Kleene’s theorem revisited

Abstract

The analysis of the famous Kleene's theorem shows that it consists indeed in two different propositions that are better distinguished when one tries to generatize the result. The first one relates rational expressions and a suitable generalization of finite automata. It holds in any monoid or, even better, in the semiring of formal power series on any monoid. It is shown that several classical results in formal language theory, for instance Elgot and Mezei characterization of rational relations by transducers and Chomsky normal form for context-free grammars, can thus be seen as particular cases of this first half of Kleene's theorem.