Invertibility of Finite Group Homomorphic Sequential Systems

T h e question of invertibil i ty of a dynamical system-i .e . , when does the output sequence (function) uniquely determine the input sequence ( funct ion)--has received a good deal of attention in the literature and has applications to problems in coding theory and functional controllabili ty (the dual of invertibility). Massey and Sain (1968), Sain and Massey (1969), Brockett and Mesarovic (1965), and Forney (1970) have all dealt with invertibil i ty conditions for linear systems. In this highly s tructured setting one can obtain quite detailed and explicit invertibili ty conditions. At the other structural extreme, Olson (1970) has obtained invertibil i ty results for general finite state machines. Of course the results in this genera! setting are not nearly as detailed as the linear results. Our work falls in between these structural extremes. Brockett and Willsky (1972, 1974) and Willsky (1973a) have introduced a class of finite state systems that evolve homomorphical ly on finite groups. As their results indicate, this class of systems, although much broader than the class of linear sequential