This paper begins with Mealy's model of a sequential machine and introduces a ``connection matrix'' which describes the machine completely. The ``equivalence'' of states of such a machine may be analyzed systematically by an iterative technique, the validity of which is rigorously established. Once equivalence is completely analyzed, it is a simple matter to write the connection matrix for the simplest equivalent machine. The process is not difficult to execute, even in complex cases, and could be programmed for a computer.

This paper presents the general concept of a weighted directed graph which we call a net. Illustrative examples leading up to the definition of a net indicate its applicability to a wide variety of problems in communication and network theory. A number of theorems concerning the proper paths of a net are established. A non-arithmetic matrix calculus is developed to facilitate computations and formalize proofs. In later papers, the techniques presented here will be exploited in the study of the theory of sequential machines.