Properties of Boolean functions/soft

Studies of Boolean functions

Properties of Boolean functions
hard	soft
binary	binary
integer	integer
permutation	permutation

Soft properties depend on the arity of the BF.

ultra equivalence classes

The two families on the left form a super-family. The family on the right is a super-family on its own. Together these two super-families form an ultra-family.

Families and clans can be merged with their complements, which creates super-families and super-clans.
Further merging them with their half-complements creates ultra-families and ultra-clans.
This family is a complete ultra-family: 1100 1010     (So its matrix of truth tables is mirror symmetric.)
(Factions do not have a unique half-complement, so there are no ultra-factions.)

tribe

For a given arity the tribe is the next biggest equivalence class after the ultra-clan.
Blunt BF belong to tribes denoted by their consul weight. All sharp BF belong to a tribe on their own.

tribes for arities 1, 2, 3

1-ary truth tables Walsh spectra consuls (A253315) tribe 0 1 2 3 [0 0] [1 0] [0 1] [1 1] [ 0 0] [ 1 1] [ 1 -1] [ 2 0] [0 0] [0 0] [0 1] [0 1] 0 0 1 1 0 s s 1 2-ary truth tables Walsh spectra consuls (A253315) tribes sharp: white 0: dark blue 1: light blue 2: green Calculated with this Python script 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [0 0 0 0] [1 0 0 0] [0 1 0 0] [1 1 0 0] [0 0 1 0] [1 0 1 0] [0 1 1 0] [1 1 1 0] [0 0 0 1] [1 0 0 1] [0 1 0 1] [1 1 0 1] [0 0 1 1] [1 0 1 1] [0 1 1 1] [1 1 1 1] [ 0 0 0 0] [ 1 1 1 1] [ 1 -1 1 -1] [ 2 0 2 0] [ 1 1 -1 -1] [ 2 2 0 0] [ 2 0 0 -2] [ 3 1 1 -1] [ 1 -1 -1 1] [ 2 0 0 2] [ 2 -2 0 0] [ 3 -1 1 1] [ 2 0 -2 0] [ 3 1 -1 1] [ 3 -1 -1 -1] [ 4 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 1] [0 1 0 1] [0 0 1 1] [0 0 1 1] [0 1 1 0] [0 1 1 0] [0 1 1 0] [0 1 1 0] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 0 1] [0 0 0 0] [0 0 0 0] 0 0 1 1 2 2 3 3 3 3 2 2 1 1 0 0 0 s s 1 s 1 2 s s 2 1 s 1 s s 0 3-ary Blunt tribes are gray or red:        0        1        2        3 Sharp tribes are white.

principalities and dominions

A principality is a set of n-ary BF whose (n+1)-ary noble equivalents form a faction. Dominions are closely related.

squad, platoon and company

squad: union of a super-faction and its partner   (always two super-factions, i.e. four factions)
platoon: union of a squad and its twin   (one or two squads, i.e. four or eight factions)
company: union of a platoon and its friend   (one or two platoons)

images for arity 3
platoon of a single squad platoon of two squads (horizontal with dark and vertical with light borders) See overview on Commons.