrint, nd we include constnt 1 denoting the successfully terminted process. The first difference is, of course, cosmetic, nd with the ddition of the constnt 1 the unry nd binry vrints of Kleene s str re interdefinble. So, our results pertining to the reltive expressiveness of BPA 0,1 (A), PA 0,1 (A) nd ACP 0,1 (A,γ) extend the expressiveness hierrchy of [3, 4] with the constnt 1. In [4] the expressiveness proofs re bsed on identifying cycles nd exit trnsitions from these cycles. There re two resons why the proofs in [3] nd [4] cnnot esily be dpted to setting with 1. First, in setting with 1 nd Kleene str there re cycles without ny exit trnsitions. Second, the inclusion of the empty process 1 gives intermedite termintion, which, combined with the previously described different behviour of cycles, forces us to consider the more generl structure of strongly connected component. 2 Preliminries In this section, we present the relevnt definitions for the process theory ACP 0,1 (A,γ) nd its subtheories PA 0,1 (A) nd BPA 0,1 (A). We give their syntx nd opertionl semntics, nd the notion of (strong) bisimilrity. We lso introduce some uxiliry technicl notions tht we need in the reminder of the pper, most notbly tht of strongly connected component. The expressions of the process theory BPA 0,1 (A) re precisely the well-known regulr expressions from the theory of utomt nd forml lnguges, but we shll consider the utomt ssocited with them modulo bisimilrity insted of modulo lnguge equivlence. The process theory ACP 0,1 (A,γ) is prmetrised by non-empty set A of ctions, nd commu-
Beten, Luttik, Muller & Vn Tilburg 3 niction function γ on A, i.e., n ssocitive nd commuttive binry prtil opertion γ : A A A. ACP 0,1 (A,γ) incorportes form of synchronistion between the components of prllel composition by llowing certin ctions to engge in communiction resulting in nother ction. The communiction function γ then defines which ctions my communicte nd wht is the result. The detils of this feture will become cler when we present the opertionl semntics of prllel composition. The set of ACP 0,1 (A,γ) expressions P ACP 0,1 (A,γ) is generted by the following grmmr: p ::= 0 1 p p p+ p p p p H (p), with rnging over A nd H rnging over subsets ofa. The process theory ACP(A,γ) (excluding the constnts 0 nd 1, but including constnt δ with exctly the sme behviour s 0, nd without the opertion ) origintes with [5]. The extension of ACP(A,γ) with constnt 1 ws investigted by [9, 2, 14] (in these rticles, the constnt ws denoted ε). The extension of ACP(A,γ) with the binry version of the Kleene str ws first proposed in [3]. The reder lredy fmilir with the process theory ACP 0,1 (A,γ) will hve noticed tht the opertions (left merge) nd (communiction merge) re missing from our syntx definition. In [5], these opertions re included s uxiliry opertions necessry for finite xiomtistion of the theory. They do not, however, dd expressiveness in our setting with Kleene str insted of generl form of recursion. We hve omitted them to chieve more efficient presenttion of our results. The constnts 0 nd 1 respectively stnd for the dedlocked process nd the successfully terminted process, nd the constnts A denote processes of which the only behviour is to execute the ction. An expression of the form p q is clled sequentil composition, n expression of the form p+q is clled n lterntive composition, nd n expression of the form p is clled str expression. An expression of the form p q is clled prllel composition, nd n expression of the form H (p) is clled n encpsultion. From the nmes for the constructions in the syntx of ACP 0,1 (A,γ), the reder probbly hs lredy n intuitive understnding of the behviour of the corresponding processes. We proceed to formlise the opertionl behviour by mens of collection of opertionl rules (see Tble 1) in the style of Plotkin s Structurl Opertionl Semntics [13]. Note how the communiction function in rule 14 is employed to model form of communiction between prllel components: if one of the components of prllel composition cn execute trnsition lbelled with, the other cn execute trnsition lbelled with b, nd the communiction function γ is defined on nd b, then the prllel composition cn execute trnsition lbelled with γ(,b). (It my help to think of the ction s stnding for the event of sending some dtum d, the ction b s stnding for the event of receiving dtum d, nd the ction γ(, b) s stnding for the event tht two components communicte dtum d.) The A-lbelled trnsition reltion ACP 0,1 (A,γ) nd the termintion reltion ACP 0,1 (A,γ) on P ACP 0,1 (A,γ) re the lest reltions P ACP 0,1 (A,γ) A P ACP 0,1 (A,γ) nd P ACP 0,1 (A,γ) stisfying the rules in Tble 1. The triple T ACP 0,1 (A,γ) = (P ACP 0,1 (A,γ), ACP 0,1 (A,γ), ACP 0,1 (A,γ)), consisting of the ACP 0,1 (A,γ) expressions together with the A-lbelled trnsition reltion nd the termintion predicte ssocited with them, is n exmple of n A-lbelled trnsition system spce. In generl, n A-lbelled trnsition system spce (S,, ) consists of (non-empty) set S, the elements of which re clled sttes, together with n A-lbelled trnsition reltion S A S nd subset S. We shll in this pper consider two more exmples of trnsition system spces, obtined by restricting the syntx of ACP 0,1 (A,γ) nd mking specil ssumptions bout the communiction function. Next, we define the A-lbelled trnsition system spce T PA 0,1 (A) = (P PA 0,1 (A), PA 0,1 (A), PA 0,1 (A)) corresponding with the process theory PA 0,1 (A). The set of PA 0,1 (A) expressions P PA 0,1 (A) consists of
4 Regulr Expressions with Prllel Composition modulo Bisimilrity 1 7 15 1 2 p p p q p q p p 1 12 8 3 p p p+q p 4 p q q p q q 9 p p p q p q 13 q q p+q q 5 p q p q q q q b q γ(,b) is defined p q γ(,b) 16 p q 10 p q p q 14 p p+q p p 6 p p p 11 p q p q p p H H (p) H (p ) 17 q p+q p p H (p) Tble 1: Opertionl rules for ACP 0,1 (A,γ), with A nd H A. the ACP 0,1 (A,γ) process expressions without occurrences of the construct H( ). The PA 0,1 (A) trnsition reltion PA 0,1 (A) on P PA 0,1 (A) nd the termintion predicte PA 0,1 (A) on P PA 0,1 (A) re the trnsition reltion nd termintion predicte induced on PA 0,1 (A) expressions by the opertionl rules in Tble 1 minus the rules 15 17. Alterntively (nd equivlently) the trnsition reltion PA 0,1 (A) cn be defined s the restriction of the trnsition reltion ACP 0,1 (A,/0), with /0 denoting the communiction function tht is everywhere undefined, top PA 0,1 (A). To define the A-lbelled trnsition system spce T BPA 0,1 (A) =(P BPA 0,1 (A), BPA 0,1 (A), BPA 0,1 (A)) ssocited with the process theory BPA 0,1 (A), let P BPA 0,1 (A) consist of ll PA 0,1 (A) expressions without occurrences of the construct. The BPA 0,1 (A) trnsition reltion BPA 0,1 (A) nd the BPA 0,1 (A) termintion predicte BPA 0,1 (A) re the trnsition reltion nd the termintion predicte induced on BPA 0,1 (A) expressions by the opertionl rules in Tble 1 minus the rules 12 17. Tht is, BPA 0,1 (A) nd BPA 0,1 (A) re obtined by restricting ACP 0,1 (A,γ) nd ACP 0,1 (A,γ) to P BPA 0,1 (A). Henceforth, we shll omit the subscripts ACP 0,1 (A,γ), PA 0,1 (A) nd BPA 0,1 (A) from trnsition reltions nd termintion predictes whenever it is cler from the context which trnsition reltion or termintion predicte is ment. Furthermore, we shll often use ACP 0,1 (A,γ), PA 0,1 (A) nd BPA 0,1 (A), respectively, to denote the ssocited trnsition system spces T ACP 0,1 (A,γ), T PA 0,1 (A) nd T BPA 0,1 (A). Let T =(S,, ) be na-lbelled trnsition system spce. If s,s S, then we write s s if there exists Asuch tht s s, nd s s if there exists no such A. We denote by + the trnsitive closure of, nd by the reflexive-trnsitive closure of. If s s then we sy tht s is rechble from s; the set of ll sttes rechble from s is denoted by [s]. We sy tht stte s is normed if there exists s such tht s s nd s. T is clled regulr if [s] is finite for ll s S. Lemm 2.1. The trnsition system spces ACP 0,1 (A,γ), PA 0,1 (A), nd BPA 0,1 (A) re ll regulr. With every stte s in T we cn ssocite n utomton (or: trnsition system) ([s], ([s] A [s] ), [s], s). Its sttes re the sttes rechble from s, its trnsition reltion nd termintion predicte re obtined by restricting nd ccordingly, nd the stte s is declred s the initil stte of the utomton. If trnsition system spce is regulr, then the utomton ssocited with stte in it is finite, i.e., it is finite utomton in the terminology of utomt theory. Thus, we get by Lemm 2.1 tht the opertionl semntics of ACP 0,1 (A,γ), nd, fortiori, tht of PA 0,1 (A) nd BPA 0,1 (A), ssocites finite utomton with every process expression.
Beten, Luttik, Muller & Vn Tilburg 5 In utomt theory, utomt re usully considered s lnguge cceptors nd two utomt re deemed indistinguishble if they ccept the sme lnguges. Lnguge equivlence is, however, rgubly too corse in process theory, where the prevlent notion is bisimilrity [11, 12]. Definition 2.2. LetT 1 =(S 1, 1, 1 ) ndt 2 =(S 2, 2, 2 ) be trnsition system spces. A binry reltion R S 1 S 2 is bisimultion betweent 1 ndt 2 if it stisfies, for ll A nd for ll s 1 S 1 nd s 2 S 2 such tht s 1 R s 2, the following conditions: (i) if there exists s 1 S 1 such tht s 1 1 s 1, then there exists s 2 S 2 such tht s 2 2 s 2 nd s 1 R s 2 ; (ii) if there exists s 2 S 2 such tht s 2 2 s 2, then there exists s 1 S 1 such tht s 1 1 s 1 nd s 1 R s 2 ; nd (iii) s 1 1 if, nd only if, s 2 2. Sttes s 1 S 1 nd s 2 S 2 re bisimilr (nottion: s 1 s 2 ) if there exists bisimultion R between T 1 nd T 2 such tht s 1 R s 2. To chieve sufficient level of generlity, we hve defined bisimilrity s reltion between trnsition system spces; to obtin suitble notion of bisimultion between utomt one should dd the requirement tht the initil sttes of the utomt be relted. Bsed on the ssocited trnsition system spces, we cn now define wht we men when some trnsition system spce is, modulo bisimilrity, less expressive thn some other trnsition system spce. Definition 2.3. Let T 1 nd T 2 be trnsition system spces. We sy tht T 1 is less expressive thn T 2 (nottion: T 1 T 2 ) if every stte in T 1 is bisimilr to stte in T 2, nd, moreover, there is stte in T 2 tht is not bisimilr to some stte int 1. When we investigte the expressiveness of ACP 0,1 (A,γ), we wnt to be ble to choose γ. So, we re ctully interested in the expressiveness of the (disjoint) union of ll trnsition system spces ACP 0,1 (A,γ) with γ rnging over ll communiction functions. We denote this trnsition system spce by γ ACP 0,1 (A,γ). In this pper we shll then estblish tht BPA 0,1 (A) PA 0,1 (A) γ ACP 0,1 (A,γ). We recll below the notion of strongly connected component (see, e.g., [6]) tht will ply n importnt rôle in estblishing tht the bove hierrchy of trnsition system spces is strict. Definition 2.4. A strongly connected component in trnsition system spce T =(S,, ) is mximl subset C of S such tht s s for ll s,s C. A strongly connected component C is trivil if it consists of only one stte, sy C={s}, nd s s; otherwise, it is non-trivil. Note tht every element of trnsition system spce is n element of precisely one strongly connected component of tht spce. Furthermore, if s is n element of non-trivil strongly connected component, then s + s. Since in strongly connected component from every element every other element cn be reched, we get s corollry to Lemm 2.1 tht strongly connected components in ACP 0,1 (A,γ), PA 0,1 (A) nd BPA 0,1 (A) re finite. Let T = (S,, ) be trnsition system spce, let s S, nd let C S be strongly connected component in S. We sy tht C is rechble from s if s s for ll s C. Lemm 2.5. Let T 1 = (S 1, 1, 1 ) nd T 2 = (S 2, 2, 2 ) be regulr trnsition system spces, nd let s 1 S 1 nd s 2 S 2 be such tht s 1 s 2. If s 1 is n element of strongly connected component C 1 in T 1, then there exists strongly connected component C 2 rechble from s 2 stisfying tht for ll s 1 C 1 there exists s 2 C 2 such tht s 1 s 2.
6 Regulr Expressions with Prllel Composition modulo Bisimilrity 3 Reltive Expressiveness of BPA 0,1 (A) nd PA 0,1 (A) In [3] it is proved tht BPA 0 (A) is less expressive thn PA 0 (A). The proof in [3] is by rguing tht the PA 0 (A) expression ( b) c d is not bisimilr with BPA 0 (A) expression. (Actully, the PA 0 (A) expression employed in [4] uses only single ction, i.e., considers the PA 0 (A) expression ( ) ; we use the ctions b, c nd d for clrity.) An lterntive nd more generl proof tht the PA 0 (A) expression bove is not expressible in BPA 0 (A) is presented in [4]. There it is estblished tht the PA 0 (A) expression bove fils the following generl property, which is stisfied by ll BPA 0 (A)-expressible utomt: If C is cycle in n utomton ssocited with BPA 0 (A) expression, then there is t most one stte p C tht hs n exit trnsition. (A cycle is sequence (p 1,..., p n ) such tht p i p i+1 (1 i<n) nd p n p 1 ; n exit trnsition from p i is trnsition p i p i such tht no element of the cycle is rechble from p i.) The following exmple shows tht utomt ssocited with BPA 0,1 (A) expressions do not stisfy the property bove. Exmple 3.1. Consider the utomton ssocited with the BPA 0,1 (A) expression 1 ( (+ 1)) b (see Figure 1) with cycle; both sttes on the cycle hve b-trnsition off the cycle. 1 ( (+1)) b 1 b b (+1) ( (+1)) b Figure 1: A trnsition system in BPA 0,1 (A) with cycle with multiple exit trnsitions. In this section we shll estblish tht BPA 0,1 (A) is less expressive thn PA 0,1 (A). As in [4] we prove tht BPA 0,1 (A)-expressible utomt stisfy generl property tht some utomton expressible in PA 0,1 (A) fils to stisfy. We find it techniclly convenient, however, to bse our reltive expressiveness proofs on the notion of strongly connected component, insted of cycle. Note, e.g., tht every process expression is n element of precisely one strongly connected component, while it my reside in more thn one cycle. Furthermore, if p q nd p nd q re in distinct strongly connected components, then we cn be sure tht p q is n exit trnsition, while if p nd q re on distinct cycles, then it my hppen tht p is rechble from q. 3.1 Strongly Connected Components in BPA 0,1 (A) We shll now estblish tht non-trivil strongly connected component in BPA 0,1 (A) is either of the form {p 1 q,..., p n q } with p i (0 i n) rechble from q nd {p 1,..., p n } not strongly connected component, or of the form {p 1 q,..., p n q} where {p 1,..., p n } is strongly connected component. To this end, let us first estblish, by resoning on the bsis of the opertionl semntics, tht process expressions in non-trivil strongly connected component re necessrily sequentil compositions; t the hert of the rgument will be the following mesure on process expressions. Definition 3.2. Let p BPA 0,1 (A) expression; then #(p) is defined with recursion on the structure of p by the following cluses:
Beten, Luttik, Muller & Vn Tilburg 7 (i) #(0)=#(1)= 0, nd #()=1; (ii) #(p q)=0 if q is str expression, nd #(p q)=#(q)+1 otherwise; (iii) #(p+q)=mx{#(p),#(q)}+1; nd (iv) #(p )=1. We estblish tht #( ) is non-incresing over trnsitions, nd, in fct, in most cses decreses. Lemm 3.3. If p nd p re BPA 0,1 (A) expressions such tht p + p, then #(p) #(p ). Moreover, if #(p)=#(p ), then p= p 1 q nd p = p 1 q for some p 1, p 1 nd q. Proof. First, the specil cse of the lemm in which p p is estblished with induction on derivtions ccording to the opertionl rules for BPA 0,1 (A). Then, the generl cse of the lemm follows from the specil cse with strightforwrd induction on the length of trnsition sequence from p to p. Let P be set of process expressions, nd let q be process expression; by P q we denote the set of process expressions P q={p q p P}. Lemm 3.4. If C is non-trivil strongly connected component in BPA 0,1 (A), then there exist set of process expressions C nd process expression q such tht C= C q. We proceed to give n inductive description of the non-trivil strongly connected components in BPA 0,1 (A). The bsis for the inductive description is the following notion of bsic strongly connected component. Definition 3.5. A non-trivil strongly connected component C = {p 1,..., p n } in BPA 0,1 (A) is bsic if there exist BPA 0,1 (A) expressions p 1,..., p n nd BPA 0,1 (A) expression q such tht p i = p i q (1 i n) nd {p 1,..., p n} is not strongly connected component in BPA 0,1 (A). Proposition 3.6. Let C be non-trivil strongly connected component in BPA 0,1 (A). Then either C is bsic, or there exist non-trivil strongly connected component C nd BPA 0,1 (A) expression q such tht C = C q. Proof. By Lemm 3.4 there exists set of sttes C nd BPA 0,1 (A) expression q such tht C = C q. If C is non-trivil strongly connected component, then the proposition follows, so it remins to prove tht if C is not non-trivil strongly connected component, then C is bsic. Note tht if C is not strongly connected component, then there re p, p C such tht p + p. Since C is non-trivil strongly connected component nd C = C q, it holds tht p q + p q. Using tht p + p, it cn be estblished with induction on the length of the trnsition sequence from p q to p q tht q + p q. It follows by Lemm 3.3 tht #(q) #(p q), nd therefore, ccording to the definition of #( ), q must be str expression. We conclude tht C is bsic. 3.2 BPA 0,1 (A) PA 0,1 (A) The crucil tool tht will llow us to estblish tht BPA 0,1 (A) is less expressive thn PA 0,1 (A) will be specil property of sttes with trnsition out of their strongly connected component in BPA 0,1 (A). Roughly, if C is strongly connected component in BPA 0,1 (A), then ll sttes with trnsition out of C hve the sme trnsitions out of C. Definition 3.7. Let C be strongly connected component in the trnsition system spce T = (S,, ) nd let s C. An exit trnsition from s is pir (,s ) such tht s s nd s C. We denote by ET(s) the set of ll exit trnsitions from s, i.e., ET(s)={(,s ) s s s C}. An element s C is clled n exit stte if s or there exists n exit trnsition from s.
8 Regulr Expressions with Prllel Composition modulo Bisimilrity Exmple 3.8. Consider the utomton ssocited with the BPA 0,1 (A) expression 1 ( b (c+ 1)) d, (see Figure 2). It hs strongly connecting component with two exit sttes, both with one exit trnsition (d,1). 1 ( b (c+1)) d b (c+1) ( b (c+1)) b d (c+1) ( b (c+1)) d 1 d c d Figure 2: A non-trivil strongly connected component in BPA 0,1 (A) with multiple exit trnsitions. Non-trivil strongly connected components in BPA 0,1 (A) rise from executing the rgument of Kleene str. An exit stte of strongly connected component in BPA 0,1 (A) is then stte in which the execution hs the option to terminte. Due to the presence of 0 in BPA 0,1 (A) this is, however, not the only type of exit stte in BPA 0,1 (A) strongly connected components. Exmple 3.9. Consider the utomton ssocited with the BPA 0,1 (A) expression 1 ( ((b 0)+ 1)) c (see Figure 3). The strongly connected component contins two exit sttes nd two (distinct) exit trnsitions. One of these exit trnsitions leds to dedlocked stte. 1 ( ((b 0)+1)) c 1 c c ((b 0)+ 1) ( ((b 0)+1)) c b 0 ( ((b 0)+1)) c Figure 3: A strongly connected component with normed exit trnsitions. The preceding exmple illustrtes tht the specil property of strongly connected components in BPA 0,1 (A) tht we re fter, should exclude from considertion ny exit trnsition rising from n occurrence of 0. This is chieved in the following definitions. Definition 3.10. Let C be strongly connected component nd let s C. An exit trnsition (,s ) from s is normed if s is normed. We denote by ET n (s) the set of normed exit trnsitions from s. An exit stte s C is live if s or there exists normed exit trnsition from s. Lemm 3.11. If p q r, then either there exists p such tht p p nd r = p q or there exist p nd q such tht p p, p, q q, nd r=q q. Lemm 3.12. If C is bsic strongly connected component, then ET n (p)= /0 for ll p C. Lemm 3.13. Let C be non-trivil strongly connected component in BPA 0,1 (A), let p C, nd let q be BPA 0,1 (A) process expression such tht C q is strongly connected component. Then p q is n live exit stte in C q iff p is n live exit stte in C nd q is normed. For chrcteristion of the set of normed exit trnsitions of sequentil composition, it is convenient to hve the following nottion: if E is set of exit trnsitions E nd p is BPA 0,1 (A) expression, then E p is defined by E p={(,q p) (,q) E}.
Beten, Luttik, Muller & Vn Tilburg 9 Lemm 3.14. Let C be non-trivil strongly connected component in BPA 0,1 (A), let p C, nd let q be normed BPA 0,1 (A) process expression such tht C q is strongly connected component. Then { ETn (p) q {(,r) r C q r is normed q r} if p ; nd ET n (p q)= ET n (p) q if p. Proposition 3.15. Let C be non-trivil strongly connected component in BPA 0,1 (A). If p 1 nd p 2 re live exit sttes in C, then ET n (p 1 )=ET n (p 2 ). Proof. Suppose tht p 1 nd p 2 re live exit sttes; we prove by induction on the structure of non-trivil strongly connected components in BPA 0,1 (A) s given by Proposition 3.6 tht ET n(p 1 ) = ET n (p 2 ) nd p 1 iff p 2. If C is bsic, then by Lemm 3.12 ET n (p 1 )= /0=ET n (p 2 ), nd, since p 1 nd p 2 re live exit sttes, it lso follows from this tht both p 1 nd p 2. Suppose tht C = C q, with C non-trivil strongly connected component, nd let p 1, p 2 C be such tht p 1 = p 1 q nd p 2 = p 2 q. Since p 1 nd p 2 re live exit sttes, by Lemm 3.13 so re p 1 nd p 2. Hence, by the induction hypothesis, ET n(p 1 )=ET n(p 2 ) nd p 1 iff p 2. From the ltter it follows tht p 1 iff p 2. We now pply Lemm 3.14: if, on the one hnd, p 1 nd p 2, then ET n (p 1 )=ET n (p 1 ) q {(,r) r C r. q r r } = ET n (p 2) q {(,r) r C r. q r r }=ET n (p 2 ), nd if, on the other hnd, p 1 nd p 2, then ET n (p 1 )=ET n (p 1 ) q=et n(p 2 ) q=et n(p 2 ). p 0 p 1 c b c p 2 b p 3 Figure 4: A PA 0,1 (A)-expressible utomton tht is not expressible in BPA 0,1 (A). The PA 0,1 (A) expression p 0 = 1 ( b) c gives rise to the utomton shown in Figure 4. It hs strongly connected component C ={p 0, p 1 } of which the live exit sttes hve different normed exit trnsitions. Hence, by Proposition 3.15, p 0 is not BPA 0,1 (A)-expressible. Theorem 3.16. BPA 0,1 (A) is less expressive thn PA 0,1 (A). 4 Reltive Expressiveness of PA 0,1 (A) nd ACP 0,1 (A,γ) The proof in [4] tht PA δ (A) is less expressive thn ACP (A,γ) uses the sme expression s the one showing tht BPA δ (A) is less expressive thn PA δ (A), but it presupposes tht γ(c,d)=e. It is climed tht the ssocited utomton fils the following generl property of cycles in PA δ (A): If C is cycle rechble from PA 0 (A) process term nd there is stte in C with trnsition to terminting stte, then ll other sttes in C hve only successors in C. The clim, however, is incorrect, s illustrted by the following exmple. (We present the exmple in the syntx of PA 0,1 (A), but it hs strightforwrd trnsltion into the syntx of PA δ (A).)
10 Regulr Expressions with Prllel Composition modulo Bisimilrity Exmple 4.1. Consider the PA 0,1 (A) expression ( (b+b b)) d, from which the cycle C={1 ( (b+b b)) d, (b+b b) ( (b+b b)) d} is rechble. Clerly, the first expression in C cn perform d-trnsition to 1. Then, ccording to the property bove, every other expression only hs trnsitions to expressions in C. However, (b+b b) ( (b+b b)) d b b ( (b+b b)) d C. If we replce, in the property bove, the notion of cycle by the notion of strongly connected component, then the resulting property does hold for PA 0 (A), but it still fils for PA 0,1 (A). Exmple 4.2. Consider the PA 0,1 (A) expression ( b) cit gives rise to the following non-trivil strongly connected component: {1 ( b) c, b ( b) c}. The expression 1 ( b) c cn do c-trnsition to 1 ( b) 1, for which the termintion predicte holds, but t the sme time b ( b) c hs n exit trnsition (c,b ( b) 1). In this section we shll estblish tht PA 0,1 (A) is less expressive thn ACP 0,1 (A,γ). To this end, we pply the sme method s in Section 3. First, we syntcticlly chrcterise the non-trivil strongly connected components ssocited with PA 0,1 (A) expressions. Then, we conclude tht wekened version of the forementioned property for strongly connected components holds in PA 0,1 (A), nd present n (A,γ) expression tht does not stisfy it. ACP 0,1 4.1 Strongly Connected Components in PA 0,1 (A) To give syntctic chrcteristion of the non-trivil strongly connected components in PA 0,1 (A), we reson gin bout the opertionl semntics. First, we extend the mesure #( ) from Section 3 to (A) expressions. PA 0,1 Definition 4.3. Let p be PA 0,1 (A) expression; #(p) is defined with recursion on the structure of p by the cluses (i) (iv) in Definition 3.2 with the following cluse dded: (v) #(p q)=0. With the extension, the non-incresing mesure #( ) still in most cses decreses over trnsitions. Lemm 4.4. If p nd p re PA 0,1 (A) expressions such tht p + p, then #(p) #(p ). Moreover, if #(p) = #(p ), then either p= p 1 q nd p = p 1 q, or p = p 1 p 2 nd p = p 1 p 2 for some process expressions p 1, p 2, p 1, p 2, nd q. Lemm 4.5. Let p, q nd r be PA 0,1 (A) process expressions such tht p q r. Then there exist PA 0,1 (A) process expressions p nd q such tht r= p q, p p nd q q. Let P nd Q be sets of process expressions; by P Q we denote the set of process expressions P Q={p q p P q Q}. We lso write P qnd p Qfor P {q} nd {p} Q, respectively. The proof of the following lemm, chrcterising the syntctic form of non-trivil strongly connected components in PA 0,1 (A), is strightforwrd dpttion nd extension of the proof of Lemm 3.4, using Lemm 4.4 nd Lemm 4.5 insted of Lemm 3.3. Lemm 4.6. If C is non-trivil strongly connected component in PA 0,1 (A), then either there exist set of process expressions C nd process expression q such tht C=C q, or there exist strongly connected components C 1 nd C 2 in PA 0,1 (A), t lest one of them non-trivil, such tht C= C 1 C 2.
Beten, Luttik, Muller & Vn Tilburg 11 The notion of bsic strongly connected component in PA 0,1 (A) is obtined from Definition 3.5 by replcing BPA 0,1 (A) by PA 0,1 (A) everywhere in the definition. In Proposition 3.6 we gve n inductive chrcteristion of non-trivil strongly connected components in BPA 0,1 (A). There is similr inductive chrcteristion of non-trivil strongly connected components in PA 0,1 (A), obtined by dding cse for prllel composition. Proposition 4.7. Let C be non-trivil strongly connected component in PA 0,1 (A). Then one of the following holds: (i) C is bsic strongly connected component; or (ii) there exist non-trivil strongly connected component C nd PA 0,1 (A) expression q such tht C= C q; or (iii) there exist strongly connected components C 1 nd C 2, t lest one of them non-trivil, such tht C= C 1 C 2. Note tht, in the bove proposition, one of the strongly connected components C 1 nd C 2 my be trivil in which cse it consists of single PA 0,1 (A) expression. 4.2 PA 0,1 (A) ACP 0,1 (A,γ) In Section 3 we deduced, from our syntctic chrcteristion of strongly connected components in BPA 0,1 (A), the property tht ll live exit sttes of strongly connected component hve the sme sets of normed exit trnsitions. This property my fil for strongly connected components in PA 0,1 (A): the utomton in Figure 4 is PA 0,1 (A)-expressible, but the live exit sttes p 0 nd p 1 of the strongly connected component {p 0, p 1 } hve different normed exit trnsitions. Note, however, tht these normed exit trnsitions both end up in nother strongly connected component {p 2, p 3 }. It turns out tht we cn relx the requirement on normed exit trnsitions from strongly connected components in BPA 0,1 (A) to get requirement tht holds for strongly connected components in PA 0,1 (A). The ide is to identify exit trnsitions if they hve the sme ction nd end up in the sme strongly connected component. Definition 4.8. Let T =(S,, ) be n A-lbelled trnsition system spce. We define binry reltion ona S by(,s) (,s ) iff = nd s nd s re in the sme strongly connected component int. Since the reltion of being in the sme strongly connected component is n equivlence on sttes in trnsition system spce, it is cler tht is n equivlence reltion on exit trnsitions. The following lemm will give some further properties of the reltion ssocited with PA 0,1 (A). Lemm 4.9. Let p nd q be PA 0,1 (A) expressions, nd let nd b be ctions. If (, p) (b,q), then (, p r) (b,q r), (, p r) (b,q r), nd (,r p) (b,r q). To formulte strightforwrd corollry of this lemm we use the following nottion: if E is set of exit trnsitions E nd p is PA 0,1 (A) expression, then E p, E p nd p E re defined by E p={(,q p) (,q) E}, nd p E ={(, p q) (,q) E}. We re now in position to estblish property of strongly connected components in PA 0,1 (A) tht will llow us to prove tht PA 0,1 (A) is less expressive thn ACP 0,1 (A,γ): strongly connected component C in PA 0,1 (A) lwys hs specil exit stte from which, up to, ll exit trnsitions re enbled.
12 Regulr Expressions with Prllel Composition modulo Bisimilrity Lemm 4.10. Let C 1 nd C 2 be sets of PA 0,1 (A) expressions. Then C 1 C 2 is strongly connected component iff both C 1 nd C 2 re strongly connected components. Moreover, C 1 C 2 is non-trivil iff t lest one of C 1 nd C 2 is non-trivil. Lemm 4.11. Let C 1 nd C 2 be strongly connected components in PA 0,1 (A), both with live exit sttes. Then C 1 C 2 is strongly connected component with live exit sttes too, nd, for ll p C 1 nd q C 2, ET n (p q)=(et n (p) q) (p ET n (q)). To formulte the specil property of strongly connected components in PA 0,1 (A) tht will llow us to prove tht some ACP 0,1 (A,γ) expressions do not hve counterprt in PA 0,1 (A), we need the notion of mximl live exit stte. Definition 4.12. Let T =(S,, ) be n A-lbelled trnsition system spce, let A S be the equivlence reltion ssocited with T ccording to Definition 4.8, let C be strongly connected component in T, nd let s C be n live exit stte. We sy tht s is mximl (modulo ) if for ll live exit sttes s C nd for ll e ET n (s ) there exists n exit trnsition e ET n (s) such tht e e. The following proposition estblishes the property with which we shll prove tht PA 0,1 (A) is less expressive thn ACP 0,1 (A,γ). Proposition 4.13. If C is strongly connected component in PA 0,1 (A) nd C hs n live exit stte, then C hs mximl live exit stte. p 4 p 0 p 1 d b c c e c p 5 p 2 p 3 d b Figure 5: An ACP 0,1 (A,γ)-expressible utomton tht is not expressible in PA 0,1 (A). Suppose γ(b,c)=e; then the ACP 0,1 (A,γ) expression p 0 = 1 ( b) d c gives rise to the utomton shown in Figure 5. It hs strongly connected component C = {p 0, p 1 }, nd none of its live exit sttes is mximl. Hence, by Proposition 4.13, p 0 is not PA 0,1 (A)-expressible. Theorem 4.14. PA 0,1 (A) is less expressive thn γ ACP 0,1 (A,γ). 5 Every Finite Automton is ACP 0,1 (A,γ)-expressible Milner observed in [10] tht there exist finite utomt tht re not bisimilr to the finite utomton ssocited with BPA 0,1 (A) expression. Our proof of Theorem 3.16 hs Milner s observtion s n immedite consequence: the finite utomton ssocited with the PA 0,1 (A) expression used in the proof is not BPA 0,1 (A)-expressible. Similrly, by Theorem 4.14, there re finite utomt tht re not expressible in PA 0,1 (A). In this section we shll prove tht every finite utomton is expressible in ACP 0,1 (A,γ), for suitble choices of A nd γ, even up to isomorphism. Before we formlly prove the result, let us first explin the ide informlly, nd illustrte it with n exmple. The ACP 0,1 (A,γ) expression tht we shll ssocite with finite utomton will hve one prllel component per stte of the utomton, representing the behviour in tht stte (i.e., which outgoing trnsitions it hs to which other sttes nd whether it is terminting). At ny time, one of those prllel components, the one corresponding with the current
Beten, Luttik, Muller & Vn Tilburg 13 stte, hs control. An -trnsition from tht current stte to next stte corresponds with communiction between two components. We mke essentil use of ACP 0,1 (A,γ) s fcility to let the ction be the result of communiction. Exmple 5.1. Consider the finite utomton in Figure 6. 0 s 0 s 1 1 0 2 2 s 1 2 s 3 1 Figure 6: A finite utomton. We ssocite with every stte s i n ACP 0,1 (A,γ) expression p i s follows: ( ) ( ) p 0 = enter 0 (leve 0,1 + leve 1,1 ), p 2 = enter 2 (leve 0,0 + leve 1,3 + 1) ( ) ( ) p 1 = enter 1 1 (leve 2,2 ), p 3 = enter 3 0 Every p i hs n enter i trnsition to gin control, nd by executing leve k, j it my then relese control to p j with ction k s effect. We define the communiction function so tht n enter i ction communictes with leve k,i ction, resulting in the ction k. Loops in the utomton (such s the loop on stte s 1 ) require specil tretment s they should not relese control. Let p 0 be the result of executing the enter 0-trnsition from p 0. We define the ACP 0,1 (A,γ) expression tht simultes the finite utomton in Figure 6 s the prllel composition of p 0, p 1, p 2 nd p 3, encpsulting the control ctions enter i nd leve k,i, i.e., s {enteri,leve k,i 0 i 3, 0 k 2}(p 0 p 1 p 2 p 3 ). We now present the technique illustrted in the preceding exmple in full generlity. Let F = (S,,s 0, ) be finite utomton, let S = {s 0,...,s n }, nd let A = { 1,..., m } be the set of ctions occurring on trnsitions in F. We shll ssocite with F n ACP 0,1 (A,γ) expression p F tht hs precisely one prllel component p i for every stte s i in S. To llow prllel component to gin nd relese control, we use collection of control ctions C, ssumed to be disjoint from A, nd defined s C={enter i 1 i n} {leve k,i 1 i n, 1 k m}. Gining nd relesing control is modelled by the communiction function γ stisfying: { k if i= j; nd γ(enter i,leve k, j )= undefined otherwise. For the specifiction of the ACP 0,1 (A,γ) expressions p i we need one more definition: for 1 i, j n we denote by K i, j the set of indices of ctions occurring s the lbel on trnsition from s i to s j, i.e., k K i, j ={k s i s j }. Now we cn specify the ACP 0,1 (A,γ) expressions p i (1 i n) by ( ) p i = 1 leve k,i (+ 1) si ). k K i, j enter i ( k K i,i k ) ( 1 j n j i.,
14 Regulr Expressions with Prllel Composition modulo Bisimilrity By(+ 1) si we men tht the summnd + 1 is optionl; it is only included if s i. The empty summtion denotes 0. (We let p i strt with 1 to get tht the finite utomton ssocited with p F is isomorphic nd not just bisimilr with F.) Note tht, in ACP 0,1 (A,γ), every p enter i hs unique outgoing trnsition; specificlly p i i p i, where p i denotes: p i =(1 ( k K i,i k ) ( 0 j n j i k K i, j leve k,i (+ 1) si )) p i. We now define p F = C (p 0 p 1 p n ). Clerly, the construction of p F works for every finite utomton F. The bijection defined by s i C (p 0 p i 1 p i p i+1 p n ) is n isomorphism from F to the utomton ssocited with p F by the opertionl semntics. We shll refer to p F s the ACP 0,1 (A,γ) expression ssocited with F. Theorem 5.2. Let F be finite utomton, nd let p F be its ssocited ACP 0,1 (A,γ) expression. The utomton ssocited with p F by the opertionl rules for ACP 0,1 (A,γ) is isomorphic to F. Corollry 5.3. For every finite utomtonf there exists n instnce of ACP 0,1 (A,γ) with suitble finite set of ctions A nd hndshking communiction function γ such tht F is ACP 0,1 (A,γ)-expressible up to isomorphism. 6 Conclusion In this pper we hve investigted the effect on the expressiveness of regulr expressions modulo bisimilrity if different forms of prllel composition re dded. We hve estblished n expressiveness hierrchy tht cn be briefly summrised s: BPA 0,1 (A) PA 0,1 (A) γ ACP 0,1 (A,γ). Furthermore, while not every finite utomton cn be expressed modulo bisimilrity with regulr expression, it suffices to dd form of ACP(A, γ)-style prllel composition, with hndshking communiction nd encpsultion, to get lnguge tht is sufficiently expressive to express ll finite utomt modulo bisimilrity. This result should be contrsted with the well-known result from utomt theory tht every non-deterministic finite utomton cn be expressed with regulr expression modulo lnguge equivlence. As n importnt tool in our proof, we hve chrcterised the strongly connected components in BPA 0,1 (A) nd PA 0,1 (A). An interesting open question is whether the two given chrcteristions re complete, in the sense tht finite utomton is expressible in BPA 0,1 (A) or PA 0,1 (A) iff ll its strongly connected components stisfy our chrcteristion. If so, then our chrcteristion would constitute useful complement to the chrcteristion of [1] nd perhps led to more efficient lgorithm for deciding whether non-deterministic utomton is expressible. In [4] it is proved tht every finite trnsition system without intermedite termintion cn be denoted in ACP 0,τ (A,γ) up to brnching bisimilrity [7], nd tht ACP 0 (A,γ) modulo (strong) bisimilrity is strictly less expressive thn ACP 0,τ (A,γ). In contrst, we hve estblished tht every finite utomton (i.e., every finite trnsition system not excluding intermedite termintion) is denoted by n ACP 0,1 (A,γ) expression. It follows tht ACP 0,1 (A,γ) nd ACP 0,1,τ (A,γ) re eqully expressive. An interesting question tht remins is whether it is possible to omit constructions from ACP 0,1 (A,γ) without losing expressiveness. We conjecture tht H ( ) cnnot be omitted without losing expressiveness: encpsulting c in the ACP 0,1 (A,γ) expression 1 ( b) b c, which is used in Section 4 to show tht PA 0,1 (A) is less expressive thn ACP 0,1 (A,γ), yields trnsition system tht we think cnnot be expressed in ACP 0,1 (A,γ) without encpsultion.
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