Petri Nets and Regular Processes

Transcription

1 Uppsl Computing Science Reserch Report No. 162 Mrch 22, 1999 ISSN Petri Nets nd Regulr Processes Petr Jnčr y Deprtment of Computer Science, Technicl University of Ostrv 17. listopdu 15, CZ Ostrv-Porub, Czech Republic Jvier Esprz z Institut für Informtik, Technische Universität München Arcisstrsse 21, D München, Germny Fron Moller x Computing Science Deprtment, Uppsl University P.O. Box 311, S Uppsl, Sweden Uppsl Computing Science Reserch Report Series Computing Science Deprtment Uppsl University ftp://ftp.csd.uu.se/pub/ppers/reports Box 311, S Uppsl SWEDEN Copyright c the uthors

2 Abstrct We consider the following problems: () Given lbelled Petri net nd finite utomton, re they equivlent? (b) Given lbelled Petri net, is it equivlent to some (unspecified) finite utomton? These questions re studied within the frmework of trce nd bisimultion equivlences, in both their strong nd wek versions. (In the wek version specil ction likened to n "-move in utomt theory is considered to be non-observble.) We demonstrte tht () is decidble for strong nd wek trce equivlence nd for strong bisimultion equivlence, but undecidble for wek bisimultion equivlence. On the other hnd, we show tht (b) is decidble for strong bisimultion equivlence, nd undecidble for strong nd wek trce equivlence, s well s for wek bisimultion equivlence. To pper in Journl of Computer nd System Sciences. y Prtilly supported by the Grnt Agency of the Czech Republic, Grnt No. 201/97/0456, nd by grnt from the Swedish STINT Fellowship progrmme. z Prtilly supported by the Sonderforschungsbereich 342 of the DFG. x Prtilly supported by TFR grnt No

3 1 Introduction In the specifiction nd verifiction of distributed systems, it is typiclly the cse tht one considers specific mthemticl model for the description of processes, long with some equivlence relting processes which demonstrte the sme semntic behviour. One of the first questions to sk then for the purpose of (utomtic) verifiction is: (to wht extent) is the equivlence decidble? In this pper we consider the clss of processes generted by lbelled plce/trnsition Petri nets, clled just Petri nets in the sequel. Petri nets constitute populr nd importnt formlism for modelling distributed systems, s exemplified by the widely-used textbooks by Peterson [22] nd Olderog [21] nd by the Advnces in Petri Nets volumes of the series Lecture Notes in Computer Science. We consider trce equivlence nd bisimultion equivlence two equivlences in the forefront of the study of these systems nd study both their strong nd wek versions. (In the strong versions, ll the lbels crried by the trnsitions of the net re ssumed to be visible ctions. In the wek versions, some trnsitions my be lbelled with specil silent ction, which plys similr role to "-moves in finite utomt. The firing of these trnsitions is ssumed to be unobservble.) Unfortuntely, lredy the strong versions (long with the strong versions of ll resonble behviourl equivlences) re undecidble for generl Petri nets [10, 11, 12], in fct even for Petri nets hving t most two unbounded plces. Fced with such negtive result, nturl step then is to restrict the problem in some wy. For exmple, for the clss of Petri nets in which every trnsition hs single input plce the so-clled Bsic Prllel Processes strong bisimultion equivlence is decidble [1], wheres ll other stndrd equivlences (such s trce equivlence) re undecidble, even in the strong cse [6, 8]. If on the other hnd we compre two bounded Petri nets, then these equivlences ll become decidble, s such nets describe behviours relized by finite utomt. We consider here the problem of restricting just one of the two Petri nets to be bounded, thus compring generl Petri nets ginst finite utomt. Within this frmework, we consider both the equivlence problem, s well s the question concerning the finiteness of given net, tht is, the question s to whether or not there is some (unspecified) finite utomton which is equivlent to the Petri net. We ddress these questions for both trce nd bisimultion equivlence. We show tht the strong nd wek trce equivlence problems re decidble, while the finiteness question for the trces of net is undecidble, even in the strong cse. In the bisimultion cse, both the equivlence nd finiteness questions re decidble for strong bisimilrity, yet undecidble for wek bisimilrity. Our results extend nd complement previous results by Vlk nd Vidl-Nquet [23] on the finiteness question for trce equivlence, which they referred to s the regulrity question s they were only interested in deciding if the trces describe regulr lnguges. They showed tht the regulrity of the terminl lnguge of net tht is, the set of trces corresponding to the firing sequences leding to fixed set of mrkings is undecidble, wheres the regulrity of 1

4 the set of ll trces of net in which ech trnsition crries different lbel is decidble. These problems cn be ddressed with respect to ny semntic equivlence, for exmple for ny of the observtion-bsed equivlences ctlogued by vn Glbbeek [3], or ny of vriety of non-interleving semntic equivlences proposed for Petri nets. We restrict our present study to two of the most importnt observtion-bsed equivlences, which hppen to lie t opposite ends (with respect to distinguishing power) of vn Glbbeek s spectrum. The pper is structured s follows. In Section 2 we define the concepts which we use, in prticulr the notion of Petri net, s well s the equivlences which we study. We lso present ctlogue of technicl results both old nd new which we exploit in our decision procedures nd undecidbility proofs. Of prticulr importnce re results bsed on the decidbility of the rechbility problem for Petri nets, nd relevnt vritions of Higmn s Theorem. In Section 3 we consider trce equivlence, nd demonstrte first the decidbility of the equivlence problem (in both the strong nd wek cses) by showing tht the trce inclusion problem in ech direction is decidble. We follow this by demonstrting the undecidbility of the finiteness problem in the strong cse. The proof is crried out by reduction from the hlting problem for Minsky mchines. In Section 4 we turn our ttention to bisimultion equivlence, nd demonstrte tht both problems re decidble in the strong cse, yet both problems re undecidble in the wek cse. The first undecidbility result follows from reduction from the continment problem for Petri nets, while the second relies on specil form of the continment problem to which the hlting problem for Minsky mchines cn be reduced. The results presented here elborte on those presented by the uthors in [17] nd [14]. 2 Preliminries Here we define some bsic notions nd introduce vrious results which will prove useful. By N we denote the set of nonnegtive integers: N = f0; 1; 2; : : :g. For set A, A denotes the set of finite sequences of elements of A; the empty sequence is denoted by " 2 A. For u 2 A nd k 2 N, we denote by u k the k-fold conctention of u; nd by juj we denote the length of u. 2.1 Lbelled Trnsition Systems nd Equivlences We define n utomton to be lbelled trnsition system (LTS), which is tuple L = h S; ; f?!g2 i where S is set of sttes, is finite set of ctions, nd ech?! is binry trnsition reltion on S, tht is,?!.?! S S; we write E?! F for h E; F i 2 By E?! F we men tht E?! F for some ; nd?! denotes the reflexive nd trnsitive 2

5 closure of the reltion?!. We write E sttes E 1 ; E 2 ; : : : ; E n-1 such tht E 1 tht E?! u F for some F. In prticulr, E " the difference between E?! F nd E?! " F.)?! u F for u = 1 2 n 2 to men tht there re?! E 2 1?! n-1?! E n n-1?! F. We write E u?! E for every E, nd E "?! to men?! F only if E = F. (Note We sy tht set of sttes S S is rechble from E, written E?! S, iff E?! F for some F 2 S. The rechbility set for stte E of L is defined by R L (E) = ff : E?! Fg. (We generlly omit the subscript L when the underlying LTS is cler from the context.) An LTS L = h S; ; f?!g2 i is finite-stte iff S is finite. L is imge finite iff succ (E) = f F : E?! F g is finite for every E 2 S nd every 2. By process E we refer to stte in trnsition system; when necessry, we denote the underlying trnsition system by L(E). By referring to finite-stte process E, we men tht L(E) is finite; similr convention holds for n imge finite process. We use the symbols R; R 0 ; : : : to denote finite-stte LTSs, nd the symbols r; r 0 ; : : : to denote sttes in finite-stte systems, tht is, finite-stte processes. A binry reltion B between processes is strong bisimultion provided tht whenever h E; F i 2 B, for ech 2, if E if F?! E 0 then F?! F 0 for some F 0 such tht h E 0 ; F 0 i 2 B; nd?! F 0 then E?! E 0 for some E 0 such tht h E 0 ; F 0 i 2 B. Two processes E nd F re strongly bisimultion equivlent or strongly bisimilr, written E F, iff there is strong bisimultion B relting them. A decresing chin of equivlence reltions between processes is defined inductively s follows. E 0 F for ll processes E nd F; nd E n+1 F iff for ech 2, if E if F?! E 0 then F?! F 0 then E?! E 0 for some E 0 such tht E 0 n F 0.?! F 0 for some F 0 such tht E 0 n F 0 ; nd The fct tht these reltions do form decresing chin of equivlences ll contining is esily confirmed (by induction on n). The next two Propositions re lso esily-confirmed folklore. Proposition 2.1 For imge finite processes E nd F, E F iff E n F for ll n 0. Proof: The forwrd impliction cn be proved by induction on n; nd the reverse impliction is proved by demonstrting tht the reltion B = bisimultion. h E; F i : E n F for ll n is strong Let us cll L = h S; ; f?!g2 i n dmissible system iff the stte set S is finite or countbly infinite (identified with set of sequences over finite lphbet), L is imge finite, nd ll of 3

6 the successor functions succ : S?! 2 S re effectively computble. (Recll tht is finite, so there re only finitely mny of these.) With this restriction in plce, the following result is immedite. Proposition 2.2 Considering only dmissible systems, ll of the reltions E n F re decidble. Therefore the nonequivlence problem E 6 F is semidecidble. Proof: To decide E n F, we need simply resort to the definition of the reltions n. E 6 F cn then be confirmed by deciding ech E n F for n = 0; 1; 2; : : : until we discover tht E 6n F for some n. We hve s yet delt only with definitions nd results concerning utomt without silent trnsitions. To introduce these trnsitions, we interpret distinguished symbol 2 s silent ction, nd modify our definitions ccordingly. (We follow this frmework dopted from process theory rther thn the utomt theoretic technique of directly llowing "-moves s we wnt to be ble to distinguish, for exmple, between?! nd?!; wheres " =, 6=.) Given ny 2 with 6=, we let E =) F represent E?! u F for some u = k ` (k; ` 0); tht is, =) = (?!)?! (?!). We then let E =) F represent E?! u F for some u = k (k 0); note tht we llow u = ", so for exmple E =) E for ll E. 1 The reltions E =) u F nd E =), u where u 2, re then the obvious generliztions: E =) " F iff E =) F, nd for u = 1 2 n 6= ", E =) u F iff there re sttes E 1 ; E 2 ; : : : E n-1 such tht E 1 =) E 2 1 =) n-1 =) E n n-1 =) F; nd E =) u iff E =) u F for some F. Finlly, we introduce u one further bit of nottion: given set S of sttes, we let =)(S) = ff : E =) u F for some E 2 Sg. The reltion of wek bisimultion equivlence, denoted by, s well s the reltions n (n = 0; 1; 2; : : :), re defined in the sme wy s for the strong reltions, n but with?! replced everywhere by =). The strong trce set of stte E of n LTS L is defined by ST (E) = fw 2 : E?!g. w Two processes E nd F re strongly trce equivlent iff ST (E) = ST (F). The wek trce set, or just trce set of stte E is defined by T (E) = fw 2 ( n fg) : E =)g. w Two processes E nd F re wekly trce equivlent, or just trce equivlent, iff T (E) = T (F). Notice tht two -free trnsition systems re wekly trce equivlent iff they re strongly trce equivlent, nd they re wekly bisimilr iff they re strongly bisimilr. As n esy consequence, decidbility of problem in the wek cse implies decidbility in the strong cse. Moreover, undecidbility of problem in the strong cse cn be shown by proving undecidbility in the wek cse for -free systems. We mke free use of these fcts. 1 This is somewht nonstndrd in process theory; our reltions =) b should be written s =) in order to fit into the process theory frmework [19], but in our presenttion we omit the extr decortion. 4

7 2.2 Petri Nets A (finite, lbelled, plce/trnsition Petri) net is tuple N = h P; T; F; ; ` i where P, T nd re finite disjoint sets of plces, trnsitions nd ctions, respectively; F : (P T) [ (T P)?! f0; 1g defines the set of rcs; h x; y i is n rc iff F(x; y) = 1; ` : T?! is lbelling, which ssocites n ction from to ech trnsition. In the Petri net literture, multiple rcs re often llowed (in which cse the rnge of F is given s N). For technicl convenience, we tret only ordinry nets; nevertheless ll of our rguments cn be esily modified to hold for these more generl nets. We disply nets grphiclly using circles for plces nd boxes for trnsitions; when lbels of trnsitions re importnt, we write them inside the boxes. A mrking of net is mpping M : P?! N ssociting number of tokens to ech plce. We denote the zero mrking, tht is, the mrking tht mps ech plce to 0, by 0. A trnsition t is enbled t mrking M, written M [ti, iff M(p) F(p; t) for every p 2 P. If trnsition t is enbled t mrking M it my fire or occur yielding the mrking M 0, denoted M [ti M 0, where M 0 (p) = M(p) - F(p; t) + F(t; p) for ll p 2 P. We extend this firing rule to sequences of trnsitions, thus writing M [t 1 t 2 t n i M 0 when M [t 1 i M 1 [t 2 i M n-1 [t n i M 0 for some M 1 ; M 2 ; : : : ; M n-1 (nd M [t 1 t 2 t n i when M [t 1 t 2 t n i M 0 for some M 0 ). We interpret net N = h P; T; F; ; ` i s n LTS where mrkings ply the role of sttes. The trnsition reltions?! re provided by the firings of the enbled trnsitions of the net: M?! M 0 iff M [ti M 0 for some t with `(t) =. Notions like M?! M 0, M =) u M 0, T (M), M 1 M 2 re then inherited from the respective notions given in the generl setting. In prticulr, we hve the notion of the rechble mrkings of mrking M of net N; in this cse we write either R N (M) or R(M) if the underlying net N is cler from the context. Observe tht the LTS derived from net is n dmissible system, so Proposition 2.2 is pplicble. We now recll some known results from Petri net theory, in prticulr the decidbility of the rechbility problem. Theorem 2.3 (Myr [18]) Given two mrkings M nd M 0 of Petri net N, it is decidble whether or not M?! M 0, tht is, whether or not M 0 2 R(M). We lso use the notion of n!-mrking; it extends the notion of mrking by llowing n infinite number of tokens to be ssocited to the plces. Formlly we set N! = N [ f!g where we suppose! stisfies n! nd! + n =! - n =! for ll n 2 N. An!-mrking, for which we reserve symbols M, c M c 0 ; : : :, is then simply mpping M c : P?! N!. Notions such s M c [tic M 0, M c?! M c 0 nd T (c M) re then nturlly defined s extensions of the previous definitions. 5

8 We define the ordering pointwise on the set N P! of!-mrkings of net with plce set P, thus writing c M c M 0 iff c M(p) c M 0 (p) for every p 2 P; this is prtil order on!- mrkings (it is reflexive, trnsitive, nd ntisymmetric). Moreover, it stisfies the finite bsis property (fbp): every infinite sequence of elements hs n infinite (not necessrily strictly) scending subsequence. This result is known s Dickson s Lemm [2]. Lemm 2.4 (Dickson) The collection of!-mrkings of net ordered by stisfies the fbp. Specificlly, given n infinite sequence of!-mrkings c M1 ;c M2 ;c M3 ; : : :, there re indices i 1 < i 2 < i 3 < such tht c Mi1 c Mi2 c Mi3. Proof: By induction on the number of plces: for ech plce p in turn, we choose n infinite subsequence Mi1 c ;c Mi2 ;c Mi3 ; : : : such tht Mi1 c (p) Mi2 c (p) Mi3 c (p). Finlly, we extend the ordering to sets of!-mrkings by defining M M 0 iff for every cm 2 M there exists c M0 2 M 0 with c M c M 0 ; this reltion is preorder on sets of!-mrkings (it is reflexive nd trnsitive, but not ntisymmetric). We my then observe the following. Lemm 2.5 () If M c0 [tic M 0 t nd M c M c0 then M c [tic Mt (nd hence Mt c M c0 t ). (b) If M c0 =) M c0 nd M c M c0 then M c =) M c with M c M c0. (c) If M c0 [ui nd M(p) c min( juj; M c0 (p) ) for ll plces p then M c [ui. (d) If M c0 =) w nd M c M c0 then M c =). w (e) If M M 0 then =)(M) =)(M 0 ). Proof: Prt () is esily proved directly; prt (b) is proved from prt () by induction on the number of unobservble trnsitions involved in the trnsition M c0 =) M c0 ; prt (c) is proved by induction on juj; prt (d) follows from prt (c); nd prt (e) follows from prt (b). Every incresing chin c M1 c M2 c M3 of!-mrkings hs unique lest upper bound c M defined by c M(p) = lim n!1 c Mn (p) for ech plce p. For set M of!-mrkings we define its completion C(M) to be M enriched by such lest upper bounds; nd we use mx(m) to refer to the subset of mximl elements of M. Formlly, these re defined s follows. C(M) = mx(m) = cm : c M is the lest upper bound of (possibly constnt-vlued) chin M1 c M2 c in M. cm 2 M : for ll M0 c 2 M, either M c = M c 0 or M c 6 M c 0. 6

9 The first importnt observtion we cn mke is tht there cn only be finitely mny mximl elements in ny set of!-mrkings. Lemm 2.6 For every set M of!-mrkings, mx(m) is finite. Proof: Immedite from Lemm 2.4. We cn then mke the following sequence of observtions. Lemm 2.7 () M C(M). (b) C(C(M)) = C(M). (c) mx(m) M. (d) If M M 0 then M M 0. (e) If M M 0 then =)(M) =)(M 0 ). (f) If mx(m) mx(m 0 ) mx(m) then mx(m) = mx(m 0 ). Proof: Immedite. Lemm 2.8 For every c M 2 C(M) there is c M 0 2 mx(c(m)) such tht c M c M 0. Proof: By induction on the number plces p with M(p) c < 1. If there is no chin M c cm 1 M2 c of distinct!-mrkings in C(M), then the result redily follows. Otherwise let M1 c 2 C(M) be the lest upper bound of such chin. Then M c M1 c, nd M1 c must hve fewer finitely-mrked plces thn M, c from which the result follows by induction. Lemm 2.9 If M M 0 then mx(c(m)) mx(c(m 0 )). Proof: Let M c 2 mx(c(m)). Then by Lemm 2.7(c), M c 2 C(M), so M c is the lest upper bound of chin M1 c M2 c in M. From M M 0 we cn find sequence cm 1;c 0 M 0 2 ; : : : in M 0 with Mi c M c0 i for ech i 1. By Lemms 2.7() nd 2.8, we cn find cm i 00 2 mx(c(m 0 )) such tht M c0 i M c00 i for ech i 1. By Lemm 2.6, there must be single cm 0 2 mx(c(m 0 )) such tht M c0 i M c 0 for infinitely mny i 1, nd hence Mi c M c 0 for ll i 1. Thus c M c M 0. 7

10 Lemm 2.10 C( S 1in M i ) = S 1in C(M i ). Proof: For inclusion in the forwrd direction, ny c M 2 C( S1in M i ), must be the lest upper bound of chin c M1 c M2 of mrkings tken from S 1in M i ; but since there re only finitely mny M i, we must be ble to tke subchin c Mi1 c Mi2 of mrkings from single M i, from which we deduce tht c M 2 C(M i ) S 1in C(M i ). For inclusion in the reverse direction, ny c M 2 S1in C(M i ) must come from some C(M i ), from which we get c M 2 C( S1in M i ). Lemm 2.11 mx( S i M i ) = mx( S i mx(m i)). Proof: c M 2 mx( Si M i ) iff M c 2 Mi for some i, nd M c 6 M c0 for every M c 0 6= M c in Si M i iff M c 2 mx(m i ) for some i, nd M c 6 M c0 for every M c 0 6= M c in Si mx(m i) iff c M 2 mx( Si mx(m i)). Lemm 2.12 =)(C(M)) C( =)(M)). Proof: Let c M 2 =)(C(M)). Then there is chin M1 c M2 c in M with lest upper bound M c such tht M c?! w M c, where w = k ` (k; ` 0) if 6= nd w = k (k 0) if =. We cn ssume (by dropping sufficient initil segment of the chin) tht cm 1 (p) = M(p) c whenever M(p) c < 1, nd tht M1 c (p) jwj whenever M(p) c = 1. By Lemm 2.5(c) we hve sequence M c0 1 ;c M 0 2 ; : : : in =)(M) such tht Mi c?! w M c0 i for ech i 1, ll using the sme sequence of net trnsistions s M c?! w M c, thus hving the sme effect on the mrkings; in prticulr, M c0 1 M c0 2. This chin hs lest upper bound M c0 in C( =)(M)); it remins to demonstrte tht M c = M c 0, tht is, tht M c (p) = M c0 (p) for ll plces p. If c M(p) < 1 then c M(p) = c Mi (p) for ll i 1, so c M (p) = c M 0 i (p) for ll i 1, so cm (p) = lim i!1 c M 0 i (p) = c M 0 (p). If M(p) c = 1 then for ech n 0 there is n i 1 such tht Mi c (p) > n + jwj, so cm i 0 (p) > n, nd hence M c0 (p) = lim i!1 M c0 i (p) = 1. Lemm 2.13 () T (c M 0 ) T (c M) for ny!-mrkings c M nd c M 0 with c M c M 0. 8

11 (b) Given n incresing chin c M1 c M2 c M3 of!-mrkings with lest upper bound cm, we hve tht S i1 T (c Mi ) = T (c M): (c) S c M2M T (c M) = S cm2mx(c(m)) T (c M). Proof: Prt () follows directly from Lemm 2.5(d). For prt (b), inclusion in the forwrd direction follows from prt () since M c Mi c for ech i. Inclusion in the reverse direction holds since whenever M c?! w we must hve some Mi c such tht Mi c (p) min jwj; M(p) c for ll plces p; the result then follows from Lemm 2.5(c). Finlly for prt (c), to show inclusion in the forwrd direction, let c M 2 M. Lemms 2.7() nd 2.8 there is c M 0 2 mx(c(m)) with c M c M 0. Then by prt (), T (c M) T (c M 0 ). To show inclusion in the reverse direction let c M 2 mx(c(m)). Thus by Lemm 2.7(c) c M 2 C(M), so there is chin c M1 c M2 in M with lest upper bound c M. By prt (b), T ( c M) = Si1 T (c Mi ) S c M2M T (c M). By Lemm 2.14 mx(c( =)(mx(c(m))))) = mx(c( =)(M))). Proof: By Lemm 2.7(f), it suffices to demonstrte 1. mx(c( =)(mx(c(m))))) mx(c( =)(M))); nd 2. mx(c( =)(M))) mx(c( =)(mx(c(m))))). For 1. we hve: so so mx(c(m)) C(M) =)(mx(c(m))) =)(C(M)) C( =)(M)) =)(mx(c(m))) C( =)(M)) so mx(c( =)(mx(c(m))))) mx(c(c( =)(M)))) For 2. we hve: so =)(M) M mx(c(m)) = mx(c( =)(M))) =)(mx(c(m))) so mx(c( =)(M))) mx(c( =)(mx(c(m))))) by Lemm 2.7(c) by Lemm 2.7(e) by Lemm 2.12 by Lemm 2.7(d) by Lemm 2.9 by Lemm 2.7(b) by Lemms 2.7() nd 2.8 by Lemm 2.5(e) by Lemm 2.9 9

12 Given n!-mrking c M of net N, we cn effectively find the (finitely mny) mximl elements of C(R(c M)); this cn be chieved by the technique of coverbility trees [22]. Similrly we cn get the following. Lemm 2.15 Given n!-mrkingc M nd n ction symbol 2, we cn effectively construct mx(c( =)fc Mg)). Hence we cn effectively construct mx(c( =)(M))) for ny finite set M of!-mrkings. Proof: Assume tht 6=. Construct net N 0 from N by removing ll trnsitions with lbel different from or, nd dding new plce p which is n input plce to ll trnsitions with lbel ; then let c M 0 hve single token on plce p nd be the sme s c M on the remining plces. Compute mx(c(r N 0(c M 0 ))), tke only the vectors in which the p-component is 0, nd drop this 0-component from the vectors. The cse where = is simpler: we construct N 0 by removing the non- trnsitions nd compute mx(c(r N 0(c M))). We shll lso need n extension of Lemm 2.4 bsed on Higmn s Theorem [5]. Theorem 2.16 (Higmn) If preorder h A; i stisfies the fbp then so does h A ; i, where = h 1 2 n ; v 0 b 1 v 1 b 2 v n-1 b n v n i : i ; b i 2 A; v i 2 A ; i b i. Corollry 2.17 The collection P fin (N P! ) of ll finite sets of!-mrkings for net with plce set P stisfies the fbp with respect to. Proof: By Lemm 2.4, h N P!; i stisfies the fbp. Hence h (N P! ) ; i lso stisfies the fbp. The corollry is then cler from the fct tht ny finite set M cn be viewed s string of its elements. Finlly, we shll need the following dditionl technicl result (for Theorem 4.8). Let us cll set of mrkings M N P simple if there is disjoint prtition P = P 1 [ P 2, fixed mpping fix : P 1! N, nd constnt n such tht M = M : M(p 1 ) = fix(p 1 ) nd M(p 2 ) n for ll p 1 2 P 1 nd p 2 2 P 2. The next result shows tht it is semidecidble if simple set M 2 is rechble vi mrkings from simple set M 1 whose nonfixed vlues cn be rbitrrily lrge. Lemm 2.18 The following problem is semidecidble: 10

13 Instnce: A mrking M 0 of net N, nd two simple sets M 1 nd M 2 of mrkings of N, where P = P 1 [ P 2 is the prtition relevnt to M 1. Question: For every m 2 N, does there exist M m 2 M 1 with M m (p 2 ) > m for every p 2 2 P 2 such tht M 0?! M m?! M 2? Proof: Consider n instnce of the problem s given bove. We use u; v; w to denote trnsition sequences of N; nd u to denote the mpping (of plces to integers) indicting the chnge in the mrking upon performing u, tht is, if M [ui M 0 then M 0 = M + u. We define preorder h D; i, where D = h u; M1 ; v; M 2 i : M 1 2 M 1 ; M 2 2 M 2 ; nd M 0 [ui M 1 [vi M 2 nd h u; M 1 ; v; M 2 i h u 0 ; M 0 1; v 0 ; M 0 2 i iff we cn write u = u 1 u 2 u 3 u k v = v 1 v 2 v 3 v` u 0 = u 1 w 1 u 2 w 2 u 3 u k w k v 0 = v 1 w k+1 v 2 w k+2 v 3 v`w k+` so tht w 0 for every prefix w = w 1 w 2 w i (0 i k+`) of w 1 w 2 w i w k+`. By vrition of the proof of Theorem 6.5 in [9] (nd using Theorem 2.16 s in tht proof), we cn show tht h D; i stisfies the fbp. (Briefly, the proof works by providing strightforwrd order-preserving encoding of the elements of D s sequences of vectors over N nd then reclling tht vectors over N stisfy the fbp.) Given h u; M 1 ; v; M 2 i h u 0 ; M1 0; v0 ; M2 0 i with the sequences s bove, for ny prefix s (j) of u 1 w j 1 u 2w j 2 u k w j k v 1w j k+1 v 2w j k+2 v`w`k we get (from Lemm 2.5(c) pplied repetedly) tht M 0 [s (j) i M (j) for some M (j), nd tht M (j) M (k) for j k. Thus we cn pump the w i s; tht is, for every j 0: M 0 [u 1 w j 1 u 2w j 2 u k w j ki M (j) 1 [v 1 w j k+1 v 2w j k+2 v`w j k+`i M (j) 2 Note tht M (j) for every j 2 N; similrly, M (j) 1 = M 1 + j w1 w w k, so w1 w 2 w k (p 1 ) = 0 for every p 1 2 P 1, so M (j) 1 2 M M 2 for every j 2 N. We cll pir h u; M 1 ; v; M 2 i h u 0 ; M 0 1; v 0 ; M 0 2 i useful if M 1 (p 2 ) < M 0 1 (p 2) for every p 2 2 P 2 (tht is, w1 w 2 w k (p 2 ) > 0 for every p 2 2 P 2 ). The existence of useful pir is obviously sufficient condition for positive nswer to the considered problem instnce (since the relevnt w i s cn be pumped indefinitely). Now observe tht if we hve positive nswer to our problem, then there must be n infinite sequence h u (1) ; M (1) 1 ; v(1) ; M (1) 2 i; h u (2) ; M (2) 1 ; v(2) ; M (2) 2 i; h u (3) ; M (3) 1 ; v(3) ; M (3) 2 i; : : : 11

14 of elements of D such tht for every n nd every p 2 2 P 2 we hve M (n) 1 (p 2) < M (n+1) 1 (p 2 ). Thus the existence of useful pir is gurnteed due to the fbp, so it is lso necessry condition for positive nswer. The desired semidecidbility is then cler due to the possibility of generting ll pirs h u; M 1 ; v; M 2 i, h u 0 ; M1; 0 v 0 ; M2 0 i 2 D nd checking if ny constitutes useful pir. 3 Trce equivlence In this section we demonstrte the decidbility of the trce equivlence problem nd the undecidbility of the trce finiteness problems. 3.1 Decidbility of (strong nd wek) trce equivlence Here we demonstrte the decidbility of the following: Given mrking M 0 of net N lbelled by ction set, nd stte r 0 of finitestte LTS R defined over the sme ction set, is T (M 0 ) = T (r 0 )? To do this, we show decidbility for the trce inclusion problem in both directions: T (M 0 ) T (r 0 ) nd T (r 0 ) T (M 0 ). Without loss of generlity we suppose tht R hs no lbels nd is?! r 0 ; deterministic, tht is, for ech stte r nd ech lbel there is t most one r 0 such tht r this cn be chieved using the stndrd "-move elimintion nd subset construction lgorithms for nondeterministic finite utomt (cf., e.g., [7]). Theorem 3.1 T (M 0 ) T (r 0 ) is decidble. Proof: Firstly we cn observe the semidecidbility of the complementry problem T (M 0 ) 6 T (r 0 ). For this, it suffices to generte ll sequences from ( n fg) nd to stop when some w 2 T (M 0 ) n T (r 0 ) is found; semidecidbility of this finl test is obvious. Now define the binry reltion S = h c M; r i : T (c M) T (r) between!-mrkings of the net nd sttes of the LTS, nd define the ordering on S by hc M; r i h c M 0 ; r 0 i iff c M cm 0 nd r = r 0. By Lemm 2.6, S hs finitely mny mximl elements, nd by Lemms 2.8 nd 2.13(), S is the downwrds closure of these mximl elements. Now observe the following simple fct. If set X of pirs hc M; r i stisfies the condition For ny (?) h M; c r i 2 X nd ny ; M c 0 such tht M c?! M c 0 there is r 0 such tht r?! r 0 nd hc M 0 ; r 0 i 2 X (we put r 0 = r when = ). 12

15 then X S. It is lso cler from Lemm 2.5() (long with Lemms 2.7() nd 2.8) tht if X is the downwrds closure of its mximl elements then it suffices to verify (?) only for its mximl elements. Since S stisfies (?) (recll tht for ech r nd there is t most one r 0 such tht r?! r 0 ), to demonstrte T (M 0 ) T (r 0 ) it suffices to generte (finite) set S 0 of pirwise incomprble elements hc M; r i such tht its downwrds closure stisfies (?) nd contins h M 0 ; r 0 i; this lst condition is obviously decidble. Thus we hve demonstrted the semidecidbility, nd therefore the decidbility, of the trce inclusion problem T (M 0 ) T (r 0 ). Theorem 3.2 T (r 0 ) T (M 0 ) is decidble. Proof: We describe terminting lgorithm for constructing tree of the following description. Ech node of the tree is lbelled by pir h r; M i where r is stte of R nd M is set of pirwise incomprble!-mrkings of N (nd hence is finite). The edges in the tree re lbelled by n fg. The tree is defined inductively s follows. 1. We strt with the root node which we lbel h r 0 ; fm 0 g i. 2. From given node h r; M i, we dd one -lbelled edge for ech trnsition r?! r 0. (By our ssumption on R, 6= nd r 0 is uniquely determined by.) This edge leds to node which we lbel by h r 0 ; M 0 i = h r 0 ; mx(c( =)(M))) i. Note tht by Lemm 2.15, we cn construct mx(c( =)fc Mg)) for ech M c 2 M, nd then tke the mximl elements mongst ll of these. This gives us our desired M 0, since M 0 = mx(c( =)(M))) = mx(c( S c M2M = mx( S c M2M C( =)fc Mg)) = mx( S c M2M mx(c( =)fc Mg)) (by Lemm 2.10) =)fc Mg))) (by Lemm 2.11) 3. A node h r; M i will be deemed lef (tht is, we do not dd the edges described in step 2) if M = ; (in which cse the lef is deemed to be unsuccessful); or M 6= ;, nd either r hs no successors in R, or there is n ncestor h r; M 0 i with M 0 M (in which cse the lef is deemed to be successful). By Corollry 2.17, this tree must be finite, nd thus our lgorithm is gurnteed to terminte. 13

16 Hving constructed the tree, the relevnt question cn be nswered s follows: if there is n unsuccessful lef h r; ; i then T (r 0 ) 6 T (M 0 ); otherwise T (r 0 ) T (M 0 ). To verify the correctness of this lgorithm, we first note the following. Clim: pth lbelled by w 6= ", we hve r 0 For ny node h r; M i in the tree reched from the root h r 0 ; fm 0 g i by?! w r nd M = mx(c( =) w fm 0 g)). Proof of the Clim: It is cler from the construction tht r w 0?! r. We prove tht M = mx(c( =) w fm 0 g)) by induction on jwj. For w =, the result follows from the definition of M. Suppose then tht h r 0 ; fm 0 g i Then w?! h r; M i M 0 = mx(c( =)(M))) = mx(c( =)(mx(c( =)fm w 0 g))))) (by induction) = mx(c( =)( =)fm w 0 g))) (by Lemm 2.14) = mx(c( =)(fm w 0 g)))?! h r 0 ; M 0 i. Certinly if M 6= ; then M C(M) 6= ;, so (by Lemm 2.8) mx(c(m)) 6= ;. Thus if we hve n unsuccessful lef h r; mx(c( =)fm w 0 g)) i = h r; ; i t the end of pth lbelled w, then w =)fm 0 g = ;, tht is, w =2 T (M 0 ), wheres r 2 T (r 0 ), so indeed T (r 0 ) 6 T (M 0 ). Suppose then tht ll leves re successful, nd in spite of this T (r 0 ) 6 T (M 0 ). Choose some w 2 T (r 0 ) n T (M 0 ) of miniml length. It cn be written s w = u 1 u 2 v with u u 2 6= " where the tree hs the pth h r 0 ; fm 0 g i?! S 1 h r; M 0 u i?! 2 h r; M i with M 0 M. Then we must hve tht v 2 T (r) but v 62 u M2 c 1 u 2 Lemm 2.13(c)). But then by Lemm 2.13(), v 62 S c M2M 0 T =) fm 0 T (c M) = S g cm2m T (c M) (by (c M) = S cm2 u =)fm 1 0 T (c M) g (gin by Lemm 2.13(c)), so u 1 v 2 T (r 0 ) n T (M 0 ), which contrdicts the minimlity of the length of w. 3.2 Undecidbility of strong trce finiteness In this subsection we demonstrte tht it is undecidble whether or not given -free net is trce-equivlent to some (unspecified) finite utomton. In fct, our construction shows tht the undecidbility result holds for ny equivlence which refines trce equivlence nd is refined by simultion equivlence; the construction cn lso be esily modified to extend the undecidbility to redy-simultion equivlence (see, e.g., [3] for definitions; the modifiction is described in [13]). However, trce equivlence is our only concern here. This undecidbility result contrsts with the decidbility result for bisimilrity presented in the next section; it lso contrsts with the decidbility result of Vlk nd Vidl-Nquet [23] for the regulrity of the trce set in the cse where the trnsitions re uniquely lbelled. 14

17 To demonstrte this result, we rely on the undecidbility of the hlting problem for Minsky counter mchines. To counter mchine C (zero input vlues re supposed), we construct net N C with initil mrking M 0 (inspired by [10] s modified in [6]) for which we cn demonstrte the following: 1. If the counter mchine C hlts, then M 0 is trce equivlent to some finite-stte process r; 2. If the mchine C does not hlt, then M 0 is not trce equivlent to ny finite-stte process r. Remrk: The bove mentioned extension of the undecidbility result follows from the fct tht trce equivlence cn be replced by simultion equivlence (or even redy-simultion equivlence for the modified construction) in cse 1 bove. Formlly, Minsky mchine cn be defined s sequence of lbelled instructions X 1 : comm 1 X 2 : comm 2 X n-1 : comm n-1 X n : hlt representing simple progrm which uses counters c 1, c 2, : : :, c m, where ech of the first n-1 instructions is either of the form X : c j :=c j +1; goto X 0 or of the form X : if c j =0 then goto X 0 else c j :=c j -1; goto X 00 Here we suppose tht Minsky mchine C strts executing with the vlue 0 in ech of the counters nd the control t lbel X 1. When the control is t lbel X k (1 k < n), the mchine executes instruction comm k, modifying the contents of the counters nd trnsferring the control to the pproprite lbel mentioned in the instruction. The mchine hlts if nd when the control reches the hlt instruction t lbel X n. We recll now the well-known fct tht the hlting problem for Minsky mchines is undecidble [20]: there is no lgorithm which decides whether or not given Minsky mchine hlts. Given Minsky mchine C, we define the net N C = h P; T; F; ; ` i with initil mrking M 0 s follows. The set of plces is P = f c 1 ; c 2 ; : : : ; c m ; X 1 ; X 2 ; : : : ; X n ; U g. 15

18 X 0 X? i c j ^ ) X + s - c j z z d??? X 0 U X 00 U i 6? - 6? (i) (ii) (iii) Figure 1: Constructions for N C d z The initil mrking M 0 will consist of just one token, locted on the plce X 1 ; nd in generl, mrking will hve token on some plce X i representing the Minsky mchine t tht prticulr instruction lbel, nd some number of tokens on ech of the plces c j representing those prticulr vlues for the counters. The set of ctions lbelling the trnsitions is = fi; d; zg, denoting the mchine events increment, decrement, nd zero, respectively. For every instruction of the form X : c j :=c j +1; goto X 0 the net hs trnsition lbelled by i with the single input plce X nd the two output plces X 0 nd c j ; see Figure 1(i). For every instruction of the form X : if c j =0 then goto X 0 else c j :=c j -1; goto X 00 the net hs trnsition lbelled by d with the two input plces X nd c j, nd the single output plce X 00 ; nd two trnsitions lbelled by z, the first with the single input plce X nd the single output plce X 0, nd the second with the two input plces X nd c j, nd the single output plce U; see Figure 1(ii). there re three further trnsitions ssocited with the plce U (for universl ). They ech hve U s both their single input plce nd their single output plce, nd they re lbelled by i, d, nd z, respectively; see Figure 1(iii). The net N C simultes the Minsky mchine C in wek sense: there is unique computtion of the net corresponding to the computtion of the mchine, but there cn be invlid trnsition sequences. These rise due to z-trnsitions being performed when the relevnt counter plce c j is not empty (nd the pproprite d-trnsition is in fct the vlid trnsition). Note tht invlid 16

19 d d k z z s + k 1 ; 2 ; : : : ; k 2 fi; d; zg i; d; z Figure 2: Construction for R z-trnsitions cn led eqully well to the universl plce from which ny ction is possible forevermore. Thus, T (M 0 ) consists of ll vlid computtion sequences (tht is, ll prefixes of the computtion of C) plus ll sets wzfi; d; zg such tht wd is vlid computtion sequence. From this it is cler tht T (M 0 ) is regulr if C hlts; this fct is more crefully demonstrted in the following. Lemm 3.3 If C hlts then M 0 is trce equivlent to some finite-stte process r 0. Proof: The bckbone of the LTS R contining r 0 consists of (finite) pth corresponding to the (vlid) computtion of C (which hlts by ssumption); see Figure 2. The sttes of R correspond to mrkings of N C ; nd the initil stte of this pth is r 0 nd corresponds to the initil mrking M 0 of N C. Outside of this pth there is one further stte u with three loops lbelled by i, d nd z. From ny stte on the pth which hs n outgoing rc lbelled by d, we hve further rc lbelled by z leding to the stte u. It is obvious then tht T (M 0 ) = T (r 0 ). For the opposite direction, we cn ssume without loss of generlity tht in ny infinite computtion of C we cn find for ny q 2 N subcomputtion during which some counter is decresed q times in succession. This is possible, for exmple, by including three extr counters 1, 2 nd 3, nd replcing ech originl instruction X i : comm i by the sequence of eight instructions 17

20 X i : 1 := 1 +1; goto Yi 1 increment 1 Yi 1 : if 1=0 then goto Yi 3 while 1 > 0 do else 1 := 1-1; goto Yi 2 decrement 1 Yi 2 : 2:= 2 +1; goto Yi 1 increment 2 Yi 3 : if 2=0 then goto Yi 6 while 2 > 0 do else 2 := 2-1; goto Yi 4 decrement 2 Yi 4 : 1:= 1 +1; goto Yi 5 increment 1 Yi 5 : 3:= 3 +1; goto Yi 3 increment 3 Yi 6 : if 3=0 then goto Yi 7 while 3 > 0 do else 3 := 3-1; goto Yi 6 decrement 3 Y 7 i : comm i The effect of this trnsformtion is to mintin in counter 1 the number of commnds executed by the Minsky mchine, nd before executing ech commnd to cuse the counter 3 to be set to this vlue nd then to be repetedly decremented down to 0; this clerly leds to longer nd longer sequences of decrement ctions, without chnging the (non-)hlting behviour of the originl progrm. Lemm 3.4 If C does not hlt then T (M 0 ) is different from the trce set of ny finite-stte process r 0. Proof: Suppose tht T (M 0 ) = T (r 0 ) for some finite-stte process r 0 tken from q-stte LTS R. Then r 0 lso must llow the prefix of vlid computtion sequence of C which includes contiguous sequence of q decrement ctions. Using the Pumping Lemm for finite-stte mchines [7], this mens tht r 0 must be ble to rech stte by following vlid computtion sequence of C from which it cn follow n rbitrry number of decrement ctions, which clerly is not possible for N C strting in M 0. Hence T (M 0 ) 6= T (r 0 ) which contrdicts our ssumption. Bsed on the two lemms nd the undecidbility of the hlting problem for Minsky mchines, we cn derive our undecidbility result. Theorem 3.5 It is undecidble whether or not given -free net is trce equivlent to some (unspecified) finite-stte LTS. 4 Bisimultion equivlence In this section we demonstrte the decidbility of the strong bisimultion equivlence nd finiteness problems, nd the undecidbility of the wek bisimultion equivlence nd finiteness problems. We strt by describing generl decision technique which we shll use. 18

21 Given trnsition system L = h S; ; f processes (tken from other trnsition systems) s INC L n = fe : concept defined, the following useful observtions cn be mde.?!g2 i, we define the clss of ll n-incomptible Proposition 4.1 For ny n, E F implies tht E n F nd E 6?! INC L(F) 8F 2 S : E 6 n Fg. With this n. In ddition, the reverse impliction holds under the further proviso tht n-1 coincides with n (nd hence with ) over L(F). Proof: The left-to-right impliction is obvious. For the right-to-left impliction, it is strightforwrd to verify tht, ssuming n-1 = n on L(F), the reltion h E 0 ; F 0 i : E 0 2 L(E); F 0 2 L(F); E 0 n F 0 ; E 0 6?! INC L(F) n is strong bisimultion. The crucil point to observe is tht whenever we hve tht E 00 n-1 F 00 nd E INC L(F) n we must hve tht E 00 n F 00. Corollry 4.2 For ny two sttes r nd r 0 of n n-stte LTS R, r n-1 r 0 iff r n r 0 (iff r r 0 ). Therefore, for ny process E nd ny stte r of R, E r iff E n r nd E 6?! INC R n. Proof: reltions must stbilize within the first n steps over ny n-stte LTS. As i+1 i, nd i = i+1 implies i = i+k for ny k 0, these equivlence Corollry 4.3 To demonstrte the decidbility of E r for ny specified clss of processes E for which E n r is decidble, it suffices to demonstrte the decidbility of the (non-)rechbility problem E 6?! INC R n. Proof: Immedite. Further development nd pplictions of this technique re presented in [15, 16]. Before we proceed with our decidbility proofs, we define few further useful concepts nd mke vrious importnt observtions. We sy tht mrking L of net N is n-bounded iff L(p) n for ech plce p. For every n-bounded mrking L, we define L n to be the set of ll mrkings M such tht L(p) = min(n; M(p)) for ech plce p, nd we note the following. Lemm 4.4 For every n-bounded mrking L nd every M 2 L n, L n M. 19

22 Proof: By simple induction on n. Note tht for every mrking M there is unique n-bounded mrking L M, defined by L M (p) = min(n; M(p)), such tht M 2 L n M ; tht is, M 2 Ln iff L = L M. Also, there re clerly only finitely mny n-bounded mrkings, ll of which we my effectively list. Next, given net N nd n n-stte LTS R, we let hn; Ri-INC = INC R n \ f M : M is mrking of N g. This is the set of mrkings of N which re not strongly n-bisimilr to (tht is, not in the reltion n with) ny stte of R. By Lemm 4.4, M n L M, so M 2 hn; Ri-INC iff L n M hn; Ri-INC. Hence hn; Ri-INC cn be expressed s the union hn; Ri-INC = L n 1 [ L n 2 [ [ L n k where L 1 ; L 2 ; : : : ; L k re ll of the n-bounded mrkings ppering in hn; Ri-INC; we cn effectively construct this union, since by Proposition 2.2 we cn decide if ech n-bounded mrking L is in hn; Ri-INC. 4.1 Decidbility of strong bisimultion equivlence The decidbility proof for strong bisimultion equivlence is bsed on the generl method described bove. Given mrking M 0 of net N nd stte r 0 of n n-stte LTS R, the question M 0 n r 0 is decidble (by Proposition 2.2). Therefore by Corollry 4.3, it suffices to show the decidbility of the question s to whether the set hn; Ri-INC is rechble from M 0. From the bove chrcteristion of hn; Ri-INC, it then suffices to show the decidbility s to whether the set L n is rechble from M 0, where L is n rbitrry n-bounded mrking. Theorem 4.5 The problem M 0 r 0 is decidble. Proof: From the bove considertions, it suffices to show the decidbility of the following: Given n n-bounded mrking L, is there some M 2 L n such tht M 0?! M? But this problem is esily reducible to the rechbility problem (Theorem 2.3): for ech plce p such tht L(p) = n we cn dd n extr trnsition which just removes token from p, nd then sk if L is rechble. 20

23 4.2 Decidbility of strong bisimultion finiteness We now prove tht it is decidble whether or not given mrking M 0 of given net N is strongly bisimilr to some (unspecified) finite-stte process. We refer to this problem s the strong bisimultion finiteness problem, or the strong b-finiteness problem for short. Formlly, we sy tht mrking M is b-finite iff R(M) contins only finitely mny equivlence clsses with respect to ; otherwise we sy tht M is b-infinite, tht is, if there exist infinitely mny mrkings M 1 ; M 2 ; M 3 ; : : : rechble from M such tht M i 6 M j for i 6= j. Since the strong equivlence problem is decidble, the strong b-finiteness problem is obviously semidecidble; it suffices to generte ll finite-stte processes r 0 nd to check if M 0 r 0. Therefore, it suffices to show tht b-infiniteness is semidecidble. We fix lbelled Petri net N = h P; T; F; ; ` i nd introduce some nottion. Let P = P 1 [ P 2 where P 1 ; P 2 re disjoint nd P 2 6= ;. For mppings M 1 : P 1?! N nd M 2 : P 2?! N, (M 1 ; M 2 ) denotes the mrking of N whose projection onto P 1 is M 1 while the projection onto P 2 is M 2. We sy mrking (M 1 ; M 2 ) of N insted of prtition P 1, P 2 6= ; of P nd mppings M 1 : P 1?! N, M 2 : P 2?! N. In ddition, by (M; -) we men tht there is prtition P = P 1 [ P 2 s bove but (M; -) is considered s mrking (M : P 1?! N) of the subnet of N obtined by removing ll plces from P 2, together with their djcent rcs; this is behviourlly equivlent to putting! on ll plces of P 2. By Lemm 4.4 then, for ny n 0, if M 0 (p) n for ech plce p then (M; M 0 ) n (M; -). Lemm 4.6 If (M; M 1 ) (M; M 2 ) (M; M 3 ) nd M 1 < M 2 < M 3 < (where < is defined pointwise) then (M; M 1 ) (M; -). Proof: For every n 0 there is n index i such tht M i (p) n for ech p. Then (M; -) n (M; M i ) holds, nd since (M; M 1 ) (M; M i ), we lso hve (M; -) n (M; M 1 ). Therefore, (M; -) n (M; M 1 ) for every n 0, nd so by Proposition 2.1, (M; -) (M; M 1 ). Lemm 4.7 A mrking M 0 is b-infinite iff there exists mrking (M; -) nd n incresing chin of mrkings M 1 < M 2 < M 3 < with (M; M i ) 2 R(M 0 ) for every i 1 such tht 1. (M; -) is b-infinite; or 2. (M; -) is b-finite nd (M; M i ) 6 (M; -) for every i 1. Proof: ()): If M 0 is b-infinite, then there exists n infinite set of pirwise non-bisimilr rechble mrkings. Consider ny infinite sequence of such mrkings. By Lemm 2.4, there is n infinite subsequence (M; M 1 ); (M; M 2 ); (M; M 3 ); : : : with M 1 < M 2 < M 3 <. If (M; -) is b-infinite then cse 1 holds. If (M; -) is b-finite then we cn ssume tht (M; M i ) 6 (M; -) for every i 1 (since the mrkings (M; M i ) re pirwise non-bisimilr, t most one of 21

24 them cn be bisimilr to (M; -), nd we cn simply drop this mrking from the sequence) nd thus cse 2 holds. ((): Let M = f(m; M i ) : i 1g. If M contins infinitely mny pirwise non-bisimilr mrkings, then M 0 is b-infinite, nd we re done. So ssume tht M contins infinitely mny pirwise bisimilr mrkings. By Lemm 4.6, ll of these mrkings must be bisimilr to (M; -), nd so 2 cnnot hold. Thus 1 holds, tht is, (M; -) is b-infinite, nd therefore M 0 must itself be b-infinite, since it hs rechble mrking which is bisimilr to (M; -). Theorem 4.8 It is decidble whether or not mrking M 0 of net N is strongly b-finite. Proof: By induction on the number of plces of N. If N hs no plces, then M 0 is the unique mpping M : ;?! N, nd is certinly b-finite, since R(M 0 ) = fm 0 g. Assume then tht N hs some plces. As noted bove, it suffices to show semidecidbility of the b-infiniteness problem; nd for this, Lemm 4.7 shows tht it suffices to enumerte ll mrkings (M; -) of N (for ll prtitions P 1 ; P 2 with P 2 6= ;), nd show tht it is semidecidble whether or not there exists chin s specified in the Lemm such tht one of the two conditions of the Lemm holds. Given mrking (M; -), we cn decide by the induction hypothesis if it is b-finite or b- infinite. Moreover: 1. The existence of chin M 1 < M 2 < M 3 < such tht (M; M i ) 2 R(M 0 ) for every i 1 is surely semidecidble: just put M 1 = M 2 = f (M; M 0 ) : M 0 is rbitrry g nd pply Lemm If (M; -) is b-finite, then the existence of chin M 1 < M 2 < M 3 < such tht (M; M i ) 2 R(M 0 ) nd (M; M i ) 6 (M; -) for every i 1 is lso semidecidble: if (M; -) is b-finite then (M; -) r for stte r of some finite-stte LTS R; we my ssume tht this R is known. (We my simply enumerte every finite stte LTS R nd decide whether M r for ech stte r of R.) Let n denote the number of sttes of R. Clim: There exists chin M 1 < M 2 < M 3 < such tht (M; M i ) 2 R(M 0 ) nd (M; M i ) 6 (M; -) for every i 1 iff there exists n n-bounded mrking L of N stisfying the following two conditions: () L 2 hn; Ri-INC; nd (b) there exists chin M 1 < M 2 < M 3 < nd mrkings M 0 1; M 0 2; M 0 3; : : : 2 L n such tht M 0?! (M; M i )?! M 0 i i 1. Proof of the Clim: for every ()): Let M 1 < M 2 < M 3 < be chin such tht (M; M i ) 2 R(M 0 ) nd (M; M i ) 6 (M; -) for every i 1. There exists n index i 0 such tht for every i i 0, M i (p) n for ech p. For i i 0 we hve (M; M i ) 6 (M; -) 22

25 by ssumption (nd so (M; M i ) 6 r), but (M; M i ) n (M; -) (nd so (M; M i ) n r). Thus by Corollry 4.2, there exists n n-bounded mrking. Since there re only finitely L i 2 hn; Ri-INC such tht (M; M i )?! L n i mny n-bounded mrkings, there exists n n-bounded mrking L nd infinitely mny indices i 1 < i 2 < i 3 < such tht L = L i1 = L i2 = L i3 =. Thus L stisfies condition (), nd the subchin M i1 < M i2 < M i3 < stisfies condition (b). ((): Let M i be n rbitrry mrking of the chin given by (b); we need to prove tht (M; M i ) 6 (M; -). From (b) we hve tht (M; M i )?! L n, nd from () we hve tht L n hn; Ri-INC, so (M; M i )?! hn; Ri-INC. Hence by Corollry 4.2, (M; M i ) 6 r, so (M; M i ) 6 (M; -). It remins to prove the semidecidbility of conditions () nd (b) for given n-bounded mrking L. Condition () is semidecidble by Proposition 2.2. For condition (b), set M 1 = f (M; M 0 ) : M 0 is rbitrry g nd M 2 = L n, nd pply Lemm Undecidbility of wek bisimultion equivlence We next show tht the question M 0 r 0 is undecidble. In fct, we prove tht neither of the problems M 0 r 0 nd M 0 6 r 0 is semidecidble. From the proof of this result, we ctully get fixed 7-stte trnsition system R fix with distinguished stte r fix such tht M 0 r fix is undecidble. In fct, even M 0 4 r fix is undecidble. As the bsis for our reduction, we use the following undecidble problem from Petri net theory: Continment problem: Given two Petri nets N 1 nd N 2 defined over the sme set of plces nd initil mrking M, is R N1 (M) R N2 (M)? The undecidbility of this problem ws first demonstrted by Rbin (see [4]) by mens of reduction from Hilbert s 10th problem. A reduction from the hlting problem for Minsky mchines cn be found in [10]. In the next section we shll need to describe the ltter reduction in more detil. Let two Petri nets N 1 = (P; ; T 1 ; F 1 ; `1) nd N 2 = (P; ; T 2 ; F 2 ; `2) be given, long with common initil mrking M. Without loss of generlity, we ssume tht jr N2 (M)j 2; nd 0 =2 R N1 (M) \ R N2 (M). 23

26 d r 0 e r 1 r 2 r 5 r 6 r 3 b r 4 b r 7 r 8 c c r 9 Figure 3: The finite-stte system R We shll describe the construction of new net N with initil mrking M 0 such tht 1. if R N1 (M) 6 R N2 (M) then M 0 r 1, where r 1 is tken from the finite trnsition system R shown in Figure 3; nd 2. if R N1 (M) R N2 (M) then M 0 r 5, where r 5 is gin tken from R. (The stte r 0 of R is used in the next section.) We cn note the following bout the sttes of R: r r 9, since r 4 r r 7, since r 7 r r 5, since r 5 r r 5, since r 1 In prticulr, r 1 6 r 5. 6 c c =) but r 9 =). b =) r 9 would hve to be mtched by r 3 b =) r 7 would hve to be mtched by r 2 =) r 2 would hve to be mtched by r 5 =) r 4, but r r 9. =) r 3, but r r 7. =) r 5, but r r 5. When defining N we use the following notion. A plce p is run-plce of set T of trnsitions if h p; t i nd h t; p i re both rcs for every t 2 T. In prticulr, the trnsitions of T cn occur only when p holds t lest one token. Figure 4 shows schem of the net N. To construct it, we first tke the disjoint union of N 1 nd N 2, relbelling ll trnsitions by. We ssume tht the plces of N i (for i = 1; 2) re given by P i = fp i : p 2 Pg, nd the trnsitions of N i re given by T i = ft i : t 2 Tg. As prt of the initil mrking M 0, we put M on N 1 nd on N 2. We then dd further plces nd trnsitions s indicted. The plce q 1 is run-plce of T 1 (grphiclly represented by double pointed white rrow), nd contins initilly one token. This token cn be moved by -trnsition to plce q 0 1, nd then by n -trnsition to q 2, which is run-plce of T 2. From q 2, the token cn be moved by nother -trnsition to q 0 2 nd by b-trnsition to q 3, which is run-plce of n dditionl set of trnsitions. This set contins: 24