Zhu Xinjie. On the Structure of Binary Feedforward Inverses with Delay 2. Journal of Computer Science and Technology, 1989, 4(2): 163-171.
 Citation: Zhu Xinjie. On the Structure of Binary Feedforward Inverses with Delay 2. Journal of Computer Science and Technology, 1989, 4(2): 163-171.

# 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, λα (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, yl,... yc 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, yl,..., yc in Y; or for any state t in C and y o, yl,..., yc, in Y, y o yl... yc satisfies the Dt condition. The socalled y o yl... yc 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+1Y.
Equation (1):ƒ(y0,...,yc-i,y'c-i+1,...,y'cα (t))=ƒ(j)(y 0,...,yc-i, λα (t))⊕y' c-i+1
Equation (2):ƒ(y 1,...,yc-j+1, y'c-j+2,...,y'c+1, λα (t))=ƒ(j)(y1,...,yc-j+1, λα(t))⊕ y'c-j+2 where =δα(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+1Y.
Condition C1:there exists a c-ary function g (1), such that
ƒ(y0,...,yc-2, y'c-1, y'c, λα(t))=g (1)(y0,...,yc-2α(t))⊕y'c-1y'c;
Condition C2:there exists a (c-1)-ary function g (2), such that
ƒ(y1,...,yc-2, y'c-1, y'c, y'c+1, λα(t))=g(2)(y1,...,yc-2, λα(t))⊕y'c-1y'c.where \hat t=\delta _\alpha \left(t \right) .

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