Deadlocking States in Context-free Process Algebra

Transcription

1 Dedlocking Sttes in Context-free Process Algebr JiříSrb Fculty of Informtics MU, Botnická 68, Brno, Czech Republic Abstrct. Recently the clss of BPA (or context-free) processes hs been intensively studied nd bisimilrity nd regulrity ppered to be decidble (see [CHS95, BCS95, BCS96]). We extend these processes with dedlocking stte into BPA δ systems. Bosscher hs proved tht bisimilrity nd regulrity remin decidble [Bos97]. We generlise his pproch introducing strict nd nonstrict version of bisimilrity. We show tht the BPA δ clss is more expressive w.r.t. (both strict nd nonstrict) bisimilrity but it remins lnguge equivlent to BPA. Finlly we give chrcteriztion of those BPA δ processes which cn be equivlently (up to bisimilrity) described within the pure BPA syntx. 1 Introduction This pper dels with BPA processes (Bsic Process Algebr) extended with dedlocks. BPA represents the clss of processes introduced by Bergstr nd Klop (see [BK85]), which corresponds to the trnsition systems ssocited with Greibch norml form (GNF) context-free grmmrs in which only left-most derivtions re permitted. For detiled description of the reltion between lnguge nd process theory we refer to [HM96]. We ine the clss BPA δ of BPA processes extended with dedlocks nd introduce two lterntive initions strict nd nonstrict of bisimilrity within this clss. The inition of BPA δ systems is bsed on specil vrible δ (we cll it dedlock). In the usul presenttion every vrible used in BPA system is supposed to be ined but for the dedlock vrible we llow no inition. This cuses tht if the system reches stte where the first vrible is δ, the system sticks t this stte nd no more ctions cn be performed. Bosscher hs proved in [Bos97] tht decidbility of bisimilrity nd regulrity in BPA systems extends to the BPA δ systems. The trick used for this extention is bsed on the ide tht δ cn be simulted by n unnormed vrible. The min topic this rticle dels with is the issue of the lnguge equivlence nd of describing BPA δ in bisimilr BPA syntx. We show in Section 3 tht extending the BPA systems with dedlocks does not yield ny lnguge extension. The uthor is supported by the Grnt Agency of the Czech Republic, grnt No. 201/97/0456

2 On the other hnd the clss of BPA δ systems is lrger with regrd to bisimilrity. An interesting question explored in this pper (Section 4) is concerned with deciding whether there exists n lterntive description of BPA δ system in bisimilr BPA syntx. We show tht it is decidble for the strict bisimilrity nd we find nice semntic chrcteriztion of the sitution in the nonstrict cse. Moreover we show tht the corresponding BPA syntx cn be effectively constructed. Severl proofs in this pper re just sketched nd their full version cn be obtined in [Srb98]. 2 Bsic initions When deling with processes we need some structure to describe their opertionl semntics. As the most suitble structure trnsition systems re widely used. We introduce the lbelled trnsition system in the extended version with the set of finl sttes s cn be found e.g. in [Mol96]. Definition 1. (lbelled trnsition system) A lbelled trnsition system is tuple (S, Act,,α 0,F) where S is set of sttes; Act is set of ctions (or lbels); S Act S is trnsition reltion, written α β, for (α,, β) ; α 0 S is the root (or strt stte) of the trnsition system; F S is the set of finl sttes which re terminl: forechα F there is no Act nd β S such tht α β. As usul we extend the trnsition reltion to the elements of Act.Welsowrite α β insted of α w β if w Act is irrelevnt. Definition 2. (lnguge genertion) Let (S, Act,,α 0,F) be lbelled trnsition system nd suppose tht α S. The lnguge generted by the stte α is L(α) = {w Act α F :α w α }. We sy tht two sttes α nd β re lnguge equivlent, written α = L β,iffl(α)=l(β). Two lbelled trnsition systems re lnguge equivlent iff their roots re lnguge equivlent. Definition 3. (bisimilrity) Let (S, Act,,α 0,F) be lbelled trnsition system. A binry reltion R S S is bisimultion iff whenever (α, β) R then for ech Act: if α α then β S : β β (α,β ) R if β β then α S : α α (α,β ) R α F β F Sttes α, β S re bisimilr (α β), iff (α, β) R for some bisimultion R. 2.1 BPA nd BPA δ systems Assume tht Vr nd Act re finite sets of vribles nd ctions such tht Vr Act =. We ine the clss E BPA of BPA expressions s the union of ɛ (empty process) nd set E BPA, + which is ined by the following bstrct syntx: E ::= X E 1.E 2 E 1 + E 2

3 Here rnges over Act nd X rnges over Vr. We stte E BPA = {ɛ} E +. BPA We cll the BPA expressions s processes nd lter on we ssume fixed sets Vr nd Act if no confusion is cused. As usul, we restrict our ttention to gurded expressions: BPA expression is gurded iff every vrible occurrence is within the scope of n tomic ction. Definition 4. (BPA system) A BPA system is qudruple (Vr, Act,,X 1 ) where Vr nd Act re finite sets of distinct vribles (Vr = {X 1,...,X n }) resp. ctions; X 1 Vr is the leding vrible; is finite set of recursive equtions = {X i = E i i =1,...,n} where ech E i E + BPA is gurded BPA expression with vribles drwn from the set Vr nd ctions from Act. Speking bout vribles nd ctions used in the system (Vr, Act,,X 1 ) we use the nottion Vr( ) ndact( ) nd for shorter referring to the BPA system we often identify the system (Vr, Act,,X 1 )with. Assume tht we hve BPA system (Vr, Act,,X 1 ). This system determines lbelled trnsition system (S, Act,,X 1,{ɛ}) whose sttes re BPA expressions built over Vr nd Act, Act is the set of lbels, the trnsition reltion is the lest reltion stisfying the following SOS rules, X 1 is the root nd ɛ is the only finl stte. ɛ E E E.F E.F if E ɛ E ɛ E.F F E E E + F E F F E + F F E E X E if X = E We now ine the clss BPA δ of BPA systems with dedlock. The inition is very similr to the inition of BPA systems except for new distinct vrible δ. There is no opertionl rule for δ in the BPA δ systems. Definition 5. (BPA δ system) A BPA δ system is qudruple (Vr, Act,,X 1 ) where Vr = {X 1,...,X n,δ} (δ is specil vrible clled dedlock), Act is finite set of ctions nd is finite set of recursive equtions = {X i = E i i =1,...,n} where ech E i E + is gurded BPA expression with vribles BPA drwn from the set Vr nd ctions from Act. It is obvious tht ny BPA system is trivilly BPA δ system. BPA δ lbelled (strict or nonstrict) trnsition system is ined s in the cse of BPA systems. If F = {ɛ} is the only finl stte we cll the lbelled trnsition system strict nd if the finl sttes re F = {ɛ, δ} {δ.e E E BPA} + we cll it nonstrict. This mens tht the reltion of bisimultion differs for both these pproches. Similrly, we cll the bisimultion strict resp. nonstrict (nd write s resp. )ccording to the type of the lbelled trnsition system we tke into ccount. These n two notions of bisimilrity imply tht δ n s ɛ but δ ɛ. An esy consequence of decidbility of bisimilrity in BPA δ [Bos97] is tht both s nd n re decidble. Following lemm results from the inition of s nd. n

4 Lemm 1. s n Let X Vr. We ine the norm of X s X =min{length(w) E:X w E },ifsuchwexists; or X = otherwise. We cll the vrible X normed iff X <. A process is normed iff its leding vrible is normed. Definition 6. A BPA (resp. BPA δ )system is sid to be in Greibch Norml Form (GNF) iff ll its ining equtions re of the form X = m j=1 jα j where m>0, j Act( ) nd α j Vr( ). If length(α j ) <kfor ech j then is sid to be in k GNF. Following theorem justifies the usge of 3 GNF. Theorem 1. Let be BPA δ system. We cn effectively find BPA δ system in 3 GNF such tht s resp. n. Proof. The proof is bsed on the proof of 3 GNF for BPA systems (see e.g. [Hüt91]), which hd to be modified to cpture the behviour of dedlocks. In fct we hd to use some dditionl trnsformtions exploiting (from left to right) the rules δ + E E nd δ.e δ. 3 Expressibility of BPA δ systems In this section we justify the importnce of introducing dedlocking stte into the BPA systems. We show tht dedlocks enlrge the descriptive power of BPA systems w.r.t. both strict nd nonstrict bisimilrity. On the other hnd introducing dedlocks does not llow to generte more lnguges. Theorem 2. There exists BPA δ system such tht no BPA system is strictly bisimilr to it. Proof. No BPA system cn be strictly bisimilr to the system {X = δ} since δ is rechble in this system nd there is no mtch for δ in ny BPA system. system such tht no BPA system is non- Theorem 3. There exists BPA δ strictly bisimilr to it. Proof. We ine BPA δ system nd show tht there is no BPA system such tht n. Consider = {X = XX +b+cδ} nd suppose tht there is BPA system in 3 GNF, = {Y i = E i i =1,...,n}, such tht n. Then there re infinitely mny sttes rechble from the leding vrible X of the system. They re of the form X n for n 1 nd for ech such stte there must be rechble stte E from such tht X n n E. ThestteX n still hs norm 1 wheres norm 1 for BPA processes implies tht it must be single vrible. Thus is nonstrictly bisimilr to system with finitely mny rechble sttes, which is contrdiction is system where infinitely mny nonstrictly nonbisimilr sttes re rechble.

5 In wht follows we show tht the clsses of BPA nd BPA δ systems re equivlent w.r.t. lnguge genertion. We will consider just the nonstrict cse (F = {ɛ, δ} {δ.e E E BPA}) + since it is obvious tht the strict cse cnnot bring ny lnguge extention. Definition 7. We ine clsses of lnguges generted by BPA resp. BPA δ systems s following: L(BPA) = {L( ) is BPA system} nd L(BPA δ ) = {L( δ ) δ is BPA δ system}. Theorem 4. It holds tht L(BPA) =L(BPA δ ). Proof. We show tht for BPA δ system δ there exists BPA system such tht L( δ )=L( ). The other direction is obvious. Our proof will be constructive. For ech vrible X δ we ine couple of new vribles X ɛ,x δ. The first one will simulte the lnguge behviour of X when reching the stte ɛ, the second one will simulte ending in the suffix of the form δα. We use the nottion α Y mening tht α is summnd in the ining eqution of the vrible Y. W.l.o.g. let δ be BPA δ system in 3 GNF. The vribles of the system will be Vr( ) = X Vr( δ ) {δ}{x ɛ,x δ } {X1 ɛδ} where Xɛ, X δ re distinct fresh vribles nd X1 ɛδ is the leding vrible, supposing tht X 1 ws the leding vrible of δ. Next we relize tht the summnds of the ining eqution for X Vr( δ ) {δ}re exctly of one of the following form (becuse of 3 GNF): () AB (b) bc (c) c (d) ddδ (e) eδ (1) where, b, c, d, e Act( δ )nda, B, C, D Vr( δ ) such tht A, B, C, D δ. Notice tht we cn suppose tht there is no summnd of the form δa becuse it cn be replced with δ. We now ine the vribles from. ForechX Vr( δ ) {δ}nd for the summnds of the vribles X ɛ nd X δ will hold: if AB X then A ɛ B ɛ X ɛ nd A ɛ B δ + A δ X δ if bc X then bc ɛ X ɛ nd bc δ X δ if c X then c X ɛ if ddδ X then dd ɛ + dd δ X δ if eδ X then e X δ if X ɛ 1 = E nd X1 δ = F then X1 ɛδ = E + F If it is the cse tht there is vrible Y Vr( ) such tht Y does not hve ny summnd we ine Y = Y. (This vrible cnnot generte ny nonempty lnguge becuse it is unnormed). Finlly we stte X1 ɛδ to be the leding vrible of the system. Exmple 1. Let us hve BPA δ system δ = {X = XX + b + cδ + by, Y = b}. The corresponding lnguge equivlent BPA system looks s following: = {X ɛ = X ɛ X ɛ + b + by ɛ,x δ = X ɛ X δ + X δ + c + by δ,y ɛ = b, Y δ =.Y δ ɛδ, X = X ɛ X ɛ + b + by ɛ + X ɛ X δ + X δ + c + by δ }.

6 It is not difficult to see tht the newly ined system is in 3 GNF nd we show tht L( δ )=L( ). For this we need one lemm using following nottion. Definition 8. Let be BPA (resp. BPA δ )systemin3 GNF, n 1 nd Y Vr( ). We ine L ɛ n(y ) nd L δ n(y ) s following: L ɛ n(y ) = {w Act( ) Y w ɛ length(w) n} L δ n (Y ) = {w Act( ) α Vr( ) : Y w δα length(w) n}. Lemm 2. For ll n 1 nd X Vr( δ ) {δ} holds tht L ɛ n(x) =L ɛ n(x ɛ ) nd L δ n (X) =Lɛ n (Xδ ). Proof. The proof is led by induction on n, following the subcses from (1). To finish the proof of our theorem let us ine for n 1thesetL n (Y) = {w L(Y ) length(w) n}. Notice tht becuse of the Lemm 2 we get L n (X 1 )=L ɛ n(x 1 ) L δ n(x 1 )=L ɛ n(x1) L ɛ ɛ n(x1)=l δ n (X1 ɛδ ) for ll n 1. Now it is cler tht L(X 1 )=L(X1 ɛδ) since if w L(X 1)then n:w L n (X 1 ) nd so w L n (X1 ɛδ ) which implies tht w L(Xɛδ 1 ). The other direction is similr. We hve shown tht L( δ )=L( ) nd our proof is complete. 4 Describing BPA δ in BPA syntx We hve shown tht w.r.t. bisimilrity the clss of BPA δ systems is strictly lrger thn tht of BPA. This chllenges the question whether given BPA δ system cn be equivlently described in BPA syntx. Theorem 5. Let (Vr, Act,,X 1 ) be BPA δ system. It is decidble whether there exists BPA system such tht s. Moreover if the nswer is positive, the system cn be effectively constructed. s Proof. The proof is stndrd nd is bsed on the fct tht δ ɛ. Suppose w.l.o.g. tht the system is in 3 GNF. The nottion α E mens gin tht α is summnd in the expression E. We will construct the sets M 0,M 1,... of vribles from which the dedlock is rechble s following: M 0 = {δ} nd for i 0thesetsM i+1 re ined s = M i {X Vr Act, Y Vr, Z M i :(X = E),.Z M i+1 E.Z.Y E (.Y.Z E nd Y < )}. We remind tht the norm of vrible cn be effectively computed. Let us denote the fixed point of this construction s M. Wecnseethtforech X Vr: X δ.α for some α Vr iff X M. IfX 1 Mthen cnnot be expressed by BPA syntx since the dedlocking stte is rechble from X 1. If X 1 M we cn nturlly trnsform into BPA system. The sitution for the nonstrict cse will be nicely chrcterised by the Corollry 1. In wht follows, the set of vribles from which dedlocking stte is

7 rechble will be of gret importnce. Hence we ine the set Vr δ of such vribles: Vr δ = {X Vr X δ or E E + : X BPA δ.e} {δ}nd we stte Vr ɛ = Vr {δ} Vr δ. The sets Vr δ nd Vr ɛ cn be effectively constructed s we hve demonstrted in the proof of the Theorem 5. In wht follows let the vribles U, V, X, Y, Z rnge over Vr δ nd A, B, C over Vr ɛ. Theorem 6. Let (Vr, Act,,X 1 ) be BPA δ system in 3 GNF. Suppose tht there re only finitely mny pirwise nonstrictly nonbisimilr Yα Vr δ.vr such tht X 1 Yα. Then there exists BPA system (Vr, Act,,X 1 ) such tht n. Proof. Let us suppose tht X 1 Vr ɛ. Then the system cn be trivilly trnsformed into bisimilr BPA system. Thus ssume tht X 1 Vr δ. We my suppose w.l.o.g. tht ech summnd of every ining eqution in does not contin n unnormed vrible (resp. δ) followed by nother vrible. We ine functions f α for ech α Vr. These functions tke n expression from E + in 3 GNF nd trnsform it into nother expression. Our gol is following. BPA We wnt to chieve f α (E) n Eα nd there should be no dedlock in f α (E). For ech α Vr let us lso ine function r α which returns the set of the new vribles dded by the function f α. Let us ssume tht X, Y, U Vr δ, A, B, C Vr ɛ with C =, β Vr ɛ such tht β < nd γ Vr. n n n n f α ( i α i )= f α ( i α i ) r α ( i α i )= r α ( i α i ) i=1 i=1 f α (XY ) = X Yα r α (XY ) = {X Yα } f α (Xδ) = X ɛ r α (Xδ) = {X ɛ } f α (δ) = r α (δ) = f α (X) = X α r α (X) = {X α } f α (AB) = ABβU γ r α (AB) = {U γ } if α = βuγ = ABβC = if α = βcγ = ABα = otherwise f α () = βu γ r α () = {U γ } if α = βuγ = βc = if α = βcγ = α = otherwise f α (Aδ) = A r α (Aδ) = f α (A) = AβU γ r α (A) = {U γ } if α = βuγ = AβC = if α = βcγ = Aα = otherwise f α (XA) = X Aα r α (XA) = {X Aα } f α (AX) = AX α r α (AX) = {X α } Let us now construct the nonstrictly bisimilr BPA system where Vr = Vr ɛ Added; Act = Act; = ɛ Γ ; X 1 = X1 ɛ.thesetsadded nd Γ re outputs of the following lgorithm nd ɛ contins exctly the ining equtions for vribles from Vr ɛ. i=1 i=1

8 Algorithm 1 1 Solve:={X1 ɛ} 2 Added:={X1} ɛ 3 Γ := 4 while Solve do 5 Let us fix X α Solve with (X = E) 6 Γ := Γ {X α = f α (E)} 7 Add:={Y β r α (E) Z ω n Added : Yβ Zω} 8 while Y β,z ω Add : Y β Z ω Yβ Zω n do 9 Add:= Add {Y β } 10 endwhile 11 Solve:= (Solve {X α }) Add 12 Added:= Added Add 13 for Y β r α (E) Add do 14 replce ll occurences of Y β in Γ with Z ω 15 where Z ω Added : Yβ Zω n 16 endfor 17 endwhile In the following lemms we demonstrte tht the lgorithm is correct nd yields BPA system such tht n. Lemm 3. For the loop 4 17 of Alg.1 holds the following invrint: Y β,z ω Added : Y β Z ω n Yβ Zω. Proof. An esy observtion. Lemm 4. Whenever during the execution of Alg.1 we hve Y α Added then Y Vr δ. Proof. All vribles in Added hd to be produced by the function r α (see line 7 nd 12). It is esily seen tht {Y Y β r α (E)} Vr δ for ny α Vr nd E E + such tht E is in 3 GNF. BPA Lemm 5. Whenever during the execution of Alg.1 we hve Y β Added then X 1 Yβ. Proof. By induction on the number of repetitions of the loop Bsic step: The only vrible in the set Added before the execution of the loop 4 17 strted is X1 ɛ. However X 1ɛ = X 1 nd so X 1 X 1 ɛ. Induction step: Suppose tht t line 12 we hve dded new vrible Y β into Added. So t line 7 we hd to hve Y β r α (E) forsomex α Solve nd (X = E). The induction hypothesis sys tht X 1 Xα (X α hd to be dded in some previous repetition of the min loop). It must hold tht γy β f α (E) whereγ Vr ɛ nd γ <. From the construction of f α we cn lso see tht Xα Yβ.ThuswegetX 1 Xα Yβ.

9 Lemm 6. Alg.1 cnnot loop forever (under the ssumption of the Theorem 6). Proof. Suppose tht the lgorithm loops forever which mens tht the set Solve is never empty. But in every loop we remove exctly one element from the set Solve (line 11). This implies tht the set Added will grow rbitrrily becuse the set Add is infinitely often unempty (otherwise the lgorithm would stop). The contrdiction is immedite from the Lemms 3, 4 nd 5. The following lemm is crucil for the proof of our theorem. Lemm 7. After the execution of Alg.1 we hve V α n Vαfor ll V α Added. Proof. We use the strtified bisimultion reltions [Mil89] k. By induction on k we show tht V α k Vα for ll k 0. This implies tht V α n Vα. This strightforwrd but lso long nd technicl proof cn be found in [Srb98]. Lemm 8. The system n is BPA system nd moreover X 1 X ɛ 1. Proof. Immeditely from the Lemm 7. We hve constructed BPA system such tht n. Theorem 7. Let (Vr, Act,,X 1 ) be BPA δ system. Suppose tht there re infinitely mny pirwise nonstrictly nonbisimilr Yα Vr δ.vr such tht X 1 Yα. Then there is no BPA system such tht n. Proof. The proof is bsed on the fct tht α n β implies α = β. Let us ssume w.l.o.g. tht there exists in 3 GNF such tht n. We show tht this is not possible. Since there re infinitely mny rechble sttes Y 1 α 1,Y 2 α 2,... of which re pirwise nonstrictly nonbisimilr there must be corresponding sttes β 1,β 2,... of the system such tht Y i α i n βi for i =1,2,... Let us now ine constnt N mx s N mx = mx{ Y i δ i =1,2,...} where Y δ =min{length(w) Y w δ or E E + BPA : Y w δ.e}. Notice tht the inition of N mx is correct since for ll i Y i δ < (becuse Y i Vr δ )nd there re only finitely mny different Y i s. Clerly Y i α i N mx for ll i. This implies tht the norm of β i is lso less or equl N mx for ll i. However, is BPA system nd ll vribles in re gurded. This mens tht there re only finitely mny different sttes of such tht their norm is less or equl N mx. Hence there must be two sttes β k nd β l n n with k l such tht β k = β l. This implies tht β k βl. Then lso Y k α k Yl α l, which is contrdiction. Suppose tht we hve BPA δ system nd tht there re infinitely mny nonbisimilr sttes from which, fter some short sequence of ctions, dedlocking stte is rechble. Then the corresponding (nonstrictly bisimilr) BPA system does not exists. This condition ppers to be both necessry nd sufficient s is illustrted by the following corollry. Corollry 1. Let (Vr, Act,,X 1 ) be BPA δ system. There re only finitely mny pirwise nonstrictly nonbisimilr Yα Vr δ.vr such tht X 1 Yα if nd only if there exists BPA system (Vr, Act,,X 1) such tht n. Proof. An immedite consequence of the Theorems 6 nd 7.

10 5 Conclusion remrks In this pper we hve focused on the clss of BPA processes extended with dedlocks. We hve shown tht for lnguge equivlence the extention is no cquisition. On the other hnd the BPA δ clss is lrger with regrd to the reltion of bisimultion. We introduce two notions of bisimilrity to cpture the different understnding of dedlock behviour. If we do not distinguish between ɛ nd δ, we spek bout nonstrict bisimilrity nd if we do, we cll the pproprite bisimultion s strict. We hve solved the question whether, given BPA δ system, there is n equivlent description (with regrd to bisimilrity) of in terms of BPA syntx. The solution for the strict bisimilrity is strightforwrd. However, the nswer to the problem deling with the nonstrict bisimilrity exploited nice semntic chrcteriztion of the subclss of BPA δ processes bisimilrly describble in BPA syntx: BPA δ system cn be trnsformed into BPA system (preserving nonstrict bisimilrity) if nd only if finitely mny nonbisimilr sttes strting with some in δ-ending vrible re rechble. There is still n open problem whether this semntic chrcteriztion is syntcticlly checkble. Acknowledgements: First of ll, I would like to thnk Ivn Černá forherhelp nd encourgement throughout the work. I m very grteful for her dvise nd vluble discussions. My wrm thnks go lso to Mojmír Křetínský nd Antonín Kučer for their constnt support nd comments. References [BCS95] O. Burkrt, D. Cucl, nd B. Steffen. An elementry decision procedure for rbitrry context-free processes. In Proceedings of MFCS 95, volume 969 of LNCS, pges Springer-Verlg, [BCS96] O. Burkrt, D. Cucl, nd B. Steffen. Bisimultion collpse nd the process txonomy. In Proceedings of CONCUR 96, volume 1119 of LNCS, pges Springer-Verlg, [BK85] J.A. Bergstr nd J.W. Klop. Algebr of communicting processes with bstrction. Theoreticl Computer Science, 37:77 121, [Bos97] D. Bosscher. Grmmrs Modulo Bisimultion. PhD thesis, CWI, University [CHS95] of Amsterdm, S. Christensen, H. Hüttel, nd C. Stirling. Bisimultion equivlence is decidble for ll context-free processes. Informtion nd Computtion, 121: , [HM96] Y. Hirshfeld nd F. Moller. Decidbility results in utomt nd process theory. In Logics for Concurrency: Automt vs Structure, volume 1043 of LNCS, pges F. Moller nd G. Birtwistle, [Hüt91] H. Hüttel. Decidbility, Behviourl Equivlences nd Infinite Trnsition Grphs. PhD thesis, The University of Edinburgh, [Mil89] R. Milner. Communiction nd Concurrency. Prentice-Hll, [Mol96] [Srb98] F. Moller. Infinite results. In Proceedings of CONCUR 96, volume 1119 of LNCS, pges Springer-Verlg, Jiří Srb. Compring the clsses BPA nd BPA with dedlocks. Technicl Report FIMU-RS-98-05, Fculty of Informtics, Msryk University, 1998.