Incompletely specified (Mealy type) sequential machines and their generalizations to infinite machines are discussed. Convenient algebraic techniques for the study of such machines are introduced. These techniques are based on binary relations and on an extension of the usual concept of homomorphism to multivalued mappings. The first part of the paper gives a short, unified introduction to the algebraic theory of partial automata. The second part unifies and extends the algebraic approach to generalized cascade decompositions which combine conventional decomposition techniques with state-splitting procedures. The relationship between such generalized decompositions and state reductions of automata is investigated.
The object of this paper is to study the realization of a sequential machine from several smaller machines. The basic tools in this investigation are the partitions with the substitution property and the partition pairs. It is shown that to every (loop-free) realization of a sequential machine from n smaller machines corresponds a set of n partitions with the substitution property whose product is the zero partition. Conversely, it is shown that to every such set of n partitions corresponds a realization of the given sequential machine from n smaller machines. The natural ordering of these partitions is reflected in the information flow between the corresponding component machines and the algebraic operations defined between these partitions corresponding to the realization, govern the modifications of this realization. Finally, it is shown how the amount of information flowing between the component machines in a realization can be studied by means of partition pairs.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.