Minimal negation operator

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.

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	f01101000	0 1 1 0 1 0 0 0	( p , q , r )
f148	f10010100	1 0 0 1 0 1 0 0	( p , q , (r))
f146	f10010010	1 0 0 1 0 0 1 0	( p , (q), r )
f97	f01100001	0 1 1 0 0 0 0 1	( p , (q), (r))
f134	f10000110	1 0 0 0 0 1 1 0	((p), q , r )
f73	f01001001	0 1 0 0 1 0 0 1	((p), q , (r))
f41	f00101001	0 0 1 0 1 0 0 1	((p), (q), r )
f22	f00010110	0 0 0 1 0 1 1 0	((p), (q), (r))

f233	f11101001	1 1 1 0 1 0 0 1	(((p), (q), (r)))
f214	f11010110	1 1 0 1 0 1 1 0	(((p), (q), r ))
f182	f10110110	1 0 1 1 0 1 1 0	(((p), q , (r)))
f121	f01111001	0 1 1 1 1 0 0 1	(((p), q , r ))
f158	f10011110	1 0 0 1 1 1 1 0	(( p , (q), (r)))
f109	f01101101	0 1 1 0 1 1 0 1	(( p , (q), r ))
f107	f01101011	0 1 1 0 1 0 1 1	(( p , q , (r)))
f151	f10010111	1 0 0 1 0 1 1 1	(( p , q , r ))

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 \((1, 1, \ldots , 1, 1)\) 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:

\[(x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.\]

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:

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /`````````/     \       |
|      /       \`````````o`````````/       \      |
|     /         \```````/ \```````/         \     |
|    /           \`````/   \`````/           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 1.  (p, q, r)

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

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /`````\         \`/         /`````\       |
|      /```````\         o         /```````\      |
|     /`````````\       / \       /`````````\     |
|    /```````````\     /   \     /```````````\    |
|   o`````````````o---o-----o---o`````````````o   |
|   |`````````````````|     |`````````````````|   |
|   |`````````````````|     |`````````````````|   |
|   |``````` Q ```````|     |``````` R ```````|   |
|   o`````````````````o     o`````````````````o   |
|    \`````````````````\   /`````````````````/    |
|     \`````````````````\ /`````````````````/     |
|      \`````````````````o`````````````````/      |
|       \```````````````/ \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 2.  ((p),(q),(r))

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.

• A singular proposition s : B^k → B can be expressed as a singular conjunction:

	\(s\ \ =\ e_1 e_2 \ldots e_{k-1} e_k\),
where	\(e_j\ =\ x_j\!\)
or	\(e_j\ =\ \nu (x_j)\!\),
for	\(j\ \ =\ 1\ \mbox{to}\ k\).

See also

Ampheck Anamnesis BACD Boole, George Boolean algebra Boolean function Boolean logic Boolean-valued function

Continuous predicate Differentiable manifold Differential topology Dynamical system Exclusive disjunction Leibniz, G.W. Logical connective Logical graph

Meno Multigrade operator Parametric operator Peirce, C.S. Sole sufficient operator Truth table Universal algebra Zeroth order logic

External links

• Wolfram Atlas of Simple Programs
  • ECARs (Elementary Cellular Automata Rules)
  • ECAR Index
  • ECAR Rule Icons
  • ECAR Examples
  • ECAR Boolean Forms