3-ary Boolean functions

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The 3-ary Boolean function represented as a Venn diagram
sec 53 containing 8 functions, including the one shown above
gbec 27 containing 3 secs, including the one shown above

There are = 256 3-ary Boolean functions, like set operations or logical connectives.

This Venn diagram 0000 0001, representing the intersection of 3 sets, or the conjunction of 3 statements respectively, gives an example of a 3-ary Boolean function.


Equivalence classes[edit]

The 22 becs ordered by weight (compare Sloane'sA039754)
See: Equivalence classes of Boolean functions

The set of 256 functions can be parted into various equivalence classes.
The most basic way to do that are the 46 small equivalence classes (secs). Functions belong to the same sec, when they can be expressed by each other by negating arguments. There are 3 arguments that can be negated, so there can be up to = 8 different functions in a sec.

When binarily colored cubes can be transformed into each other by mirroring and rotation, they are essentially the same.
The corresponding Boolean functions are often called equivalent and belong to the same big equivalence class (bec).


Numbers of equivalence classes
wec: 5
sec: Sloane'sA000231(3) = 46 bec: Sloane'sA000616(3) = 22
gsec: 30 gbec: 14
ggsec: 18 ggbec: 8


The following sequences show the number of secs and becs by weight:

                weight   =   0   1   2   3   4   5   6   7   8
secs: A054724(3, 0..8)   =   1   1   7   7  14   7   7   1   1
becs: A039754(3, 0..8)   =   1   1   3   3   6   3   3   1   1

The following sequences show the number of secs and becs by number of functions in each sec:

                             1    2    4    8
secs: A227725(3, 0..3)   =   2    7   14   23
becs:                        2    3    6   11


Permutations of functions in a bec[edit]

Theoretically a bec could have 3! * 23 = 6 * 8 = 48 functions, but the largest becs have only 24. (There are some becs of 4-ary functions with the highest possible number of functions, e.g. this one.)

Nonlinearity[edit]

The nonlinearity of a function is the smallest Hamming distance it has from one of the 16 linear functions, i.e. from a row of the binary Walsh matrix or its complement.
The functions in wec E0 are the linear functions, i.e. those with nonlinearity 0. The functions in wec O have nonlinearity 1, and all others have nonlinearity 2.
There are no 3-ary bent functions.

Walsh spectrum[edit]

The Boolean function
has the Walsh spectrum [1]
and the binary Walsh spectrum . The latter is row 1 of the Walsh matrix.

The Walsh spectrum of a Boolean function is the product of its binary string (as a row vector) with a Walsh matrix.
The first entry of the Walsh spectrum is the functions digit sum, and all entries have the same parity.

In the following chapter Walsh spectra are shown in the "by ggbec" sections.
Walsh spectra of the same sec (which differ only in the signs of their entries) are always shown in the same file.

The Walsh spectra of complements sum up to . [2]


Walsh matrix with row numbers

Further the binary Walsh spectra are always shown by the red squares in the background. They are always rows of a binary Walsh matrix - or, in other terms, exclusive disjunctions of unnegated arguments.

The functions with odd digit sum, which make up wec O, have each binary Walsh spectrum one time in every sec.

The functions with even digit sum, which make up the other four wecs, have always the same binary Walsh spectrum in the same sec:

= 0000 0000 in wec E0
= 0101 0101     or     = 0011 0011     or     = 0000 1111        in wec E1
= 0110 0110     or     = 0101 1010     or     = 0011 1100        in wec E2
= 0110 1001 in wec E3


Functions with the same binary Walsh spectrum[edit]

In the diagrams in this section the even functions are shown in red, and the odd ones (i.e. those in wec O) in yellow. Each odd function differs in the lowest bit (numerically: by 1) from an even function with the same binary Walsh spectrum. So in the octeract graphs each red vertex is connected to a yellow vertex by an edge of the direction corresponding to the lowest bit (like the edge connecting vertices 0 and 1). The pattern of the red vertices is always symmetrical to the main axes. Surprisingly the yellow vertices are also symmetrical, but to axes tilted to the left or to the right (indicated by \ or / in the table).

Nested equivalence classes[edit]

Overview[edit]

3-ary Boolean functions; Walsh equivalence classes.svg

The matrix on the right shows the 256 functions arranged in a way similar to the Hasse diagram, which is an octeract graph.
The colors indicate the five wecs, as shown in the table below.

The two following matrices are examples of a gbec and a ggsec. Both contain the sec 26. Files like these are in the collapsible boxes below in boxes labeled Positions.

These matrices are symmetric to the main diagonal, because they contain complete secs. They are also symmetric to the antidiagonal, because they contain all the complements, and complements are symmetric about the center of the matrix. The ggsec matrices are also symmetric about the central axes, because they contain all the half-complements. Left half-complements are vertically, and right ones are horizontally symmetric to each other.

gbec 25 with 6 secs (sec 26 is light green)
ggsec 21 with 4 secs (sec 26 is light red)


O
nonlinearity 1
  E0  
nonlinearity 0
  E1  
nonlinearity 2
  E2  
nonlinearity 2
  E3  
nonlinearity 2
by ggbec

Venn 0000 0001.svg        Venn 0000 0111.svg
ggbec 1 with 64 functions

Venn 0001 0110.svg        Venn 0001 1001.svg
ggbec 22 with 64 functions
by ggbec

Venn 0000 0000.svg
ggbec 0 with 2 functions

Venn 0000 1111.svg
ggbec 15 with 6 functions

Venn 0110 0110.svg        Venn 0110 1001.svg
ggbec 60 with 8 functions
by ggbec

Venn 0001 0001.svg        Venn 0001 1110.svg
ggbec 3 with 48 functions
by ggbec

Venn 0000 1001.svg        Venn 0101 0011.svg
ggbec 6 with 48 functions
Venn 1000 0001.svg        Venn 0001 0111.svg
ggbec 23 with 16 functions
by ggsec by ggsec by ggsec by ggsec
1 22
1 7 22 25
1 127 7 31 22 107 25 61
1 1 1 8
127 8
7 7 8
31 8
19 19 19 8
55 8
28 28 8
61 8
21 21 21 8
87 8
26 26 8
91 8
22 22 22 8
107 8
25 25 8
103 8
0 15 60
0 15 60 105
0 255 15 60 105
0 0 0 1
255 1
15 15 2
51 51 51 2
60 60 2
85 85 85 2
90 90 2
102 102 102 2
105 105 2
3
3 30
3 63 30
3 3 3 4
63 4
5 5 5 4
95 4
17 17 17 4
119 4
30 30 8
54 54 54 8
86 86 86 8
6
6 27
6 111 27
6 6 6 4
111 4
18 18 18 4
123 4
29 29 8
20 20 20 4
125 4
27 27 8
53 53 53 8
23
23 24
23 24 126
23 23 23 8
24 24 4
126 4

The tables above show how the equivalence classes are nested in each other. Each table shows the equivalence classes in a wec.
The vertical axes show ggsecs, gsecs and secs, and the horizontal ones show ggbecs, gbecs and becs.
The entries in the matrices show which secs belong to which becs.
The numbers in these fields show how many functions are in each sec. A dark red background tells that the sec contains a monotonic function.


wec O with 128 functions[edit]

wec O ordered by ggbec[edit]

wec O ordered by ggsec[edit]

wec E0 with 16 functions[edit]

wec E0 ordered by ggbec[edit]

wec E0 ordered by ggsec[edit]

wec E1 with 48 functions[edit]

wec E1 ordered by ggbec[edit]

wec E1 ordered by ggsec[edit]

wec E2 with 48 functions[edit]

wec E2 ordered by ggbec[edit]

wec E2 ordered by ggsec[edit]

wec E3 with 16 functions[edit]

Monotonic functions[edit]

contradiction A and B and C A and B A and C B and C (A and B) or (A and C) (A and B) or (B and C) (A and C) or (B and C) A B C (A or B) and (A or C) and (B or C) <====> (A and B) or (A and C) or (B and C) (A or B) and (A or C) (A or B) and (B or C) (A or C) and (B or C) A or B A or C B or C A or B or C tautology
Formulas appear when the mouse is moved over the diagram, and the sec files are linked from them. In the sec files the monotonic functions are always in the top right position.


The right Hasse diagram in the file on the right shows the 20 monotonic 3-ary Boolean functions.

If a sec has a monotonic function, all the secs in the gbec and gsec it belongs to also have one. So one can call secs, gbecs and gsecs monotonic, when they contain monotonic functions. There are 6 gbecs and 12 gsecs that are monotonic in this sense. Their collapsible boxes are marked with Tudor roses Tudor Rose.svg.

In the above tables that show the nesting of the equivalence classes the monotonic secs have a dark red instead of a black background.

gbec weight gsec
0 0, 8 0
1 1, 7 1
3 2, 6 3, 5, 17
7 3, 5 7, 19, 21
15, 23 4 15, 23, 51, 85

All wecs except E2 contain monotonic functions.


Subgroups of nimber addition[edit]

See: Subgroups of nimber addition#Z23

16 secs with in all 8 + 7*4 + 7*2 + 1 = 51 functions are related to subgroups of nimber addition:

      
1000 0000 1100 0000 1010 0000 1000 1000 1000 0010 1000 0100 1001 0000 1000 0001 1010 1010 1100 1100 1111 0000 1100 0011 1010 0101 1001 1001 1001 0110 1111 1111
These cubes correspond to the matrices' top lines. In the linked files they are always in the bottom left position.


In each sec matrix its leading function appears in a sona pattern. This pattern is 0 in N(0) iff the sec contains 8 different functions. All the other different sona patterns appear only in the sec matrices of the actual sonas between 1 and 15 and their Boolean complements. Each gbec is marked in green with its sona-bec, and each gsec is marked with its sona-sec.

Complements in secs[edit]

There are two kinds of secs that are their own complements (i.e. complete gsecs): Those with sona rank 2, marked as N(2, ...), and the following seven secs containing 8 functions each:

Gbecs by similarity to 4-ary sec matrix[edit]

In the octeract matrices of the gbecs it can be seen, that they belong together in a way that is independent of ggbecs and wecs. Their pattern is part of a sec matrix of one of the five 4-ary exact-value functions. On the left the functions in these gbecs are shown in the Hasse diagram.


0[edit]

3-ary Boolean functions in octeract graph; related to 4-ary functions with weight 1; 00.svg 4-ary Boolean functions; sec matrix of 1000 0000 0000 0000.svg
0
3-ary Boolean functions; gbec 0.svg
gbec 0 Venn 0000 0000.svg wec E0
3-ary Boolean functions; gbec 60.svg
gbec 60 Venn 0110 0110.svg wec E0
3-ary Boolean functions; gbec 24.svg
gbec 24 Venn 1000 0001.svg wec E3

1[edit]

3-ary Boolean functions in octeract graph; related to 4-ary exact-value functions; 1.svg 4-ary Boolean functions; sec matrix of 0110 1000 1000 0000.svg
1, 2, 4, 8
3-ary Boolean functions; gbec 1.svg
gbec 1 Venn 0000 0001.svg wec O
3-ary Boolean functions; gbec 25.svg
gbec 25 Venn 0001 1001.svg wec O


2[edit]

3-ary Boolean functions in octeract graph; related to 4-ary exact-value functions; 2.svg 4-ary Boolean functions; sec matrix of 0001 0110 0110 1000.svg
3, 5, 6, 9, 10, 12
3-ary Boolean functions; gbec 3.svg
gbec 3 Venn 0001 0001.svg wec E1
3-ary Boolean functions; gbec 30.svg
gbec 30 Venn 0001 1110.svg wec E1
3-ary Boolean functions; gbec 6.svg
gbec 6 Venn 0000 1001.svg wec E2
3-ary Boolean functions; gbec 27.svg
gbec 27 Venn 0101 0011.svg wec E2

3[edit]

3-ary Boolean functions in octeract graph; related to 4-ary exact-value functions; 3.svg 4-ary Boolean functions; sec matrix of 0000 0001 0001 0110.svg
7, 11, 13, 14
3-ary Boolean functions; gbec 7.svg
gbec 7 Venn 0000 0111.svg wec O
3-ary Boolean functions; gbec 22.svg
gbec 22 Venn 0001 0110.svg wec O


4[edit]

3-ary Boolean functions in octeract graph; related to 4-ary functions with weight 1; 15.svg 4-ary Boolean functions; sec matrix of 0000 0000 0000 0001.svg
15
3-ary Boolean functions; gbec 15.svg
gbec 15 Venn 0000 1111.svg wec E0
3-ary Boolean functions; gbec 105.svg
gbec 105 Venn 0110 1001.svg wec E0
3-ary Boolean functions; gbec 23.svg
gbec 23 Venn 0001 0111.svg wec E3

References[edit]

  1. Compare Figure 1 in Walsh Spectrum Computations Using Cayley Graphs, by W. J. Townsend and M. A. Thornton
  2. Compare Property 1 (page 3 of the PDF) in The spectral test of the Boolean function linearity by P. Porwik