This note deals with the multiple realization of synchronous sequential machines for which the initial states are given. The results given by Smith and Kohavi, who have studied the problem for machines fed by the same input, are generalized to the case of machines having the same clock but different inputs. For these machines, a maximum common factor is defined, and it is shown how, if it exists, it can be obtained. The problem of internal state or input state splitting for machines that do not have a maximum factor and that are incomplete is not faced.

In this paper the possibility of removing in synchronous sequential machines some of the delay elements which store the internal state is discussed. Theorems and methods are given for eliminating all removable delay elements from a given realization of a synchronous machine and for selecting the assignments leading to a realization which requires the least number of delay elements. It is shown how the existence of these assignments is related to the existence of a pair of partitions on the set of states of the given machine. The internal performance of the sequential networks in which at least one but not all delay elements are removed is studied, and a "hybrid operating mode" is defined. A reclassification of sequential machines in terms of the algebraic structure of machines which relates to the number of removable delay elements is proposed. Sequential machines are, therefore, divided in three categories: synchronous, hybrid, and asynchronous, and some problems of decomposition for the three types of machines are discussed.