R. Valk and M. Jantzen. The residue vector sets with applications to decidability problems in Petri nets. Acta Informatica, 21:643--674, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....prompt if there exists a number k such that no occurrence sequence contains more than k consecutive silent transitions. Promptness is thus strongly related to the notion of divergence in process algebras. The promptness and strong promptness problems were shown to be decidable by Valk and Jantzen [74]. It follows easily from a result of [71] that the promptness problem is polynomial for live and bounded free choice Petri nets. Persistence The persistence problem (to decide if a given Petri net is persistent) was shown to be decidable by Grabowsky [24] Mayr [53] and Muller [60] It is not ....

R. Valk and M Jantzen. The Residue of Vector Sets with Applications to Decidabilty problems in Petri Nets. Acta Informatica 21, 643--674 (1985).

....algorithm. Only recently, Bouziane discovered an EXPSPACE algorithm [14] which matches the long known complexity lower bound of Lipton [103] The existence of a run in (2. containing infinitely many occurrences of transitions of a given set T was shown to be decidable by Jantzen and Valk [83]. Later Yen has shown that this can be decided also within exponential space [150] Overall, this yields an EXPSPACE algorithm. Since on the other hand the reachability problem for Petri nets can be encoded in LTL, the model checking problem is also EXPSPACE hard, proving the problem to be ....

M. Jantzen and R. Valk. The residue of vector sets with applications to decidability problems in Petri nets. Acta Informatica, 21:643--674, 1985. 87

....or in [13] where their proofs are briefly recalled: 5 1. Every upward closed set K IN p has a finite basis. 2. Any strictly increasing sequence K 0 ae K 1 ae K 2 Delta Delta Delta of upward closed sets in IN p is finite. We will require another result from Valk and Jantzen: Theorem 2. 1 [29] A finite basis of an upward closed set K IN p is effectively computable iff for any vector u 2 IN p the predicate #u K 6= is decidable. Transition systems. A transition system (TS) is a structure S = hS; i where S = fs; t; g is a set of states, and S Theta S is any ....

R. Valk and M. Jantzen, The residue of vector sets with applications to decidability problems in Petri nets, Acta Informatica 21 (1985), pp. 643--674. 25

....oe 1 (t) 0) 9 1 2 9oe 0 ; oe 1 W A2A W t2T ( V r2T v A (r) # oe 1 (r) # oe 1 (t) 0) 25 As a consequence, the nontermination problem for each of the above notions of fairness can be solved in P. 3. Promptness Detection Problem. The concept of promptness was introduced in [30] as a way to deal with systems communicating with the environment. Let T I and TE be two disjoint sets of transitions such that T I [ TE = T . T I and TE can be viewed as the sets of internal and external transitions, respectively. A Petri net (P; T; 0 ) is said to be prompt (with respect to ....

R. Valk, and M. Jantzen, The residue of vector sets with applications to decidability problems in Petri nets, Acta Informa., 21 (1985), 643--674.

....effectively constructible. Proof The set [ S is upward closed, because all i are upward closed. Thus, this set is characterized by the finite set of its minimal elements (see 13 Lemma 3.2 and 3. 3) To find the minimal elements, we use a construction that was described by Valk and Jantzen in [VJ85]. The important point here is that we can use Lemma 6.4 to check the existence of configurations that satisfy . For example, if hq; x 2 ; x 3 )i j= then we can check if hq; n 1 ; x 2 ; x 3 )i j= for n 1 = 0, n 1 = 1, n 1 = 2, until we find the minimal n 1 s.t. hq; n 1 ; x 2 ; x 3 ....

R. Valk and M. Jantzen. The Residue of Vector Sets with Applications to Decidability Problems in Petri Nets. Acta Informatica, 21, 1985.

....S is UC definable and effectively constructible. Proof. S is upward closed, because all i are upward closed. Thus, it is characterized by the finite set of its minimal elements (see Lemma 3) To find the minimal elements, we use a construction that was described by Valk and Jantzen in [VJ85]. The important point here is that we can use Lemma 22 to check the existence of configurations that satisfy . For example, if hq; x 2 ; x 3 )i j= then we can check if hq; n 1 ; x 2 ; x 3 )i j= for n 1 = 0, n 1 = 1, n 1 = 2, until we find the minimal n 1 s.t. hq; n 1 ; x 2 ; x 3 ....

R. Valk and M. Jantzen. The Residue of Vector Sets with Applications to Decidability Problems in Petri Nets. Acta Informatica, 21, 1985.

....= J 3 [ P red(J 2 ) P red( p 1 p 3 ) J 3 [ p 2 1 p 2 J 5 = J 4 [ P red(J 3 [ p 2 1 p 2 ) J 3 [ p 2 1 p 2 [ P red(p 2 1 p 2 ) z = J 4 For Petri nets, computing P red (I) is quite simple as we just saw. We obtain a finite basis (or a set of residuals according to [VJ85] s terminology) It can be used to answer coverability questions because it is possible to cover M starting from M 0 iff M 0 2 P red ( M ) This algorithmic idea can be generalized for an arbitrary WSTS S: Definition 13. hS; i has effective pred basis when there is an algorithm computing a ....

R. Valk and M. Jantzen. The residue of vector sets with applications to decidability problems in Petri nets. Acta Informatica, 21:643--674, 1985.

....ti; k) n j ) e j for j = 1; k and l[hc; ti; k) n new ) e. This construction generates nite causal automata for the nite nets which are n safe for some n. We emphasize that it is decidable whether a nite P T net is n safe for some n. A possible procedure can be found in [44]. Theorem 6.27 Given a nite P T net N , the HD automaton AN is nite if and only if N is n safe for some n. Proof We show that AN has a nite number of states. Since a nite number of steps is possible in a net from a particular marking, also the number of transitions exiting from each state of ....

R. Valk and M. Jantzen. The residue vector sets with applications to decidability problems in Petri nets. Acta Informatica, 21:643-674, 1985.

....over Sigma 0 generated by sequences of transitions in e N ThetaS A reaching markings from which there are infinite sequences of transitions including infinitely often transitions in T1 . The set of such markings is actually semilinear and effectively constructible using the result proved in [18]: Lemma 7.1 Let N be a Petri net, and t one of its transitions. Then, the set M1 (N ; t) is semilinear and can be effectively constructed. Let us denote by L1 the semilinear set S t2T1 M1 ( e N Theta SA ; t) Then, by (9) g L(N ) b OE i ] b OE # i ] 6= iff 9oe 2 ( Sigma ....

R. Valk and M. Jantzen. The Residue of Vector Sets with Applications to Decidability Problems in Petri Nets. Acta Informatica, 21, 1985.

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R. Valk and M. Jantzen. The residue vector sets with applications to decidability problems in Petri nets. Acta Informatica, 21:643--674, 1985.

No context found.

R. Valk and M. Jantzen. The Residue of Vector Sets with Applications to Decidability Problems in Petri Nets. Acta Informatica, 21, 1985.

No context found.

R. Valk, M. Jantzen. The Residue of Vector Sets with Applications to Decidability in Petri Nets, Acta Informatica, 21, 643-674, 1985.

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