Verification of Nonblockingness in Bounded Petri Nets With a Semi-Structural Approach

Abstract

This paper proposes a semi-structural approach to verify the nonblockingness of a Petri net. We provide an algorithm to construct a novel structure, called minimax basis reachability graph (minimax-BRG): it provides an abstract description of the reachability set of a net while preserving all information needed to test if the net is blocking. We prove that a bounded deadlock-free Petri net is nonblocking if and only if its minimax-BRG is unobstructed, which can be verified by solving a set of integer linear programming problems (ILPPs). For Petri nets that are not deadlock-free, one needs to determine the set of deadlock markings. This can be done with an efficient approach based on the computation of maximal implicit firing sequences enabled by the markings in the minimax-BRG. The approach we developed does not require exhaustive exploration of the state space and therefore achieves significant practical efficiency, as shown by means of numerical simulations.

Verification of Nonblockingness in Bounded Petri Nets With Min-Max Basis Reachability Graphs

“…On the other hand, in our previous work [33] we show that the standard BRG cannot be directly used to solve the NB-V problem due to the possible presence of livelocks and deadlocks. In particular, livelocks describe an undesirable non-dead repetitive behavior such that the system is bound to evolve along a particular subset of its reachability space.…”

“…However, the set of markings that form a livelock is usually hard to characterize and is not encoded in the classical BRG of the system. As a countermeasure, preliminary results are presented in [33] to show how it is possible to tailor the BRG to detect livelocks. In more detail, a structure named the expanded BRG is proposed, which expands the BRG so that all markings in R(N, M 0 ) reached by firing a sequence of transitions ending with an explicit transition are included.…”

Non-Blockingness Verification of Bounded Petri Nets Using Basis Reachability Graphs

“…Given a plant net, in general there exist several valid basis partitions, each of which leads to a different BRG. According to Example 1 presented in [10], a BRG constructed with a randomly selected basis partition may not encode all information needed to test if a plant is non-blocking. The reason lies in the fact that in a BRG there may exist some livelocks among a set of non-basis markings; thus, the blocking behavior of the plant net cannot be detected by checking the structure of the BRG.…”

“…Although the BRG-based techniques have been proved to be operative and efficient, in [10], [11] we have shown that a conventional BRG is in general not applicable to tackle the non-blockingness verification problem. The reason is that in a BRG there may exists some livelocks among a set of non-basis markings.…”

“…In such a case, the blocking behavior of the plant net cannot be detected by inspecting the structure of the BRG. To overcome such a problem, in [10], [11] two augmented version of BRGs called expanded BRG and minimax-BRG are developed. Although these new structures exhibit practical efficiency in solving the nonblockingness verification problem, unlike the basis-markingbased approach [17], currently there exist no analysis methods for addressing with these models on state estimation and supervisory control problems, in which non-blockingness analysis plays a key role.…”