Petri Automata

TL;DR

This paper introduces Petri automata, a new finite automata model inspired by Petri nets, to recognize graph languages in Kleene allegories, and proves a Kleene theorem for these automata.

Contribution

It defines Petri automata for recognizing graph languages in Kleene allegories and establishes a Kleene theorem linking automata and extended regular expressions.

Findings

Petri automata recognize graph languages definable by extended regular expressions.

Decidability of identity-free relational Kleene lattices is established.

The decision problem is proven to be EXPSPACE-complete.

Abstract

Kleene algebra axioms are complete with respect to both language models and binary relation models. In particular, two regular expressions recognise the same language if and only if they are universally equivalent in the model of binary relations. We consider Kleene allegories, i.e., Kleene algebras with two additional operations and a constant which are natural in binary relation models: intersection, converse, and the full relation. While regular languages are closed under those operations, the above characterisation breaks. Putting together a few results from the literature, we give a characterisation in terms of languages of directed and labelled graphs. By taking inspiration from Petri nets, we design a finite automata model, Petri automata, allowing to recognise such graphs. We prove a Kleene theorem for this automata model: the sets of graphs recognisable by Petri automata are precisely the sets of graphs definable through the extended regular expressions we consider. Petri automata allow us to obtain decidability of identity-free relational Kleene lattices, i.e., the equational theory generated by binary relations on the signature of regular expressions with intersection, but where one forbids unit. This restriction is used to ensure that the corresponding graphs are acyclic. We actually show that this decision problem is EXPSPACE-complete.