Abstract Processes of Place/Transition Systems

TL;DR

This paper proposes a new abstract concept of process for place/transition systems in Petri net theory, providing a fully satisfying solution for certain classes of nets, especially structural conflict nets.

Contribution

It introduces BD-processes as an abstract process notion, resolving issues in existing definitions for specific classes of Petri nets.

Findings

BD-processes form a fully satisfying solution for one-safe nets.

In structural conflict nets, BD-processes yield a unique maximal abstract process for conflict-free nets.

The approach separates conflict and concurrency due to token multiplicity.

Abstract

A well-known problem in Petri net theory is to formalise an appropriate causality-based concept of process or run for place/transition systems. The so-called individual token interpretation, where tokens are distinguished according to their causal history, giving rise to the processes of Goltz and Reisig, is often considered too detailed. The problem of defining a fully satisfying more abstract concept of process for general place/transition systems has so-far not been solved. In this paper, we recall the proposal of defining an abstract notion of process, here called BD-process, in terms of equivalence classes of Goltz-Reisig processes, using an equivalence proposed by Best and Devillers. It yields a fully satisfying solution for at least all one-safe nets. However, for certain nets which intuitively have different conflicting behaviours, it yields only one maximal abstract process. Here we identify a class of place/transition systems, called structural conflict nets, where conflict and concurrency due to token multiplicity are clearly separated. We show that, in the case of structural conflict nets, the equivalence proposed by Best and Devillers yields a unique maximal abstract process only for conflict-free nets. Thereby BD-processes constitute a simple and fully satisfying solution in the class of structural conflict nets.