Home Spaces and Invariants to Analyze Parameterized Petri Nets

Abstract

This article focuses on comparing the notions of home spaces and invariants, in Transition Systems and more particularly, in Petri Nets as well as a variety of derived Petri Nets. After recalling basic notions of Petri Nets and semiflows, we then discuss important characteristics of finite generating sets for F, the set of all semiflows with integer coordinates of a given Petri Net. Then, we particularly focus on F+ the set of semiflows with non-negative coordinates. Minimality of semiflows and minimality of supports are critical to develop effective analysis of invariants and behavioral properties of Petri Nets such as boundedness or even liveness. We recall known decomposition theorems considering N, Q+, or Q. The result over N is being improved into a necessary and sufficient condition. In addition, we present general new results about the topology and the behavioral properties of a Petri Net, illustrating the importance of considering semiflows with non-negative coordinates. Then, we regroup a number of results around the notion of home space and home state applied to transition systems. Home spaces and semiflows are used to efficiently support the analysis of behavioral properties. In this regard, we present a methodology to analyze a Petri Nets by successive refinement of home spaces directly deduced from semiflows and apply it to analyze a parameterized example drawn from the telecommunication industry underlining the efficiency brought by using minimal semiflows of minimal supports as well as the new results on the topology of the model. This methodology is better articulated than in previous papers, and brings us closer to an automated analysis.