This Venn diagram, representing the intersection of 3 sets, or the conjunction of 3 statements respectively, gives an example of a 3-ary Boolean function.
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 $2^{3}$ = 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).
Theoretically a bec could have 3! * 2^{3} = 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.)
permuting all 3 arguments
3
4
permutations of functions in a sec
Cycle graph of S_{3}
Each of the six files corresponds to a permutation of (A, B, C).
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.
Nonlinearity
The 8 functions in the Walsh matrix and their complements are linear.
$\scriptstyle (A\lor B)\land \neg C$ in sec 7 has nonlinearity 1.
$\scriptstyle (\neg A\lor \neg C)\land (\neg B\lor C)$ in sec 53 has nonlinearity 2.
The Boolean function $\scriptstyle (1,0,1,0,0,1,1,0)$
has the Walsh spectrum $\scriptstyle (4,2,0,-2,0,2,0,2)$^{[1]}
and the binary Walsh spectrum $\scriptstyle (0,1,0,1,0,1,0,1)$. 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 $\scriptstyle (8,0,0,0,0,0,0,0,)$. ^{[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:
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).
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.
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.
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 20monotonic 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 .
In the above tables that show the nesting of the equivalence classes the monotonic secs have a dark red instead of a black background.
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.
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:
Complements in 8-element secs in a cube
The position of the complement in the sec cube corresponds to the position of the sec cube in the big cube.
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-aryexact-value functions. On the left the functions in these gbecs are shown in the Hasse diagram.