Sequential Functions

A sequential machine receives input symbols in a sequence, works on this sequence in some way, and yields a sequence of output symbols. The sequenceto-sequence transformation which the machine performs will here be called a sequential function. If the transformation is such tha t at any stage the output symbol depends only on the sequence of input symbols which have already been received we will call the sequential function retrospective. When the machine has only a finite number of possible internal configurations, its function is further restricted. The machine can hold only a certain amount of information, and cannot always make use of all of the information contained in tha t portion of the input sequence which it has received. The retrospective sequential function performed by such a machine will be called finitary. Machines which perform finitary sequential functions have been stucaed by Moore [1], Mealy [2], and Ginsburg [3]. The present note differs in approach from the papers of these authors by associating the concept of state with the function rather than with the structure of the sequential machine. This approach facilitates derivation of some fundamental results on the composition and inversion of sequential functions.