On the characterization of linear and linearizable automata by a superposition principle

Abstract

This paper deals with the superponability of linear and linearizable automata. Automata linear in GF(p),p2, with input and output sets {0, 1,⋯,p-1} and with arbitrary initial state are characterized as the class of automata superponable, i.e. permutable, with respect to a functionax+(1-a)x′. The casep=2 requires special considerations. The inputoutput behaviour of this automata can be described by generalized impulse responses. Using instead of GF(p) an arbitrary commutative field on the same basic set {0,1,⋯,p-1} some of the functions nonlinear in GF(p) are linear in this field. These functions are called linearizable. Linearizable functions are characterized by a functional equation and solvability conditions and as polynomials in GF(p). Thus the well known theory of linear automata can be extended to linearizable automata.