Studies of Euler diagrams/NP tables


 * These tables are in category .

The natural subdivisions of an NP equivalence class are permutation and negation equivalence classes.

These tables show all functions of an NP-EC (hereafter just EC) with the P-ECs as rows and the N-ECs as groups of columns.

The functions are represented by their Zhegalkin indices, and the order of the rows and columns is based on these numbers. The chosen representative is highlighted in violet, and its name is shown in the header. The transformations, denoted by number icons, refer to this representative. (Otherwise it has no influence on the table.)

The following table shows the EC of  and its complement. It has three N-ECs, each with 8 functions. Its six P-ECs contain 3 functions if the Venn diagrams are symmetric, or 6 functions if they are not.

The number icons denote the transformations from the representative to each particular function. A transformation is a signed permutation, i.e. a negator pattern paired with a permutation &mdash; both of which can be represented by a natural number. While the permutation index numbers are unambiguous, there is some potential confusion about numbering the negator patterns. Generally it is more useful to refer to which values are negated. (See this 8×6 matrix.) But functions in the same P-EC have the same negated places in common &mdash; which are thus used as row labels in these tables. To avoid confusion, the icons referring to negated values are round, and those referring to negated places are square.

So each function corresponds to a set of transformations. To such a set corresponds a set of keyneg indices, a set of valneg indices and a set of perm indices. P-ECs (rows) have the same set of keyneg indices, and N-ECs (column groups) have the same set of perm indices in common. The sets of perm indices are of The  $$S_n$$, where $$n$$ is the arity. (See .) The valneg set is shown under each Zhegalkin index. Hovering over it will show a table with the whole set of transformations.

The representative function of the table above is. The hover table is shown for. Below is shown how the two transformations turn the Euler diagram of the former into (mirrored) Euler diagrams of the latter.

Often the set of valneg indices of a function contains only a single element. Two are also common:

There is only one 3-ary EC with three:

Eight are an extreme case. This is very rare:

(This table is wider than most screens. The hover tables on the right can be seen by keeping the mouse on them, and pressing the right arrow key.)