Kleene Algebra with Converse
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.
Unable to display preview. Download preview PDF.
- [BP14]Brunet, P., Pous, D.: Extended version of this abstract, with omitted proofs. Technical report, LIP - CNRS, ENS Lyon (2014), http://hal.archives-ouvertes.fr/hal-00938235
- [Con71]Conway, J.H.: Regular algebra and finite machines. Chapman and Hall Mathematics Series (1971)Google Scholar
- [Kle51]Kleene, S.C.: Representation of Events in Nerve Nets and Finite Automata. Memorandum. Rand Corporation (1951)Google Scholar
- [Koz91]Kozen, D.: A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events. In: LICS, pp. 214–225. IEEE Computer Society (1991)Google Scholar
- [MS73]Meyer, A., Stockmeyer, L.J.: Word problems requiring exponential time. In: Proc. ACM Symposium on Theory of Computing, pp. 1–9. ACM (1973)Google Scholar
- [Mil89]Milner, R.: Communication and Concurrency. Prentice Hall (1989)Google Scholar
- [Red64]Redko, V.N.: On defining relations for the algebra of regular events. In: Ukrainskii Matematicheskii Zhurnal, pp. 120–126 (1964)Google Scholar