As a starting point we study finite-state automata, which represent the simplest devices for recognizing languages. The theory of finite-state automata has been described in numerous textbooks both from a computational and an algebraic point of view. Here we immediately look at the more general concept of a monoidal finite-state automaton, and the focus of this chapter is general constructions and results for finite-state automata over arbitrary monoids and monoidal languages. Refined pictures for the special (and more standard) cases where we only consider free monoids or Cartesian products of monoids will be given later.

Classical finite-state automata represent the most important class of monoidal finite-state automata. Since the underlying monoid is free, this class of automaton has several interesting specific features. We show that each classical finite-state automaton can be converted to an equivalent classical finite-state automaton where the transition relation is a function. This form of ‘deterministic’ automaton offers a very efficient recognition mechanism since each input word is consumed on at most one path. The fact that each classical finite-state automaton can be converted to a deterministic automaton can be used to show that the class of languages that can be recognized by a classical finite-state automaton is closed under intersections, complements, and set differences. The characterization of regular languages and deterministic finite-state automata in terms of the ‘Myhill–Nerode equivalence relation’ to be introduced in the chapter offers an algebraic view on these notions and leads to the concept of minimal deterministic automata.

In this chapter we present C(M) implementations of the main automata constructions. Our aim is to provide full descriptions of the implementations that are clear and easy to follow. In some cases the simplicity of the implementation is achieved at the expense of some inefficiency.