In this chapter we consider the characterization of finite-state machines and the sets of sequences that they accept. We investigate a number of generalized forms of finite-state machines and prove that these forms are equivalent, with respect to the sets of sequences that they accept, to the basic deterministic finite-state model. In Sections 16.2 and 16.3 we study the properties of nondeterministic state diagrams, called transition graphs, which will prove to be a useful tool in the study of regular expressions. Procedures are developed whereby any transition graph can be converted into a deterministic state diagram.

Section 16.4 presents the language of regular expressions, which provides a precise characterization of the sets of sequences accepted by finite-state machines. In the following two sections we prove that any finite-state machine can be characterized by a regular expression and that every regular expression can be realized by a finite-state machine. Finally, in Section 16.7 we will be concerned with a generalized form of finite-state machines known as two-way machines.

Deterministic recognizers

So far, we have regarded a finite-state machine as a transducer that transforms input sequences into output sequences. In this chapter we shall view a machine as a recognizer that classifies input strings into two classes, those that it accepts and those that it rejects. The set consisting of all the strings that a given machine accepts is said to be recognized by that machine.