Use of multiple index matrices in generalized automata theory

A generalization of automata theory is constructed using multivalued functions of several arguments and with several results rather than the single valued functions of a single argument which appear in conventional automata theory. Algebraic rules are developed for composing such functions which enable one to specify an element of a semigroup which represents a composite function and a convenient method is described for visualizing the constructions of the system. Some of the results of conventional automata theory and language theory are extended to systems of the present type.