# A completeness theorem for Kleene algebras and the algebra of regular events

@article{Kozen1991ACT, title={A completeness theorem for Kleene algebras and the algebra of regular events}, author={Dexter Kozen}, journal={[1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science}, year={1991}, pages={214-225} }

A finitary axiomatization of the algebra of regular events involving only equations and equational implications that is sound for all interpretations over Kleene algebras is given. Axioms for Kleene algebra are presented, and some basic consequences are derived. Matrices over a Kleene algebra are considered. The notion of an automaton over an arbitrary Kleen algebra is defined and used to derive the classical results of the theory of finite automata as a result of the axioms. The completeness…

## 407 Citations

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This paper presents an axiomatization of Kleene algebras and a proof that they do, in fact, describe the behavior of regular expressions and finds that n× n matrices over a Kleene algebra are also a KleENE algebra.

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A weak variant of omega algebra, where one of the usual star induction axioms is absent, is studied in the context of recursive regular equations, which yields a sound and complete axiomatisation in which the "regular" axiOMs are weaker than Kleene algebra.

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Kleene algebra (an Horn axiomatization of Kleene's algebra of regular events) has proved to be an eeective tool for reasoning about programs. Within the algebra, we can reason succinctly about both…

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This work uses Isabelle/HOL for a detailed systematic study of regular algebras given by Boffa, Conway, Kozen and Salomaa, and formalises a soundness proof for the smallest class and obtains completeness of the largest one relative to a deep result by Krob.