A bialgebraic approach to automata and formal language theory

A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are compatible. Bialgebras are usually studied over a commutative ring. In this paper, we apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring. We then treat automata as certain representation objects of algebras and formal languages as elements of dual algebras of coalgebras. Using this perspective, we demonstrate many analogies between the two theories. Finally, we show that there is an adjunction between the category of “algebraic” automata and the category of deterministic automata. Using this adjunction, we show that K-linear automaton morphisms can be used as the sole rule of inference in a complete proof system for automaton equivalence.