Kleene Algebra with Converse

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

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

The equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik, and Stefanescu in 1995. We reformulate some of their proofs in syntactic and elementary terms, and we provide a new algorithm to decide the corresponding theory. This algorithm is both simpler and more efficient; it relies on an alternative automata construction, that allows us to prove that the considered equational theory lies in the complexity class PSPACE. Specific regular languages appear at various places in the proofs. Those proofs were made tractable by considering appropriate automata recognising those languages, and exploiting symmetries in those automata.

Mots clés: regular expression; PSPACE; complexity; decidability; automata; converse; relations; Kleene algebra

Domaines: Algorithme et structure de données [cs.DS]; Complexité [cs.CC]; Logique en informatique [cs.LO]; Mathématique discrète [cs.DM]

Paul Brunet, Damien Pous. Kleene Algebra with Converse. RAMiCS, Apr 2014, Marienstatt im Westerwald, Germany. pp.101-118. ⟨hal-00938235⟩