Zhegalkin matrix

The Zhegalkin matrix is an infinite binary matrix.

It is closely related to the infinite integer matrix of Gray code permutation powers ( A195467) and to the algebraic normal form of Boolean functions.
(Ivan Zhegalkin was the inventor of the ANF. The naming choices made here are new.)

Its colums are the truth tables of all Boolean functions. The column number is the Zhegalkin index of the respective Boolean function.

Its rows are a subset of the Walsh functions, namely the XORs of atoms forming a Sierpiński triangle ( A001317).

The columns of the 8×256 Zhegalkin matrix are the truth tables of the 256 3-ary Boolean functions.
The 8×256 matrix above are the colum indices in little-endian binary. On the left are the Sierpiński triangle and the row indices.

as rows of a binary Walsh matrix
This is a 256×256 binary Walsh matrix. Each row is the variadic XOR of the atoms shown in the 256×8 matrix on the left.

Zhegalkin permutations

sizes of finite Zhegalkin matrices

$n$	0	1	2	3	4
$2^{n}\times 2^{2^{n}}$	1×2	2×4	4×16	8×256	16×65536

For arity $n$ the map from ANFs to truth tables gives a finite Zhegalkin matrix of size $2^{n}\times 2^{2^{n}}$. (It is the top left corner of the infinite matrix.)

It can be interpreted as a permutation of the integers $0~...~2^{2^{n}}-1$, which shall be called Zhegalkin permutation $\Pi _{n}$.

It is a self-inverse Walsh permutation of degree 2ⁿ. The corresponding element of GL(2ⁿ, 2) is the lower Sierpiński triangle.
Π₂ is a Walsh permutation of degree 4, and permutes the integers 0 ... 15. The corresponding element of GL(4, 2) is the 4×4 lower Sierpiński triangle.

$\Pi$ = triangle  A197819:     (fixed points in parentheses)

     k   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n 
0       (0) (1)
1       (0)  3  (2)  1
2       (0) 15  10   5  12   3  (6)  9  (8)  7   2  13   4  11 (14)  1

row 3 of length 256:     0, 255, 170, 85 ... 84, 171, 254, 1

[0, 255, 170, 85, 204, 51, 102, 153, 136, 119, 34, 221, 68, 187, 238, 17, 240, 15, 90, 165, 60, 195, 150, 105, 120, 135, 210, 45, 180, 75, 30, 225, 160, 95, 10, 245, 108, 147, 198, 57, 40, 215, 130, 125, 228, 27, 78, 177, 80, 175, 250, 5, 156, 99, 54, 201, 216, 39, 114, 141, 20, 235, 190, 65, 192, 63, 106, 149, 12, 243, 166, 89, 72, 183, 226, 29, 132, 123, 46, 209, 48, 207, 154, 101, 252, 3, 86, 169, 184, 71, 18, 237, 116, 139, 222, 33, 96, 159, 202, 53, 172, 83, 6, 249, 232, 23, 66, 189, 36, 219, 142, 113, 144, 111, 58, 197, 92, 163, 246, 9, 24, 231, 178, 77, 212, 43, 126, 129, 128, 127, 42, 213, 76, 179, 230, 25, 8, 247, 162, 93, 196, 59, 110, 145, 112, 143, 218, 37, 188, 67, 22, 233, 248, 7, 82, 173, 52, 203, 158, 97, 32, 223, 138, 117, 236, 19, 70, 185, 168, 87, 2, 253, 100, 155, 206, 49, 208, 47, 122, 133, 28, 227, 182, 73, 88, 167, 242, 13, 148, 107, 62, 193, 64, 191, 234, 21, 140, 115, 38, 217, 200, 55, 98, 157, 4, 251, 174, 81, 176, 79, 26, 229, 124, 131, 214, 41, 56, 199, 146, 109, 244, 11, 94, 161, 224, 31, 74, 181, 44, 211, 134, 121, 104, 151, 194, 61, 164, 91, 14, 241, 16, 239, 186, 69, 220, 35, 118, 137, 152, 103, 50, 205, 84, 171, 254, 1]

fixed points

number of fixed points

$n$	1	2	3	4	5
$2^{2^{n-1}}$	2	4	16	256	65536

$\Phi$ = triangle  A358167:

     k  0   1   2   3   4   5   6    7    8    9   10   11   12   13   14   15
n
0       0   1
1       0   2
2       0   6   8  14
3       0  30  40  54  72  86  96  126  128  158  168  182  200  214  224  254 

row 4 of length 256:     0, 510, 680, 854 ... 64680, 64854, 65024, 65534

[0, 510, 680, 854, 1224, 1334, 1632, 1950, 2176, 2430, 2600, 3030, 3144, 3510, 3808, 3870, 4320, 4382, 4680, 5046, 5160, 5590, 5760, 6014, 6240, 6558, 6856, 6966, 7336, 7510, 7680, 8190, 8320, 8574, 8744, 9174, 9288, 9654, 9952, 10014, 10240, 10750, 10920, 11094, 11464, 11574, 11872, 12190, 12384, 12702, 13000, 13110, 13480, 13654, 13824, 14334, 14560, 14622, 14920, 15286, 15400, 15830, 16000, 16254, 16512, 16766, 16936, 17366, 17480, 17846, 18144, 18206, 18432, 18942, 19112, 19286, 19656, 19766, 20064, 20382, 20576, 20894, 21192, 21302, 21672, 21846, 22016, 22526, 22752, 22814, 23112, 23478, 23592, 24022, 24192, 24446, 24576, 25086, 25256, 25430, 25800, 25910, 26208, 26526, 26752, 27006, 27176, 27606, 27720, 28086, 28384, 28446, 28896, 28958, 29256, 29622, 29736, 30166, 30336, 30590, 30816, 31134, 31432, 31542, 31912, 32086, 32256, 32766, 32768, 33278, 33448, 33622, 33992, 34102, 34400, 34718, 34944, 35198, 35368, 35798, 35912, 36278, 36576, 36638, 37088, 37150, 37448, 37814, 37928, 38358, 38528, 38782, 39008, 39326, 39624, 39734, 40104, 40278, 40448, 40958, 41088, 41342, 41512, 41942, 42056, 42422, 42720, 42782, 43008, 43518, 43688, 43862, 44232, 44342, 44640, 44958, 45152, 45470, 45768, 45878, 46248, 46422, 46592, 47102, 47328, 47390, 47688, 48054, 48168, 48598, 48768, 49022, 49280, 49534, 49704, 50134, 50248, 50614, 50912, 50974, 51200, 51710, 51880, 52054, 52424, 52534, 52832, 53150, 53344, 53662, 53960, 54070, 54440, 54614, 54784, 55294, 55520, 55582, 55880, 56246, 56360, 56790, 56960, 57214, 57344, 57854, 58024, 58198, 58568, 58678, 58976, 59294, 59520, 59774, 59944, 60374, 60488, 60854, 61152, 61214, 61664, 61726, 62024, 62390, 62504, 62934, 63104, 63358, 63584, 63902, 64200, 64310, 64680, 64854, 65024, 65534]

illustration of fixed points $\Phi _{3}$ in permutation $\Pi _{3}$

row sums:     1, 2, 28, 2032, 8388352...
For $n\geq 1$ this is $~~~2^{2^{n-1}}\cdot (2^{(2^{n}-1)}-1)~~~$ =      A147537( A000225) n sum of row n 0 1 1 $2^{1}\cdot (2^{1}-1)$ 2 2 $2^{2}\cdot (2^{3}-1)$ 28 3 $2^{4}\cdot (2^{7}-1)$ 2032 4 $2^{8}\cdot (2^{15}-1)$ 8388352

illustrations of rows 3 and 4

row 3 row 4 The images show how $\Phi _{n}$ is derived from $\Pi _{n-1}$, the permutation of the same length. The long matrices have two halves: In the upper half the bit pattern in column $k$ is that of $\Xi _{n-1,~k}~=~\Pi _{n-1,~k}\oplus k$.    (Compare triangle Ξ in the next section.) The bit pattern in the lower half is identical to that of the gray column indices. $\Phi _{n,~k}~~=~~\Xi _{n-1,~k}+{\bigl (}2^{2^{n-1}}\cdot k{\bigr )}~~=~~{\bigl (}\Pi _{n-1,~k}\oplus k{\bigr )}+{\bigl (}2^{2^{n-1}}\cdot k{\bigr )}$ The small matrices on the left are divided in the same way: The upper half is a Sierpiński triangle without the main diagonal, and the lower half is the main diagonal.

Let $\mathrm {P} (n)=2^{(2^{n}-1)}$ =  A058891(n+1).
Each row contains the unique power of two P(n) in position P(n−1):     $\Pi _{n,~\mathrm {P} (n-1)}=\mathrm {P} (n)$

value XOR place

The triangle of fixed points has a second relationship with the triangle of Zhegalkin permutations.

$\Xi _{n,~k}~~=~~\Pi _{n,~k}\oplus k$:

     k   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n 
0        0   0
1        0   2   0   2
2        0  14   8   6   8   6   0  14   0  14   8   6   8   6   0  14

row 3 of length 256:     0, 254, 168, 86 ... 168, 86, 0, 254

[0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254, 0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254]

For $n\geq 1$ the values in row n are repetitions of the fixed points. E.g. row 2 contains the values 0, 6, 8, 14 (each repeated four times).

The set of places $I_{n,~k}$ where row n of this triangle has entries $\Phi _{n,~k}$ can be calculated by XORing the entry $\Pi _{n-1,~k}$ with the entries of $\Phi _{n}$.

$I_{n,~k}~~=~~\{i\mid \Pi _{n,~i}\oplus i=\Phi _{n,~k}\}~~=~~\{\Pi _{n-1,~k}\oplus f\mid f\in \Phi _{n}\}$

example for n = 3

The left column is $\Pi _{2}$, and the top row is $\Phi _{3}$. The entries are their bitwise XORs. [[ 0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254], [ 15, 17, 39, 57, 71, 89, 111, 113, 143, 145, 167, 185, 199, 217, 239, 241], [ 10, 20, 34, 60, 66, 92, 106, 116, 138, 148, 162, 188, 194, 220, 234, 244], [ 5, 27, 45, 51, 77, 83, 101, 123, 133, 155, 173, 179, 205, 211, 229, 251], [ 12, 18, 36, 58, 68, 90, 108, 114, 140, 146, 164, 186, 196, 218, 236, 242], [ 3, 29, 43, 53, 75, 85, 99, 125, 131, 157, 171, 181, 203, 213, 227, 253], [ 6, 24, 46, 48, 78, 80, 102, 120, 134, 152, 174, 176, 206, 208, 230, 248], [ 9, 23, 33, 63, 65, 95, 105, 119, 137, 151, 161, 191, 193, 223, 233, 247], [ 8, 22, 32, 62, 64, 94, 104, 118, 136, 150, 160, 190, 192, 222, 232, 246], [ 7, 25, 47, 49, 79, 81, 103, 121, 135, 153, 175, 177, 207, 209, 231, 249], [ 2, 28, 42, 52, 74, 84, 98, 124, 130, 156, 170, 180, 202, 212, 226, 252], [ 13, 19, 37, 59, 69, 91, 109, 115, 141, 147, 165, 187, 197, 219, 237, 243], [ 4, 26, 44, 50, 76, 82, 100, 122, 132, 154, 172, 178, 204, 210, 228, 250], [ 11, 21, 35, 61, 67, 93, 107, 117, 139, 149, 163, 189, 195, 221, 235, 245], [ 14, 16, 38, 56, 70, 88, 110, 112, 142, 144, 166, 184, 198, 216, 238, 240], [ 1, 31, 41, 55, 73, 87, 97, 127, 129, 159, 169, 183, 201, 215, 225, 255]] Which places in row 3 of the XOR triangle have the entry $\Phi _{3,~10}=168$? Row 10 starts with $\Pi _{2,~10}=2$. $I_{3,~10}~~=~~\{i\mid \Pi _{3,~i}\oplus i=168\}~~=~~\{2\oplus f\mid f\in \Phi _{3}\}~~=~~\{2,28,42,52,74,84,98,124,130,156,170,180,202,212,226,252\}$

Python code

functions

from sympy import binomial def sierpinski(n): return int(sum([(binomial(n, i) % 2) * 2 ** i for i in range(n + 1)])) def zhegalkin_perm(n, k): row_length = 1 << (1 << n) # 2 ** 2 ** n assert k < row_length string_length = 1 << n # 2 ** n k_binary = "{0:b}".format(k).zfill(string_length) reflected_result = 0 for i, binary_digit in enumerate(k_binary): if binary_digit == '1': s = sierpinski(i) reflected_result ^= s reflected_result_binary = "{0:b}".format(reflected_result).zfill(string_length) result_binary = reflected_result_binary[::-1] return int(result_binary, 2) def zhegalkin_fixed(n, k): if n == 0: assert k < 2 return k else: row_length = 1 << (1 << (n-1)) # 2 ** 2 ** (n-1) assert k < row_length p = zhegalkin_perm(n-1, k) p_xor_k = p ^ k shifted_k = row_length * k return p_xor_k + shifted_k

Walsh permutations and Sierpiński triangles

The classes used here are part of a project not yet published. A link will be added later. from discretehelpers.walsh_perm import WalshPerm from discretehelpers.binv import Binv def anf_int_to_boolf(input_integer): from discretehelpers.boolf import Boolf input_indices = Binv(intval=input_integer).indices if input_indices == set(): return Boolf(False) result_boolf = Boolf(False) for i in input_indices: loop_indices = Binv(intval=i).indices loop_boolf = Boolf(multi_and=loop_indices) result_boolf = result_boolf ^ loop_boolf return result_boolf def create_sequence(arity): sequence = [] for i in range(2 ** 2 ** arity): boolf = anf_int_to_boolf(i) binv = Binv(boolf.truth_table(arity)) sequence.append(binv.intval) return sequence sequence_0 = create_sequence(0) # [0, 1] sequence_1 = create_sequence(1) # [0, 3, 2, 1] sequence_2 = create_sequence(2) # [0, 15, 10, 5, 12, 3, 6, 9, 8, 7, 2, 13, 4, 11, 14, 1] sequence_3 = create_sequence(3) # [0, 255, 170, 85 ... 84, 171, 254, 1] sequence_4 = create_sequence(4) # [0, 65535, 43690, 21845 ... 21844, 43691, 65534, 1] wp_0 = WalshPerm(perm=sequence_0) wp_1 = WalshPerm(perm=sequence_1) wp_2 = WalshPerm(perm=sequence_2) wp_3 = WalshPerm(perm=sequence_3) wp_4 = WalshPerm(perm=sequence_4) wp_0.matrix() # array([[1]]) wp_1.matrix() # array([[1, 0], # [1, 1]]) wp_2.matrix() # array([[1, 0, 0, 0], # [1, 1, 0, 0], # [1, 0, 1, 0], # [1, 1, 1, 1]]) wp_3.matrix() # array([[1, 0, 0, 0, 0, 0, 0, 0], # [1, 1, 0, 0, 0, 0, 0, 0], # [1, 0, 1, 0, 0, 0, 0, 0], # [1, 1, 1, 1, 0, 0, 0, 0], # [1, 0, 0, 0, 1, 0, 0, 0], # [1, 1, 0, 0, 1, 1, 0, 0], # [1, 0, 1, 0, 1, 0, 1, 0], # [1, 1, 1, 1, 1, 1, 1, 1]]) wp_4.matrix() # array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], # [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], # [1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0], # [1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], # [1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0], # [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0], # [1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0], # [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], # [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]])