Process Algebra with Recursive Operations

Transcription

1 Process Algebr with Recursive Opertions Jn Bergstr 2,3 Wn Fokkink 1 Albn Ponse 1,3 1 CWI Kruisln 413, 1098 SJ Amsterdm, The Netherlnds url: 2 Utrecht University, Deprtment of Philosophy Heidelbergln 8, 3584 CS Utrecht, The Netherlnds url: 3 University of Amsterdm, Progrmming Reserch Group Kruisln 403, 1098 SJ Amsterdm, The Netherlnds url: Contents 1 Introduction 2 2 Preliminries: Axioms nd Opertionl Semntics ACP-Bsed Systems Trnsition Rules nd Opertionl Semntics Recursive Specifictions Axiomtistion of the Binry Kleene Str Preliminries Completeness Irredundncy of the Axioms Negtive Results Extensions of BPA (A) Axiomtistions of Other Itertive Opertions Axiomtistion of No-Exit Itertion Axiomtistion of Multi-Exit Itertion Axiomtistion of Prefix Itertion Axiomtistion of String Itertion Axiomtistion of Flt Itertion

2 5 Expressivity Results Expressivity of the Binry Kleene Str Expressivity of Multi-Exit Itertion Expressivity of String Itertion Expressivity of Flt Itertion Non-Regulr Recursive Opertions Process Algebr with Push-Down Opertion Expressing Stck Undecidbility Results Specil Constnts Silent Step nd Firness Empty Process Introduction In process lgebr, (potentilly) infinite process is usully represented s the solution of system of gurded recursive equtions, nd proof theory nd verifiction tend to focus on resoning bout such recursive systems. Although specifiction nd verifiction of concurrent processes defined in this wy serve their purposes well, recursive opertions give more direct representtion nd re esier to comprehend. The Kleene str cn be considered s the most fundmentl recursive opertion. In process lgebr, the defining eqution for the binry Kleene str reds x y = x (x y) + y where models sequentil composition nd + models non-deterministic choice ( binds stronger thn +). In terms of opertionl semntics, the process x y chooses between x nd y, nd upon termintion of x hs this choice gin. For exmple, the expression b for tomic ctions nd b cn be depicted by b b where expresses successful termintion. This chpter discusses the Kleene str in the setting of process lgebr, nd considers some derived recursive opertions, with focus on xiomtistions nd expressiveness hierrchies. In the summer of 1951, S.C. Kleene ws supported by the RAND Corportion, leding to Reserch Memorndum Representtion of Events in Nerve Nets nd

3 3 Finite Automt [53]. The mteril in tht pper, bsed on the fundmentl pper [65] 1, ws republished under the sme title five yers lter [54]. In this seminl work, Kleene introduced the binry opertion for describing regulr events. He defined regulr expressions, which correspond to finite utomt, nd gve lgebric trnsformtion rules for these, notbly E F = E(E F ) F (E F being the iterte of E on F ). Kleene noted the correspondence with conventions of lgebr, treting E F s the nlogue of E + F, nd EF s the product of E nd F. In 1958, Copi, Elgot, nd Wright [34] showed interest in the results from [54]. However, they judged Kleene s theorems on nlysis 2 nd synthesis 3 obscured both by the complexity of his bsic concepts nd by the nture of the elements used in his nets. They introduced simpler nd stronger nets (in sense wekening Kleene s synthesis result, but stting tht this brings the essentil nture of the result into shrper focus ), nd simpler opertions. In prticulr they introduced unry opertion [...] becuse the opertion Kleene uses seems essentilly singulry nd becuse the singulry opertion simplifies the lgebr of regulr events. It should be noted tht the singulry nd binry str opertions re interdefinble. [34, pge 195] This contrdicts Kleene s originl rgument in [53, pge 50] tht the length of n event is t lest one, nd tht for this reson he did not define E s unry opertion. Four yers lter, Redko [69] proved tht there does not exist sound nd complete finite equtionl xiomtistion for regulr expressions. (This proof ws simplified nd corrected by Pilling; see [35, Chpter 11].) In 1966, Slom [70] presented sound nd complete finite xiomtistion for regulr expressions, with s bsic ingredient n implictionl xiom dting bck to Arden [9], nmely (in process lgebr nottion): x = (y x) + z = x = y z if y does not hve the so-clled empty word property. According to Kozen [56], this lst property is not lgebric, in the sense tht it is not preserved under ction refinement; he proposed two lterntive implictionl xioms tht do not hve this 1 Kleene judged the theory for nerve nets with circles in [65] (McCulloch nd Pitts, 1943) to be obscure, nd proceeded independently. He cme up with regulr events nd the mjor correspondence results in utomt theory. 2 Theorem 5, stting tht finite utomt model regulr events. 3 Theorem 3, stting tht ech regulr event cn be described by finite utomton ( nerve net ).

4 4 1 INTRODUCTION drwbck. Krob [57] settled two conjectures by Conwy [35], to obtin n infinite sound nd complete equtionl xiomtistion for regulr expressions. Bloom nd Ésik [14] developed n lterntive infinite equtionl xiomtistion for regulr expressions, within the frmework of itertion theories. In 1984, Milner [61] ws the first to consider the unry Kleene str in process lgebr, modulo strong bisimultion equivlence [66]. In contrst with regulr expressions, this setting is not sufficiently expressive to describe ll regulr (i.e., finitestte) processes. Moreover, the merge opertor x y [59] tht executes its two rguments in prllel, nd which is fundmentl to process lgebr, cnnot lwys be eliminted from process terms in the presence of the Kleene str. Milner presented n xiomtistion for the unry Kleene str in strong bisimultion semntics, being subset of Slom s xiomtistion, nd sked whether his xiomtistion is complete. Fokkink [40] solved this question for no-exit itertion x δ, where the specil constnt δ, clled dedlock, does not exhibit ny behviour; since δ blocks the exit, x δ executes x infinitely often. Bloom, Ésik, nd Tubner [15] presented complete xiomtistion for regulr synchronistion trees modulo strong bisimultion equivlence, within the frmework of itertion theories. Milner lso sked for chrcteristion of those recursive specifictions tht cn be described modulo strong bisimultion semntics in process lgebr with the unry Kleene str. Bosscher [24] solved this question in the bsence of the dedlock. The two questions by Milner in their full generlity remin unsolved. The unry Kleene str nturlly gives rise to the empty process [74], which does not combine well with the merge opertor. Therefore, Bergstr, Bethke, nd Ponse [12, 13] returned in 1993 to the binry version x y of the Kleene str in process lgebr, nming the opertor fter its discoverer. Troeger [73] introduced process specifiction lnguge with itertion, in which he introduced striking xiom for the binry Kleene str, here presented in process lgebr nottion: x (y ((x + y) z) + z) = (x + y) z. Fokkink nd Zntem [39, 44] gve n ffirmtive nswer to n open question from [13], nmely, tht the defining xiom nd Troeger s xiom for the binry Kleene str, together with x (y z) = (x y) z nd the five stndrd xioms on + nd for bsic process lgebr [18], provide n equtionl chrcteristion of strong bisimultion equivlence. Corrdini, De Nicol, nd Lbell studied the unry Kleene str in the presence of dedlock modulo resource bisimultion equivlence. In [31, 32] they presented complete xiomtistion bsed on Kozen s conditionl xioms. In [33] they cme up with complete equtionl xiomtistion including Troeger s xiom. Aceto, Fokkink, nd Ingólfsdóttir [4] proved tht whole rnge of process semntics corser thn strong bisimultion do not llow finite equtionl chrcteristion of the binry Kleene str. Furthermore, Sewell [72] showed tht there

5 5 does not exist finite equtionl chrcteristion of the binry Kleene str modulo strong bisimultion equivlence in the presence of the dedlock δ, due to the fct tht ( k ) δ is strongly bisimilr to δ for positive integers k. Severl vritions of the binry Kleene str were introduced, to obtin prticulr desirble properties. In order to increse the expressive power of the binry Kleene str in strong bisimultion semntics, Bergstr, Bethke, nd Ponse [12] proposed multiexit itertion (x 1,..., x k ) (y 1,..., y k ) for positive integers k, with s defining eqution (x 1,..., x k ) (y 1,..., y k ) = x 1 ((x 2,..., x k, x 1 ) (y 2,..., y k, y 1 )) + y 1. Aceto nd Fokkink [1] presented n xiomtic chrcteristion of multi-exit itertion in bsic process lgebr, modulo strong bisimultion equivlence. Prefix itertion (similr to the dely opertion from Hennessy [51]) is obtined by restricting the left-hnd side of the binry Kleene str to tomic ctions. Aceto, Fokkink, nd Ingólfsdóttir [5, 37] presented finite equtionl chrcteristions of prefix itertion in bsic CCS [59], modulo whole rnge of process semntics. String itertion [7] is obtined by restricting the left-hnd side of the binry Kleene str to non-empty finite strings of tomic ctions. Aceto nd Groote [7] presented n equtionl chrcteristion of string itertion in bsic CCS, modulo strong bisimultion equivlence. Bergstr, Bethke, nd Ponse [12] introduced flt itertion, which is obtined by restricting the left-hnd side of the binry Kleene str to sums of tomic ctions. Unlike the binry Kleene str, the merge opertor cn be eliminted from process terms in the presence of flt itertion. In [48], vn Glbbeek presented complete finite equtionl chrcteristion of flt itertion in bsic CCS extended with the silent step τ [59], modulo four rooted wek bisimultion semntics tht tke into ccount the silent nture of the specil constnt τ (identifying xτ nd x). This chpter lso concerns expressivity of (subsystems of) the lgebr of communicting processes (ACP) [18] extended with the constnt τ nd bstrction opertors [19], nd enriched with the binry Kleene str or vrints thereof. The τ, which represents silent step, in combintion with bstrction opertors, which renme ctions into τ, enbles one to bstrct from internl behviour. In weker semntics tht identify xτ nd x, ech regulr process cn be specified in ACP τ (i.e., ACP with bstrction) nd the binry Kleene str, using only hndshking (i.e., two-prty communiction) [21] nd some uxiliry ctions. Furthermore, in such semntics, ech computble process cn be specified in ACP τ with bstrction nd

6 6 2 PRELIMINARIES: AXIOMS AND OPERATIONAL SEMANTICS single recursive, non-regulr opertion, using only hndshking nd some uxiliry ctions. This chpter is set up s follows. Section 2 introduces the preliminries. Section 3 contins n exposition on xiomtistions for the binry Kleene str, while Section 4 discusses xiomtistions for relted itertive opertions. Section 5 compres the expressivity of these itertive opertions, nd Section 6 studies the expressivity of some non-regulr recursive opertions. Finlly, Section 7 touches upon topics such s firness nd the empty process. Acknowledgements. We thnk Fron Moller for providing useful comments. 2 Preliminries: Axioms nd Opertionl Semntics In this section we recll vrious process lgebr xiom systems, structurl opertionl semntics, behviourl equivlences, nd recursive specifictions. For detiled introduction to these mtters see, e.g., Beten nd Weijlnd [28]. 2.1 ACP-Bsed Systems Let A be finite set of tomic ctions, b, c,..., nd let δ be constnt not in A. We write A δ for A {δ}. Let furthermore : A δ A δ A δ be communiction function tht is commuttive, b = b for ll, b A δ, ssocitive, (b c) = ( b) c for ll, b, c A δ, nd tht stisfies δ = δ for ll A δ. The communiction function will be used to define communiction ctions: in the cse tht b = c A, the simultneous execution of ctions nd b results in communiction ction c. The ctions in A nd the communiction function cn be regrded s prmeters of the process lgebr xiom systems tht re presented below. The process lgebric frmework ACP(A,, τ) stnds for prticulr signture over fixed A nd communiction function, nd set of xioms over this signture. Let

7 2.1 ACP-Bsed Systems 7 P denote the set of process terms over this signture: sorts: A (the given, finite set of tomic ctions), P (the set of process terms; A P), opertions: + : P P P (non-deterministic choice or sum), : P P P (sequentil composition), : P P P (merge), : P P P (left merge), : P P P (communiction merge, extending H : P P τ I : P P the given communiction function), (encpsultion, H A), (bstrction, I A), constnts: δ P (dedlock), τ P (silent step). Intuitively, n ction represents indivisible behviour, δ represents inction, nd τ represents invisible internl behviour. Moreover, P + Q executes either P or Q, P Q first executes P nd t its successful termintion proceeds to execute Q, nd P Q executes P nd Q in prllel llowing communiction of ctions from P nd Q. The opertors P Q nd P Q both cpture prt of the behviour of P Q: P Q tkes its first trnsition from P, while the first trnsition of P Q is communiction of ctions from P nd Q. Finlly, in H (P ) ll ctions from H in P re blocked, while in τ I (P ) ll ctions from I in P re renmed into τ. We tke to be the opertion tht binds strongest, nd + the one tht binds wekest. As usul in lgebr, we tend to write xy insted of x y. For {+,,, } we will ssume tht expressions P 0 P n ssocite to the right. Furthermore, for k 1 we define x k+1 s x x k, nd x 1 s x. In Tble 1 the xioms of the system ACP(A,, τ) re collected. Note tht + nd re ssocitive, nd tht + is moreover commuttive nd idempotent. In the cse tht (b c) = δ for ll, b, c A, while itself defines communiction, we spek of hndshking. We will study the following subsystems of ACP(A,, τ): BPA(A). The signture of BPA(A) contins the elements of A, non-deterministic choice, nd sequentil composition. The xioms of BPA(A) re (A1) (A5). BPA δ (A). The signture of BPA δ (A) is the signture of BPA(A) extended with the dedlock. The xioms of BPA δ (A) re (A1) (A7). PA(A). The signture of PA(A) is the signture of BPA(A) extended with the merge nd left merge. The xioms of PA(A) re (A1) (A5), (CM2) (CM4)

8 8 2 PRELIMINARIES: AXIOMS AND OPERATIONAL SEMANTICS (with rnging over A), nd (M1) x y = x y + y x. PA δ (A). The signture of PA δ (A) is the signture of PA(A) extended with the dedlock. The xioms of PA(A) re (A1) (A7), (M1), nd (CM2) (CM4) (with rnging over A δ ). ACP(A, ). The signture of ACP(A, ) is the signture of ACP(A,, τ) without the silent step nd bstrction opertors. The xioms of ACP(A, ) re (A1) (A7), (CF1) (CF2), (CM1) (CM9), nd (D1) (D4) (with, b rnging over A δ ). We note tht in PA(A) nd PA δ (A), commuttivity of the merge (x y = y x) cn be derived from xioms (A1) nd (M1). The binry equlity reltion = on process terms induced by n xiom system is obtined by tking ll closed instntitions of xioms, nd closing it under equivlence (i.e., under reflexivity, symmetry, nd trnsitivity) nd under contexts. 2.2 Trnsition Rules nd Opertionl Semntics We define structurl opertionl semntics in the style of Plotkin [68], to relte ech process term to lbelled trnsition system. Then we define strong bisimultion s n equivlence between lbelled trnsition systems, which crries over to process terms. The opertionl semntics nd strong bisimultion re used in the proofs on xiomtistions in Sections 3 nd 4, nd on clssifiction results in Section 5. A lbelled trnsition system (LTS) is tuple S, {, A}, s, where S is set of sttes, for A is binry reltion between sttes, for A is unry predicte on sttes, s S is the initil stte. Expressions s s nd s re clled trnsitions. Intuitively, s s denotes tht from stte s one cn evolve to stte s by the execution of ction, while s denotes tht from stte s one cn terminte successfully by the execution of ction ( is pronounced tick ). Consider one of the process lgebr xiom systems BPA(A) ACP(A,, τ), nd let P represent ll process terms given by its signture. We wnt to relte ech process term in P to lbelled trnsition system. We tke the process terms in P s the set of sttes, nd the tomic ctions in A s the set of lbels. (Note tht tomic ctions cn denote both sttes nd lbels.) Exploiting the syntctic structure of process terms, the trnsition reltions nd for A re defined by mens of inductive proof rules clled trnsition rules S c. Vlidity of the premises in

9 2.2 Trnsition Rules nd Opertionl Semntics 9 Tble 1: The xioms for ACP(A,, τ), where, b A δτ nd H, I A. (A1) x + y = y + x (A2) x + (y + z) = (x + y) + z (A3) x + x = x (A4) (x + y)z = xz + yz (A5) (xy)z = x(yz) (A6) x + δ = x (A7) δx = δ (C1) b = b (C2) ( b) c = (b c) (C3) δ = δ (CM1) x y = (x y + y x) + x y (CM2) x = x (CM3) x y = (x y) (CM4) (x + y) z = x z + y z (CM5) x b = ( b)x (CM6) bx = ( b)x (CM7) x by = ( b)(x y) (CM8) (x + y) z = x z + y z (CM9) x (y + z) = x y + x z (D1) H () = if H (D2) H () = δ if H (D3) H (x + y) = H (x) + H (y) (D4) H (xy) = H (x) H (y) (B1) xτ = x (TI1) τ I () = if I (TI2) τ I () = τ if I (TI3) τ I (x + y) = τ I (x) + τ I (y) (TI4) τ I (xy) = τ I (x)τ I (y)

10 10 2 PRELIMINARIES: AXIOMS AND OPERATIONAL SEMANTICS S, under certin substitution, implies vlidity of the conclusion c under the sme substitution. The trnsition rules in Tble 2 define the lbelled trnsition system ssocited to process term in ACP(A,, τ). The signture nd prmeters of P (possibly including communiction function ) determine which trnsition rules re pproprite. For exmple, the lst two trnsition rules for (i.e., with x y in the left-hnd sides of their conclusions) re not relevnt for PA δ (A). Note tht the dedlock δ hs no outgoing trnsitions. The lbelled trnsition system relted to process term P hs P itself s initil stte. Often we will write simply P for the lbelled trnsition system relted to P. Intuitively, strong bisimultion reltes two sttes if the LTSs rooted t these sttes hve the sme brnching structure. This semntics does not tke into ccount the silent nture of τ. Definition [Strong bisimultion] A strong bisimultion is binry, symmetric reltion R over the set of sttes tht stisfies P R Q P P R Q P P = Q (Q Q P R Q ) = Q Two sttes P nd Q re strongly bisimilr, nottion P Q, if there exists strong bisimultion reltion R with P R Q. Theorem (Equivlence) It is not hrd to see tht strong bisimultion is n equivlence reltion over ACP(A, ). 2. (Congruence) Strong bisimultion equivlence is congruence reltion up to ACP(A, ), mening tht P 0 Q 0 nd P 1 Q 1 implies P 0 + P 1 Q 0 + Q 1, P 0 P 1 Q 0 Q 1, P 0 P 1 Q 0 Q 1, P 0 P 1 Q 0 Q 1, P 0 P 1 Q 0 Q 1, nd H (P 0 ) H (Q 0 ). This follows from the fct tht the trnsition rules in Tble 2 re in pth formt; see [27, 36, 49, 6]. 3. (Soundness) Up to nd including ACP(A, ), ll xiom systems re sound with respect to strong bisimiltion equivlence, mening tht P = Q implies P Q. Since strong bisimultion is congruence, soundness follows from the fct tht ll closed instntitions of xioms in ACP(A, ) re vlid in strong bisimultion semntics. 4. (Completeness) Up to nd including ACP(A, ), ll xiom systems re complete with respect to strong bisimiltion equivlence, mening tht P Q implies P = Q; see, e.g., [28, 43].

11 2.2 Trnsition Rules nd Opertionl Semntics 11 Tble 2: Trnsition rules for ACP(A,, τ), where, b A τ, H, I A., A τ x x y y x x + y x x + y y x + y y x + y x x xy x y x xy y x x x y x y y y x x y x y x y y y x y x x x y b y x y b x y if b A x y b x y b if b A x y b y x y b y if b A x x y b x y b x if b A x x x y x y x x y y x x y b y x y b x y x y b y x y b y x x y b x y b x x y b x y b x x H (x) H (x ) if H x H (x) if H x x τ τ I (x) τ I (x ) if I x τ τ I (x) if I x x τ I (x) τ I (x ) if I x τ I (x) if I

12 12 2 PRELIMINARIES: AXIOMS AND OPERATIONAL SEMANTICS We proceed to define some more semntic equivlence reltions on sttes in lbelled trnsition system tht do not tke into ccount the silent nture of τ. These definitions use the following notions, ssuming n underlying LTS. Let P σ Q for σ A denote tht stte P cn evolve to stte Q by the execution of the sequence of ctions σ. This binry reltion on sttes is defined s follows (with ɛ denoting the empty string): - P ɛ P ; - P σ Q Q R P σ ; R - P σ Q Q P σ. A string σ A is trce of P if P σ Q for some stte Q or P σ. Definition [Substte] Q is substte of P if P σ Q for some σ A. Q is proper substte of P if P σ Q for some σ A \ {ɛ}. Definition [Redy simultion] A simultion is binry reltion R over the set of sttes tht stisfies: P R Q P P R Q P P = Q (Q Q P R Q ) = Q A simultion R is redy simultion if whenever P R Q nd is trce of Q, then is trce of P. Two sttes P nd Q re simultion equivlent, nottion P S Q, if P R 1 Q nd Q R 2 P for simultions R 1 nd R 2. Two sttes P nd Q re redy simultion equivlent, nottion P RS Q, if P R 1 Q nd Q R 2 P for redy simultions R 1 nd R 2. Definition [Lnguge equivlence] Two sttes P nd Q re lnguge equivlent, nottion P L Q, if for ech trce P σ there is trce Q σ, nd vice vers. Definition [Trce equivlence] Two sttes P nd Q re trce equivlent, nottion P T Q, if they give rise to the sme set of trces.

13 2.3 Recursive Specifictions 13 In [45], vn Glbbeek gve comprison of wide rnge of behviourl equivlences nd preorders (i.e., reltions tht re in generl not symmetric) tht do not tke into ccount the silent nture of τ. Aprt from the equivlence reltion discussed bove, he studied completed trce preorder nd vriety of decorted trce preorders, which re bsed on decorted versions of trces. These preorders re corser thn strong bisimultion nd redy simultion, but more refined thn lnguge nd trce equivlence. In Section 4.3, prefix itertion is xiomtised with respect to some of the decorted trce preorders. The reder is referred to [45] for the definitions of these preorders. Four stndrd semntic equivlences tht tke into ccount the silent nture of τ (nd tht constitute congruence reltions over ACP(A,, τ)) re rooted brnching bisimultion [50], rooted dely bisimultion [60], rooted η-bisimultion [16], nd rooted wek bisimultion [62]. Of these four semntics, rooted brnching bisimultion constitutes the finest reltion, while rooted wek bisimultion constitutes the corsest reltion. In ll four semntics the xiom (B1) (i.e., xτ = x) is vlid, nd tht is ll tht is needed for the expressivity results concerning ACP(A,, τ) discussed in this chpter. In [47], vn Glbbeek gve comprison of wide rnge of process semntics tht tke into ccount the silent nture of τ. 2.3 Recursive Specifictions Although itertive opertions re recursive by nture, we will use recursive specifictions nd ssocited notions to prove some of the results concerning these opertions. Definition [Recursion] We ssume set of recursion vribles {X j j J} for some index set J. A recursive specifiction E is set of recursive equtions X j = T j for j J, where T j is n ACP(A,, τ)-term in which the recursion vribles X i for i J my occur. Given some process semntics, processes P j (for j J) form solution of E if substitution of P j for X j in the recursive equtions of E yields equtions tht hold in this semntics. If recursive specifiction hs solutions, then these solutions re often referred to by the nmes of the corresponding recursion vribles in E. Tble 3 presents the trnsition rules for recursive specifictions. T X x x Tble 3: Trnsition rules for recursion. if X = T E T X if X = T E

14 14 2 PRELIMINARIES: AXIOMS AND OPERATIONAL SEMANTICS If for instnce E = {X = X + b}, then X X b by the second trnsition rule. X by the first trnsition rule, nd Recursive specifictions need not hve unique solutions in ny resonble process semntics, exmples being {X = X} nd {X = τx}. The following two definitions re sufficient to remedy this imperfection. Definition [τ-gurdedness] Let P be n expression contining recursion vrible X. An occurrence of X in P is τ-gurded if P hs subexpression Q where A τ nd Q contins this occurrence of X. A recursive specifiction E = {X j = T j j J} is τ-gurded if by repetedly substituting T j expressions for occurrences of X j, nd by pplying xioms of ACP(A,, τ), one cn obtin the sitution tht ll occurrences of recursion vribles in right-hnd sides of recursive equtions re τ-gurded. Definition [τ-convergence] A recursive specifiction E = {X j = T j j J} σ is τ-convergent if X j P (for j J) implies tht there is no infinite τ-trce P τ P τ P τ. Recursive specifictions tht re τ-gurded nd τ-convergent hve unique solution modulo ny resonble process semntics. The existence of solution underlies the soundness of the recursive definition principle [10]. We proceed to introduce the recursive specifiction principle (RSP), discussed in for instnce [20, 10]. This principle sttes tht the recursive specifiction E = {X j = T j j J} hs t most one solution per recursion vrible, modulo the process semntics under considertion. (RSP) y j = T j {y i /X i i J} for j J (for k J) X k = y k (RSP) is sound with respect to the process equivlences mentioned thus fr, provided E is both τ-convergent nd τ-gurded. Note tht (RSP) is not sound with respect to the recursive specifictions {X = X} nd {X = τx}. Definition [Regulr process] A process P is regulr if P is bisimilr to process τ I (Q) where Q is solution for recursion vrible in finite recursive specifiction {X i = n α i,j X j + β i i = 1,..., n} j=1 where α i,j nd β j re finite sums of ctions in A (the empty sum representing δ).

15 15 3 Axiomtistion of the Binry Kleene Str In Section 1 we introduced the binry Kleene str (BKS), nottion. This opertion is defined by the eqution x y = x(x y) + y. This section considers its finite equtionl xiomtistion in strong bisimultion semntics, nd presents negtive result on the finite equtionl xiomtisbility of BKS in vriety of other process semntics. 3.1 Preliminries BPA (A) is obtined by extending BPA(A) with BKS. Bergstr, Bethke, nd Ponse [13] introduced n xiomtistion for BPA (A) modulo strong bisimultion equivlence, which consists of xioms (A1) (A5) for BPA(A), extended with the xioms (BKS1) (BKS3) for BKS in Tble 4. The xiom (BKS3) stems from Troeger [73]. Tble 4: Axioms for BKS. (BKS1) x(x y) + y = x y (BKS2) x (yz) = (x y)z (BKS3) x (y((x + y) z) + z) = (x + y) z In order to provide process terms over BPA (A) with n opertionl semntics, we introduce trnsition rules for BKS. The trnsition rules for BKS in Tble 5 express tht x y repetedly executes x until it executes y. Together with the trnsition rules for BPA(A) in Tble 2 they provide lbelled trnsition systems to process terms over BPA (A). Note tht by the first two trnsition rules in Tble 5, stte P cn hve itself s proper substte. For exmple, b b. Tble 5: Trnsition rules for BKS. x x x y x (x y) x x y x y y y y x y y x y The trnsition rules for BKS re in pth formt [27, 49]. Hence, strong bisimultion equivlence is congruence with respect to BPA (A); see [36, 6]. Furthermore, its

16 16 3 AXIOMATISATION OF THE BINARY KLEENE STAR xiomtistion is sound for BPA (A) modulo strong bisimultion equivlence. Since strong bisimultion equivlence is congruence, this cn be verified by checking soundness for ech xiom seprtely. It cn be esily shown tht the BKS xioms re vlid in strong bisimultion. Fokkink nd Zntem [44] proved tht the xiomtistion for BPA (A) is complete modulo strong bisimultion equivlence. Their proof is bsed on term rewriting nlysis (see, e.g., [22]), in quest to reduce bisimilr process terms to the sme ground norml form, which does not reduce ny further. Since this im cnnot be fulfilled for BKS, this opertor is replced by n opertor representing x(x y), nd the BKS xioms re dopted to fit this new opertor. Those xioms re turned into conditionl rewrite rules, which re pplied modulo ssocitivity nd commuttivity of the + (see, e.g., [67]). Knuth-Bendix completion [52] is pplied to mke the conditionl rewrite system wekly confluent. Termintion of the resulting conditionl term rewriting system is obtined by mens of the technique of semntic lbelling from Zntem [75]. Hence, ech process term is provbly equl to norml form. Finlly, creful cse nlysis lerns tht if two norml forms re strongly bisimilr, then they re syntcticlly equl modulo ssocitivity nd commuttivity of the +. This observtion yields the desired completeness result. An lterntive completeness proof ws proposed in [39], bsed on induction on the structure of process terms. Tht proof method is more generl, nd ws lter on pplied to obtin completeness results for xiomtistions of itertion opertions in [1, 33, 40]. In the light of the generic pplicbility of this proof method nd the significnce of the completeness result in the relm of this chpter, we present the proof from [39] in some detil. Following Milner [63] (see lso [46]), the ltter proof strtegy cn lso be used to derive ω-completeness of the xiomtistion for BPA (A). Tht is, if P nd Q re open terms over BPA (A), which my contin vribles, nd if σ(p ) = σ(q) holds for ll closed instntitions σ, then P = Q cn be derived from the xioms. Often, ω-completeness cn be proved by providing vribles with n opertionl semntics, such tht P Q holds with respect to this new opertionl semntics if nd only if σ(p ) σ(q) holds for ll closed instntitions σ with respect to the originl opertionl semntics. In [39], completeness of the xiomtistion for BPA (A) modulo bisimultion equivlence is derived for open terms, which immeditely implies ω- completeness of the xiomtistion. Here, we present the proof from [39] for closed (insted of open) terms. The reder is referred to [39] for proof of ω-completeness. (The motivtion to refrin from this generlistion here is clrity of presenttion; we prefer to work in n unmbiguous semntic frmework throughout this chpter.) 3.2 Completeness We note tht ech process term over BPA (A) hs only finitely mny substtes. In the sequel, process terms re considered modulo ssocitivity nd commuttivity of the +, nd we write P = AC Q if P nd Q cn be equted by xioms (A1) nd

17 3.2 Completeness 17 (A2). As usul, n i=1 P i represents P 1 + +P n. We tke cre to void empty sums (where 0 i=1 P i + Q is not considered empty nd equls Q). For ech process term P, its collection of possible trnsitions is non-empty nd finite, sy {P i P i i = 1,..., m} {P b j j = 1,..., n}. We cll m i P i + i=1 n j=1 b j the HNF-expnsion of P (Hed Norml Form expnsion, cf. [28]). The process terms i P i nd b j re the summnds of P. Lemm Ech process term is provbly equl to its HNF-expnsion. Proof. Strightforwrd, by structurl induction, using (A4), (A5), nd (BKS1). Process terms in BPA (A) re normed, which mens tht they re ble to terminte in finitely mny trnsitions. The norm [11] of process yields the length of the shortest termintion trce of this process. Norm cn be defined inductively by = 1 P + Q = min{ P, Q } P Q = P + Q P Q = Q. We note tht strongly bisimilr processes hve the sme norm. The following lemm, due to Cucl [29], is typicl for normed processes. Lemm Let P Q RS. By symmetry we my ssume Q S. We cn distinguish two cses: either P R nd Q S; or there is proper substte P of P such tht P RP nd P Q S. Proof. This lemm follows from the following fcts A nd B. A. If P Q RS nd Q S, then either Q S, or there is proper substte P of P such tht P Q S. We prove fct A by induction on P. First, let P = 1. Then P for some, so P Q Q. Since P Q RS, nd R S > S Q for ll substtes R of R, it follows tht R nd Q S. Then we re done. Next, suppose we hve proved the cse for P n, nd let P = n + 1. Then there is P with P = n nd P P, which implies P Q P Q. Since P Q RS, we hve two options:

18 18 3 AXIOMATISATION OF THE BINARY KLEENE STAR 1. R nd P Q S. Then we re done. 2. R R nd P Q R S. Since P = n, induction yields either Q S or P Q S for proper substte P of P. Agin, we re done. This concludes the proof of fct A. B. If P Q RQ, then P R. Define binry reltion B on process terms by T B U iff T Q UQ. We show tht B constitutes strong bisimultion reltion between P nd R. Since is symmetric, so is B. P Q RQ, so P B R. Suppose T B U nd T. Then T Q Q. Since T Q U Q, nd U Q > Q for ll substtes U of U, it follows tht UQ Q. In other words, U. Suppose T B U nd T T. Then T Q T Q. Since T Q UQ, nd Q < T Q, it follows tht there is trnsition U U with T Q U Q. Hence, T B U. This concludes the proof of fct B. Finlly, we show tht fcts A nd B together prove the lemm. Let P Q RS with Q S. According to fct A we cn distinguish two cses. Q S. Then P Q RS RQ, so fct B yields P R. P Q S for some proper substte P of P. Then P Q RS RP Q, so fct B yields P RP. We construct set B of bsic terms, such tht ech process term is provbly equl to bsic term. The completeness theorem is proved by showing tht strongly bisimilr bsic terms re provbly equl. (x + y)z xz + yz (xy)z x(yz) (x y)z x (yz) The term rewriting system bove consists of directions of the xioms (A4), (A5), nd (BKS2), pointing from left to right. Its rewrite rules re to be interpreted modulo

19 3.2 Completeness 19 ssocitivity nd commuttivity of the +. The term rewriting system is terminting, mening tht there re no infinite reductions. This follows from the following weight function w in the nturl numbers: w() = 2 w(p + Q) = w(p ) + w(q) w(p Q) = w(p ) 2 w(q) w(p Q) = w(p ) + w(q). It is not hrd to see tht if P reduces to Q in one or more rewrite steps, then w(p ) > w(q). Since the ordering on the nturl numbers is well-founded, we conclude tht the term rewriting system is terminting. Let G denote the collection of ground norml forms, i.e., the collection of process terms tht cnnot be reduced by ny of the three rewrite rules. Since the term rewriting system is terminting, nd since its rewrite rules re directions of xioms, it follows tht ech process term is provbly equl to process term in G. The elements in G re defined by: P ::= P + P P P P. G is not yet our desired set of bsic terms, due to the fct tht there exist process terms in G which hve substte outside G. We give n exmple. Exmple Let A = {, b, c}. Clerly, ( b) c G, nd ( b) c ( b)(( b) c). The substte ( b)(( b) c) is not in G, becuse the third rewrite rule in R reduces this process term to (b(( b) c)). In order to overcome this compliction, we introduce the following collection of process terms: H = {P Q, P (P Q) P Q G nd P is proper substte of P }. We define n equivlence reltion = on H by putting P (P Q) = P Q for proper substtes P of P, nd tking the reflexive, symmetric, trnsitive closure of =. The set B of bsic terms is the union of G nd H. Lemm If P B nd P P, then P B. Proof. By induction on the structure of P. If P H\G, then it is of the form Q (Q R) for some Q R G. So P is of the form either Q R or Q (Q R) for some proper substte Q of Q. In both cses, P B. If P G, then it is of the form i iq i + j R j S j + k b k, where the Q i, R j, nd S j re in G. So P is of the form Q i, R j Sj, R j (R j S j ), or S j, which re ll bsic terms (in the lst cse, this follows by structurl induction).

20 20 3 AXIOMATISATION OF THE BINARY KLEENE STAR The L-vlue [44] of process term is defined by L(P ) = mx{ P P proper substte of P }. L(P ) < L(P Q) nd L(P ) < L(P Q), becuse for ech proper substte P of P, P Q is proper substte of P Q nd P (P Q) is proper substte of P Q. Since norm is preserved under strong bisimultion, it follows tht the sme holds for L-vlue; i.e., if P Q then L(P ) = L(Q). We define n ordering on B s follows: P Q if L(P ) < L(Q); P Q if P is substte of Q but Q is not substte of P ; if P Q nd Q R, then P R. Note tht if P, Q H with P = Q, then P nd Q hve the sme proper substtes, nd so L(P ) = L(Q). These observtions imply tht the ordering on B respects the equivlence = on H, tht is, if P = Q R = S, then P S. Lemm is well-founded ordering on B. Proof. If P is substte of Q, then ll proper substtes of P re proper substtes of Q, so L(P ) L(Q). Hence, if P Q then L(P ) L(Q). Assume, towrd contrdiction, tht there exists n infinite bckwrd chin P 2 P 1 P 0. Since L(P n+1 ) L(P n ) for ll n N, there is n N such tht L(P n ) = L(P N ) for ll n > N. Since P n P N for n > N, it follows tht P n is substte of P N for n > N. Ech process term hs only finitely mny substtes, so there re m, n > N with n > m nd P n = AC P m. Then P n P m, so we hve found contrdiction. Hence, is well-founded. In the proofs of the next two lemms, we need weight function g in the nturl numbers, which is defined inductively by g() = 0 g(p + Q) = mx{g(p ), g(q)} g(p Q) = mx{g(p ), g(q)} g(p Q) = mx{g(p ), g(q) + 1}. It is not hrd to see, by structurl induction, tht if P P, then g(p ) g(p ). Lemm Let P Q B. If Q is proper substte of Q, then Q P Q.

21 3.2 Completeness 21 Proof. Q is substte of Q, so g(q ) g(q) < g(p Q), which implies tht P Q cnnot be substte of Q. On the other hnd, Q is substte of P Q, so Q P Q. Lemm If P B nd P P, then either P P, or P, P H nd P = P. Proof. This lemm follows from the following fcts A nd B. A. If P B nd P P, then either P H or P hs smller size thn P. We prove fct A by induction on the structure of P. Let Since P P = AC i Q i + i j R j Sj + k b k. P, P is of one of the following forms. P = AC Q i for some i. Then P hs smller size thn P. P = AC R j (R j S j ) or P = AC R j Sj for some j. Then P H. S j P for some j. Then induction yields tht either P H, or P hs smller size thn S j, which in turn hs smller size thn P. This concludes the proof of fct A. B. If P H nd P P, then either g(p ) > g(p ), or P H nd P = P. Since P H, either P = AC Q (Q R) or P = AC Q R for some Q nd R. Hence, P = AC Q (Q R), P = AC Q R, or P = AC R for proper substte R of R. In the first two cses P H nd P = P, nd in the lst cse g(p ) = g(r ) g(r) < g(q R) = g(p ). This concludes the proof of fct B. Finlly, we show tht fcts A nd B together prove the lemm. Let P P P ; we prove tht P, P H nd P = P. P with Since P is substte of P nd P P, P is substte of P. So there exists sequence of trnsitions P P1 n Pn (n 1) where P 0 = AC P, P 1 = AC P, nd P n = AC P. Suppose P k H for ll k. Then ccording to fct A, P k+1 hs smller size thn P k for k = 0,..., n 1, so P n hs smller size thn P 0. This contrdicts P 0 = AC P = AC P n. Hence, P l H for some l.

22 22 3 AXIOMATISATION OF THE BINARY KLEENE STAR Since ech P k is substte of ech P k, g(p k ) must be the sme for ll k. Then it follows from fct B, together with P l H, tht P k H for ll k nd P 0 = P1 = = P n. Now we re redy to prove the desired completeness result for BPA (A). Theorem (A1) (A5), (BKS1) (BKS3) is complete for BPA (A) modulo strong bisimultion equivlence. Proof. Ech process term is provbly equl to bsic term, so it is sufficient to show tht strongly bisimilr bsic terms re provbly equl. Assume P, Q B with P Q; we show tht P = Q, by induction on the ordering. To be precise, we ssume tht we hve lredy delt with strongly bisimilr pirs R, S B with R P nd S Q, or R P nd S = Q, or R = P nd S Q. First, ssume tht P or Q is not in H, sy P H. By the induction hypothesis, together with Lemm 3.2.7, for ll trnsitions P P nd Q Q with P Q we hve P = Q. Since P Q, xiom (A3) cn be used to dpt the HNF-expnsions of P nd Q to the form P = m n i P i + b j Q = i=1 j=1 m n i Q i + b j, i=1 j=1 where P i = Q i for i = 1,..., m. Hence, P = Q. Next, ssume P, Q H. We distinguish three cses. 1. Let P = AC R S nd Q = AC T U. We prove R S = T U. We spell out the HNF-expnsions of R nd T : R = R i, T = T j, i I j J where the R i nd the T j re of the form either V or. Since R S T U, ech R i (R S) for i I is strongly bisimilr either to T j (T U) for j J or to summnd of U. We distinguish these two cses. () R i (R S) T j (T U) for j J. Then R i (R S) T j (R S), so by Lemm R i T j. (b) R i (R S) U for trnsition U U. Thus, I cn be divided into the following, not necessrily disjoint, subsets: I 0 = {i I j J (R i T j )} I 1 = {i I U U (R i (R S) U )}.

23 3.2 Completeness 23 Similrly, J cn be divided: J 0 = {j J i I (T j R i )} J 1 = {j J S S (T j (T U) S )}. If both I 1 nd J 1 re non-empty, then U R S for proper substte U of U, nd S T U for proper substte S of S, nd so U S. By Lemm 3.2.6, S R S nd U T U, so induction yields R S = U = S = T U, nd we re done. Hence, we my ssume tht either I 1 or J 1 is empty, sy J 1 =. We proceed to derive i I 1 R i (R S) + S = U. (1) We show tht ech summnd t the left-hnd side of the equlity sign is provbly equl to summnd of U, nd vice vers. By definition of I 1, for ech R i (R S) with i I 1 there is summnd U of U such tht R i (R S) U. By Lemm U T U, so induction yields R i (R S) = U. Consider summnd S of S. Since R S T U nd J 1 =, it follows tht S is strongly bisimilr to summnd U of U, so induction yields S = U. Finlly, summnds of S correspond with summnds of U. By the converse rguments it follows tht ech summnd of U is provbly equl to summnd t the left-hnd side of the equlity sign. This concludes the derivtion of (1). Since J 1 =, it follows tht J 0, so I 0. By the definitions of I 0 nd J 0 = J, ech R i with i I 0 is strongly bisimilr to T j with j J, nd vice vers. Since L(R i ) L(R) < L(R S), induction yields R i = T j. Hence, i I 0 R i = T. (2) Finlly, we derive R S (A3) = (BKS3),(A3) = ( i I 0 R i + i I 1 R i ) S ( i I 0 R i ) ( i I 1 R i (R S) + S) (1),(2) = T U.

24 24 3 AXIOMATISATION OF THE BINARY KLEENE STAR 2. Let P = AC R (R S) nd Q = AC T U. We prove R (R S) = T U. U = T U = R (R S) 2 implies tht U does not hve tomic summnds, so its HNF-expnsion is of the form i iu i. Since R (R S) T U, ech U i is strongly bisimilr to R S or to R (R S) for proper substte R of R. According to Lemm U i T U, so induction yields U i = R S or U i = R (R S). This holds for ll i, so U = i iu i = V (R S) for some process term V. Then R (R S) T U (T V )(R S), so Lemm implies R T V. Since L(R ) < L(R (R S)), induction yields R = T V. Hence, R (R S) = (T V )(R S) (BKS2) = T (V (R S)) = T U. 3. Let P = AC R (R S) nd Q = AC T (T U). We prove R (R S) = T (T U). By symmetry we my ssume R S T U. Lemm distinguishes two possible cses. R T nd R S T U. Since L(R ) < L(R (R S)), induction yields R = T, nd cse 1 pplied to R S T U yields R S = T U. Hence, R (R S) = T (T U). R T R nd R (R S) T U for proper substte R of R. Since L(R ) < L(R (R S)), induction yields R = T R. Furthermore, cse 1 pplied to R (R S) T U yields R (R S) = T U. Hence, R (R S) = (T R )(R S) (BKS2) = T (R (R S)) = T (T U). This finishes the derivtion of P = Q. 3.3 Irredundncy of the Axioms Fokkink [39] showed tht ech of the BKS xioms is essentil for the obtined completeness result. Theorem If one of the BKS xioms is skipped from (A1) (A5), (BKS1) (BKS3), then this xiomtistion is no longer complete for BPA (A) modulo strong bisimultion equivlence. Proof. We pply stndrd technique for proving tht n eqution e cnnot be derived from n equtionl theory E, which prescribes to define model for E in which e is not vlid. In order to show tht (BKS1) cnnot be derived from (A1) (A5), (BKS2), (BKS3), we define n interprettion function φ of open terms in the nturl numbers: φ() = 0 φ(x) = 0 φ(p + Q) = mx{φ(p ), φ(q)} φ(p Q) = φ(p ) φ(p Q) = mx{φ(p ) + 1, φ(q) + 1}.

25 3.4 Negtive Results 25 It is esy to see tht this interprettion is model for (A1) (A5), (BKS2), (BKS3). However, φ(( ) + ) = 0, while φ( ) = 1. Hence, ( ) + = cnnot be derived from (A1) (A5), (BKS2), (BKS3). In order to show tht (BKS2) cnnot be derived from (A1) (A5), (BKS1), (BKS3) we define n interprettion function ψ of open terms in the nturl numbers: ψ() = 0 ψ(x) = 0 ψ(p + Q) = mx{ψ(p ), ψ(q)} ψ(p Q) = ψ(q) ψ(p Q) = mx{ψ(p ) + 1, ψ(q)}. It is esy to see tht this interprettion is model for (A1) (A5), (BKS1), (BKS3). However, ψ(( )) = ψ() = 0, while ψ( ()) = mx{ψ() + 1, ψ()} = 1. Hence, ( ) = () cnnot be derived from (A1) (A5), (BKS1), (BKS3). In order to show tht (BKS3) cnnot be derived from (A1) (A5), (BKS1), (BKS2), we define n interprettion function η of open terms in sets of nturl numbers: η() = η(x) = η(p + Q) = η(p ) η(q) η(p Q) = η(p ) η(q) η(p Q) = η(p ) η(q) { P }. It is esy to see tht this interprettion is model for (A1) (A5), (BKS1), (BKS2). However, η(() (((+) )+)) = {, + } = {1, 2} while η((+) ) = { + } = {1}. Hence, () ((( + ) ) + ) = ( + ) cnnot be derived from (A1) (A5), (BKS1), (BKS2). 3.4 Negtive Results In contrst with the positive result on the finite equtionl xiomtisbility of BPA (A) modulo strong bisimultion equivlence, Aceto, Fokkink, nd Ingólfsdóttir [4] showed tht BPA (A) is not finitely bsed modulo ny process semntics in between redy simultion (see Definition 2.2.4) nd lnguge equivlence (see Definition 2.2.5). In the cse of singleton lphbet, this nswered problem in regulr lnguges rised by Slom in [71]; see [3]. Crvenković, Dolink, nd Ésik [30] provided more elegnt nswer to the ltter question. Redy simultion nd lnguge equivlence constitute congruence reltions over BPA (A) (in the cse of lnguge equivlence this follows from the fct tht the trnsition rules for BPA (A) re in L cool formt [41]). Process semntics in the liner/brnching time spectrum [45] tht re finer thn lnguge equivlence nd corser thn redy simultion, nd which constitute congruence reltions over

26 26 3 AXIOMATISATION OF THE BINARY KLEENE STAR BPA (A), re filure semntics, redy semntics, filure trce semntics, nd redy trce semntics. The result bove follows from the existence of n infinite set of equtions tht cnnot ll be proved by mens of ny finite set of equtions tht is sound modulo lnguge equivlence. This fmily of equtions consists of E.n ( n ) + ( n ) ( + + n ) = ( n ) ( + + n ) for n 1, where is some ction. Redy simultion is the finest semntics in the liner/brnching time spectrum in which the E.n re sound. Note tht for n > 1, none of the equtions E.n is sound in strong bisimultion equivlence. Given finite set of equtions tht is sound with respect to lnguge equivlence, Aceto, Fokkink, nd Ingólfsdóttir construct model A p for these equtions in which eqution E.p fils, for some prime number p. The model tht is used for this purpose is bsed on n dpttion of construction due to Conwy [35], who used it to obtin new proof of theorem, originlly due to Redko [69], sying tht BPA (A) is not finitely bsed modulo lnguge equivlence. Let be n ction. For p prime number, the crrier A p of the lgebr A p consists of non-empty forml sums of 0, 1,..., p 1, together with the forml symbol, tht is, { i I i I {0,..., p 1}} { }. The syntx of A p contins three more opertors, which re semntic counterprts of the binry function symbols in BPA (A). In order to void confusion, circled symbols denote the opertors in the lgebr A p :,, nd represent the semntic counterprts of +,, nd, respectively. Tble 6 presents n xiomtistion for A p. Tble 6: Axiomtistion for the lgebr A p x = x = i I i j J j = h I J h x = x = i I i j J j = h {(i+j) mod p (i,j) I J} h x y = { y if x = 0 otherwise Process terms over BPA (A) re mpped to A p s expected: every ction in A is mpped to the symbol 1, while +,, nd re mpped to,, nd, respectively.

27 3.5 Extensions of BPA (A) 27 For process term P, the denottion of P in the lgebr A p is represented by A p [[P ]. We note tht eqution E.p fils in A p. Nmely, A p [[ ( p ) + ( p ) ( + + p )] = i = A p [[( p ) ( + + p )]. The following theorem is the key to the nonxiomtisbility result from [4]. Theorem For every finite set E of equtions tht re sound with respect to L, there exists prime number p such tht ll equtions in E re vlid in A p. Corollry No congruence reltion over BPA (A) tht is included in L nd stisfies E.n for ll n 1 hs complete finite equtionl xiomtistion. A process semntics tht is corser thn lnguge equivlence is trce equivlence T, where two process terms re considered equivlent if they give rise to the sme (not necessrily terminting) trces (see Definition 2.2.6). If A > 1, then trce equivlence is not congruence reltion over BPA(A); e.g., + T, but ( + )b T b. However, if the set A of ctions is singleton {}, then trce equivlence constitutes congruence reltion over BPA (A). In contrst with their negtive results on the finite xiomtisbility of BPA (A) modulo process semntics between redy simultion nd lnguge equivlence, Aceto, Fokkink, nd Ingólfsdóttir showed tht BPA ({}) modulo trce equivlence is xiomtised completely by the five xioms for BPA({}) together with the three xioms in Tble 7. p 1 i=0 Tble 7: Axioms for trce equivlence (A = {}) (T1) x + (y z) = (T2) x + xy = xy (T3) xy = yx Theorem equivlence. (A1) (A5), (T1) (T3) is complete for BPA ({}) modulo trce 3.5 Extensions of BPA (A) The signture of BPA δ (A) is obtined by extending BPA δ(a) with BKS. Its xioms re those of BPA (A) nd of BPA δ (A), i.e., (A1) (A7) in Tble 1 for BPA δ (A),

28 28 4 AXIOMATISATIONS OF OTHER ITERATIVE OPERATIONS nd (BKS1) (BKS3) for BKS. Sewell [72] showed tht there does not exist complete equtionl xiomtistion for BPA δ (A) modulo strong bisimultion equivlence. This motivtes the introduction of the implictionl xiom (RSP ) x = yx + z x = y z It remins n open question, dting bck to Milner [61], whether (A1) (A7), (BKS1), (RSP ) is complete modulo strong bisimultion equivlence. We note tht (BKS2) nd (BKS3) cn be derived from this xiomtistion. The signture of PA (A) is obtined by extending PA(A) with BKS. The xioms of PA (A) re those of PA(A) nd (BKS1) (BKS3). The system PA δ (A) cn be extended in similr wy to PA δ (A). The system ACP (A, ) is defined by inclusion of (BKS1) (BKS4); see Tble 8. Note tht (BKS4) cn be derived using (RSP ). Finlly, ACP (A,, τ) is obtined by inclusion into ACP(A,, τ) of (BKS1) (BKS5); see Tble 8. Note tht (RSP ) is not sound for ACP (A,, τ); e.g., τ = ττ + δ, but τ = τ δ is not desirble identity in ny process semntics. Tble 8: Axioms for BKS with encpsultion nd bstrction. (BKS4) H (x y) = H (x) H (y) (BKS5) τ I (x y) = τ I (x) τ I (y) 4 Axiomtistions of Other Itertive Opertions This section considers four restricted versions nd one generlised version of BKS. We discuss the different dvntges of ech of these opertors, nd formulte vrious xiomtistions nd completeness results. 4.1 Axiomtistion of No-Exit Itertion No-exit itertion (NEI) x ω is bisimilr to x δ. No-exit itertion cn be used to formlly describe progrms tht repet certin procedure without end. Mny communiction protocols cn be expressed, nd shown correct, using no-exit itertion.