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

Abstract

Liveness, (non-)deadlockability and reversibility are behavioral properties of Petri nets that are fundamental for many real-world systems. Such properties are often required to be monotonic, meaning preserved upon any increase of the marking. However, their checking is intractable in general and their monotonicity is not always satisfied. To simplify the analysis of these features, structural approaches have been fruitfully exploited in particular subclasses of Petri nets, deriving the behavior from the underlying graph and the initial marking only, often in polynomial time.In this paper, we further develop these efficient structural methods to analyze deadlockability, liveness, reversibility and their monotonicity in weighted Petri nets. We focus on the join-free subclass, which forbids synchronizations, and on the homogeneous asymmetric-choice subclass, which allows conflicts and synchronizations in a restricted fashion.For the join-free nets, we provide several structural conditions for checking liveness, (non-)deadlockability, reversibility and their monotonicity. Some of these methods operate in polynomial time. Furthermore, in this class, we show that liveness, non-deadlockability and reversibility, taken together or separately, are not always monotonic, even under the assumptions of structural boundedness and structural liveness. These facts delineate more sharply the frontier between monotonicity and nonmonotonicity of the behavior in weighted Petri nets, present already in the join-free subclass.In addition, we use part of this new material to correct a flaw in the proof of a previous characterization of monotonic liveness and boundedness for homogeneous asymmetric-choice nets, published in 2004 and left unnoticed.

Checking marking reachability with the state equation in Petri net subclasses

“…Subsystems play a fundamental role in the analysis of Petri nets, typically leading to characterizations relating the system's behavior to properties of its subsystems; this approach yielded polynomialtime checking methods in various subclasses, e.g. [14,4,6]. We exploit such subsystems in this paper to obtain some of our new results on reachability.…”

“…Polynomial-time sufficient conditions of liveness and reversibility also exist for wellformed HFC nets and join-free (JF) nets (i.e. without synchronizations) [14,4,6].…”

“…To avoid such a costly computation, subclasses are often considered, allowing to derive efficiently their behavior from their structure only. This approach has led to various polynomial-time checking methods dedicated to several subclasses, the latter being defined by structural restrictions in many cases [1,2,3,4,5,6].…”

Efficient Synthesis of Weighted Marked Graphs with Circular Reachability Graph, and Beyond

“…Next result is a specialisation of Theorem 4.5 in [27] to circuit Petri nets, using the fact that the liveness of circuits is monotonic, i.e. preserved upon any addition of initial tokens (see [34,13,27] and Theorem 7.10 in [29] with its proof).…”

“…Proposition 1 (Sufficient condition of liveness [31,27,29]). Consider a conservative circuit system S = (N, M 0 ), with N = (P, T, W ).…”