Issue No.09 - September (1969 vol.18)
K. Vairavan , IEEE
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  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 ? -. 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.
Finite memory, finite state machines, lower bound on memory, minimal canonical realizations.
K. Vairavan, "On the Lower Bound to the Memory of Finite State Machines", IEEE Transactions on Computers, vol.18, no. 9, pp. 856-861, September 1969, doi:10.1109/T-C.1969.222782