Properties of Boolean functions/hard/integer

Hard properties can be assigned to a BF, without referencing its Template:Boolf-prop.
Integer properties have all non-negative integers as values.

Template:Anchorvalency ≤ adicity ≤ arity

Valency is the number of Template:Boolf-prop, i.e. the number of arguments acutally used. It is the number of circles in the Euler diagram.

Adicity follows from the biggest atom. 2^adicity is the required length of the truth table, or the period length of the infinite truth table.

In this project arity is not a property of a BF. The term is used in two slightly different ways:

• arity n as a shorthand for adicity ≤ n   (as in: the 16 2-ary Boolean functions)
• arity n as an argument 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.

This can be explained in terms of object-oriented programming: Valency and adicity are attributes of a BF class. Arity is an argument to some of its methods.

strength

2^strength = family size

Template:Anchorgravity and depth

2^gravity and 2^depth are the cardinalities of legion and cohort.
The result for the contradiction is infinity. Soft equivalents have been defined to avoid this problem. Soft gravity is twin Template:Boolf-prop. Soft depth is related to Template:Boolf-prop.