Studies of Boolean functions/terminology

• Template:Anchor Boolean function, BF Template:Spaces usually meant as BF with infinite arity and periodic truth table Template:Spaces similar to Template:W

• Template:Anchor truth table, TT Template:Spaces usually meant as a truth table of finite length, determined by an Template:Sect

• Template:Anchor weight Template:Spaces The weight of a BF is a fraction between 0 and 1. The weight of a TT is a non-negative integer.

• Template:Anchor valency ≤ adicity ≤ 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^adicity is the required TT length, or the period length of the infinite truth table.
  • The term arity is used in two slightly different ways:
    • arity n as an argument (e.g. of a class method) to get a finite truth table of length $2^n$   E.g. $B\wedge C$ can be shown as 3-ary 0000 0011 or as 4-ary 0000 0011 0000 0011.
    • arity n as a shorthand for $adicity\le n$   (as in: the 16 2-ary Boolean functions)
    • (For a while there may also be a third way, namely the erroneous use instead of valency or adicity.)

• Template:Anchor atom Template:Spaces Atoms are also called sets or arguments of a BF. Template:Spaces what is usually shown by a circle and labeled A, B, C...

• Template:Anchor atomvals Template:Spaces the vector of atoms of a BF

• Template:Anchor dense Template:Spaces A dense BF has no gaps before or between the Template:Sect. Its Template:Sect are equal. Template:Spaces often called non-degenerate Template:Spaces

• Template:Anchor spread Template:Spaces not dense

• Template:Anchor root Template:Spaces The root of a BF is its dense equivalent.   The term is also used for the TT of the root BF. It is a binary vector of length 2^valency without repeating patterns.

• Template:Anchor segment Template:Spaces geometric element of an Euler diagram, e.g. its cells and the walls between them Template:Spaces The number of segments in a Venn diagram is 3^valency.

Template:Collapsible START A segment has a dimension, namely the number of zeros in its ternary label.
(From the perspective of Euler diagrams the term is counter-intuitive, because it corresponds to the Template:W, rather than to the Template:W. For the Template:Sect it is used in the same way.)

The relationships between segments that differ in only one digit are important:

• Another segment with a 0 in the differing place is a superior. (Points are superior to lines, lines to areas, etc.)
• Another segment with + or − in the differing place is an inferior.
• Another segment with the opposite sign in the differing place is a neighbor.

2D example Template:Boolflink
Cell 4 is a gapspot.

3D example Template:Boolflink

• Template:Anchor spot Template:Spaces cell of an Euler diagram Template:Spaces defined as Template:Sect with dimension 0 Template:Spaces The number of spots in a Venn diagram is 2^valency.

• Template:Anchor fullspot Template:Spaces corresponds to true place in TT

• Template:Anchor gapspot Template:Spaces corresponds to false place in TT, but necessary for geometrically sound Euler diagram

• Template:Anchor link Template:Spaces connection between neighboring spots, i.e. wall between cells Template:Spaces defined as Template:Sect with dimension 1

• Template:Anchor border Template:Spaces set of links that belong to the same Template:Sect, i.e. all walls of the same color

• Template:Anchor split Template:Spaces set without the notion of inside and outside Template:Spaces usually the same as a partition into two blocks

• Template:Anchor hypersplit Template:Spaces generalization of a split Template:Spaces partitions space into 2ⁿ orthants

• Template:Anchor filtrate Template:Spaces reduction of a BF to a subset of its Template:Sect, i.e. what remains when some circles are removed from the Euler diagram

• Template:Anchor bundle Template:Spaces part of an Euler diagram that is connected by crossing borders Template:Spacessee e.g. decompose, multi-bundle 3-2-2-1, 4-ary bundles

• Template:Anchor blighted Template:Spaces arity can be reduced Template:Spaces bloated or blotted Template:Spaces (blight, blightless)

• Template:Anchor bloated Template:Spaces some arguments are equal or complementary to each other Template:Spaces (bloat, bloatless)

• Template:Anchor blotted Template:Spaces some arguments are equal or complementary to niverse or empty set Template:Spaces (blot, blotless)

• Template:Anchor transformation Template:Spaces signed permutation that turns elements of the same Template:Sect into each other

• Template:Anchor (Zhegalkin) twin Template:Spaces Zhegalkin index interpreted as TT of the same length Template:Spaces (E.g. all bits true and only left bit true are always twins, because the Zhegalkin index of the tautology is 1.)

• Template:Anchor Zhegalkin index, Ж Template:Spaces non-negative integer identifying a Boolean function Template:Spaces related to algebraic normal form

• Template:Anchor representative Template:Spaces some Boolean function that represents its whole equivalence class Template:Spaces typically the smallest Zhegalkin index of a Template:Sect

• Template:Anchor junior (senior) Template:Spaces Boolean functions of Template:Sect n−1 are junior to those of arity n (and those of arity n+1 are senior) Template:Spaces

• Template:Anchor quadrant Template:Spaces = Template:Spaces Template:Sect + 2 · Template:Sect Template:Spaces ∈ {0, 1, 2, 3}

• Template:Anchor reverse Template:Spaces BF with reversed TT

• Template:Anchor junarity (senarity) Template:Spaces arity − 1 (arity + 1)

• Template:Anchor noble Template:Spaces TT is noble, iff identical to its Template:Sect

• Template:Anchor patron Template:Spaces XOR of twins, a noble TT

• Template:Anchor gentle Template:Spaces set of TTs is gentle, iff identical to set of twins

properties

see Properties of Boolean functions and of truth tables

• Template:Anchor Template:BF-prop Template:Spaces negation and permutation equivalence class Template:Spaces partitioned into families and factions

• Template:Anchor Template:BF-prop Template:Spaces negation EC

• Template:Anchor Template:BF-prop Template:Spaces permutation EC

• Template:Anchor splinter Template:Spaces intersection of family and faction

• Template:Anchor Template:TT-prop Template:Spaces Template:Sect TT form tribes, denoted by their Template:Sect weight.

• Template:Anchor Template:BF-prop Template:Spaces Prefix used for EC of BF extended by complement.

• Template:Anchor Template:TT-prop Template:Spaces Prefix used for EC of TT extended by half-complement.

foibles

• Template:Anchor foible Template:Spaces The foibles are seven properties, that correspond to the vertices of a Fano plane.   (see here for an overview)

• • Template:Anchor even/odd Template:Spaces foible of a BF, equal to first digit of TT Template:Spaces oddness also called parity

• • Template:Anchor evil/odious Template:Spaces foible of a BF, equal to last digit of TT Template:Spaces odiousness also called depravity Template:Spaces BF is odious, iff Template:Sect has odd weight

• • Template:Anchor pretty/ugly Template:Spaces foible of a BF, XOR of odd and odious Template:Spaces uglyness

• • Template:Anchor blunt/sharp Template:Spaces foible of a TT, equal to parity of TT weight Template:Spaces sharpness

• • Template:Anchor obtuse/acute Template:Spaces foible of a TT, similar to sharpness Template:Spaces acuteness

• • Template:Anchor polite/rude and smooth/rough Template:Spaces foibles of a TT, similar to sharpness and acuteness Template:Spaces rudeness and roughness

• Template:Anchor female/male Template:Spaces BF is male, iff its Template:Sect is Template:Sect. Template:Spaces gender

• Template:Anchor Template:BF-prop Template:Spaces BF ↦ linear BF

• Template:Anchor sub-prefect Template:Spaces BF ↦ senior linear BF   (category)

• Template:Anchor Template:TT-prop Template:Spaces TT ↦ linear TT   (basically twin prefect)

• Template:Anchor Template:TT-prop Template:Spaces XOR of left and right half of TT

• Template:Anchor Template:TT-prop Template:Spaces XOR of left and reversed right half of TT

• Template:Anchor Template:TT-prop Template:Spaces   Template:Sect corresponding to Template:Sect Template:Sect

provisional

• Template:Anchor partner/friend Template:Spaces BF whose TT differ only in least/most significant bit. Applies also to factions.

• Template:Anchor squad Template:Spaces Union of a faction with its complement, its partner and its complement partner. Always four factions.

• Template:Anchor platoon Template:Spaces Union of a squad and its twin. May consist of one of two squads, i.e. four or eight factions.

• Template:Anchor company Template:Spaces The name would make sense for the union of a platoon and its friend. May consist of one or two platoons.

• Template:Anchor cluster Template:Spaces That name is still used for platoon.

• Template:Anchor federation Template:Spaces related to clusters and great dominions

general terms

These words are sometimes used in variable names.

• Template:Anchor Template:W Template:Spaces inside and outside of a set   (see also Template:Sect)

• Template:Anchor powers of two and factorial

$n$	$2^n$	$2^{2^n}$	$n!$
paltry	plenty	plethora	galore