User:Prime.mover/Sandbox/Minimal Negation Operator

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

General Case
Let the function $\neg_j: \mathbb B^k \to \mathbb B$ be defined for each integer $j$ in the interval $\closedint 1 k$ as follows:


 * $\map {\neg_j} {x_1, \ldots, x_j, \ldots, x_k} := x_1 \land \ldots \land x_{j - 1} \land \neg x_j \land x_{j + 1} \land \ldots \land x_k$

Then $\nu_k : \mathbb B^k \to \mathbb B$ is defined as:


 * $\map {\nu_k} {x_1, \ldots, x_k} := \map {\neg_1} {x_1, \ldots, x_k} \lor \ldots \lor \map {\neg_j} {x_1, \ldots, x_k} \lor \ldots \lor \map {\neg_k} {x_1, \ldots, x_k}$

If we think of the point:
 * $x = \tuple {x_1, \ldots, x_k} \in \mathbb B^k$

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

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

This makes $\map {\nu_k} {x_1, \ldots, x_k}$ 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.

The remainder of this discussion proceeds on the algebraic boolean convention that the plus sign $+$ and the summation symbol $\sum$ both refer to addition modulo $2$.

Unless otherwise noted, the boolean domain $\mathbb B = \set {0, 1}$ 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 $\ds \sum_{j \mathop = 1}^k x_j$ maps the bit sequence $\tuple {x_1, \ldots, x_k}$ to its parity.

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


 * The function $\map {\nu_2} {x, y}$ is the same as that associated with the operation $x + y$ and the relation $x \ne y$.
 * In contrast, $\map {\nu_3} {x, y, z}$ is not identical to $x + y + z$.
 * More generally, the function $\map {\nu_k} {x_1, \dots, x_k}$ for $k > 2$ is not identical to the boolean sum $\ds \sum_{j \mathop = 1}^k x_j$.
 * 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.

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.

Logical Boundaries and Their Complements

 * $\begin {array} {|c|c|c|c|} \hline

\LL_1 & \LL_2 & \LL_3 & \LL_4 \\ \hline & p : & 1 1 1 1 0~0~0~0 & \\ \hline & q : & 1 1 0~0~1~1~0~0 & \\ \hline & r : & 1 0 1~0~1~0~1~0 & \\ \hline f_{104} & f_{01101000} & 0~1~1~0~1~0~0~0 & \mathsf {(~p~,~q~,~r~)} \\ f_{148} & f_{10010100} & 1~0~0~1~0~1~0~0 & \mathsf {(~p~,~q~,(r))} \\ f_{146} & f_{10010010} & 1~0~0~1~0~0~1~0 & \mathsf {(~p~,(q),~r~)} \\ f_{97} & f_{01100001} & 0~1~1~0~0~0~0~1 & \mathsf {(~p~,(q),(r))} \\ f_{134} & f_{10000110} & 1~0~0~0~0~1~1~0 & \mathsf {((p),~q~,~r~)} \\ f_{73} & f_{01001001} & 0~1~0~0~1~0~0~1 & \mathsf {((p),~q~,(r))} \\ f_{41} & f_{00101001} & 0~0~1~0~1~0~0~1 & \mathsf {((p),(q),~r~)} \\ f_{22} & f_{00010110} & 0~0~0~1~0~1~1~0 & \mathsf {((p),(q),(r))} \\ \hline \end {array}$

Graphical Presentation
Two ways of visualizing the space $\mathbb B^k$ of $2^k$ points are the hypercube picture and the Venn diagram picture.

Each point of $\mathbb B^k$ is the unique point in the fiber of truth $\sqbrk {\size s}$ of a singular proposition $s : \mathbb B^k \to \mathbb B$.

Thus it is can be represented on a graphical presentation as the unique point where a singular conjunction of $k$ literals is $1$.

Hypercube Presentation
The hypercube picture associates each point of $\mathbb B^k$ with a unique point of the $k$-dimensional hypercube.

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


 * The point $\tuple {1, 1, \ldots, 1, 1}$ with all $1$s as coordinates is the point where the conjunction of all posited variables evaluates to $1$, namely, the point where:


 * $x_1 \land x_2 \land \dotsm \land x_{n - 1} \land x_n = 1$


 * The point $\tuple {0, 0, \ldots, 0, 0}$ with all $0$s as coordinates is the point where the conjunction of all negated variables evaluates to $1$, namely, the point where:
 * $\neg x_1 \land \neg x_2 \land \dotsm \land \neg x_{n - 1} \land \neg x_n = 1$

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:
 * $s = e_1 \land e_2 \land \ldots \land e_{k - 1} \land e_k$

Then the proposition $\map \nu {e_1, e_2, \ldots, e_{k - 1}, e_k}$ is $1$ on the points adjacent to the point where $s$ is $1$, and $0$ everywhere else on the hypercube.

Venn Diagram Presentation
The Venn diagram picture associates each point of $\mathbb B^k$ with a unique cell of the Venn diagram on $k$ circles.

For example, consider the case where $k = 3$.

Then the minimal negation operation $\map \nu {p, q, r}$ has the following Venn diagram:


 * Venn Diagram (P,Q,R).jpg

Figure $2$: $\map \nu {p, q, r}$

For a contrasting example, the boolean function expressed by the form $\map \nu {\neg p, \neg q, \neg r}$ has the following Venn diagram:


 * Venn Diagram ((P),(Q),(R)).jpg

Figure $3$: $\map \nu {\neg p, \neg q, \neg r}$