On the Structure of Binary Feedforward Inverses with Delay 2

Let M'=S(M _α, ƒ) be a semi-input-memory finite automaton with input alphabet Y and output alpha-bet X. If X=Y=0, 1, then M' is a feedforward inverse with delay 2 if and only if there exists a cycle C of state diagram of M _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 state t in C and y _o, y_l,... y_c in Y; or of ƒ⁽²⁾ (y ₀,..., y _c-2, y _α (t)) ⊕ y _c-1 for any state t in C and y _o, y_l,..., y_c in Y; or for any state t in C and y _o, y_l,..., y_c, in Y, y _o y_l... y_c satisfies the Dt condition. The socalled y _o y_l... y_c satisfying the Dt condition is that:for some i, j, (i, j)∈(1,2), (1, 3), (2,1), (2,2), (3,1), (3,2), there exists a (c+2-k)-ary function ƒ ^(k), k=1, 2, 3, such that the Equation (1) and Equation (2) hold simultaneously for all y'_c-2,...,y'_c+1∈Y.
Equation (1):ƒ(y₀,...,y_c-i,y'_c-i+1,...,y'_c,λ_α (t))=ƒ^(j)(y ₀,...,y_c-i, λ_α (t))⊕y' _c-i+1
Equation (2):ƒ(y ₁,...,y_c-j+1, y'_c-j+2,...,y'_c+1, λ_α (t))=ƒ^(j)(y₁,...,y_c-j+1, λ_α(t))⊕ y'_c-j+2 where =δ_α(t); and if (i, j)=(1,2), then one and only one of the following conditions C₁ and C₂ holds for all y'_c-1, y'_c, y'_c+1∈Y.
Condition C₁:there exists a c-ary function g ⁽¹⁾, such that
ƒ(y₀,...,y_c-2, y'_c-1, y'_c, λ_α(t))=g ⁽¹⁾(y₀,...,y_c-2,λ_α(t))⊕y'_c-1⊕y'_c;
Condition C₂:there exists a (c-1)-ary function g ⁽²⁾, such that
ƒ(y₁,...,y_c-2, y'_c-1, y'_c, y'_c+1, λ_α(t))=g⁽²⁾(y₁,...,y_c-2, λ_α(t))⊕y'_c-1⊕y'_c.where \hat t=\delta _\alpha \left(t \right) .