Properties of Boolean functions/hard

Studies of Boolean functions

Properties of Boolean functions
hard	soft
binary	binary
integer	integer

Hard properties can be assigned to a BF, without referencing its arity.

weight

Weight is the quotient of true and all places of the truth table. It is usually shown as an integer for a given arity (soft weight).
This triangle shows the numbers of BF by weight.

nonlinearity

Nonlinearity is the extent to which a BF is not linear. It is usually shown as an integer for a given arity (soft nonlinearity).

atoms

Atoms or atomvals are the relevant arguments of the BF. They correspond to the circles of an Euler diagram. The number of atoms is the valency.
I this project the letters A, B, C... correspond to atoms 0, 1, 2...   E.g. $B\land C$ has atoms 1 and 2.

root

The root of a BF created by replacing its atoms with the set {0, ..., valency−1}.
When it is its own root, the BF is dense.

equivalence classes

based on input negation and permutation: family (negation), faction (permutation), clan (both)   (Typically represented by Smallest Zhegalkin index.)

extended by complement: super-family, super-faction, super-clan   (Families and clans can be self-complementary. Factions can not.)

(Further extension by half-complement leads to ultra-famlies and -clans, which are soft properties.)

prefect

The prefect is a way to assign each BF to a linear BF. A linear is assigned to itself.    (3-ary images)

calculating prefect from Zhegalkin index
Ж 123 to prefect 5	Ж 38 to prefect ¬6	Ж 142 to prefect ¬7	The yellow matrix has the pattern of a XAND matrix.
			The possible results are the linear functions.

legion and cohort

These properties assign every almost every BF to a finite set of integers. The result for the contradiction is infinite. Soft equivalents have been defined to avoid this problem.
The cardinalities of the results are gravity and depth.

complement

all places of the truth table negated, e.g. 0001 and 1110       (LSB of the Zhegalkin index negated)
The complement of a set of BF is the set of complements. (Families and clans can be self-complementary. Factions can not.)

reverse

truth table reversed, e.g. 0001 and 1000

dual

complement of the reverse, e.g. 0001 and 0111