Continuity and Realizability of Sequence Transformations

In this paper we study some relations between the continuity of sequence transformations and their realizability by logic nets. The main results discussed include: the Curtis-Hedlund-Lyndon theorem which states that, if a sequence transformation is continuous and unitary (commutative with the shift transformation), then it can be realized by a net without feedback, and, the extension of this theorem to the finitary case. We find that unitary transformations are realized by definite automata, and finitary transformations are realized by indefinite automata. The term ``automata´´ is used here in a modification of its usual sense, and we explore the relation between the conventional and modified notions. Many of the concepts and more significant mathematical results in this report can be found in another context in works by Hedlund and others. What may be new and of interest to computer scientists, we believe, is their application to the theory of sequential circuits.