Properties of Boolean functions/soft/integer

Properties of Boolean functions
hard	soft
binary	binary
integer	integer
permutation	permutation

Soft properties depend on the arity of the BF.
Integer properties have consecutive non-negative integers as values.

softened

weight

the number of true places in the truth table of length 2^arity (hard weight is a fraction)

nonlinearity

extent to which a BF is not linear (hard nonlinearity is a fraction)

gravity and depth

2^gravity and 2^depth are the cardinalities of legion and cohort. Equivalently for the softened properties. Only the result for the contradiction differs from hard gravity and depth.

consul (binary Walsh spectrum)

The Walsh spectrum is the product of a truth table with a Walsh matrix.
The binary Walsh spectrum is the product of a truth table with a binary Walsh matrix, using F₂ operations (mod 2).
It is always a Walsh function, and shall be called consul. The term is also used for the integer denoting the Walsh function.

One could refine the consul with a sign. That signed consul would be the twin prefect.
So the (unsigned) consul is the Walsh index of the twin prefect. (See an overview for arity 3.)

The consul weight is the binary weight of the consul integer. (E.g. consul 6 has consul weight 2.)

sharp family with consuls 0...7	blunt family with consul 2 (compare 4-ary family with consul 6)
3-ary families with Walsh spectra (integers) and consuls (red backgrounds on the right)

patron

The patron of a BF is the XOR of itself and its twin. It is a noble. (3-ary images)

praetor

XOR of left and right half of the truth table. (3-ary images)

quaestor

XOR of left and reversed right half of the truth table (i.e. of the coordinates in the Hasse matrix) (3-ary images)

faction XOR and crest

The XOR of all BF in a faction is a symmetric BF. It can be converted to an integer $0...2^{arity+1}-1$ . E.g. for 3-ary BF to integers 0...15.
Often the FX of a BF is complementary to that of its complement. These BF are crested, and the BF is called crest. The crested BF are partitioned into blocks of equal size.
The FX of a BF can also be equal to that of its complement. These BF are uncrested.
faction XOR, is crested, crest, great crest, super great crest