3 - Classical Finite-State Automata and Regular Languages

Summary

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.