View source for Families of Boolean functions

{{Boolf header}}

Boolean functions belong to the same family, when they can be [[Studies of Euler diagrams/transformations|transformed]] into each other by negating arguments.<br>
The size of a family is always a power of two. The maximal size is 2<sup>''{{Boolf-hard|adicity}}''</sup>, i.e. the period length of the truth table.<br>
<small style="opacity: .5;">(While the size of {{Boolf-hard|faction}} and {{Boolf-hard|clan}} increases with the chosen arity, the size of a family is fixed.)</small>

The simplest property of a family is its weight, i.e. the number of true entries in each truth table.<br>
An important property is the parity of the weight. Those with odd weight shall be called '''sharp''', those with even weight '''blunt'''.<br>
<small style="opacity: .5;">(It is the same as the parity of the quaestor weight.)</small>

A family belongs to a {{boolf-hard|clan}} (where the arguments are not just negated, but also permuted).<br>
<small>Together with its complement, it forms a super-family.</small>
<small style="opacity: .5;">Adding a half-complement forms an {{Boolf-soft|ultra-family}}.</small><br>

==family matrix==

A family matrix is a {{w|Bitwise operation#XOR|bitwise XOR}} table with integers replaced by binary values.
Each row (or column) is the truth table of a Boolean functions in the family.

{|
|- style="vertical-align: top;"
| [[File:Pattern of a 4-ary family matrix.svg|thumb|x250px|bitwise XOR table with labels on the left showing how the arguments are negated]]
| [[File:Family of nitako with twins.svg|thumb|x250px|red family matrix of a 4-ary function, containing each function twice]]
| [[File:Boolean functions like 1110 0000.svg|thumb|x250px|eight members of a family as Venn diagrams and rows of a family matrix]]

| 
{{multiple image
 | align = right  | total_width = 300
 | image1 = Setpart of size 8; intpart 4·2; blocks (0, 3), (1, 2), (4, 7), (5, 6).svg
 | image2 = Setpart of size 8; intpart 2·4; blocks (0, 3, 5, 6), (1, 2, 4, 7).svg
 | footer = two family matrices of 3-ary [[Seal (discrete mathematics)|seals]], one with 4 functions (left), the other with 2 (right)
}}
|}

==sequences==

{{Families of Boolean functions/sequences/arity to count}}

{{Collapsible START|family representatives|collapsed gap-below}}
The representative is the [[smallest Zhegalkin index]].<br>
<small>The length of the sequence for arity ''n'' is {{oeislink|A000231}}(''n''), the number of ''n''-ary families.</small>
{{Integer sequences related to Boolean functions/reps of families/nested}}
{{Collapsible END}}

{{oeislink|A227725}} <math>T(n, k)</math> is the number of <math>n</math>-ary families of '''size''' <math>2^k</math>.<br>
{{oeislink|A227724}} <math>T(n, k)</math> is the number of '''balanced''' <math>n</math>-ary families of '''size''' <math>2^k</math>.<br>
{{oeislink|A054724}} <math>T(n, w)</math> is the number of <math>n</math>-ary families with '''weight''' <math>w</math>. See [[Template:Boolf weight triangle; families|here]].

==blunt and sharp==

Sharp families always have the maximal size.<br>
Each member of sharp family has a unique {{boolf-soft|consul}}, while all members of a blunt family have the same consul.<br>
<small>All sharp families belong to the same {{Boolf-soft|tribe}}. The tribe of a blunt family is denoted by the binary weight of the consul.</small>

It is easy to calculate a family representative of a sharp BF <small style="opacity: .7;">(a [[Gender of Boolean functions|male]] to be precise)</small>,
namely the one with consul 0.

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

An important fact <small style="opacity: .7;">(visible in the table of sequences above)</small> 
is that the number of blunt families is equal to the number of super-families.<br>
The numbers of sharp and introverted <small>(self-complementary)</small> families always sum up to the number of blunt families <small>(= the number of super-families)</small>.<br>
<small>
<small>Arity 1 is a special case, because the one sharp and the one introverted family are the same.</small>
Apart from that, no sharp familiy is introverted.
</small>

{{Families of Boolean functions/sharpness}}

==super-clans==

The following table shows the 46 3-ary families within the 14 super-clans. That means, each family is shown in its {{Boolf-hard|clan}}, and together with its complement.<br>
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).<br>
<small style="opacity: .5;">(The number of families in a super-clan is 1, 2, 3 or 6.)</small>

{{Collapsible START|tables of super-clans|collapsed wide}}
{{Families of Boolean functions/box with sharp families by quaestor weight}}
{{Families of Boolean functions/super-clans}}
{{Collapsible END}}

==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.<br>
An important property <small>(that links the 3-ary to 2-ary families)</small> is the {{Boolf prop 3-ary|village}} &ndash; seen as columns in the matrices of Zhegalkin indices.<br>
Blunt families have a consul and a {{Boolf prop 3-ary binary|is solid|solidity}}. <small>(Sharp families have four solid and four fluid members.)</small>

{{Collapsible START|tables of super-families|collapsed wide}}
{{Families of Boolean functions/table of super-families}}
{{Collapsible END}}

==misc.==

'''3-ary partitions:''' {{Boolf prop 3-ary|family}} (46), {{Boolf prop 3-ary|reverse family}} (46), {{Boolf prop 3-ary|super family}} (30), {{Boolf prop 3-ary|ultra family}} (18), {{Boolf prop 3-ary|family size}} (4), {{Boolf prop 3-ary|quaestor weight}} (5), {{Boolf prop 3-ary|tribe}} (5), {{Boolf prop 3-ary binary|is sharp}} (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)]]


[[Category:Families of Boolean functions]]