In this paper, we theoretically analyze a certain extension of a finite automaton, called a linear separation automaton (LSA). An LSA accepts a sequence of real vectors, and has a weight function and a threshold sequence at every state, which determine the transition from some state to another at each step. Transitions of LSAs are just corresponding to the behavior of perceptrons. We develop the theory of minimizing LSAs by using Myhill-Nerode theorem for LSAs. Its proof is performed as in the proof of the theorem for finite automata. Therefore we find that the extension to an LSA from the original finite automaton is theoretically natural.