View source for Properties of Boolean functions/hard

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

Hard properties can be assigned to a BF, without referencing its {{boolf-prop|arity}}.

==weight==

'''Weight''' is the quotient of true and all places of the truth table. <small>It is usually shown as an integer for a given arity ({{boolf-prop|soft weight}}).</small><br>
<small>[[Template:Boolf weight triangle; all|This triangle]] shows the numbers of BF by weight.</small>

==nonlinearity==

'''[[Nonlinearity of Boolean functions|Nonlinearity]]''' is the extent to which a BF is not [[Linear Boolean functions|linear]]. <small>It is usually shown as an integer for a given arity ({{boolf-prop|soft nonlinearity}}).</small>

=={{anchor|atom|atoms|atomvals}}atoms==

'''Atoms''' or '''atomvals''' are the relevant arguments of the BF. They correspond to the circles of an Euler diagram. The number of atoms is the {{boolf-prop|valency}}.<br>
<small>I this project the letters A, B, C... correspond to atoms 0, 1, 2... &nbsp; E.g. <math>B \land C</math> has atoms 1 and 2.</small>

==root==

The '''root''' of a BF created by replacing its atoms with the set {0, ..., {{boolf-prop|valency}}&minus;1}.<br>
When it is its own root, the BF is {{boolf-prop|dense}}.

==equivalence classes==

{{anchor|family|faction|clan}}
based on input negation and permutation: '''[[Families of Boolean functions|family]]''' (negation), '''faction''' (permutation), '''clan''' (both) &nbsp; <small>(Typically represented by [[Smallest Zhegalkin index]].)</small>

{{anchor|super|super-family|super-faction|super-clan}}
extended by complement: '''super-family''', '''super-faction''', '''super-clan''' &nbsp; <small>(Families and clans can be self-complementary. Factions can not.)</small>

<small style="opacity: .5;">(Further extension by half-complement leads to {{boolf-prop|ultra-family|ultra-famlies}} and {{boolf-prop|ultra-clan|-clans}}, which are soft properties.)</small>

==prefect==

The prefect is a way to assign each BF to a [[Linear Boolean functions|linear]] BF. A linear is assigned to itself. &nbsp;&nbsp; <small style="opacity: .7;">[[c:Category:3-ary Boolean functions in octeract matrix; great prefects|(3-ary images)]]</small>

{| class="collapsible-with-classes collapsible collapsed"
!colspan="4"| calculating prefect from Zhegalkin index
|-
|rowspan="2"| [[File:Zhe 1101 1110 (123) to prefect 5.svg|300px|thumb|center|{{color|#00a900|Ж 123}} to {{color|#eb0000|prefect 5}}]]
|rowspan="2"| [[File:Zhe 0110 0100 (38) to prefect ~6.svg|300px|thumb|center|{{color|#00a900|Ж 38}} to {{color|#eb0000|prefect ¬6}}]]
|rowspan="2"| [[File:Zhe 0111 0001 (142) to prefect ~7.svg|300px|thumb|center|{{color|#00a900|Ж 142}} to {{color|#eb0000|prefect ¬7}}]]
| [[File:XAND 8×8 with row numbers.svg|thumb|The yellow matrix has the pattern of a {{w|Logical biconditional|XAND}} matrix.]]
|-
|
{{multiple image
 | align = right  | total_width = 230
 | image1 = Binary Walsh matrix of order 8 with row numbers.svg
 | image2 = Binary Walsh matrix of order 8 with row numbers, complement.svg
 | footer = The possible results are the linear functions.
}}
|}

=={{anchor|legion|cohort}}[[Legion and cohort of Boolean functions|legion and cohort]]==

These properties assign every almost every BF to a finite set of integers. <small style="opacity: .5;">The result for the contradiction is infinite. Soft equivalents have been defined to avoid this problem.</small><br>
The cardinalities of the results are {{boolf-prop|gravity}} and {{boolf-prop|depth}}.

==complement==

all places of the truth table negated, e.g. <code>0001</code> and <code>1110</code> {{spaces|5}} <small style="opacity: .5;">(<abbr title="least significant bit">LSB</abbr> of the [[Zhegalkin index]] negated)</small><br>
The complement of a set of BF is the set of complements. <small>({{boolf-prop|family|Families}} and {{boolf-prop|clan}}s can be self-complementary. {{boolf-prop|faction|Factions}} can not.)</small>

==reverse==

truth table reversed, e.g. <code>0001</code> and <code>1000</code>

==dual==

complement of the reverse, e.g. <code>0001</code> and <code>0111</code>



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[[Category:Properties of Boolean functions]]