Properties of truth tables
Studies of Boolean functions 
Studies of Boolean functions 
 
The number of Boolean functions with arity (short for adicity ) is ( A001146).
This article is about properties of finite truth tables of Boolean functions, that change with the arity.
simple
[edit  edit source] weight: number of true places
 sharpness: weight parity sharp: odd weight dull: even weight
subsets
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equivalence classes based on similarity
[edit  edit source]see also: Extended families and clans of Boolean functions
ultra (extension with halfcomplement)
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The two families on the left form a superfamily.  
Each Boolean function has two halfcomplements. (The truth table is complemented on the left or on the right side.)
A superfamily has a unique halfcomplement. Merging them gives an ultrafamily.
As superclans consist of superfamilies, they can be extended in the same way.
(Factions do not have a unique halfcomplement, so it does not seem useful to define ultrafactions.)
ultrafamily (negation, complement, halfcomplement)
[edit  edit source]The functions in an ultrafamily have symmetric positions in a hypercube graph (or the related matrix). See matrices.
This family is a complete ultrafamily: 1100 1010 (better seen in its matrix)
ultraclan (negation, permutation, complement, halfcomplement)
[edit  edit source]An ultraclan is the merge of a superclan and its halfcomplement.
It can also be seen as a merge of ultrafamilies, that are permutations of each other.
See here for a table of the 39 ultrafamilies of 4ary Boolean functions.
partitions into blocks of equal size
[edit  edit source]consul (binary Walsh spectrum)
[edit  edit source]The Walsh spectrum of a TT is its product with a Walsh matrix.
The binary Walsh spectrum of a TT is its product with a binary Walsh matrix, using F_{2} 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.
The consul integer is the Walsh index of the prefect of the twin. The consul is essentially the prefect of the twin, but without the negation.
(One could also define a sign for the consul, by using a negated binary Walsh matrix. But the sign would just be the sharpness.)
3ary families with Walsh spectra (integers) and consuls (red backgrounds on the right) 
tribe  

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

3ary_Boolean_functions#Walsh spectrum, 4ary_Boolean_functions#wec
patron
[edit  edit source]The patron of a truth table is the XOR of itself and its twin. It is a noble. (3ary images)
praetor
[edit  edit source]XOR of left and right half of the TT. (3ary images)
quaestor
[edit  edit source]XOR of left and reversed right half of the truth table (i.e. of the coordinates in the Hasse matrix) (3ary images)
principalities and dominions
[edit  edit source]truth tables  Zhegalkin indices  

principality  
dominion 
A principality is a set of nary truth tables whose (n+1)ary noble equivalents form a faction.
Dominions are closely related to them. The reversed truth tables of the principalities are the Zhegalkin indices of the dominion.
The following table shows the six members of the red principality, which are also shown in the matrices on the right.
A representative of the 4ary noble faction can be seen in the bottom left corner of this image.
(It is easily seen, that this tetrahedron can be permuted into six different positions.)
3ary  4ary noble  

TT  Ж  
0001 0000  8  136  0000 0001 0001 0000  2176 
0000 0100  32  160  0000 0001 0000 0100  8320 
0001 0100  40  40  0000 0000 0001 0100  10240 
0000 0010  64  192  0000 0001 0000 0010  16512 
0001 0010  72  72  0000 0000 0001 0010  18432 
0000 0110  96  96  0000 0000 0000 0110  24576 
An overview of all 11 3ary principalities and dominions can be seen here.
Interesting subsets are those with entries on the diagonal and in the top row of the matrix of Zhegalkin indices.