On Causal Semantics of Petri Nets

TL;DR

This paper explores causal semantics of Petri nets, comparing detailed and abstract process notions, and introduces a subclass called structural conflict nets where a unique maximal abstract process exists.

Contribution

It defines a new class of Petri nets, structural conflict nets, and proposes an equivalence-based approach to abstract processes within this class.

Findings

In structural conflict nets, a unique maximal abstract process exists if and only if the net is conflict-free.

The paper provides an overview of existing approaches to causal semantics of Petri nets.

It clarifies the relationship between conflict, concurrency, and token multiplicity in Petri nets.

Abstract

We consider approaches for causal semantics of Petri nets, explicitly representing dependencies between transition occurrences. For one-safe nets or condition/event-systems, the notion of process as defined by Carl Adam Petri provides a notion of a run of a system where causal dependencies are reflected in terms of a partial order. A well-known problem is how to generalise this notion for nets where places may carry several tokens. Goltz and Reisig have defined such a generalisation by distinguishing tokens according to their causal history. However, this so-called individual token interpretation is often considered too detailed. A number of approaches have tackled the problem of defining a more abstract notion of process, thereby obtaining a so-called collective token interpretation. Here we give a short overview on these attempts and then identify a subclass of Petri nets, called structural conflict nets, where the interplay between conflict and concurrency due to token multiplicity does not occur. For this subclass, we define abstract processes as equivalence classes of Goltz-Reisig processes. We justify this approach by showing that we obtain exactly one maximal abstract process if and only if the underlying net is conflict-free with respect to a canonical notion of conflict.