Studies of Boolean functions/terminology
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- Boolean function, BF usually meant as BF with infinite arity and periodic truth table similar to Boolean expression
- truth table, TT usually meant as a truth table of finite length, determined by an arity
- weight The weight of a BF is a fraction between 0 and 1. The weight of a TT is a non-negative integer.
- 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. 2adicity 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 E.g. can be shown as 3-ary
0000 0011
or as 4-ary0000 0011 0000 0011
. - arity n as a shorthand for (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.)
- arity n as an argument (e.g. of a class method) to get a finite truth table of length E.g. can be shown as 3-ary
- atom Atoms are also called sets or arguments of a BF. what is usually shown by a circle and labeled A, B, C...
- atomvals the vector of atoms of a BF
- dense A dense BF has no gaps before or between the atoms. Its valency and adicity are equal. often called non-degenerate
- spread not dense
- root 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 2valency without repeating patterns.
- segment geometric element of an Euler diagram, e.g. its cells and the walls between them The number of segments in a Venn diagram is 3valency.
dimension of a segment | ||||||||
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A segment has a dimension, namely the number of zeros in its ternary label. The relationships between segments that differ in only one digit are important:
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- spot cell of an Euler diagram defined as segment with dimension 0 The number of spots in a Venn diagram is 2valency.
- fullspot corresponds to true place in TT
- gapspot corresponds to false place in TT, but necessary for geometrically sound Euler diagram
- link connection between neighboring spots, i.e. wall between cells defined as segment with dimension 1
- border set of links that belong to the same atom, i.e. all walls of the same color
- split set without the notion of inside and outside usually the same as a partition into two blocks
- hypersplit generalization of a split partitions space into 2n orthants
- filtrate reduction of a BF to a subset of its atoms, i.e. what remains when some circles are removed from the Euler diagram
- bundle part of an Euler diagram that is connected by crossing borders see e.g. decompose, multi-bundle 3-2-2-1, 4-ary bundles
- blighted arity can be reduced bloated or blotted (blight, blightless)
- bloated some arguments are equal or complementary to each other (bloat, bloatless)
- blotted some arguments are equal or complementary to niverse or empty set (blot, blotless)
- transformation signed permutation that turns elements of the same clan into each other
- (Zhegalkin) twin Zhegalkin index interpreted as TT of the same length (E.g. all bits true and only left bit true are always twins, because the Zhegalkin index of the tautology is 1.)
- Zhegalkin index, Ж non-negative integer identifying a Boolean function related to algebraic normal form
- representative some Boolean function that represents its whole equivalence class typically the smallest Zhegalkin index of a clan
- junior (senior) Boolean functions of arity n−1 are junior to those of arity n (and those of arity n+1 are senior)
- reverse BF with reversed TT
- junarity (senarity) arity − 1 (arity + 1)
- patron XOR of twins, a noble TT
- gentle set of TTs is gentle, iff identical to set of twins
properties
editsee Properties of Boolean functions and of truth tables
- clan negation and permutation equivalence class partitioned into families and factions
- family negation EC
- faction permutation EC
- splinter intersection of family and faction
- super Prefix used for EC of BF extended by complement.
- ultra Prefix used for EC of TT extended by half-complement.
- foible The foibles are seven properties, that correspond to the vertices of a Fano plane. (see here for an overview)
- even/odd foible of a BF, equal to first digit of TT oddness also called parity
- evil/odious foible of a BF, equal to last digit of TT odiousness also called depravity BF is odious, iff Ж has odd weight
- pretty/ugly foible of a BF, XOR of odd and odious uglyness
- dull/sharp foible of a TT, equal to parity of TT weight sharpness
- obtuse/acute foible of a TT, similar to sharpness acuteness
- polite/rude and smooth/rough foibles of a TT, similar to sharpness and acuteness rudeness and roughness
- prefect BF ↦ linear BF
- sub-prefect BF ↦ senior linear BF (category)
- consul TT ↦ linear TT (basically twin prefect)
- praetor XOR of left and right half of TT
- quaestor XOR of left and reversed right half of TT
- principality and dominion juniors corresponding to noble factions
provisional
edit- cluster A cluster contains four or eight factions, that are complements and twins.
- federation abandoned term related to clusters
general terms
editThese words are sometimes used in variable names.
- recto and verso inside and outside of a set (see also split)
- powers of two and factorial
paltry | plenty | plethora | galore |