A coalgebraic semantics for causality in Petri nets

TL;DR

This paper develops a coalgebraic framework for modeling causality in Petri nets, providing both set-theoretic and coalgebraic models that capture causal dependencies and enable state reduction.

Contribution

It introduces a novel coalgebraic semantics for Petri nets with causality, linking set-theoretic and coalgebraic models, and offers a compact, minimal representation of causal behavior.

Findings

Set-theoretic causal models correspond with behavior structure semantics.

Finitely-branching models with symmetries enable minimal state representations.

Coalgebraic models can be reduced to compact automata via presheaf-based equivalences.

Abstract

In this paper we revisit some pioneering efforts to equip Petri nets with compact operational models for expressing causality. The models we propose have a bisimilarity relation and a minimal representative for each equivalence class, and they can be fully explained as coalgebras on a presheaf category on an index category of partial orders. First, we provide a set-theoretic model in the form of a a causal case graph, that is a labeled transition system where states and transitions represent markings and firings of the net, respectively, and are equipped with causal information. Most importantly, each state has a poset representing causal dependencies among past events. Our first result shows the correspondence with behavior structure semantics as proposed by Trakhtenbrot and Rabinovich. Causal case graphs may be infinitely-branching and have infinitely many states, but we show how they can be refined to get an equivalent finitely-branching model. In it, states are equipped with symmetries, which are essential for the existence of a minimal, often finite-state, model. The next step is constructing a coalgebraic model. We exploit the fact that events can be represented as names, and event generation as name generation. Thus we can apply the Fiore-Turi framework: we model causal relations as a suitable category of posets with action labels, and generation of new events with causal dependencies as an endofunctor on this category. Then we define a well-behaved category of coalgebras. Our coalgebraic model is still infinite-state, but we exploit the equivalence between coalgebras over a class of presheaves and History Dependent automata to derive a compact representation, which is equivalent to our set-theoretical compact model. Remarkably, state reduction is automatically performed along the equivalence.