View source for Studies of Boolean functions/terminology

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{{Shortcut|Boolf-term}}

* {{anchor|BF}} '''Boolean function''', '''BF''' {{spaces|4}} usually meant as BF with infinite arity and periodic truth table {{spaces|4}} <small>similar to {{w|Boolean expression}}</small>

* {{anchor|TT}} '''truth table''', '''TT''' {{spaces|4}} usually meant as a truth table of finite length, determined by an {{sect|arity}}

* {{anchor|weight}} '''weight''' {{spaces|4}} The weight of a BF is a fraction between 0 and 1. The weight of a TT is a non-negative integer.

* {{anchor|valency|adicity|arity}} '''valency &le; adicity &le; arity'''
** Valency is the number of arguments actually used. It is the number of circles in the Euler diagram.
** Adicity follows from the biggest atom. 2<sup>''adicity''</sup> is the required TT length<span style="opacity: .5;">, or the period length of the infinite truth table.</span>
** The term ''arity'' is used in two slightly different ways:
*** arity ''n'' as an argument <small>(e.g. of a class method)</small> to get a finite truth table of length <math>2^n</math> &nbsp; <small>E.g. <math>B \land C</math> can be shown as 3-ary <code>0000 0011</code> or as 4-ary <code>0000 0011 0000 0011</code>.</small>
*** arity ''n'' as a shorthand for <math>adicity \le n</math> &nbsp; <small>(as in: ''the 16 2-ary Boolean functions'')</small>
***<small style="opacity: .5;">(For a while there may also be a third way, namely the erroneous use instead of valency or adicity.)</small>

* {{anchor|atom}} '''atom''' {{spaces|4}} Atoms are also called sets or arguments of a BF. {{spaces|4}} <small>what is usually shown by a circle and labeled A, B, C...</small>

* {{anchor|atomvals}} '''atomvals''' {{spaces|4}} the vector of atoms of a BF

* {{anchor|dense}} '''dense''' {{spaces|4}} A dense BF has no gaps before or between the {{sect|atom|atoms}}. Its {{sect|valency|valency and adicity}} are equal. {{spaces|4}} <small>often called non-degenerate</small> {{spaces|4}}

* {{anchor|spread}} '''spread''' {{spaces|4}} not dense

* {{anchor|root}} '''root''' {{spaces|4}} The root of a BF is its dense equivalent. &nbsp; <small style="opacity: .7;">The term is also used for the TT of the root BF. It is a binary vector of length 2<sup>''valency''</sup> without repeating patterns.</small>

* {{anchor|segment}} '''segment''' {{spaces|4}} geometric element of an Euler diagram, e.g. its cells and the walls between them {{spaces|4}} <small>The number of segments in a Venn diagram is 3<sup>''valency''</sup>.</small>

{{Collapsible START|dimension of a segment|collapsed wide}}
A segment has a '''dimension''', namely the number of zeros in its ternary label.<br>
<small>(From the perspective of Euler diagrams the term is counter-intuitive, because it corresponds to the {{w|hypercube}}, rather than to the {{w|cross polytope}}. For the {{sect|hypersplit|hypersplits}} it is used in the same way.)</small>

<div style="font-size: 80%;">
The relationships between segments that differ in only one digit are important:
* Another segment with a 0 in the differing place is a '''superior'''. <small>(Points are superior to lines, lines to areas, etc.)</small>
* Another segment with + or − in the differing place is an '''inferior'''.
* Another segment with the opposite sign in the differing place is a '''neighbor'''.
</div>

{| class="collapsible-with-classes collapsible collapsed light followed wide center gap-above"
!colspan="2"| 2D example {{boolflink|kinide}}
|-
| [[File:EuDi; 2x3 kinide.svg|300px]]<br>Cell 4 is a gapspot.
| [[File:EuDi; 2x3 ternary.svg|600px]]
|}

{| class="collapsible-with-classes collapsible collapsed light wide center"
!colspan="2"| 3D example {{boolflink|medusa}}
|-
| [[File:EuDi; medusa.png|400px]]
| [[File:EuDi; medusa ternary.svg|800px]]
|}
{{Collapsible END}}

* {{anchor|spot}} '''spot''' {{spaces|4}} cell of an Euler diagram {{spaces|4}} <small>defined as {{sect|segment}} with dimension 0</small> {{spaces|4}} <small>The number of spots in a Venn diagram is 2<sup>''valency''</sup>.</small>

* {{anchor|fullspot}} '''fullspot''' {{spaces|4}} corresponds to true place in TT

* {{anchor|gapspot}} '''gapspot''' {{spaces|4}} corresponds to false place in TT, but necessary for geometrically sound Euler diagram

* {{anchor|link}} '''link''' {{spaces|4}} connection between neighboring spots, i.e. wall between cells {{spaces|4}} <small>defined as {{sect|segment}} with dimension 1</small>

* {{anchor|border}} '''border''' {{spaces|4}} set of links that belong to the same {{sect|atom}}, i.e. all walls of the same color

* {{anchor|splits}} [[Studies of Euler diagrams/splits|'''split''']] {{spaces|4}} set without the notion of inside and outside {{spaces|4}} <small>usually the same as a partition into two blocks</small>

* {{anchor|hypersplit}} [[Studies of Euler diagrams/splits#hypersplits|'''hypersplit''']] {{spaces|4}} generalization of a split {{spaces|4}} partitions space into 2<sup>''n''</sup> orthants

* {{anchor|filtrate}} [[Studies of Euler diagrams/filtrates|'''filtrate''']] {{spaces|4}} reduction of a BF to a subset of its {{sect|atom|atoms}}, i.e. what remains when some circles are removed from the Euler diagram

* {{anchor|bundle}} '''bundle''' {{spaces|4}} part of an Euler diagram that is connected by crossing borders {{spaces|4}}<small>see e.g. [[Studies of Euler diagrams/decompose|decompose]], [[Studies of Euler diagrams/gap variants/multi-bundle 3-2-2-1|multi-bundle 3-2-2-1]], [[Studies of Euler diagrams/blightless/4-ary/bundles|4-ary bundles]]</small>

* {{anchor|blighted}} [[Studies of Euler diagrams/blighted|'''blighted''']] {{spaces|4}} arity can be reduced {{spaces|4}} <small>bloated or blotted</small> {{spaces|4}} <small style="opacity: .5;">(blight, blightless)</small>

* {{anchor|bloated}} [[Studies of Euler diagrams/bloated|'''bloated''']] {{spaces|4}} some arguments are equal or complementary to each other {{spaces|4}} <small style="opacity: .5;">(bloat, bloatless)</small>

* {{anchor|blotted}} [[Studies of Euler diagrams/blotted|'''blotted''']] {{spaces|4}} some arguments are equal or complementary to niverse or empty set {{spaces|4}} <small style="opacity: .5;">(blot, blotless)</small>

* {{anchor|transformation}} [[Studies of Euler diagrams/transformations|'''transformation''']] {{spaces|4}} signed permutation that turns elements of the same {{sect|clan}} into each other

* {{anchor|Zhegalkin_twin|twin}} [[Zhegalkin twins|'''(Zhegalkin) twin''']] {{spaces|4}} Zhegalkin index interpreted as TT of the same length {{spaces|4}} <small style="opacity: .5;">(E.g. ''all bits true'' and ''only left bit true'' are always twins, because the Zhegalkin index of the tautology is 1.)</small>

* {{anchor|zhe}} [[Zhegalkin matrix|'''Zhegalkin index''']], '''Ж''' {{spaces|4}} non-negative integer identifying a Boolean function {{spaces|4}} <small>related to [[algebraic normal form]]</small>

* {{anchor|representative}} '''representative''' {{spaces|4}} some Boolean function that represents its whole [[Equivalence classes of Boolean functions|equivalence class]] {{spaces|4}} <small>typically the [[smallest Zhegalkin index]] of a {{sect|clan}}</small>

* {{anchor|junior|senior}} '''junior <small>(senior)</small>''' {{spaces|4}} Boolean functions of {{sect|arity}} ''n''&minus;1 are junior to those of arity ''n'' <small>(and those of arity ''n''+1 are senior)</small> {{spaces|4}} 

* {{anchor|quadrant}} '''quadrant''' {{spaces|2}} = {{spaces|2}} {{sect|odd|parity}} + 2 &middot; {{sect|odious|depravity}} {{spaces|2}} <small>&in; {0, 1, 2, 3}</small>

* {{anchor|reverse}} '''reverse''' {{spaces|2}} BF with reversed TT

* {{anchor|junarity|senarity}} '''junarity <small>(senarity)</small>''' {{spaces|4}} arity &minus; 1 <small>(arity + 1)</small>

* {{anchor|noble}} '''[[Noble Boolean functions|noble]]''' {{spaces|4}} TT is noble, iff identical to its {{sect|twin}}

* {{anchor|patron}} '''[[Noble Boolean functions#patrons|patron]]''' {{spaces|4}} XOR of twins, a noble TT

* {{anchor|gentle}} '''[[Gentle sets of Boolean functions|gentle]]''' {{spaces|4}} set of TTs is gentle, iff identical to set of twins


==properties==

<small>see [[Properties of Boolean functions]] and [[Properties of truth tables|of truth tables]]</small>

* {{anchor|clan}} '''{{BF-prop|clan}}''' {{spaces|4}} negation and permutation equivalence class {{spaces|4}} <small>partitioned into families and factions</small>

* {{anchor|family}} '''{{BF-prop|family}}''' {{spaces|4}} negation EC

* {{anchor|faction}} '''{{BF-prop|faction}}''' {{spaces|4}} permutation EC

* {{anchor|splinter}} '''splinter''' {{spaces|4}} intersection of family and faction

* {{anchor|tribe}} '''{{TT-prop|tribe}}''' {{spaces|4}} {{sect|sharp|Blunt}} TT form tribes, denoted by their {{sect|consul}} weight.

* {{anchor|super}} '''{{BF-prop|super}}''' {{spaces|4}} Prefix used for EC of BF extended by complement.

* {{anchor|ultra}} '''{{TT-prop|ultra}}''' {{spaces|4}} Prefix used for EC of TT extended by half-complement.

[[File:Foibles of Boolean functions.svg|thumb|160px|foibles]]
* {{anchor|foible}} '''foible''' {{spaces|4}} The foibles are seven properties, that correspond to the vertices of a Fano plane. &nbsp; <small style="opacity: .5;">(see [[commons:Category:Quadrant colors in hypercube matrices (image set)|here]] for an overview)</small>

** {{anchor|even|odd|parity}} '''<span style="opacity: .5;">even/</span>odd''' {{spaces|4}} <span style="opacity: .5;">foible of a BF,</span> equal to first digit of TT {{spaces|4}} <small>'''oddness''' also called '''parity'''</small>

** {{anchor|evil|odious|depravity}} '''<span style="opacity: .5;">evil/</span>odious''' {{spaces|4}} <span style="opacity: .5;">foible of a BF,</span> equal to last digit of TT {{spaces|4}} <small>'''odiousness''' also called '''depravity'''</small> {{spaces|4}} <small>BF is odious, iff {{sect|zhe|Ж}} has odd weight</small>

** {{anchor|ugly}} '''<span style="opacity: .5;">pretty/</span>ugly''' {{spaces|4}} <span style="opacity: .5;">foible of a BF,</span> XOR of odd and odious {{spaces|4}} <small>'''uglyness'''</small>

** {{anchor|sharp|blunt|dull}} '''<span style="opacity: .5;">blunt/</span>sharp''' {{spaces|4}} <span style="opacity: .5;">foible of a TT,</span> equal to parity of TT weight {{spaces|4}} <small>'''sharpness'''</small>

** {{anchor|obtuse|acute}} '''<span style="opacity: .5;">obtuse/</span>acute''' {{spaces|4}} <span style="opacity: .5;">foible of a TT,</span> similar to sharpness {{spaces|4}} <small>'''acuteness'''</small>

** {{anchor|rude|rough}} '''<span style="opacity: .5;">polite/</span>rude''' and '''<span style="opacity: .5;">smooth/</span>rough''' {{spaces|4}} <span style="opacity: .5;">foibles of a TT,</span> similar to sharpness and acuteness  {{spaces|4}} <small>'''rudeness''' and '''roughness'''</small>

* {{anchor|male|female|gender}} '''<span style="opacity: .5;">female/</span>male''' {{spaces|4}} BF is male, iff its {{sect|root}} is {{sect|sharp}}. {{spaces|4}} <small>'''[[Gender of Boolean functions|gender]]'''</small>

* {{anchor|prefect}} '''{{BF-prop|prefect}}''' {{spaces|4}}  BF ↦ linear BF

* {{anchor|subprefect|sub-prefect}} '''sub-prefect''' {{spaces|4}}  BF ↦ senior linear BF &nbsp; <small>([[commons:Category:3-ary Boolean functions in octeract matrix; great subprefects|category]])</small>

* {{anchor|consul}} '''{{TT-prop|consul}}''' {{spaces|4}} TT ↦ linear TT &nbsp; <small style="opacity: .5;">(basically twin prefect)</small>

* {{anchor|praetor}} '''{{TT-prop|praetor}}''' {{spaces|4}} XOR of left and right half of TT

* {{anchor|quaestor}} '''{{TT-prop|quaestor}}''' {{spaces|4}} XOR of left and reversed right half of TT

* {{anchor|principality|dominion}} {{TT-prop|principalities and dominions|'''principality''' and '''dominion'''}} {{spaces|4}} &nbsp; {{sect|junior|juniors}} corresponding to {{sect|noble}} {{sect|faction|factions}} 

<div style="opacity: .5;">
===provisional===

* {{anchor|partner|friend}} '''partner/friend''' {{spaces|4}} BF whose TT differ only in least/most significant bit. Applies also to factions.

* {{anchor|squad}} '''squad''' {{spaces|4}} Union of a faction with its complement, its partner and its complement partner. Always four factions.

* {{anchor|platoon}} '''platoon''' {{spaces|4}} Union of a squad and its twin. May consist of one of two squads, i.e. four or eight factions.

* {{anchor|company}} '''company''' {{spaces|4}} The name would make sense for the union of a platoon and its friend. May consist of one or two platoons.

* {{anchor|cluster}} '''cluster''' {{spaces|4}} That name is still used for platoon.

* {{anchor|federation}} '''federation''' {{spaces|4}} related to clusters and great dominions

</div>

==general terms==

These words are sometimes used in variable names.

* {{anchor|recto|verso}} '''{{w|recto and verso}}''' {{spaces|2}} inside and outside of a set &nbsp; <small>(see also {{sect|split}})</small>

* {{anchor|paltry|plenty|plethora|galore}} powers of two and factorial

{| class="wikitable" style="text-align: center;"
| <math>n</math> || <math>2^n</math> || <math>2^{2^n}</math> || <math>n!</math>
|- style="font-weight: bold;"
| paltry || plenty || plethora || galore
|}


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[[Category:Properties of Boolean functions]]