On shift register realizations for sequential machines

This paper considers the problem of determining whether a sequential machine, given by its flow table, can be realized in the form of binary shift registers. In general we are interested in a realization using the smallest number of shift registers possible. One-to-one assignments and many-to-one assignments are considered. Initially we use partitions for one-to-one assignments, then extend the methods to use set systems for many-to-one assignments. Binary partitions are introduced, and shift. registers are characterized in terms of binary partitions. An algorithm is given for the determination of the required partitions for a shift register assignment. Binary set systems are introduced and shift registers are characterized in terms of binary set systems. It is proven that there exists a sequential machine that cannot be realized by a single binary shift register of finite length. Methods for determining the set systems necessary for a single shift register assignment are given.