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.