Minimal negation operator

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In logic and mathematics, the minimal negation operator $\nu\!$ is a multigrade operator $(\nu_{k})_{k \in \mathbb{N}}$ where each $\nu_{k}\!$ is a k-ary boolean function defined in such a way that $\nu_{k}(x_1, \ldots , x_k) = 1$ if and only if exactly one of the arguments $x j$ is 0.

In contexts where the initial letter $\nu\!$ is understood, the minimal negation operators can be indicated by argument lists in parentheses. The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.

$\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}$

It may also be noted that $(x, y)\!$ is the same function as $x + y\!$ and $x \ne y$ , and that the inclusive disjunctions indicated for $(x, y)\!$ and for $(x, y, z)\!$ may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function $(x, y, z)\!$ is not the same thing as the function $x + y + z\!$ .

The minimal negation operator (mno) has a legion of aliases: logical boundary operator, limen operator, threshold operator, or least action operator, to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.

The next section discusses two ways of visualizing the operation of minimal negation operators. A few bits of terminology will be needed as a language for talking about the pictures, but the formal details are tedious reading, and may already be familiar to many. As a result, the full definitions of the terms marked in bold are relegated to a Glossary at the end of the article.

1 Truth tables
2 Charts and graphs
3 Glossary of basic terms
4 References
5 See also
6 External links

[edit] Truth tables

Table 1 is a truth table for the sixteen boolean functions of type f : B³ → B, each of which is either a boundary of a point in B³ or the complement of such a boundary.

Table 1. Logical Boundaries and Their Complements
L₁	L₂	L₃	L₄
Decimal	Binary	Sequential	Parenthetical
	p =	1 1 1 1 0 0 0 0
	q =	1 1 0 0 1 1 0 0
	r =	1 0 1 0 1 0 1 0

f₁₀₄	f_01101000	0 1 1 0 1 0 0 0	( p , q , r )
f₁₄₈	f_10010100	1 0 0 1 0 1 0 0	( p , q , (r))
f₁₄₆	f_10010010	1 0 0 1 0 0 1 0	( p , (q), r )
f₉₇	f_01100001	0 1 1 0 0 0 0 1	( p , (q), (r))
f₁₃₄	f_10000110	1 0 0 0 0 1 1 0	((p), q , r )
f₇₃	f_01001001	0 1 0 0 1 0 0 1	((p), q , (r))
f₄₁	f_00101001	0 0 1 0 1 0 0 1	((p), (q), r )
f₂₂	f_00010110	0 0 0 1 0 1 1 0	((p), (q), (r))

f₂₃₃	f_11101001	1 1 1 0 1 0 0 1	(((p), (q), (r)))
f₂₁₄	f_11010110	1 1 0 1 0 1 1 0	(((p), (q), r ))
f₁₈₂	f_10110110	1 0 1 1 0 1 1 0	(((p), q , (r)))
f₁₂₁	f_01111001	0 1 1 1 1 0 0 1	(((p), q , r ))
f₁₅₈	f_10011110	1 0 0 1 1 1 1 0	(( p , (q), (r)))
f₁₀₉	f_01101101	0 1 1 0 1 1 0 1	(( p , (q), r ))
f₁₀₇	f_01101011	0 1 1 0 1 0 1 1	(( p , q , (r)))
f₁₅₁	f_10010111	1 0 0 1 0 1 1 1	(( p , q , r ))

[edit] Charts and graphs

Two common ways of visualizing the space B^k of 2^k points are the hypercube picture and the venn diagram picture. Depending on how literally or figuratively one regards these pictures, each point of B^k is either identified with or represented by a point of the k-cube and also by a cell of the venn diagram on k "circles".

In addition, each point of B^k is the unique point in the fiber of truth $[ | s | ]$ of a singular proposition s : B^k → 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 whose coordinates are all 1 is the unique point where the conjunction of all posited variables $x_1\ x_2\ \ldots\ x_{n-1}\ x_n$ is 1.

The point whose coordinates are all 0 is the unique point where the conjunction of all negated variables $(x_1)(x_2)\ldots(x_{n-1})(x_n)$ is 1.

To pass from these limiting examples to the general case, observe that a singular proposition s : B^k → 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 $\nu (p, q, r)\!$ , when there is no risk of confusion written more simply as $(p, q, r)\!$ , has the following Venn diagram:

Figure 1. (p, q, r) shaded

For a contrasting example, the boolean function expressed by the form $((p),(q),(r))\!$ has the following Venn diagram:

Figure 2. ((p),(q),(r)) shaded

[edit] Glossary of basic terms

A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, usually but not invariably with 0 = false and 1 = true.

A boolean variable x is a variable that takes its value from a boolean domain, as x ∈ B.

In situations where boolean values are interpreted as logical values, a boolean-valued function f : X → B or a boolean function g : B^k → B is frequently called a proposition.

Given a sequence of k boolean variables, x₁, …, x_k, each variable x_j may be treated either as a basis element of the space B^k or as a coordinate projection x_j : B^k → B.

This means that the k objects x_j for j = 1 to k are just so many boolean functions x_j : B^k → B , subject to logical interpretation as a set of basic propositions that generate the complete set of $2^{2^k}$ propositions over B^k.

A literal is one of the 2k propositions x₁, …, x_k, (x₁), …, (x_k), in other words, either a posited basic proposition x_j or a negated basic proposition (x_j), for some j = 1 to k.

In mathematics generally, the fiber of a point y under a function f : X → Y is defined as the inverse image $f - 1 (y)$ .

In the case of a boolean function f : B^k → B, there are just two fibers:
- The fiber of 0 under f, defined as $f - 1 (0)$ , is the set of points where f is 0.
- The fiber of 1 under f, defined as $f - 1 (1)$ , is the set of points where f is 1.

When 1 is interpreted as the logical value true, then $f - 1 (1)$ is called the fiber of truth in the proposition f. Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation $[|f|] = f^{-1}(1)\!$ for the fiber of truth in the proposition f.

A singular boolean function s : B^k → B is a boolean function whose fiber of 1 is a single point of B^k.

In the interpretation where 1 equals true, a singular boolean function is called a singular proposition.

Singular boolean functions and singular propositions serve as functional or logical representatives of the points in B^k.

A singular conjunction in B^k → B is a conjunction of k literals that includes just one conjunct of the pair $\{ x_j,\ \nu (x_j) \}$ for each j = 1 to k.