Lector and mentor of Boolean functions

Studies of Boolean functions

The mentor is a rather dubious soft property of a BF. But it seems interesting.
It is found in three steps: Creating a family matrix, getting the senior nobles of its rows, and getting their prefects.
The first digits of the prefects form the mentor.

Ж 24 ↦ Ж 25

Ж 93 ↦ Ж 93

Ж 128 ↦ Ж 151

Ж 249 ↦ Ж 239

The following images are the 4-ary equivalents of those above. The results are either the same as above, or the complement.

Ж 24 ↦ Ж 25

Ж 93 ↦ Ж 93

Ж 128 ↦ Ж 150

Ж 249 ↦ Ж 238

Walsh permutations

A Boolean function has a unique mentor for a given arity.
The four integer permutations corresponding to that BF permutation are Walsh, so they can be represented by invertible binary matrices, called compression matrices.

compression matrices for arities 1 to 8

lector

arities 1 and 2

arity 1
	from T	from Z
to T
to Z

arity 2
	from T	from Z
to T
to Z

arities 3 and 4

arity 3
	from T	from Z
to T
to Z

arity 4
	from T	from Z
to T
to Z

arities 5 and 6

arity 5
	from T	from Z
to T
to Z

arity 6
	from T	from Z
to T
to Z

arities 7 and 8

arity 7
	from T	from Z
to T
to Z

arity 8
	from T	from Z
to T
to Z

mentor

arities 1 and 2

arity 1
	from T	from Z
to T
to Z

arity 2
	from T	from Z
to T
to Z

arities 3 and 4

arity 3
	from T	from Z
to T
to Z

arity 4
	from T	from Z
to T
to Z

arities 5 and 6

arity 5
	from T	from Z
to T
to Z

arity 6
	from T	from Z
to T
to Z

arities 7 and 8

arity 7
	from T	from Z
to T
to Z

arity 8
	from T	from Z
to T
to Z

full permutations for arity 3

lector

from T to T

from T to Z

from Z to T

from Z to Z

mentor

from T to T

from T to Z

from Z to T

from Z to Z

The mentor permutation is almost hard. For arity n the permutation of Zhegalkin indices is M_n. The beginning of M_n+1 is L_n.
M_n and L_n are very similar: They are equal in the first half, and differ by exchanged neighbors in the second.
The hard property corresponding to the new permutation shall be called lector. It is simply the mentor for a higher arity.
The compression matrix of the lector permutation is part of a top right Sierpinski triangle. Its diagonals follow a negated XOR pattern. (See image.)
Its vector is the sequence 1, 2, 4, 9, 16, 33, 65, 150, 256, 513, 1025, 2310, 4097, 8466, 16660, 38505, 65536...

short Walsh permutation (seminar)

15×15 matrix corresponding to the Walsh permutation for arity 4. The matrix is always that of M or L without the top row and left column.

Neighboring Zhegalkin indices (i.e. 2·n and 2·n+1) denote complements.
So although there is no mentor bijection between Boolean functions, there is one between pairs of complements.
Complement and mentor partition the set of all Boolean functions into blocks of size 4 or 2. Such a block shall be called (big or small) seminar.
The Zhegalkin indices in a big seminar are $\{a,a+1,b,b+1\}$ with even $a$ and $b$ , so it can be represented by the pair $(a,b)$ .
An example of a seminar is {138, 139, 156, 157}, represented as (138, 156). See image. In the short permutation it is represented by the pair (69, 78).

The pairs $\left({\frac {a}{2}},{\frac {b}{2}}\right)$ are the cycles of a self-inverse Walsh permutation of degree $2^{arity}-1$ . (For arity 3 the degree is 7, and the permuted integers are 0...127.)
For arities 1 and 2 this permutation is neutral. For arity 3 is has 64 fixed points (of 128 places, i.e. 1/2). For arity 5 it has 1024 fixed points (of 32768 places, i.e. 1/32).

The first 64 entries of the sequence are the fixed points. The next 64 entries form these 32 cycles:

[ 64,  75], [ 65,  74], [ 66,  73], [ 67,  72], [ 68,  79], [ 69,  78], [ 70,  77], [ 71,  76], 
[ 80,  91], [ 81,  90], [ 82,  89], [ 83,  88], [ 84,  95], [ 85,  94], [ 86,  93], [ 87,  92], 
[ 96, 107], [ 97, 106], [ 98, 105], [ 99, 104], [100, 111], [101, 110], [102, 109], [103, 108],
[112, 123], [113, 122], [114, 121], [115, 120], [116, 127], [117, 126], [118, 125], [119, 124]

The permutation for arity 4 corresponds to the 15×15 matrix shown on the right.
It is described by this vector: (1, 2, 4, 8, 16, 32, 75, 128, 256, 512, 1155, 2048, 4233, 8330, 19252)

For arity 3 the Boolean functions in big seminars are the sharp ones (i.e. those with odd weight). See images.

For a given arity, each seminar is part of a chunky seminar. For arity 3 they all have size 16. E.g. (142, 152), (230, 240), 14, 24, 102, 112 are all part of this.

chains

The permutations ’M and M’ have fewer fixed points and longer cycles than ’M’ and M.
A cycle of ’M shall be called chain. (Cycles of M’ are the same, but reversed.)
The XOR of all entries of a chain is one of the fixed points, and shall be called anchor. The fixed points are nobles.

3-ary partitions: chain, chain length, chain quadrants, reduced chain quadrants, chunky chain, anchor

relationship to serration

The matrix marked with Я is a permutation matrix.

Among the 3-ary BF there are 16 equal to their serrator. (See serrator splice.)
Take the truth tables of the mentors of their twins. Show them in a Hasse matrix (similar to a Hasse diagram).
They form the permutation matrix of Walsh permutation (2, 3, 8, 12). For arity 4 it is (189, 84, 231, 81, 219, 69, 126, 21).
With lector instead of mentor the WP are (6, 1, 9, 4) and (232, 84, 178, 81, 142, 69, 43, 21).
All these matrices are symmetric.