Decidability of Weak Bisimilarity for a Subset of Basic Parallel Processes

Transcription

1 Decidbility of We Bisimilrity for Subset of Bsic Prllel Processes Colin Stirling Division of Informtics University of Edinburgh emil: 1 Introduction In the pst decde there hs been vriety of results showing decidbility of bisimultion equivlence between infinite stte systems. The initil result, due to Beten, Bergstr nd Klop [1], proved decidbility for normed BPA processes, described using irredundnt context-free grmmrs. This ws extended to ll BPA processes nd then to pushdown utomt [5, 16, 14]. Decidbility of bisimilrity ws lso shown for Bsic Prllel (BP) processes, restricted subset of Petri nets, [4]. For full Petri nets Jnčr proved tht bisimultion equivlence is undecidble [11]. An open question is the dividing line between decidbility nd undecidbility of bisimilrity in the cse of sequentil systems. For instnce, is bisimultion equivlence decidble for the generl clss of prefix-recognisble trnsition grphs introduced by Cucl [2]? A poignnt problem is tht these grphs exhibit infinite brnching. Fmilies of infinite stte systems for which bisimilrity is nown to be decidble re finitely brnching. For ech lbel nd for ech configurtion the set of its -successors is finite nd esily computble. Therefore if two systems re not bisimultion equivlent then there is lest pproximnt n > 0 such tht they re not equivlent t level n, nd for ech n the equivlence t level n is decidble. But if processes re infinite brnching then inequivlence my be mnifested t higher ordinls, nd therefore new technique is required to estblish semidecidbility of inequivlence. Insted of exmining richer fmilies of infinite stte systems one cn loo t the problem of deciding we bisimultion equivlence for restricted clsses. We bisimilrity bstrcts from silent ctivity, with the consequence tht BPA nd BP processes re infinitely brnching. We bisimultion inequivlence is then generlly not finitely pproximble. In this pper we exmine the decision problem of we bisimilrity for normed BP processes. Esprz [6] observes tht we bisimilrity is semidecidble, becuse positive witness is semiliner. Decidbility ws proved for restricted subclss, the totlly normed processes, by Hirshfeld [7]. And Jnčr, Kučer nd Myr show decidbility of we bisimilrity between generl (PA) processes which includes BP processes nd finite stte processes [12]. However in both these cses inequivlence is finitely pproximble. In this pper we prove decidbility

2 of we bisimilrity for subset of normed BP processes for which inequivlence need not be finitely pproximble. Underpinning this result is finite symbolic chrcteristion of the infinite brnching of normed BP processes. Indeed we believe tht the technique will estblish decidbility of we bisimilrity for ll normed BP processes, but the combintorics become wesome. In section 2 we define normed Bsic Prllel processes nd we bisimultion equivlence. Section 3 is devoted to the finite chrcteristion of the infinite trnsition reltions. Then in section 4 we utilise the chrcteristion to prove the decidbility result using the tbleu method. Proofs of two crucil lemms re given in section 5. 2 Normed Bsic Prllel processes Ingredients of Bsic Prllel (BP) processes re finite set Γ = {X 1,..., X n } of toms, finite set A = { 1,..., } of ctions nd finite set T of bsic trnsitions, ech of the form X α where X is n tom, A {τ} nd α is multiset of toms whose size is t most 2. A BP process, or configurtion, is prllel composition of toms. We let α, β,... rnge over such processes. A process therefore hs the form X Xn n, which is the prllel composition of 1 copies of X 1,... nd n copies of X n where ech i 0. We let be the empty composition, where ech i = 0. If α nd β re two processes then αβ is their multiset union (nd we often write Xα or αx s n bbrevition for the multiset union of {X} nd α). The behviour of BP process is determined by the following extension rule: if X α T then Xβ αβ. The silent ction τ A is included s possible ction. We ssume the usul expnsion of the trnsition reltion to words, α w β where w (A {τ}). Exmple 1 The toms Γ re {A, Y, Z} nd A is the singleton set {}. The bsic trnsitions re A, Y A, Z τ A, Y Y A, Z Z, Z τ ZA. ZA 3 hs the following trnsitions, ZA 3 A 4, ZA 3 ZA 2, ZA 3 ZA 3 nd ZA 3 τ ZA 4. Exmple 2 The toms re {C, D, U, V }, A = {c, d} nd the bsic trnsitions re U τ UD, U c, U c C, U c τ c c U, V V D, V, V C, C c C, C c, D d, D τ. For ech n 0 there is the extended trnsition U τ n c UD n. BP processes re communiction free Petri nets, where the plces re the toms, nd trnsition X α is firing rule. A configurtion X Xn n represents the mring when there re i toens on plce X i. They re communiction free becuse ech trnsition requires just one plce to fire. As is usul in process clculi when there re silent trnsitions, the we trnsition reltions = nd = for A re defined s follows. α = β iff n 0. α τ n β α = β iff α 1, β 1. α = α 1 β 1 = β

3 There cn be infinite brnching with respect to these trnsition reltions, s illustrted by Exmples 1 nd 2, Y = Y A n nd V = c CD n for ll n 0. An tom X is normed if there is word w A such tht X = w. A BP definition is normed if ll its toms re normed. Both Exmples 1 nd 2 re normed. The norm of tom X, written N(X), is the length of shortest word w such tht X = w. N(Y ) = 2 nd N(D) = 0 where Y nd D re from the exmples bove. Norm extends to configurtions α, written N(α), which is the length of shortest word w such tht α = w. If BP definition is normed then so is ny process α = X Xn n nd N(α) is 1 i n ( i N(X i )). A subset of normed BP processes is the totlly normed processes, s introduced by Hüttel nd exmined by Hirshfeld [9, 7]. A BP process definition is totlly normed if ll its toms re normed nd hve norm greter thn 0. Exmple 1 is totlly normed but Exmple 2 is not becuse N(D) = 0. Our interest is with deciding when two normed BP processes re we bisimultion equivlent. There is more thn one wy to define this equivlence. First we strt with the nturl (nd symmetric ) version. Definition 1 A binry reltion B between BP processes is we bisimultion reltion provided tht whenever αbβ nd (A {}) if α = α then there is β such tht β = β nd α Bβ if β = β then there is n α such tht α = α nd α Bβ Two processes α nd β re wely bisimilr, written α β, if there is we bisimultion reltion B such tht αbβ. Importnt properties of we equivlence re tht it is congruence for BP processes, see [7] for instnce, nd tht it preserves norm. Fct 1 If α β then αδ βδ nd N(α) = N(β). An lterntive (nd equivlent) bsis for the definition of we bisimilrity is s follows, where if A then â is nd if = τ then â is, [13]. Definition 2 A binry reltion B between BP processes is wb reltion provided tht whenever αbβ nd (A {τ}) if α α then there is β such tht β â = β nd α Bβ if β β then there is n α such tht α â = α nd α Bβ Fct 2 B is wb reltion iff B is we bisimultion reltion. To estblish tht α β it therefore suffices to exhibit binry reltion contining α nd β nd prove tht it is wb reltion. In generl such reltion will be infinite. The reltion {(Y A n, ZA n ), (A n, A n ) : n 0} over processes of Exmple 1 is wb reltion, which proves tht Y Z. The symmetric definition of equivlence supports we bisimultion pproximnts, o for ny ordinl o, which re themselves equivlence reltions. Definition 3 The reltions o for ordinls o re defined inductively s follows, where we ssume tht l is limit ordinl (such s ω).

4 α 0 β α o+1 β iff for (A {}) if α = α then β. β = β nd α o β if β = β then α. α = α nd α o β α l β iff o < l. α o β Fct 3 α β iff for ll ordinls o. α o β. Exmple 2 illustrtes the need for ordinls beyond ω. Although U V for ny n 0, U n V. The inequivlence is due to the trnsition U = c U. Process c V does not hve similr trnsition. However for ny n 0, V = CD n nd U n CD n but U n+1 CD n becuse of the trnsition U = d UD n. Therefore it follows tht U ω+1 V. It is conjectured by Stříbrná [15] tht one only needs ordinls which re less thn ω 2 to estblish inequivlence between ll BP processes, including the unnormed. She proves this in the specil cse when there re no toms of norm 0 nd A is singleton set. And in the cse of totlly normed processes she shows tht the closure ordinl is ω: if α β then for some n 0, α n β. Esprz observes tht wb reltion which witnesses the equivlence α β is semiliner [6], which estblishes semidecidbility of we bisimilrity for ll BP processes (nd therefore decidbility for totlly normed processes). But the problem is estblishing semidecidbility of inequivlence. A new pproch to deciding we equivlence is now developed. First we finitely chrcterise the infinite brnching of normed BP process, nd then we use the chrcteristion to show tht equivlence nd inequivlence cn be cptured by exmining only boundedly mny trnsitions. However we re only ble to prove decidbility for subset of normed BP processes which includes the totlly normed processes. The fmily lso includes Exmple 2 where inequivlence is not finitely pproximble. The subset is given by technicl restriction, whose nottion is now developed. 3 Strtifiction nd genertors In this section we symboliclly chrcterise the we trnsition reltions of normed BP processes. Assume fixed normed BP process definition with toms Γ, ction set A nd trnsitions T. The initil step is to strtify the bsic trnsitions in T, by including numericl index on the trnsition reltion which represents the chnge in norm produced by the trnsition. If X α T then we re-write it s X n α where n = N(α) N(X). The index n is bounded, 1 n 2M, where M is the mximum norm of ny tom in Γ. Either trnsition is norm reducing, but then by t most 1, or it is nondecresing nd becuse α 2 the increse in norm is t most 2M. An importnt, but simple, observtion is tht for strtified τ-trnsition, X τ n α, the index n must be nondecresing, n 0. A selection of strtified trnsitions from Exmples 1

5 nd 2 of the previous section is A 1, Z 0 Z, Y τ 1 Y A, U c 1, D τ 0, V τ 0 V D. The definition of strtifiction is extended to the we trnsition reltions s follows. τ τ = 0 β iff m > 0. α 1,...,α m. α = α α m = β = n+1 β iff α, β. α = j α τ +1 β = l β where n = j + + l = n β iff α, β. α = j α β = l β where n = j + + l α α α For instnce, U = 0 UD 64 nd U = c 1 D 80 re strtified we trnsitions rising from Exmple 2 of the previous section. We bisimultion equivlence cn be redefined using the strtified we trnsition reltions. Assume tht K is the lrgest increse in norm of the BP process definition, mx{n : X n α T}. For instnce, K = 1 for Exmple 1 of the previous section. Definition 1 A reltion B between normed BP processes is strtified we bisimultion reltion provided tht whenever αbβ nd n K nd (A {}) if α = n α then there is β such tht β = n β nd α Bβ if β = n β then there is n α such tht α = n α nd α Bβ Proposition 1 B is strtified we bisimultion reltion iff B is wb reltion. Hence α β iff there is strtified we bisimultion reltion B which contins the pir α nd β. In the next section we shll lso define ssocited strtified we pproximnts. The crucil feture of totlly normed processes is tht they re finitely brnching with respect to the strtified we trnsition reltions. Fct 1 If α is totlly normed then for ll nd n {δ : α = n δ} is finite. This is not generlly the cse for normed BP processes. For instnce V = c 1 D n for ll n 0. The crucil component of infinite brnching is the reltion = 0, to which we now direct our nlysis. The following result is useful. Proposition 2 If α = 0 β nd β = 0 δ nd α δ then α β. Consequently if X = 0 Y nd Y = 0 X then X Y. In this circumstnce, if X Y then we sy tht tom Y is redundnt becuse of X. A BP definition cn therefore be replced with n equivlent definition which does not contin redundnt toms. If Y Γ is redundnt becuse of X Γ then we chnge Γ to Γ {Y } nd we replce ll trnsitions Y α T with X α nd ll trnsitions Z Y α T with Z Xα. It is cler tht this trnsformtion of BP definition preserves we bisimultion equivlence. We therefore ssume tht toms of BP definition dhere to the following condition: (1) if X Y nd X = 0 Y then not(y = 0 X). The reson tht the trnsition reltion = 0 cn be infinite brnching is becuse toms cn generte other toms. If X = 0 XA then we sy tht X

6 genertes A. And for ech tom X, the set of toms generted by X, written G(X), is {A : X = 0 XA}. In Exmple 2 of the previous section, G(D) = nd G(U) = G(V ) = {D}. Proposition 3 1. If A G(X) then N(A) = 0 2. If A G(X) nd A = 0 B then B G(X) 3. If A G(B) nd B G(X) then A G(X) 4. If α G(X) then X = 0 Xα 5. If α G(X) then X Xα If A G(X) then A n+1 Xα Xα. Hence ny configurtion α cn be reduced to n equivlent miniml norml form nf(α). Definition 2 If α = X Xn n then nf(α) = Xl Xln n where 1. if j i nd X j G(X i ) nd i > 0 then l j = 0, 2. if X i G(X i ) nd i > 0 nd j i. j = 0 or X i G(X j ) then l i = if j = 0 or for ll i such tht i 0, X j G(X i ) then l j = j. Proposition 4 1. If X l Xln n = nf(α) nd Xj1 1...Xjn n = nf(α) then l i = j i for ech i 2. α nf(α) Assume BP process definition which obeys condition (1) nd let Γ 0 be the set of generble toms, {A Γ : X. A G(X)}. An extended configurtion either hs the form β where β = nf(β), or hs the form βa 1...A where β = nf(β) nd ech A i Γ 0 nd A i G(X) for ny X β, nd A i β. Theorem 1 For ny configurtion α nd (A {}) nd n there is finite set of extended configurtions E(α,, n) such tht 1. if α = n δ then either nf(δ) E(α,, n) or δ = δ 1 A l1 1...Al nd ech l i 0 nd β = nf(δ 1 ) nd βa 1...A E(α,, n), 2. if β E(α,, n) then α = n β, 3. if βa 1... A E(α,, n) then l l 0. α = n βa l Al Proof: Assume configurtion α nd ssume A {} nd n 1. Any trnsition α = n δ cn be decomposed s follows α = j α 1 δ = l δ where j + + l = n. Clerly for the set {δ : α = n δ} there re only finitely mny different indices j, nd l which cn be involved in decomposition (becuse K nd both j nd l re t lest 0). In turn trnsition λ = m λ cn lso be decomposed. If m = 0 then λ τ τ λ nd if m > 0 then λ = τ 0 λ 1 λ 1 = m λ where > 0. Hence ny trnsition = n is built from only finitely mny compositions of trnsitions = 0, nd τ m where m > 0. And for ech λ the sets {λ : λ λ } nd {λ : λ τ m λ } re finite.

7 nd bounded, from the BP process definition. Hence the importnt trnsitions involved in decomposition re the = 0 trnsitions, which we now concentrte on. A trnsition of the form X τ 0 XA is generting trnsition, nd trnsition X τ 0 X is useless. Consider ny configurtion β 0 nd derivtion τ τ τ d = β 0 0 β β n such tht no trnsition in the derivtion is either generting trnsition or useless trnsition. For fixed β 0 there re only finitely mny such derivtions, nd therefore only finitely mny configurtions ppering in ny such derivtion, {β 0,...,β m }. This follows from condition (1) erlier: if X τ+ 0 Y α nd Y τ+ 0 Xδ nd X Y then Y is redundnt becuse of X (s N(α) = 0 = N(δ)). In fct crude upper bound on the number of such finl configurtions 1 is β 0 2 Γ. For ech derivtion d of β i let d(β i ) Γ be the subset of toms which occur nywhere within the derivtion, nd let G(d(β i )) be the set {G(X) : X d(β i )}. There re only finitely mny different such sets ssocited with ech β i. For ech such subset we introduce preliminry extended configurtion s follows. First if G(d(β i )) = for some derivtion d then one preliminry configurtion is β i. Next if G(d(β i )) = {A 1,..., A } then nother preliminry configurtion is β i A 1... A where β i is the result of removing ll occurrences of A j from β i. There re only finitely mny such preliminry extended configurtions ssocited with ech β i. Preliminry extended configurtions re preliminry becuse they my not yet be in norml form. However it is esy to see tht if β 0 = 0 δ then either δ = β i or δ = β i Al1 1...Al for some l 1 0,..., l 0. Moreover if β i A 1...A is preliminry extended configurtion then by Proposition 3.4 for ll l 1 0,..., l 0, β 0 = 0 β i Al Al. We now complete the rgument of the theorem. First if = nd n = 0 nd α = β 0 then we merely tidy the preliminry extended configurtions. If β i is such configurtion then we let β i = nf(β i ), nd if β i A 1... A is configurtion then we let β i = nf(β i ) nd we remove ech A j such tht A j G(X) when X β i. By the resoning bove the resulting finite set of extended configurtions, E(α,, 0), obey the theorem. Otherwise or n 0. Assume the finite set of preliminry extended forms ssocited with {α 1 : α = 0 α 1 }. Consider the possible trnsitions from preliminry extended form where τ or 0. There re two cses. First if the form is β i then the required finite set is {β ij : β i β ij }. Second is tht the preliminry form is β i A 1... A. We now te the following finite set {β ij A 1...A : β i β ij } {β iδa 1... A : A j δ} nd then tidy their elements by removing ny occurrences of A j from β ij nd δ. The result is finite set of preliminry extended forms, with the crucil property tht if α = 0 α 1 α 1 then either α 1 is preliminry form or α 1 = β ij Al Al for some l 1 0,..., l 0 nd β ij A 1... A is preliminry form nd for ech such form nd l 1 0,..., l 0 there is n α 1 such tht α = 0 α 1 β ij Al1 1...Al. The rgument is now repeted, so tht there is finite set of preliminry extended forms which chrcterise the set {α 2 : α = 0 1 The size of configurtion δ, δ, is its number of occurrences of toms.

8 α 1 α 1 = 2 }, nd so on. Becuse there cn be only finitely mny different indices involved in decomposition of the trnsition = n it follows tht there is finite set of preliminry extended forms which chrcterise {δ : α = n δ}. This finite set is then tidied into set of extended forms, E(α,, n), s described erlier. Theorem 1 offers finite symbolic chrcteristion of the infinite brnching of normed BP processes. Moreover its proof shows how finite set of extended configurtions E(α,, n) which chrcterises {δ : α = n δ} is computed. Exmple 1 Consider E(U, c, 0) where U is from Exmple 2 of the previous section. There is only one decomposition of the trnsition = c 0 in this exmple, U = c 0 α 1 0 α 1 = 0 δ. First consider the preliminry extended configurtions for {α 1 : U = 0 α 1 }. This consists of the singleton set {UD }. Next c we exmine the 0 trnsitions, nd there re two possibilities U c 0 U nd U c 0 C. Hence {UD, CD } contins the preliminry extended configurtions for {α 1 : α = c 0 α 1 0 α 1}. This is the sme set of preliminry extended configurtions for {δ ; α = c 0 δ}. Now we tidy this set. Becuse D G(U) the resulting set E(U, c, 0) is {U, CD }. We show tht this set obeys the three conditions of Theorem 1. Suppose U = c 0 δ. There re two cses. First U = c 0 UD n nd becuse D G(U) it follows tht U = nf(ud n ) nd U E(U, c, 0). Second is tht U = c 0 CD n nd UD E(U, c, 0). This estblishes condition 1 of Theorem 1. For 2 note tht U = c 0 U. And for condition 3, U = c 0 CD n for ll n 0. In contrst the set E(V, c, 0), where V is lso from Exmple 2 of the previous section, is the singleton set {CD }. 4 The decidbility result Given normed BP process definition it is esy to find its redundnt toms, nd to remove them. The sets G(X) for ech tom X is esily computble. Moreover for ech configurtion α, nd n K, one cn compute finite set of extended configurtions E(α,, n) which chrcterises {δ : α = n δ}, using Theorem 1 of the previous section. The min problem with deciding whether or not α β is their infinite brnching. The technique for overcoming this is to use the finite chrcteristion to show tht we only need to exmine boundedly mny trnsitions of α nd β. However we re only ble to show this for subset of normed BP processes which includes Exmples 1 nd 2 of section 2. We restrict to the subset of normed BP process definitions which obey the following condition. If G(X) nd X τ 0 α then α = Xα Effectively this imposes the constrint tht genertors re pure, if X is genertor then ny trnsition X τ 0 α is generting trnsition or is useless (X τ 0 X). Both Exmples 1 nd 2 of section 2 obey this condition. The next result relies on this constrint.

9 Proposition 1 1. If G(X) nd Xα = 0 β then X β. 2. For ech α the set {nf(δ) : α = 0 δ} is finite. The restriction on BP processes does not imply tht the sets {δ : α = 0 δ} nd {nf(δ) : α = n δ nd or n > 0} re finite 2. Assume tht E(α,, n) is the finite set of extended configurtions given by Theorem 1 of the previous section, which chrcterises {δ : α = n δ}. Elements of this set re either finite, of the form β, or infinite, of the form βa 1... A. It follows from the restriction on BP processes tht if β is finite element nd β = 0 β then nf(β ) is lso finite element becuse if β contins genertor X then β lso contins X. Furthermore if β = 0 β nd nf(β ) β nd β = β then nf(β ) β (becuse otherwise β would contin redundnt tom). Therefore the following holds becuse of the restriction on processes. Fct 1 If β 0 is finite element of E(α,, n) whose size E(α,, n) = m nd = 0 β 1 = 0... = 0 β m then for some i < m, nf(β i ) = nf(β i+1 ). β 0 There is similr property in the cse of infinite elements. Assume tht δ = βa n An is n instnce of n infinite element βa 1... A E(α,, n) nd δ = 0 δ. Then δ λa l Al where ech l i n i nd either nf(λ) = β or nf(λ) β nd β = 0 λ. Moreover in the cse tht nf(λ) β nd δ = 0 λ A l A l then nf(λ ) β. Fct 2 If δ 0 = βa n An nd βa 1... A E(α,, n) nd E(α,, n) = m nd δ 0 = 0 δ 1 = 0... = 0 δ m then for some i < m, δ i λ i A l1 1...Al nd δ i+1 λ i Al A l nd nf(λ i ) = nf(λ i ) nd l i l i. Strtified we bisimultion pproximnts, o, re now defined s follows. Definition 1 The reltions o for ordinls o re defined inductively s follows, where we ssume tht l is limit ordinl (such s ω). α 0 β α o+1 β iff if α = 0 α then β. β = 0 β nd α o+1 β if β = 0 β then α. α = 0 α nd α o+1 β nd for ( A nd n K) or ( = nd 1 n K) if α = n α then β. β = n β nd α o β if β = n β then α. α = n α nd α o β α l β iff o < l. α o β 2 It is possible to define hierrchies of BP processes ccording to finiteness of these sets. At the lowest level re the totlly normed processes, where for ny α, nd n the set {δ : α = n δ} is finite.

10 The definition of o is unusul becuse we distinguish between = 0 trnsitions nd the remining trnsitions = n for resons tht will become clerer in the decision procedure. Becuse of Proposition 1 the definition is well-defined, s there cn only be finitely mny different elements in the set {δ : α = 0 δ}. For ech ordinl o, the reltion o is n equivlence reltion. However for prticulr ordinls o the reltion o my differ from o s defined in section 2. Proposition 2 1. α β iff for ll ordinls o. α o β. 2. For ll ordinls o, α o nf(α). 3. If α o β then αδ o βδ. The procedure for deciding whether α β is given by tbleu proof system, which is gol directed. One strts with the initil gol nf(α) = nf(β), to be understood s is α β?, nd then one reduces it to subgols using smll number of rules. Gol reduction continues until one reches either obviously true gols (such s δ = δ) or obviously flse subgols (such s γ = δ nd one of these processes hs n = n trnsition which the other does not hve). Such procedure ws used for deciding strong bisimilrity between rbitrry BP processes [4], nd we will me use of this decidbility proof. Gols hve the form α = γ where α = nf(α) nd γ = nf(γ). There re four reduction rules, reducing gols to subgols. Let n (δ) = {nf(δ ) : δ = n δ } where n K. The first tbleu rule is for = 0 trnsitions, nd by Proposition 1.2 the set 0 (δ) is finite set. where C is the following condition α = γ α 1 = γ 1... α l = γ l C ( α 0 (α). i. α = α i ) ( i. α i 0 (α)) ( γ 0 (γ). i. γ = γ i ) ( i. γ i 0 (γ)) The rule is sound, if ll the subgols re true then so is the gol. A finer ccount shows soundness with respect to pproximnts, if α i o γ i for ll subgols then α o γ. The rule is lso complete in the sense tht if the gol α = γ is true then there is n ppliction of the rule such tht ll subgols α i = γ i re lso true. Next we wnt similr rule which covers the remining trnsitions = n, when n K nd either or n > 0. A similr rule would reduce gol α = γ to only finite set of subgols. However set n (δ) my be infinite. Therefore we need mechnism which shows tht we only need to consider bounded finite subsets of elements of n (α) nd n (γ) for ech = n. Lemms 1 nd 2 below estblish this. These results re quite involved, nd so we dely their proofs until the next section where they re presented in full. They constitute the hert of the decidbility result. Although the set n (δ) my be infinite, the set E(δ,, n) is not only finite but lso bounded. We show tht we need only exmine smll elements of n (δ).

11 The first lemm covers the cse when the gol α = γ is not true. Without loss of generlity ssume α o+1 γ nd α = n α nd for ll γ such tht γ = n γ, α o γ. Lemm 1, the bounded inequivlence lemm, shows tht there is smll α n (α) with this property. A smll element of n (α) is either finite element of E(α,, n) or is n element βa l Al where ech l i E(γ,, n) +1 nd βa 1... A E(α,, n). Lemm 1 Let α o+1 γ nd ssume tht α = n α nd for ll γ such tht γ = n γ, α o γ. Then either 1. nf(α ) is finite element in E(α,, n) or 2. α = δa n An nd βa 1... A E(α,, n) nd nf(δ) = β nd there exists l 1,..., l such tht ech l i E(γ,, n) +1 nd for ll γ such tht γ = n γ, βa l Al o γ. Lemm 2, the bounded equivlence lemm, covers the cse when α = γ is true. It shows tht smll elements of n (α) cn be mtched with elements of n (γ) which hve bounded size. Lemm 2 Assume α γ. Then for ech nd n 1. if α = n α nd nf(α ) E(α,, n) then () either γ = n γ nd nf(γ ) E(γ,, n) nd nf(α ) nf(γ ) (b) or γ = n λb s Bsm m nd λ B1... Bm E(γ,, n) nd nf(λ) = λ nd ech s i E(α,, n) nd nf(α ) λ B s Bsm m, 2. if α = n δa n An nd βa 1... A E(α,, n) nd β = nf(δ) nd ech n i E(γ,, n) + 1 then () either γ = n γ nd nf(γ ) E(γ,, n) nd βa n1 1...An nf(γ ) (b) or γ = n λb s Bsm m nd λ B1... B m E(γ,, n) nd nf(λ) = λ nd ech s i n i + E(α,, n) nd βa n An λ B s1 1...Bsm m. = 0 Consequently the second tbleu proof rule hs similr form to the trnsition rule except tht the condition C is tht for ll n, n K nd or n > 0, smllα n (α). i. α = α i i. n. α i n (α) nd α i hs bounded size smllγ n (γ). i. γ = γ i i. n. γ i n (γ) nd γ i hs bounded size where the precise notion of bounded size is given in Lemm 2.2 prt (b). Lemm 1 gurntees tht the rule is sound, if the gol α = γ is flse nd so α o+1 γ then for t lest one of the subgols α i = γ i it is the cse tht α i o γ i (where the pproximnt index decreses). Lemm 2 justifies completeness, if the gol α = γ is true then there is n ppliction of the rule such tht ll subgols α i = γ i re lso true. The finl two rules SUB(L) nd SUB(R) re ten from the tbleu decision procedure for strong bisimultion equivlence for rbitrry BP processes [4]. We ssume fixed liner ordering < on the toms Γ which is defined so tht if tom Y G(X) nd Y X then X < Y. The ordering < is extended to

12 configurtions, s the lexicogrphicl ordering. Assume X 1 < X 2 <... < X n. Consequently X Xn n < X l1 1...Xln n iff there is n i 1 such tht i < l i nd for ll j < i. j = l j. Clerly, nf(α) α nd if α < γ then nf(αβ) < nf(γβ). The SUB rules re given below, where SUB(L) is the left rule nd SUB(R) is the right rule. α = γ. γ < α nd t lest one. ppliction of αδ = λ nf(γδ) = λ γ = α. γ < α nd t lest one = n. ppliction of λ = αδ λ = nf(γδ) = n We explin the rule SUB(L). If the current gol is αδ = λ nd in the proof tree bove the gol on the pth to the root there is the gol α = γ nd γ < α nd there is t lest one ppliction of the rule = n where or n > 0 long this pth then we reduce the gol to the subgol nf(γδ) = λ. Note tht this hs the effect of reducing the size of the left configurtion, s nf(γδ) < αδ. The SUB rules re sound nd complete. Fct 3 1. If αδ o λ nd α o γ then nf(γδ) o λ 2. If α γ nd αδ λ then nf(γδ) λ One builds proof tree strting from n initil gol α = γ nd repetedly pplying the tbleu proof rules s follows. First one pplies the = 0 rule nd then one pplies the = n rule to ll the resulting subgols. Cll this simple bloc. And then one repetedly pplies the SUB rules to the subgols of bloc, until they no longer pply. At which point the whole process is repeted. One pplies the = 0 rule to ll subgols nd so on. There is lso the importnt notion of when gol is finl gol. Finl gols re either successful or unsuccessful. A successful finl gol hs the form α = α nd n unsuccessful finl gol hs the form δ = λ nd for some nd n either n (δ) = nd n (λ) or n (δ) nd n (λ) =. Clerly successful finl gol is true nd n unsuccessful finl gol is flse. A successful tbleu proof for α = γ is finite proof tree whose root is α = γ nd ll of whose inner subgols re the result of n ppliction of one of the rules, nd ll of whose leves re successful finl gols. The following results estblish decidbility of between restricted BP processes. The proofs of these results re minor vrints of proofs in [3, 4]. Theorem 1 Every tbleu for α = γ is finite nd there is only finite number of tbleu for α = γ.

13 Theorem 2 α γ iff there is successful tbleu for α = γ. Therefore the min result follows tht is decidble between restricted BP processes. We now exmine how this result shows equivlence nd inequivlence for the exmples from section 2. Exmple 1 We show tht Y Z where these toms re from Exmple 1 of section 2. Assume Y < Z. We build tbleu with root Y = Z. First we pply the = rule which results in the sme gol Y = Z. Then we pply the = n rule. In this exmple K = 1 nd so we need to consider the four trnsitions = 1, = 1, = 0 nd = 1. 0 (Y ) = {Y A} 1 (Y ) = {A} 0 (Y ) = {A, Y } 1 (Y ) = {Y A, AA} 0 (Z) = {ZA} 1 (Z) = {A} 0 (Z) = {A, Z} 1 (Z) = {ZA, AA} So the gol Y = Z reduces to the following subgols, (1)Y A = ZA, (2)A = A, (3)Y = Z nd (4)AA = AA. Gols (2) nd (4) re successful leves. In the cses of (1) nd (3) becuse Y < Z nd Y = Z ppers on their pths to the root, nd in both cses there is t lest one ppliction of the rule = n, we cn pply SUB(R) to yield (1 ) Y A = Y A nd (3 ) Y = Y, which completes the successful tbleu. Exmple 2 We show tht U V where U, V re from Exmple 2 of section 2 even though U n V for ll n 0. There re only finitely mny tbleux for this gol nd ll re unsuccessful. Assume tht U < V. The strting gol is U = V. First we pply the = 0 rule, which yields the sme gol U = V. For this exmple K = 0. Next we pply the = n rule nd there re two possible trnsitions = 0 nd = d 0. We only exmine the first of these. E(U, c, 0) = {U, CD } c nd E(V, c, 0) = {CD }. The smll elements of c 0 (U) = {U, C, CD, CD 2 } nd the smll elements of c 0 (V ) = {C, CD, CD 2, CD 3 }. Therefore we must find mtching of smll elements with bounded elements. The esy subgols re C = C, CD = CD, CD 2 = CD 2 nd CD 3 = CD 3. The problem cse is mtch for the smll element U c 0 (U). By Lemm 2 becuse U is finite element of E(U, c, 0) the mtching element must be CD s where s 2. Thus we must hve one of the following subgols (1)U = C, (2)U = CD, (3)U = CD 2. Assume it is (3). We pply the = 0 rule to this gol, 0 (CD 2 ) = {C, CD, CD 2 } nd 0 (U) = {U}. So in fct we must hve ll the subgols (1), (2) nd (3). But now we hve n unsuccessful lef (1), becuse d 0 (U) = {U} nd d 0 (C) =. 5 Proofs of the min lemms In this section we prove the two boundedness lemms of the previous section. Proof of Lemm 1: Assume α = n α nd α o γ for ll γ such tht γ = n γ. Consider the sets E(α,, n) nd E(γ,, n). By Theorem 1 of section 3 either nf(α ) E(α,, n) or α = δa n1 1...An nd βa 1... A E(α,, n) nd nf(δ) = β. If the first holds then the result is proved. Assume therefore tht

14 βa n An o γ for ll γ such tht γ = n γ. Let β 1 = βan An, nd consider the lest m 1 such tht β 1 Am1 1 o γ for ll γ such tht γ = n γ. If m 1 E(γ,, n) + 1 then the result is proved for l 1 = m 1. The rgument is then repeted for other A i. Let β 2 = βam1 1 An An nd consider the lest m 2 such tht β 2 Am2 2 o γ for ll γ such tht γ = n γ. Therefore without loss of generlity ssume tht β 1A m1 1 o γ for ll γ such tht γ = n γ nd m 1 > E(γ,, n) + 1. Hence there is γ such tht γ = n γ nd β 1 Am1 1 1 o γ. By Theorem 1 of section 3 either nf(γ ) E(γ,, n) or γ = λb s Bsm m nd λ B1...Bm E(γ,, n) nd nf(λ) = λ. Assume the first cse, tht nf(γ ) = γ 0 E(γ,, n). Hence β 1 Am1 1 1 o γ 0. However β 1 Am1 1 1 = 0 β 1 Am1 2 1 = 0... = 0 β 1 A0 1. So therefore γ 0 = 0 γ 1 = 0... = 0 γ m 1 1 nd γ j o β 1 Am1 (j+1) 1. By Fct 1 of the previous section there is n i, nf(γ i ) = nf(γ i+1 ) nd therefore β 1 Am1 (i+2) 1 o β 1 Am1 (i+1) 1. Becuse o is congruence it follows tht β 1A m1 (i+2) 1 o β 1A m1 1 which is contrdiction. Next we consider the other cse, β 1 Am1 1 1 o γ 0 nd γ 0 = λbs Bsm m nd λ B1... B 1 E(γ,, n) nd nf(λ) = λ. The rgument proceeds s bove. β 1A m1 1 1 = 0 β 1A m1 2 1 = 0... = 0 β 1A 0 1 Therefore γ 0 = 0 γ 1 = 0... = 0 γ m 1 1 nd γ j o β 1A m1 (j+1) 1. By Fct 2 of the previous section for some i, γ i λ ib t Btm m nd γ i+1 λ i Bt Bt m nd t i t i nd nf(λ i ) = nf(λ i ). γ i+1 o β 1 Am1 (i+2) 1 nd γ i o β 1 Am1 (i+1) 1. Let η = B t1 t B tm t m m. By congruence, γ i+1 η o β 1A m1 (i+2) 1 η o β 1A m1 (i+1) 1. Therefore by congruence γ i+1 ηi+2 o β 1A m1 1 which is contrdiction. Proof of Lemm 2: Assume α γ. First lso ssume tht α = n α nd nf(α ) E(α,, n). Hence γ = n γ nd nf(α ) γ. By Theorem 1 of section 3 either nf(γ ) E(γ,, n) nd nf(α ) = nf(γ ), or γ = λb s Bsm m nd λ B1... B m E(γ,, n) nd nf(λ) = λ nd nf(α ) = λ B s1 1...Bsm m. We show tht ech s i cn be chosen so tht s i E(α,, n). The strtegy for proving this is similr to the proof of Lemm 1 bove. Let λ = λ B s Bsm m nd let m 1 be the smllest index such tht nf(α ) λ B m1 1. If m 1 E(α,, n) then we let s 1 = m 1 nd repet the rgument for the other indices s i. Therefore ssume tht m 1 > E(α,, n). However λ B m1 1 = 0 λ B m1 1 1 = 0... = 0 λ B1 0. Therefore ssuming α 0 = nf(α ), α 0 = 0 α 1 = 0... = 0 α m1 nd α j λ B m1 j 1. By Fct 1 of the previous section for some i, nf(α i ) = nf(α i+1 ) nd so λ B m1 i 1 λ B m1 (i+1) 1. Therefore by congruence, λ B m1 (i+1) 1 λ B m1 1 which is contrdiction. Next ssume tht α = n δa n An nd βa 1...A E(α,, n) nd β = nf(δ) nd ech n i E(α,, n) + 1. Be- cuse α γ it follows tht γ = n γ nd βa n An γ. By Theorem 1 of section 3 either nf(γ ) E(γ,, n) nd therefore βa n An nf(γ ), or γ = λb s Bsm m nd λ B1... B m E(γ,, n) nd nf(λ) = λ nd βa n An λ B s Bsm m. We show tht ech s i cn be chosen so tht s i n i + E(γ,, n). Let λ = λ B s Bsm m nd let m 1 be the smllest index such tht βa n1 1...An λ B m1 1. If m 1 n i + E(γ,, n) then we let

15 s 1 = m 1 nd repet the rgument for the other indices. Therefore ssume tht m 1 > n i + E(γ,, n). However λ B m1 1 = 0 λ B m1 1 1 = 0... = 0 λ B1 0. Therefore let η 0 = βa n An, nd so η 0 = 0 η 1 = 0... = 0 η m1 nd η j λ B m1 j 1. Becuse m 1 > n i + E(γ,, n) it follows vi fct 2 of the previous section tht for some i, η i η i+1 nd so λ B m1 (i+1) 1 λ B m1 i 1 nd therefore by congruence λ B m1 1 λ B m1 (i+1) 1 which contrdicts tht m 1 is lest index. References 1. Beten, J., Bergstr, J. nd Klop, J. (1993). Decidbility of bisimultion equivlence for processes generting context-free lnguges. Journl of Assocition for computing Mchinery, 40, Cucl, D. (1996). On infinite trnsition grphs hving decidble mondic theory. Lecture Notes in Computer Science, 1099, Christensen, S. (1993). Decidbility nd Decomposition in Process Algebrs. Ph.D thesis University of Edinburgh, Tech Report ECS-LFCS Christensen, S., Hirshfeld, Y. nd Moller, F. (1993). Bisimultion equivlence is decidble for ll bsic prllel processes. Lecture Notes in Computer Science, 715, Christensen, S., Hüttel, H., nd Stirling, C. (1995). Bisimultion equivlence is decidble for ll context-free processes. Informtion nd Computtion, 121, Esprz, J. (1997). Petri nets, commuttive context-free grmmrs, nd bsic prllel processes. Fundment Informtice, 30, Hirshfeld, Y. (1996). Bisimultion trees nd the decidbility of we bisimilrity. Electronic Notes in Theoreticl Computer Science, Hirshfeld, Y., Jerrum, M. nd Moller, F. (1996). A polynomil-time lgorithm for deciding equivlence of normed bsic prllel processes. Journl of Mthemticl Structures in Computer Science, 6, Hüttel, H. (1991). Silence is golden: brnching bisimilrity is decidble for contextfree processes. Lecture Notes in Computer Science, 575, Hüttel, H., nd Stirling, C. (1991). Actions spe louder thn words: proving bisimilrity for context free processes. Proceedings 6th Annul Symposium on Logic in Computer Science, IEEE Computer Science Press, Jnčr, P. (1995). Undecidbility of bisimilrity for Petri nets nd some relted problems. Theoreticl Computer Science, 148, Jnčr, P., Kučer, A. nd Myr, R. (1998). Deciding bisimultion-lie equivlences with finite-stte processes. Lecture Notes in Computer Science, 1443, Milner, R. (1989) Communiction nd Concurrency. Prentice-Hll. 14. Sénizergues, G. (1998). Decidbility of bisimultion equivlence for equtionl grphs of finite out-degree. Procs. IEEE FOCS 98, Stříbrná, J. (1999). Decidbility nd complexity of equivlences for simple process lgebrs. Ph.D thesis, University of Edinburgh, Tech. Report ECS-LFCS Stirling, C. (1998). Decidbility of bisimultion equivlence for normed pushdown processes. Theoreticl Computer Science, 195,