Circular Logic
Problem 209
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 |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
| x | y | x XOR y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 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)?