A Kleene Theorem for Petri automata

Highlights in Brussels, September 2016

Petri Automata are automata based on Petri nets, whose operational semantics is designed to recognise sets of graphs. We introduced them in a recent paper [1] with Pous to study Kleene Allegory (KAl) expressions: terms built from a finite alphabet of variables, with the constants 0 and 1, the unary operators converse and Kleene star, and the binary operators union, composition and intersection. To any such expression, one can associate a set of graphs such that two expressions are universally equivalent when interpreted as binary relations if and only if their associated sets of graphs are equal. In [1] we gave a method to build from any Kleene Allegory expression a Petri automaton recognising the appropriate set of series parallel graphs. It is then natural to wonder what is the class of graph languages recognised by Petri automata. For the usual notion of finite state automata, the well known Kleene Theorem states that the languages recognisable by automata are exactly those specifiable by regular expressions. This year we provided a similar theorem for Petri automata and Kleene Allegory expressions [2]. Indeed the graph languages specified by Petri automata are precisely the languages denoted by KAl. Moreover, this entails that decidability of the equational theory of Kleene Allegories is equivalent to that of language equivalence for Petri automata. This decidability problem is still open. It is worth mentioning that the technical part of the proof in [2] actually uses a smaller syntax than Kleene Allegories, without converse nor identity. In this fragment the considered graphs are series-parallel, so that the key technical result is actually a Kleene theorem for series-parallel regular expressions and languages. References: [1] Paul Brunet and Damien Pous. Petri automata for Kleene allegories. In Logic in Computer Science (LICS), 2015 30th Annual ACM/IEEE Symposium on, pages 68–79, July 2015. [2] Paul Brunet. Algebras of relations: from algorithms to formal proofs. PhD Thesis, October 2016.

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