Soft properties of Boolean functions

Studies of Boolean functions

**Properties of Boolean functions**
Hard	not dependent on arity
Soft	dependent on arity

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Boolf-soft

The number of Boolean functions with arity $n$ (short for adicity $\leq n$ ) is $2^{2^{n}}$ ( A001146).

This article is about properties of finite truth tables of Boolean functions, that change with the arity.

simple

weight: number of true places
sharpness: weight parity sharp: odd weight dull: even weight

subsets

noble: truth tables that are their own twins
symmetric: order of inputs does not matter

equivalence classes based on similarity

ultra (extension with half-complement)

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.

Each Boolean function has two half-complements. (The truth table is complemented on the left or on the right side.)
A super-family has a unique half-complement. Merging them gives an ultra-family.
As super-clans consist of super-families, they can be extended in the same way.
(Factions do not have a unique half-complement, so it does not seem useful to define ultra-factions.)

ultra-family (negation, complement, half-complement)

The functions in an ultra-family have symmetric positions in a hypercube graph (or the related matrix). See matrices.
This family is a complete ultra-family: 1100 1010 (better seen in its matrix)

ultra-clan (negation, permutation, complement, half-complement)

An ultra-clan is the merge of a super-clan and its half-complement.
It can also be seen as a merge of ultra-families, that are permutations of each other.
See here for a table of the 39 ultra-families of 4-ary Boolean functions.

partitions into blocks of equal size

consul (binary Walsh spectrum)

The Walsh spectrum of a TT is its product with a Walsh matrix.
The binary Walsh spectrum of a TT is its product with a binary Walsh matrix, using F₂ operations (mod 2).
It is always a Walsh function, and shall be called consul. The term is also used for the integer denoting the Walsh function.

The consul integer is the Walsh index of the prefect of the twin. The consul is essentially the prefect of the twin, but without the negation.
(One could also define a sign for the consul, by using a negated binary Walsh matrix. But the sign would just be the sharpness.)

sharp family with consuls 0...7	dull family with consul 2 (compare 4-ary family with consul 6)
3-ary families with Walsh spectra (integers) and consuls (red backgrounds on the right)

tribe

The consul weight is the binary weight of the consul integer. (E.g. consul 6 has consul weight 2.)
Dull TT belong to ultra-clans with a unique consul weight, which belong to a dull tribe denoted by that consul weight.
Sharp TT form a tribe on their own.

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
Dull tribes are gray or red: 0 1 2 3 Sharp tribes are white.

3-ary_Boolean_functions#Walsh spectrum, 4-ary_Boolean_functions#wec

patron

The patron of a truth table is the XOR of itself and its twin. It is a noble. (3-ary images)

praetor

XOR of left and right half of the TT. (3-ary images)

quaestor

XOR of left and reversed right half of the truth table (i.e. of the coordinates in the Hasse matrix) (3-ary images)

principalities and dominions

Principalities and dominions of Boolean functions

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