Maximal Memory Binary Input-Binary Output Finite-Memory Sequential Machines

Gill^[1] has shown that if there exists a finite-memory n-state sequential machine with finite memory ¿, then ¿ cannot exceed ¿n(n-1)¿Nn. He has further shown^[2] that there exists an n-state Nn input-binary output machine with memory ¿= Nn for every n. The question of whether a tighter upper bound might be placed on ¿ by the order of the input alphabet was raised by Gill. Massey^[3] recently has shown that there exists a ternary input-binary output finite-memory machine with memory ¿=Nn for every n. The primary purpose of this note is to show that for every n there exists an n-state binary input-binary output finite-memory machine with memory ¿= Nn, and thus ¿ is shown not to be limited by the order of the input alphabet. It is shown that for every n there are actually at least two different machines with memory ¿ = Nn. It will also be shown that for every n there exists a binary input-binary output n-state finite-memory machine with ¿ = Nn-1.