Properties of Boolean functions/hard/binary

Hard properties can be assigned to a BF, without referencing its arity.
Binary properties have the values true or false, so they partition all BF into a subset and the rest.

Linear Boolean functions are Walsh functions and their complements.

See Seal (discrete mathematics). Seal block is a broader property.

no gaps before or between the atoms, i.e. valency = adicity     (often called nondegenerate)

BF is strong, iff strength = adicity, so its family has the biggest possible size 2^adicity.

same number of true and false places, i.e. weight = 0.5
BF that are not balanced are light or heavy, i.e. their weight is below or above 0.5.

• BF is closed, iff complement is in same family, otherwise unclosed.       Self-complementary families are closed.
• BF is unopen, iff complement is in same clan, otherwise open.       Self-complementary clans are unopen.
• BF is ajar, iff complement is in same clan, but not in same family

no true place under false place (when places are arranged in a Hasse diagram)   (counted by Dedekind numbers)

• parity:   BF is odd/even iff first place in truth table is true/false.   Same as parity of the Zhegalkin index.

• depravity:   BF is odious/evil iff last place in truth table is true/false.   Same as parity of the binary weight of the Zhegalkin index.

• BF is ugly, iff parity and depravity are different (i.e. iff the quadrant is 1 or 2).

• quadrant = parity + 2 · depravity

images for arity 3
	odd	odious	ugly	quadrants
truth tables	1 1000 0000	128 0000 0001	129 1000 0001
Zhegalkin indices		255 1111 1111	254 0111 1111

• gender:   BF is male/female, iff its root is sharp/blunt. See Gender of Boolean functions.

BF is honest/dishonest, iff the XOR of all members of its family is the tautology/contradiction.   (Most BF are honest.)