A Modified Completeness Theorem of KAT and Decidability of Term Reducibility

Abstract

Kleene algebra with tests (KAT) was introduced by Kozen as an extension of Kleene algebra (KA). The decidability of equational formulas p = q and Horn formulas ∧ i p i = q i → p = q in KAT has been studied so far by several researchers. Continuing this line of research, this paper studies the decidability of existentially quantified equational formulas ∃ q ∈ P. (p = q) in KAT, where P is a fixed collection of KAT terms. A new completeness theorem of KAT is proved, and via the completeness theorem, the decision problem of ∃ q ∈ P. (p = q) is reduced to a certain membership problem of regular languages, to which a pseudo-identity-based decision method is applicable. Based on this reduction, an instance of the problem is studied and shown to be decidable.