Properties of Boolean functions/soft

Studies of Boolean functions

Properties of Boolean functions
hard	soft
binary	binary
integer	integer

Soft properties depend on the arity of the BF.

twin

The Zhegalkin twin of a BF is its Zhegalkin index interpreted as a truth table of length 2^arity.

half-complement

The half-complement of a BF is its XOR with x_arity−1. See e.g. the half-complements of $A\land B$ :

arity 3: $(A\land B)\oplus C$ 0001 0001 XOR 0000 1111 = 0001 1110
arity 4: $(A\land B)\oplus D$ 0001 0001 0001 0001 XOR 0000 0000 1111 1111 = 0001 0001 1110 1110

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  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
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.

partner and friend

The partner/friend of a BF differs from its truth table in the least/most significant bit.
E.g. 0110 has the partner 1110 and the friend 0111.

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

See overview on Commons.