Dense-Timed Petri Nets: Checking Zenoness, Token liveness and Boundedness

Abstract

Abstract. We consider Dense-Timed Petri Nets (TPN), an extension of Petri nets in which each token is equipped with a real-valued clock and where the semantics is lazy (i.e., enabled transitions need not fire; time can pass and disable transitions). We consider the following verification problems for TPNs.(i) Zenoness: whether there exists a zeno-computation from a given marking, i.e., an infinite computation which takes only a finite amount of time. We show decidability of zenoness for TPNs, thus solving an open problem from [dFERA00]. Furthermore, the related question if there exist arbitrarily fast computations from a given marking is also decidable.On the other hand, universal zenoness, i.e., the question if all infinite computations from a given marking are zeno, is undecidable.(ii) Token liveness: whether a token is alive in a marking, i.e., whether there is a computation from the marking which eventually consumes the token. We show decidability of the problem by reducing it to the coverability problem, which is decidable for TPNs.(iii) Boundedness: whether the size of the reachable markings is bounded. We consider two versions of the problem; namely semantic boundedness where only live tokens are taken into consideration in the markings, and syntactic boundedness where also dead tokens are considered. We show undecidability of semantic boundedness, while we prove that syntactic boundedness is decidable through an extension of the Karp-Miller algorithm.

Decidable Classes of Unbounded Petri Nets with Time and Urgency

“…The decidability of control-state reachability and boundedness for TimedArc PNU is adapted from [2,1], by defining a well-quasi order over the markings and using the theory of well structured transition systems [17]. The well quasi order is defined in the following way.…”

Petri net-based process monitoring: a workflow management system for process modelling and monitoring

“…This means that when a token t arrives to a place p it must stay there at least ts p time units although its transition fires. Finally, token-timed PN (or dense-timed PN) [7] is an extension of Petri nets in which each token is equipped with a real-valued clock so the time spent for every token can be registered. In this article, when timed Petri nets are mentioned is referring to this token-timed PN extension.…”

“…Every time a token appears in the input place i means that a business new process p has started inside the workflow. Tokens are colored so the processes can be distinguished RAPN (Definition 5) incorporate resources to high level Petri nets [7]. Resources (Definition 2) are related with sets of consecutive transitions (forming subpaths, Definition 3) where the first transition (t s ) is the one which allocates the resource and the last (t e ) is the one which releases it.…”

Comparing the Expressiveness of Timed Automata and Timed Extensions of Petri Nets

“…We then reach the configuration ( 1 , [x = d]). In the location 1 we can now delay for at most 3 − d time units because of the invariant x ≤ 3 associated with the location 1 . As soon as the value of the clock x is at least 2 we can also take the discrete transition labelled with b and return to the initial configuration ( 0 , [x = 0]) because the value of the clock x is reset to 0 by taking this transition.…”

“…On the other hand, coverability, boundedness and other problems remain decidable for TAPN [77,3,1], which are also known to offer 'weak' expressiveness, in the sense that TAPN cannot simulate Turing machines [25]. Coverability is decidable even for TAPN extended with read-arcs [30].…”