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Problem 209


Problem 209


Circular Logic

A k-input binary truth table is a map from k input bits(binary digits, 0 [false] or 1 [true]) to 1 output bit. For example, the 2-input binary truth tables for the logical AND and XOR functions are:

x y x AND y x y x XOR y
0 0 0 0 0 0
0 1 0 0 1 1
1 0 0 1 0 1
1 1 1 1 1 0

How many 6-input binary truth tables, τ, satisfy the formula

τ(a, b, c, d, e, f) AND τ(b, c, d, e, f, a XOR (b AND c)) = 0

for all 6-bit inputs (a, b, c, d, e, f)?


圆环之理

k元真值表是从k比特位(即二进制位的简称,其中0代表假,1代表真)输入到1比特位输出的映射。例如,逻辑运算符AND(和)和XOR(异或)的2元真值表如下所示:

x y x AND y x y x XOR y
0 0 0 0 0 0
0 1 0 0 1 1
1 0 0 1 0 1
1 1 1 1 1 0

有多少个6元真值表τ,对于所有的6比特位输入(a, b, c, d, e, f)始终满足下述等式?

τ(a, b, c, d, e, f) AND τ(b, c, d, e, f, a XOR (b AND c)) = 0