Kleene Algebra Completeness

This paper gives a new presentation of Kozen’s proof of Kleene algebra completeness featured in his article A completeness theorem for Kleene algebras and the algebra of regular events. A few new variants are introduced, shortening the proof. Specifically, we directly construct an ε-free automaton to prove an equivalent to Kleene’s representation theorem (implementing Glushkov’s instead of Thompson’s construction), and we bypass the use of minimal automata by directly implementing a Myhill-Nerode equivalence relation on the union of equivalent deterministic automata.