Gender of Boolean functions



A Boolean function shall be called male, iff its root is sharp (i. e. iff its compressed truth table has odd weight). (Equivalently, it is female, iff after removing all repetitions, the weight of the truth table is still even.)

For positive arities, there are more males than females. The imbalance peaks for arity 2. For higher arities, the ratio is almost balanced. The ratio is balanced for the infinite set of all Boolean functions. Both sets are countable, so there is a trivial bijection. But is there a meaningful bijection?

These matrices show the 97 females and 159 males among the first 256 Boolean functions. The matrix above shows the Zhegalkin indices. The Zhegalkin matrix below shows the truth tables. The male truth tables on the left side of the matrix contain repetitions, i.e. their pattern can be described by a shorter truth table.



The upper matrix shows the Zhegalkin indices. The one below shows atomvals of the corresponding Boolean functions. This is sequence, the bitwise ORs of the binary exponents seen above.