View source for Properties of Boolean functions/soft

{{Boolf header}}
{{Properties of Boolean functions/header}}

Soft properties depend on the {{boolf-prop|arity}} of the BF.

==twin==

The [[Zhegalkin twins|Zhegalkin twin]] of a BF is its [[Zhegalkin matrix|Zhegalkin index]] interpreted as a truth table of length 2<sup>''arity''</sup>.

==half-complement==

The half-complement of a BF is its XOR with x<sub>''arity''&minus;1</sub>. &nbsp; See e.g. the half-complements of <math>A \land B</math>:
* arity 3: {{spaces|5}} <math>(A \land B) \oplus C</math> {{spaces|5}} <small><code>0001 0001</code> XOR <code>0000 1111</code> = <code>0001 1110</code></small>
* arity 4: {{spaces|5}} <math>(A \land B) \oplus D</math> {{spaces|5}} <small><code>0001 0001 0001 0001</code> XOR <code>0000 0000 1111 1111</code> = <code>0001 0001 1110 1110</code>
</small>

=={{anchor|ultra|ultra-family|ultra-clan}}ultra equivalence classes==

{| class="wikitable" style="float:right; text-align: center; margin: 0 0 10px 10px;"
|colspan="3"|
The two families on the left form a super-family.<br>
The family on the right is a super-family on its own.<br>
Together these two super-families form an ultra-family.
|-
|colspan="2"| [[File:3-ary Boolean functions; super-family of Zhe 40.svg|200px]]
| [[File:3-ary Boolean functions; super-family of Zhe 44.svg|200px]]
|-
| [[File:Venn 0100 0001.svg|30px]]
| [[File:Venn 0111 1101.svg|30px]]
| [[File:Venn 0010 0111.svg|30px]]
|-
| [[File:Boolean functions like 1000 0010.svg|100px]]
| [[File:Boolean functions like 1011 1110.svg|100px]]
| [[File:Boolean functions like 1110 0100.svg|100px]]
|}

{{Boolf-prop|family|Families}} and {{Boolf-prop|clan}}s can be merged with their complements, which creates {{boolf-prop|super}}-families and super-clans.<br>
Further merging them with their {{boolf-prop|half-complement}}s creates '''ultra-families''' and '''ultra-clans'''.<br>
<small>This family is a complete ultra-family: [[:File:Boolean functions like 1100 1010.svg|1100 1010]] {{spaces|3}} (So [[:File:3-ary Boolean functions; super-family of Zhe 98.svg|its matrix]] of truth tables is mirror symmetric.)</small><br>
<small style="opacity: .5;">(Factions do not have a unique half-complement, so there are no ultra-factions.)</small>

==tribe==

For a given arity the tribe is the next biggest equivalence class after the {{boolf-prop|ultra-clan}}.<br>
{{boolf-prop|blunt|Blunt}} BF belong to tribes denoted by their {{boolf-prop|consul weight}}. All {{boolf-prop|sharp}} BF belong to a tribe on their own.

{{Collapsible START|tribes for arities 1, 2, 3|collapsed strong}}
{{Properties of truth tables/tribes}}
{{Collapsible END}}

=={{anchor|principality|dominion}}[[Principalities and dominions of Boolean functions|principalities and dominions]]==

A '''principality''' is a set of ''n''-ary BF whose (''n''+1)-ary [[Noble Boolean functions|noble]] equivalents form a {{boolf-prop|faction}}. '''Dominions''' are closely related.



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[[Category:Properties of Boolean functions]]