Polynomial Sufficient Conditions of Well-Behavedness and Home Markings in Subclasses of Weighted Petri Nets

Abstract

International audienceJoin-Free Petri nets, whose transitions have at most one input place, model systems without synchronizations, while Choice-Free Petri nets, whose places have at most one output transition, model systems without conflicts. These classes respectively encompass the state machines (S-systems) and the marked graphs (T-systems). Whereas a structurally bounded and structurally live Petri net is said to be "well-formed", a bounded and live Petri net is said to be "well-behaved". Necessary and sufficient conditions for the well-formedness of Join-Free and Choice-Free nets have been known for some time, yet the behavioral properties of these classes are still not well understood. In particular polynomial sufficient conditions for liveness, that is, polynomial in time and with a polynomial initial number of tokens, have not been found until now. Besides, home markings, which can be reached from every reachable marking thus allowing for the construction of systems that can return to their initial data distribution, are not well apprehended either for these subclasses. We extend results on weighted T-systems to the class of weighted Petri nets and present transformations which preserve the language of the system and reduce the initial marking. We introduce a notion of balancing that makes possible the transformation of conservative systems into so-called "token-conservative" systems, whose number of tokens is invariant, while retaining the feasible transition sequences. This transformation is pertinent for all well-formed Petri nets and leads to polynomial sufficient conditions of liveness for well-formed Join-Free and Choice-Free nets. Finally, we also provide polynomial live and home markings for Fork-Attribution systems

On Deadlockability, Liveness and Reversibility in Subclasses of Weighted Petri Nets

“…Deadlockability, liveness and related properties have been studied previously in the (weighted) join-free class, notably under the conservativeness (i.e. structural boundedness) assumption [2,9,14,17] or in more restricted subclasses [24,25]. Figure 3.…”

“…Similar techniques, sometimes in a weaker form, have been developed for other classes with weights, including JF and HAC nets [9,15,10,16,2,14,17]. Also, several polynomial-time sufficient conditions of liveness, boundedness and reversibility, taken together, have been proposed for several classes [17,18,11], some of which generalize the HJF and OFC nets.…”

“…The termination and validity of this algorithm are shown in Lemma 6.3 below, which is a variant of Lemma 2 in [21] for JF nets. Notice that the assumption of liveness of this previous result is replaced by strong connectedness and the fact that every reachable marking enables at least one place, implying liveness in the JF case [30,17]. With these new assumptions, we do not need here the notion of fairness exploited in [21], as detailed in the following proof.…”

Synthesis of Weighted Marked Graphs from Constrained Labelled Transition Systems: A Geometric Approach

“…In previous studies on the analysis or synthesis of Petri nets, structural restrictions encompassed plain nets (each weight equals 1; also called ordinary nets) [25], homogeneous nets (meaning that for each place p, all the output weights of p are equal) [28,23], free-choice nets (the net is homogeneous, and any two transitions sharing an input place have the same set of input places) [11,28], choice-free nets (each place has at most one output transition) [27,20], marked graphs (each place has at most one output transition and one input transition) [7,26,6,14], join-free nets (each transition has at most one input place) [28,10,21,23], etc.…”

On Liveness and Deadlockability in Subclasses of Weighted Petri Nets

“…Similar techniques, sometimes in a weaker form, have been developed for other classes with weights, including JF and HAC nets [22,16,11,1,20,18,9].…”