Counting Petri net markings from reduction equations

Abstract

We propose a method to count the number of reachable markings of a Petri net without having to enumerate these rst. The method relies on a structural reduction system that reduces the number of places and transitions of the net in such a way that we can faithfully compute the number of reachable markings of the original net from the reduced net and the reduction history. The method has been implemented and computing experiments show that reductions are eective on a large benchmark of models.Structural reductions are an important class of optimization techniques for the analysis of Petri Nets (PN for short). The idea is to use a series of reduction rules that decrease the size of a net while preserving some given behavioral properties. These reductions are then applied iteratively until an irreducible PN is reached on which the desired properties are checked directly. This approach, pioneered for Petri nets by Berthelot [2, 3], has been used to reduce the complexity of several problems, such as checking for boundedness of a net, for liveness analysis, for checking reachability properties [12] or for LTL model checking [7].In this paper, we enrich the notion of structural reduction by keeping track of the relation between the markings of an (initial) Petri net, N 1 , and its reduced (nal) version, N 2 . We use reductions of the form (N 1 , Q, N 2 ), where Q is a system of linear equations that relates the (markings of ) places in N 1 and N 2 . We say that Q is a set of reduction equations.In our approach, reductions are tailored so that the state space of N 1 (its set of reachable markings) can be faithfully reconstructed from that of N 2 and equations Q. In particular, when

Accelerating the Computation of Dead and Concurrent Places Using Reductions

“…Our approach is based on a combination of structural reductions with linear equations first proposed in [4,5]. Our main contribution, in the current work, is the definition of a new data-structure that precisely captures the structure of these linear equations, what we call the Token Flow Graph (TFG).…”

Structural Reductions Revisited

“…[5] has a similar rule limited to fusing two adjacent places linked by a pair of elementary transitions. This rule (and a generalization of it) is presented using a different formalization in [2].…”

“…The classical reduction rules [1] include pre and post agglomeration, for which [9,11] give broad general definitions that can be applied also to colored nets. More recently, several competitors in the Model Checking Contest have worked on the subject, [5] defines 8 reduction rules used in the tool Tapaal and [2] defines very general transition-centric reduction rules used in the tool Tina.…”

LTL under reductions with weaker conditions than stutter-invariance

“…Structural reductions take their roots in the work of Lipton [14] and Berthelot [1]. Nowadays, these reductions are still considered as an attractive way to alleviate the state explosion problem [13,2]. Structural reductions strive to fuse structurally "adjacent" events into a single atomic step, leading to less interleaving of independent events and less observable behaviors in the resulting system.…”