Difference between revisions of "Minimal negation operator"

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The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> for just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math>
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The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> and where <math>\nu_k (x_1, \ldots , x_k) = 1</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math>
  
 
In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.  In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math>
 
In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.  In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math>

Revision as of 21:28, 25 August 2009

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 \(\nu_k : \mathbb{B}^k \to \mathbb{B}\) and where \(\nu_k (x_1, \ldots , x_k) = 1\) in just those cases where 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. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, \(\texttt{(x, y, z)}\) = \(\nu (x, y, z).\!\)

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} \texttt{()} & = & \nu_0 & = & 0 & = & \operatorname{false} \'"`UNIQ-MathJax1-QINU`"' * 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:

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

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 \(\texttt{(p, q, r)}\) — has the following venn diagram:

Venn Diagram (P,Q,R).jpg

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

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

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

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

Glossary of basic terms

Boolean domain
A boolean domain \(\mathbb{B}\) is a generic 2-element set, for example, \(\mathbb{B} = \{ 0, 1 \},\) whose elements are interpreted as logical values, usually but not invariably with \(0 = \operatorname{false}\) and \(1 = \operatorname{true}.\)
Boolean variable
A boolean variable \(x\!\) is a variable that takes its value from a boolean domain, as \(x \in \mathbb{B}.\)
Proposition
In situations where boolean values are interpreted as logical values, a boolean-valued function \(f : X \to \mathbb{B}\) or a boolean function \(g : \mathbb{B}^k \to \mathbb{B}\) is frequently called a proposition.
Basis element, Coordinate projection
Given a sequence of \(k\!\) boolean variables, \(x_1, \ldots, x_k,\) each variable \(x_j\!\) may be treated either as a basis element of the space \(\mathbb{B}^k\) or as a coordinate projection \(x_j : \mathbb{B}^k \to \mathbb{B}.\)
Basic proposition
This means that the set of objects \(\{ x_j : 1 \le j \le k \}\) is a set of boolean functions \(\{ x_j : \mathbb{B}^k \to \mathbb{B} \}\) subject to logical interpretation as a set of basic propositions that collectively generate the complete set of \(2^{2^k}\) propositions over \(\mathbb{B}^k.\)
Literal
A literal is one of the \(2k\!\) propositions \(x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},\) in other words, either a posited basic proposition \(x_j\!\) or a negated basic proposition \(\texttt{(} x_j \texttt{)},\) for some \(j = 1 ~\text{to}~ k.\)
Fiber
In mathematics generally, the fiber of a point \(y \in Y\) under a function \(f : X \to Y\) is defined as the inverse image \(f^{-1}(y) \subseteq X.\)
In the case of a boolean function \(f : \mathbb{B}^k \to \mathbb{B},\) there are just two fibers:
The fiber of \(0\!\) under \(f,\!\) defined as \(f^{-1}(0),\!\) is the set of points where the value of \(f\!\) is \(0.\!\)
The fiber of \(1\!\) under \(f,\!\) defined as \(f^{-1}(1),\!\) is the set of points where the value of \(f\!\) is \(1.\!\)
Fiber of truth
When \(1\!\) is interpreted as the logical value \(\operatorname{true},\) 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 \(f.\!\)
Singular boolean function
A singular boolean function \(s : \mathbb{B}^k \to \mathbb{B}\) is a boolean function whose fiber of \(1\!\) is a single point of \(\mathbb{B}^k.\)
Singular proposition
In the interpretation where \(1\!\) equals \(\operatorname{true},\) 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 \(\mathbb{B}^k.\)
Singular conjunction
A singular conjunction in \(\mathbb{B}^k \to \mathbb{B}\) is a conjunction of \(k\!\) literals that includes just one conjunct of the pair \(\{ x_j, ~\nu(x_j) \}\) for each \(j = 1 ~\text{to}~ k.\)
A singular proposition \(s : \mathbb{B}^k \to \mathbb{B}\) can be expressed as a singular conjunction:
\(s ~=~ e_1 e_2 \ldots e_{k-1} e_k\),

\(\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}\)

See also

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External links


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