Zhegalkin twins

dummy

Ж

ANF   matrix   twins

noble   gentle

smallest Zhegalkin index

2-ary functions ordered in a Hasse diagram. Truth tables in red, Zhegalkin indices in green.

Zhegalkin twins are truth tables of the same length, that correspond to a column and its index in a finite Zhegalkin matrix.
For a given arity, one can also describe two integers or Boolean functions as Zhegalkin twins. (For truth tables the arity is implied by their length.)

Calculation

These images show how one truth table is calculated from the other. (One could describe it as a bitwise XOR of variadic ANDs.)
Here the red patterns are meant as Boolean functions, while the green patterns are meant as their Zhegalkin indices.

The red pattern represents a truth table, which continues periodically. The green bit-pattern represents an integer, and continues with zeros.

The least significant green bit of the green pattern is on top. That of the red pattern is on the left.
One can see, that the LSBs of both patterns are equal. Also that changing the green LSB will change all red bits.

Ж 143 to TT 145   (reverse)

Ж 22 to TT 150   (reverse)

Ж 30 to TT 30

Equivalence classes

See Smallest Zhegalkin index for details.

farofe

Family matrix of farofe in red. Green rows are twins of red rows, and blue columns are twins of green columns. Faction matrix of farofe in red. Green rows are twins of red rows. The resulting matrix also describes a faction. The twin relationship is not just one between Boolean functions, but between factions. Patrons of the twins on the left. This is a noble faction, i.e. it is its own twin. (This is a faction of noble Boolean functions, and each of them is its own twin.)

faradi
This example is just like the one above. But potential confusion comes from the fact, that the 4-ary function faradi has a twin that is only 2-ary. The twin relationship is really one between truth tables of a given length, and not one between Boolean functions.

Selections by equal place

These images show half the columns of the 8×256 matrix. They are selected by the truth value of one row.   (There are also images for other rows, e.g. false rows in the lower matrix)
The columns of the upper long matrix are to be understood as Zhegalkin indeces, and the columns below as truth tables.
The rows are linear Boolean functions, and the small matrices on the left show their Walsh indices. (If the row is a negated Walsh function, its first entry is true.)
The file with even/evil functions shows the complements of the odd/odious functions. (Complements have opposite parity as well as opposite depravity.)

parity
even Place 0 of truth table and Zhegalkin index is false. odd Place 0 of truth table and Zhegalkin index is true.

depravity
evil The weight of the Zhegalkin index is even. The last place of the truth table is false. odious The weight of the Zhegalkin index is odd. The last place of the truth table is true.

Variadic logical connectives

The pairs of matrices shown in the following boxes are twins. This is shorthand for the fact, that their rows are Zhegalkin twins. (Compare Zhegalkin matrix.)

The left side of each pair shows the truth tables of variadic logical connectives. (Compare these 16×16 matrices, which also show the arguments.)

Some of them are unusual, and have unusual names:
SAND (a.k.a. minimal negation operator) could be called all but one. Its reflection SNOR could be called no but one.
The name XOR is used for the parity function. The reflection of not XOR is called XAND. (Because the reflection of not OR is AND.)
GAND is SAND extended by AND. Its reflection GNOR is SNOR extended by not OR.
In ESAND and OSAND this extension happens only for an even or odd number of arguments. Their reflections are ESNOR and OSNOR.
EQ makes sense as generalization of the biconditional, and is true if all arguments have the same truth value (but not if there are no arguments).

The right side of each pair is a lower triangular matrix. Its entries are part of the Sierpinski triangle, which is the twin of not OR.

Each of these triangles is symmetric to another one (in two cases to itself). Only the triangle rows are symmetric (not the matrix rows).
Pairs with symmetric triangles are in the same box, e.g.:    XOR / OSAND,    SNOR / OSNOR

Some triangles are relative complements in the Sierpinski triangle:    TRUE / OR,    AND / EQ,    GNOR / SNOR,    OSNOR / ESNOR

TRUE		AND

not AND

not OR

OR		EQ

not XOR		ESAND

XOR		OSAND

not XAND		SAND

XAND		GAND

GNOR		ESNOR

SNOR		OSNOR

upper triangular
AND not OR These matrices were also in the last section. This section also shows pairs of matrices whose rows are Zhegalkin twins. The binary patterns are the same as in the lower triangular matrices in the section before, but they are horizontally and vertically flipped. Just like in the section before, there are pairs of matrices whose triangle rows are horizontally mirrored. Here they are marked with the same color. The labels in this section are to be understood like this: Take the lower triangular twin with that label, flip it horizontally and vertically, and make the twin of that. E.g. take the XOR twin from the last section. Flip it to get not XAND in this section. The twin of that is XOR in this section. (Half of these rows are noble, i.e. their own twins.)
OSNOR	OSAND	SAND	SNOR
not XAND	XOR	XAND	not XOR
ESNOR	ESAND	GAND	GNOR
OR	not AND	EQ