Minimal Input-Memory and Output-Memory Finite-State Machines

A finite-state machine (FSM) is said to have finite input-memory, μ
_i
(finite output-memory μ
_o
) if μ
_i
(μ
_o
) is the least integer such that y
_k
= f(X
_k
, X
_k-1
,..., X
_k-μi
), (y
_k
= f(X
_k
, y
_k-1
,..., y
_k-μo
)). If no such integer μ
_i
(μ
_o
) exists then by convention μ
_i
= ∞ (μ
₀
= ∞). It is well known that for a nondegenerate binary-input binary-output FSM M, μ
_i
≥ [log2 n] (μo ≥ [log2 n]), where n is the number of non-equivalent states in M. In this note we show that the above bound cannot be improved upon by giving a procedure to construct, for any n > 0, a binary-input binary-output FSM with n nonequivalent states and minimal input-memory μ
_i
= [log
₂
n] (minimal output-memory μ
_o
= [ log2 n]). In the process of proving the tightness of the lower bound on μ
_i
(μ
_o
), we enumerate the number of distinct minimal binary r-stage feed-forward (output feedback) shift registers.