The Algebraic Approach I: The Algebraization of the Chomsky Hierarchy

The algebraic approach to formal language and automata theory is a continuation of the earliest traditions in these fields which had sought to represent languages, translations and other computations as expressions (e.g. regular expressions) in suitably-defined algebras; and grammars, automata and transitions as relational and equational systems over these algebras, that have such expressions as their solutions. The possibility of a comprehensive foundation cast in this form, following such results as the algebraic reformulation of the Parikh Theorem, has been recognized by the Applications of Kleene Algebra (AKA) conference from the time of its inception in 2001. Here, we take another step in this direction by embodying the Chomsky hierarchy, itself, within an infinite complete lattice of algebras that ranges from dioids to quantales, and includes many of the forms of Kleene algebras that have been considered in the literature. A notable feature of this development is the generalization of the Chomsky hierarchy, including type 1 languages, to arbitrary monoids.