ABSTRACT

A finite state machine (FSM) is said to have finite memory ? if ? is the least integer such that y<inf>k</inf>= f(X<inf>k</inf>, X<inf>k</inf>-1,... X<inf>k-?</inf>, Y<inf>k-1</inf>, ... ?<inf>k-?</inf>) where y<inf>k</inf>and X<inf>k</inf>represent the output and input at time k. If no such ? 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 ? is bounded as follows: Recently, considerable attention has been devoted to the study of the upper bound on ? [2]-[5]. In this paper we examine the lower bound on ?. 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<sup>k</sup>, 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 ? there exist binary input, binary output minimal FSMs with the maximum number of states n=2<sup>2?</sup>. Finally, we enumerate the equivalence classes of these finite memory machines with memory ? and n = 2<sup>2?</sup>states.

INDEX TERMS

Finite memory, finite state machines, lower bound on memory, minimal canonical realizations.

CITATION

K. Vairavan, "On the Lower Bound to the Memory of Finite State Machines," in

*IEEE Transactions on Computers*, vol. 18, no. , pp. 856-861, 1969.

doi:10.1109/T-C.1969.222782

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