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Principalities and dominions of Boolean functions

From Wikiversity
Studies of Boolean functions
truth tables Zhegalkin indices
principality
dominion

A principality is a set of n-ary 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 4-ary 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.)

3-ary 4-ary noble
truth tables Ж
0001 0000 8 136 0000 0001 0001 0000 2176
0000 0100 32 160 0000 0001 0000 0100 8320
0000 0010 64 192 0000 0001 0000 0010 16512
0001 0100 40 40 0000 0000 0001 0100 10240
0001 0010 72 72 0000 0000 0001 0010 18432
0000 0110 96 96 0000 0000 0000 0110 24576

Overviews on Commons: 2-ary, 3-ary, 4-ary
Interesting subsets are those with entries on the diagonal and in the top row of the matrix of Zhegalkin indices.


Principalities follow directly from the concept of nobles. Dominions are less obvious, but arguably more important.

The quaestors of each faction form a dominion. Many factions with the same quadrant belong to the same dominion.
Take 0100 1011 (Ж 26) as an example. (It has quadrant 2).
For arity 3 the quaestors are {3, 5, 9} (yellow in this file) and for arity 4 they are {15, 51, 85, 153, 165, 195} (yellow in this file, also shown above in the bottom left corner).