User:Arthur/Sandbox

Definition

Let $\Bbb B$ be a Boolean domain:

$\Bbb B = \left\{{\mathrm F, \mathrm T}\right\}$

The minimal negation operator $\nu$ is a multiary operator:

$\nu_k: \Bbb B^k \to \Bbb B$

where:

• $k \in \N$
• $\nu_k$ is a boolean function defined as:

$\nu_k \left({x_1, x_2, \ldots, x_k}\right) = \begin{cases} \mathrm T & : \exists! x_j \in \left\{{x_1, x_2, \ldots, x_k}\right\}: x_j = \mathrm F \\ \mathrm F & : \text {otherwise} \end{cases}$

That is: $\nu_k \left({x_1, x_2, \ldots, x_k}\right)$ is true iff exactly one of its arguments is false.

Examples

Expressed in disjunctive normal form, the first few instances of $\nu_k$ are as follows:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \nu_0\)	\(=\)	\(\displaystyle \mathrm F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \nu_1 \left({p}\right)\)	\(=\)	\(\displaystyle \neg p\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \nu_2 \left({p, q}\right)\)	\(=\)	\(\displaystyle \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \nu_3 \left({p, q, r}\right)\)	\(=\)	\(\displaystyle \left({\neg p \land q \land r}\right) \lor \left({p \land \neg q \land r}\right) \lor \left({p \land q \land \neg r}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \nu_4 \left({p, q, r, s}\right)\)	\(=\)	\(\displaystyle \left({\neg p \land q \land r \land s}\right) \lor \left({p \land \neg q \land r \land s}\right) \lor \left({p \land q \land \neg r \land s}\right) \lor \left({p \land q \land r \land \neg s}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

It can directly be seen that:

• $\nu_0$ is the false constant, or a contradiction $\bot$
• $\nu_1$ is the same operator as the logical Not operator $\neg$

• $\nu_2$ is the same operator as the exclusive or operator $\oplus$

For $k > 2$ there is no immediate correspondence between $\nu_k$ and conventional logical operators.

Notation

The symbol $\nu$ used for this is the Greek letter nu.

Not sure whether this is an instance of the MNO or something else.

General Case

To express the general case of $\nu_k$ in terms of familiar operations, it helps to introduce an intermediary concept:

Let the function $\lnot_j: \mathbb B^k \to \mathbb B$ be defined for each integer $j\!$ in the interval $[1, k]\!$ by the following equation:

$\neg_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \neg x_j \land x_{j+1} \land \ldots \land x_k$

Then $\nu_k : \mathbb B^k \to \mathbb B$ is defined by the following equation:

$\nu_k (x_1, \ldots, x_k) = \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k)$

If we think of the point:

$x = (x_1, \ldots, x_k) \in \mathbb B^k$

by the logical conjunction $x_1 \land \ldots \land x_k$

then the minimal negation $\nu_k \left({x_1, \ldots, x_k}\right)$ indicates the set of points in $\mathbb B^k$ that differ from $x$ in exactly one coordinate.

This makes $\nu_k \left({x_1, \ldots, x_k}\right)$ a discrete functional analogue of a point omitted neighborhood in analysis, or more exactly, a point omitted distance one neighborhood.

In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, the logical boundary operator.

Not really relevant to this page, but bits can be extracted into other places

The remainder of this discussion proceeds on the algebraic boolean convention that the plus sign $(+)\!$ and the summation symbol $(\textstyle\sum)$ both refer to addition modulo 2. Unless otherwise noted, the boolean domain $\mathbb B = \{ 0, 1 \}$ is interpreted so that $0 = \operatorname{false}$ and $1 = \operatorname{true}.$ This has the following consequences:

•	The operation $x + y\!$ is a function equivalent to the exclusive disjunction of $x\!$ and $y,\!$ while its fiber of 1 is the relation of inequality between $x\!$ and $y.\!$
•	The operation $\textstyle\sum_{j=1}^k x_j$ maps the bit sequence $(x_1, \ldots, x_k)\!$ to its parity.

The following properties of the minimal negation operators $\nu_k : \mathbb B^k \to \mathbb B$ may be noted:

•	The function $\mathsf{(x, y)}$ is the same as that associated with the operation $x + y\!$ and the relation $x \ne y.$
•	In contrast, $\mathsf{(x, y, z)}$ is not identical to $x + y + z.\!$
•	More generally, the function $\nu_k (x_1, \dots, x_k)$ for $k > 2\!$ is not identical to the boolean sum $\textstyle\sum_{j=1}^k x_j.$
•	The inclusive disjunctions indicated for the $\nu_k\!$ of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint.

These truth tables also need to be extracted, rebuilt into our house style and re-interpreted.

Truth tables

Table 1 is a truth table for the sixteen boolean functions of type $f : \mathbb B^3 \to \mathbb B$ whose fibers of 1 are either the boundaries of points in $\mathbb B^3$ or the complements of those boundaries.

$\text{Table 1.}~~\text{Logical Boundaries and Their Complements}$

$\mathcal{L}_1$	$\mathcal{L}_2$	$\mathcal{L}_3$	$\mathcal{L}_4$
	$p\colon\!$	$1~1~1~1~0~0~0~0$
	$q\colon\!$	$1~1~0~0~1~1~0~0$
	$r\colon\!$	$1~0~1~0~1~0~1~0$
$\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}$	$\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}$	$\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}$	$\begin{matrix} \mathsf{(~p~,~q~,~r~)} \\[4pt] \mathsf{(~p~,~q~,(r))} \\[4pt] \mathsf{(~p~,(q),~r~)} \\[4pt] \mathsf{(~p~,(q),(r))} \\[4pt] \mathsf{((p),~q~,~r~)} \\[4pt] \mathsf{((p),~q~,(r))} \\[4pt] \mathsf{((p),(q),~r~)} \\[4pt] \mathsf{((p),(q),(r))} \end{matrix}$
$\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}$	$\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}$	$\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}$	$\begin{matrix} \mathsf{(((p),(q),(r)))} \\[4pt] \mathsf{(((p),(q),~r~))} \\[4pt] \mathsf{(((p),~q~,(r)))} \\[4pt] \mathsf{(((p),~q~,~r~))} \\[4pt] \mathsf{((~p~,(q),(r)))} \\[4pt] \mathsf{((~p~,(q),~r~))} \\[4pt] \mathsf{((~p~,~q~,(r)))} \\[4pt] \mathsf{((~p~,~q~,~r~))} \end{matrix}$

Not sure about this lot, I think they're better somewhere else

Charts and graphs

This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in italics are relegated to a Glossary at the end of the article.

Two ways of visualizing the space $\mathbb B^k$ of $2^k\!$ points are the hypercube picture and the venn diagram picture. The hypercube picture associates each point of $\mathbb B^k$ with a unique point of the $k\!$-dimensional hypercube. The venn diagram picture associates each point of $\mathbb B^k$ with a unique "cell" of the venn diagram on $k\!$ "circles".

In addition, each point of $\mathbb B^k$ is the unique point in the fiber of truth $[|s|]\!$ of a singular proposition $s : \mathbb B^k \to \mathbb B,$ and thus it is the unique point where a singular conjunction of $k\!$ literals is $1.\!$

For example, consider two cases at opposite vertices of the cube:

•	The point $(1, 1, \ldots , 1, 1)$ with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to $1,\!$ namely, the point where:
	$x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.$
•	The point $(0, 0, \ldots , 0, 0)$ with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to $1,\!$ namely, the point where:
	$\mathsf{(} x_1 \mathsf{)(} x_2 \mathsf{)} \ldots \mathsf{(} x_{n-1} \mathsf{)(} x_n \mathsf{)} ~=~ 1.$

To pass from these limiting examples to the general case, observe that a singular proposition $s : \mathbb B^k \to \mathbb B$ can be given canonical expression as a conjunction of literals, $s = e_1 e_2 \ldots e_{k-1} e_k$. Then the proposition $\nu (e_1, e_2, \ldots, e_{k-1}, e_k)$ is $1\!$ on the points adjacent to the point where $s\!$ is $1,\!$ and 0 everywhere else on the cube.

For example, consider the case where $k = 3.\!$ Then the minimal negation operation $\nu (p, q, r)\!$ — written more simply as $\mathsf{(p, q, r)}$ — has the following venn diagram:

$\text{Figure 2.}~~\mathsf{(p, q, r)}$

For a contrasting example, the boolean function expressed by the form $\mathsf{((p),(q),(r))}$ has the following venn diagram:

$\text{Figure 3.}~~\mathsf{((p),(q),(r))}$