On the Lower Bound to the Memory of Finite State Machines

A finite state machine (FSM) is said to have finite memory μ if μ is the least integer such that y k = f(X k , X k -1,... X k-μ , Y k-1 , ... μ k-μ ) where y k and X k represent the output and input at time k. If no such μ exists, then by convention the memory is said to be infinite. It has been observed [1] that if the memory of a p-nary input, q-nary output n-state minimal nondegenerate FSM is finite, then the memory μ is bounded as follows: Recently, considerable attention has been devoted to the study of the upper bound on μ [2]-[5]. In this paper we examine the lower bound on μ. We show that the lower bound is tight for all positive integers n and certain values of p and q. We also show that when n = 4 k , k a positive integer, there exist binary input, binary output minimal FSMs with minimal memory. It will be seen that this is equivalent to showing that for all positive integers μ there exist binary input, binary output minimal FSMs with the maximum number of states n=2 2μ . Finally, we enumerate the equivalence classes of these finite memory machines with memory μ and n = 2 2μ states.