Families of Boolean functions

Boolean functions belong to the same family, when they can be transformed into each other by negating arguments.
The size of a family is always a power of two. The maximal size is 2^{Template:Boolf-hard}, i.e. the period length of the truth table.
(While the size of Template:Boolf-hard and Template:Boolf-hard increases with the chosen arity, the size of a family is fixed.)

The simplest property of a family is its weight, i.e. the number of true entries in each truth table.
An important property is the parity of the weight. Those with odd weight shall be called sharp, those with even weight blunt.
(It is the same as the parity of the quaestor weight.)

A family belongs to a Template:Boolf-hard (where the arguments are not just negated, but also permuted).
Together with its complement, it forms a super-family. Adding a half-complement forms an Template:Boolf-soft.

family matrix

A family matrix is a Template:W table with integers replaced by binary values. Each row (or column) is the truth table of a Boolean functions in the family.

Error creating thumbnail: bitwise XOR table with labels on the left showing how the arguments are negated

File:Family of nitako with twins.svg red family matrix of a 4-ary function, containing each function twice

eight members of a family as Venn diagrams and rows of a family matrix

sequences

			arity $n$
			0	1	2	3	4	5
		balanced families (central values of Template:Oeislink)	0	1	3	14	870	18796230
$\frac{(2^{2^{n-1}})(2^n-1)}{2^n}$	Template:Boolf-seq	introverted families	0	1	3	14	240	63488
$\frac{2^{2^n}}{2^n}=2^{(2^n-n)}$	Template:Boolf-seq	extroverted families (BF with a particular consul)	2	2	4	32	4096	134217728
$2^n-n-1$		Eulerian numbers	0	0	1	4	11	26
$\frac{2^{2^n}}{2^{n+1}}=2^{(2^n-n-1)}$	Template:Boolf-seq	sharp families (BF with a particular prefect)	1	1	2	16	2048	67108864
$\frac{2^{2^n}+(2^{2^{n-1}+1})(2^n-1)}{2^{n+1}}$	Template:Boolf-seq	blunt families super-families	1	2	5	30	2288	67172352
$\frac{2^{2^n}+(2^{2^{n-1}})(2^n-1)}{2^n}$	Template:Boolf-seq	families	2	3	7	46	4336	134281216
	Template:Boolf-seq	sharp super-families		1	1	8	1024	33554432
	Template:Boolf-seq	blunt super-families		1	4	22	1264	33617920
	Template:Boolf-seq	families with maximal size $2^n$ (right column of Template:Boolf-seq)	2	1	2	23	3904	134156284
		blunt families with maximal size $2^n$	1	0	0	7	1856	67047420

Template:Collapsible START The representative is the smallest Zhegalkin index.
The length of the sequence for arity n is Template:Oeislink(n), the number of n-ary families. Template:Integer sequences related to Boolean functions/reps of families/nested Template:Collapsible END

Template:Oeislink $T(n,k)$ is the number of $n$-ary families of size $2^k$.
Template:Oeislink $T(n,k)$ is the number of balanced $n$-ary families of size $2^k$.
Template:Oeislink $T(n,w)$ is the number of $n$-ary families with weight $w$. See here.

blunt and sharp

Sharp families always have the maximal size.
Each member of sharp family has a unique Template:Boolf-soft, while all members of a blunt family have the same consul.
All sharp families belong to the same Template:Boolf-soft. The tribe of a blunt family is denoted by the binary weight of the consul.

It is easy to calculate a family representative of a sharp BF (a male to be precise), namely the one with consul 0.

There are many bijections between blunt and sharp Boolean functions.
It would be practical to have one, that assigns all members of a blunt family to members of the same sharp family.
So one sharp family would correspond to a set of blunt families. That would be nice to have, but may not exist.

An important fact (visible in the table of sequences above) is that the number of blunt families is equal to the number of super-families.
The numbers of sharp and introverted (self-complementary) families always sum up to the number of blunt families (= the number of super-families).
Arity 1 is a special case, because the one sharp and the one introverted family are the same. Apart from that, no sharp familiy is introverted.

super-clans

The following table shows the 46 3-ary families within the 14 super-clans. That means, each family is shown in its Template:Boolf-hard, and together with its complement.
In each matrix a family has a distinct color. Complements have the same base color (RGB or beige). Clans have the same shade (light or dark).
(The number of families in a super-clan is 1, 2, 3 or 6.)

super-families

The following table shows the 46 3-ary families within the 30 super-families. That means, each family is shown together with its complement.
An important property (that links the 3-ary to 2-ary families) is the Template:Boolf prop 3-ary – seen as columns in the matrices of Zhegalkin indices.
Blunt families have a consul and a Template:Boolf prop 3-ary binary. (Sharp families have four solid and four fluid members.)

misc.

3-ary partitions: Template:Boolf prop 3-ary (46), Template:Boolf prop 3-ary (46), Template:Boolf prop 3-ary (30), Template:Boolf prop 3-ary (18), Template:Boolf prop 3-ary (4), Template:Boolf prop 3-ary (5), Template:Boolf prop 3-ary (5), Template:Boolf prop 3-ary binary (2)

• c:Category:3-ary truth tables in octeract matrix; ultra-families
• c:Category:3-ary Boolean functions in octeract matrix; super-clans
• c:Category:3-ary Boolean functions in octeract matrix; reverse super-clans

• c:Category:3-ary Boolean functions; small equivalence classes; cubes
• c:Category:Walsh spectra of 3-ary Boolean functions; families

• c:Category:Family to senior nobles to prefects (image set)