Petri automata for Kleene allegories

Paul Brunet (1, 2) , Damien Pous (1, 2)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 PLUME - Preuves et Langages

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 algebra with two additional operations which are natural in binary relation models: intersection and converse. While regular languages are closed under those operations, the above characterisation breaks. Instead, we give a characterisation in terms of languages of directed and labelled graphs. We then design a finite automata model allowing to recognise such graphs, by taking inspiration from Petri nets. This model allows 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. The decidability of graph-language equivalence in the full model remains open.

Mots clés: converse; automata; Kleene algebra; decidability; PSPACE; Petri nets; bisimulation; algorithm

Domaines: Logique en informatique [cs.LO]

Paul Brunet, Damien Pous. Petri automata for Kleene allegories. Logic in Computer Science, Jul 2015, Kyoto, Japan. pp.68-79, ⟨10.1109/LICS.2015.17⟩. ⟨hal-01073936v3⟩