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.

This chapter describes a special construction based on finite-state automata with important applications: the Aho–Corasick algorithm is used to efficiently find all occurrences of a finite set of strings (also called pattern set, or dictionary) in a given input string, called the ‘text’. Search is ‘online’, which means that the input text is neither fixed nor preprocessed in any way. This problem is a special instance of pattern matching in strings, and other automata constructions are used to solve other pattern matching tasks. From an automaton point of view, the Aho–Corasick algorithm comes in two variants. We first present the more efficient version where a classical deterministic finite-state automaton is built for text search. The disadvantage of this first construction is that the resulting automaton can become very large, in particular for large pattern alphabets. Afterwards we present the second version, where an automaton with additional transitions of a particular kind is built, yielding a much smaller device for text search.