On Liveness and Reversibility of Equal-Conflict Petri Nets

Abstract

Weighted Petri nets provide convenient models of many man-made systems. Real applications are often required to possess the fundamental Petri net properties of liveness and reversibility, as liveness preserves all the functionalities (fireability of all transitions) of the system and reversibility lets the system return to its initial state (marking) using only internal operations.Characterizations of both behavioral properties, liveness and reversibility, are known for well-formed weighted Choice-Free and ordinary Free-Choice Petri nets, which are special cases of Equal-Conflict Petri nets. However, reversibility is not well understood for this larger class, where choices must share equivalent preconditions, although characterizations of liveness are known.In this paper, we provide the first characterization of reversibility for all live Equal-Conflict Petri nets by extending, in a weaker form, a known condition that applies to the Choice-Free and Free-Choice subclasses. We deduce the monotonicity of reversibility in the live Equal-Conflict class. We also give counter-examples for other classes where the characterization does not hold. Finally, we focus on well-formed Equal-Conflict Petri nets, for which we offer the first polynomial sufficient conditions for liveness and reversibility, contrasting with the previous exponential time conditions.

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

“…We deduce from them the non-monotonicity of non-deadlockability in this class under similar assumptions. These examples contrast with the HFC case, thus also with the HJF case [9,21,11]. Besides, they show that a previous characterization of m-reversibility for live HFC nets, developed in [21] and based on the existence of an initially feasible sequence that contains all transitions and leads to the initial markingalso known as a T-sequence-, is not sufficient to ensure m-reversibility in live, structurally bounded, inhomogeneous JF nets.…”

“…The subclass of homogeneous nets, in which each place has all its outputs weights equal, has been fruitfully studied in previous works under additional restrictions [9,10,11]. In this paper, we focus on join-free (JF) nets, which do not have synchronizations, and homogeneous asymmetric-choice (HAC) nets, in which each pair p, p of input places of any synchronization satisfies the following: all the outputs of p are also outputs of p , or conversely.…”

“…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.…”

“…However, monotonic liveness (m-liveness) and monotonic non-deadlockability are fulfilled by OFC nets and some larger classes that contain HJF nets [13,9,15,12,20,16]. M-liveness+reversibility, meaning m-liveness and m-reversibility considered together, is also fulfilled by homogeneous (weighted) free-choice (HFC, also called equal-conflict or EC) nets [13,11]. In the HFC class, under some restrictions, liveness and non-deadlockability are equivalent, allowing to deduce the same results for the assumption of non-deadlockability [9].…”

“…In the HAC class and its subclass of HFC nets, m-liveness has been expressed in terms of the mliveness of JF P-subnets induced by siphons [9,10]. Exploiting this fact in a bottom-up approach, polynomial-time sufficient conditions of m-liveness for bounded JF nets were shown to propagate to the decomposable, bounded HFC nets [11]. For the Choice-free (CF) nets (not to be confused with freechoice nets), which form a subclass of the HFC nets, similar techniques have been exploited to deduce polynomial-time sufficient conditions of m-liveness+reversibility, using particular reversible JF subnets [18].…”