On the structure of binary feedforward inverses with delay 2

Abstract

LetM′ = C(M _α, ƒ) be a semi-input-memory finite automaton with input alphabetY and output alpha-betX. IfX=Y={0, 1}, thenM′ is a feedforward inverse with delay 2 if and only if there exists a cycleC of state diagram ofM _α such that ƒ(y _o , ...,y _c , λ_α (t)) can be expressed in the form of ƒ⁽¹⁾ (y ₀, ...,y _{c − 1},y _α (t)) ⊕y _c for any statet inC andy _o, y_l, ... y_c inY; or of ƒ⁽²⁾ (y ₀, ...,y _{c − 2},y _α (t)) ⊕y _{c − 1} for any statet inC andy _o, y_l, ..., y_c inY; or for any statet inC andy _o, y_l, ..., y_c, inY, y _o y_l ... y_c satisfies the D [t] condition. The socalledy _o y_l ... y_c satisfying the D [t] condition is that: for somei, j, (i, j)∈{(1,2), (1, 3), (2,1), (2,2), (3,1), (3,2)}, there exists a (c+2−k)—ary functionf ^(k), k=1, 2, 3, such that the Equation (1) and Equation (2) hold simultaneously for ally ^−2/′ _c , ...,y ^+1/′ _c ∈Y.

where$\hat t = \delta _\alpha \left( t \right)$; and if (i, j)=(1,2), then one and only one of the following conditions C₁ and C₂ holds for ally ^−1/′ _c ,y ^′ _c ,y ^+1/′ _c ∈Y.

Condition C₁: there exists ac-ary functiong ⁽¹⁾, such that

Condition C₂: there exists a (c−1)-ary functiong ⁽²⁾, such that

, where$\hat t = \delta _\alpha \left( t \right)$.