Abstract
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.
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References
Bloom, S.L., Ésik, Z., Stefanescu, G.: Notes on equational theories of relations. Algebra Universalis 33, 98–126 (1995)
Boffa, M.: Une remarque sur les systèmes complets d’identités rationnelles. Informatique Théorique et Applications 24, 419–428 (1990)
Boffa, M.: Une condition impliquant toutes les identités rationnelles. Informatique Théorique et Applications 29, 515–518 (1995)
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
Conway, J.H.: Regular algebra and finite machines. Chapman and Hall Mathematics Series (1971)
Ésik, Z., Bernátsky, L.: Equational properties of Kleene algebras of relations with conversion. Theoretical Computer Science 137, 237–251 (1995)
Glushkov, V.M.: The abstract theory of automata. Russian Mathematical Surveys 16, 1 (1961)
Kleene, S.C.: Representation of Events in Nerve Nets and Finite Automata. Memorandum. Rand Corporation (1951)
Kozen, D.: A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events. In: LICS, pp. 214–225. IEEE Computer Society (1991)
Krob, D.: A Complete System of B-Rational Identities. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 60–73. Springer, Heidelberg (1990)
Meyer, A., Stockmeyer, L.J.: Word problems requiring exponential time. In: Proc. ACM Symposium on Theory of Computing, pp. 1–9. ACM (1973)
Milner, R.: Communication and Concurrency. Prentice Hall (1989)
Redko, V.N.: On defining relations for the algebra of regular events. In: Ukrainskii Matematicheskii Zhurnal, pp. 120–126 (1964)
Salomaa, A.: Two Complete Axiom Systems for the Algebra of Regular Events. J. ACM 13, 158–169 (1966)
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Brunet, P., Pous, D. (2014). Kleene Algebra with Converse. In: Höfner, P., Jipsen, P., Kahl, W., Müller, M.E. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2014. Lecture Notes in Computer Science, vol 8428. Springer, Cham. https://doi.org/10.1007/978-3-319-06251-8_7
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DOI: https://doi.org/10.1007/978-3-319-06251-8_7
Publisher Name: Springer, Cham
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