Identification and Minimization of Linear Machines

This paper is a study of linear machines and their submachines. Methods are presented for finding the reduced form of a given linear machine with or without fixed initial state. A technique is suggested for detecting whether a machine is linear or can be embedded as a submachine in a linear machine. In the latter case a state assignment is produced for the minimum linear realization. Encoded inputs and outputs are assumed, but given machines are not assumed reduced, nor are there any restrictions on the number of states in the given machine or in the linear realization. The method also detects machines that are linearly realizable when constants are available. The main results are that the reduced form of a linear machine is linear and that a linearly realizable machine with r-independent states has an r-dimensional realization.