Studies of Boolean functions/terminology




 * (Boolean) function, BF     usually meant as BF with infinite arity and periodic truth table      similar to


 * truth table, TT     usually meant as a truth table of finite length, determined by an


 * 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. 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 $$2^n$$  E.g. $$B \land C$$ can be shown as 3-ary   or as 4-ary.
 * 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.)


 * 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


 * root     A root BF has no gaps before or between the . Its  are equal.      often called non-degenerate      (The term dense might be more intuitive.)

A Boolean function is determined by its root and its atomvals. It is a root, iff there are no gaps before or between the atoms. (The atomvals are consecutive integers starting from 0, or they are empty.) Every BF has a root, but there is a small ambiguity:
 * as a BF: its equivalent with gapless atomvals
 * as a : binary vector of length 2 without repeating patterns


 * spread     not root


 * 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.

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, rather than to the . For the 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.


 * spot     cell of an Euler diagram      defined as  with dimension 0      The number of spots in a Venn diagram is 2valency.


 * fullspot     corresponds to true place in


 * gapspot     corresponds to false place in, but necessary for geometrically sound Euler diagram


 * link     connection between neighboring spots, i.e. wall between cells      defined as  with dimension 1


 * border     set of links that belong to the same, 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, 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  into each other


 * clan     negation and permutation equivalence class      partitioned into families and factions


 * family     negation equivalence class


 * faction     permutation equivalence class


 * cluster      A cluster contains four or eight factions, that are complements and twins. (category)


 * (Zhegalkin) twin     Zhegalkin index interpreted as  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


 * junior (senior)      Boolean functions of  n&minus;1 are junior to those of arity n (and those of arity n+1 are senior)


 * junarity (senarity)      arity &minus; 1 (arity + 1)


 * noble      is noble, iff identical to its


 * gentle     set of TTs is gentle, iff identical to set of twins


 * foible     The foibles are seven properties, that correspond to the vertices of a Fano plane. Most important are  and.


 *  even/ odd      of a, equal to first digit of       oddness also called parity


 *  evil/ odious      of a, equal to last digit of       odiousness also called depravity      BF is odious, iff  has odd weight


 * ugly      of a, XOR of odd and odious      uglyness


 *  blunt/ sharp      of a, equal to parity of  weight      sharpness


 *  obtuse/ acute      of a, similar to sharpness      acuteness


 * rude and rough      of a, similar to sharpness and acuteness       rudeness and roughness


 *  female/ male     BF is male, iff its  is .      gender


 * quadrant     =       + 2 &middot;       &in; {0, 1, 2, 3}


 * reverse     BF with reversed TT

general terms
These words are sometimes used in variable names.


 *      inside and outside of a set   (see also )


 * powers of two and factorial