Difference between revisions of "Minimal negation operator"

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In [[logic]] and [[mathematics]], 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> if and only if 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]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> if and only if 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.  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.
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In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators may 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>
 +
 
 +
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
  
 
{| align="center" cellpadding="10" style="text-align:center"
 
{| align="center" cellpadding="10" style="text-align:center"
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& = & \operatorname{false}
 
& = & \operatorname{false}
 
\\[6pt]
 
\\[6pt]
\texttt{(} x \texttt{)}
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\texttt{(x)}
& = &
+
& = & \tilde{x}
\tilde{x}
 
 
& = & x^\prime
 
& = & x^\prime
 
\\[6pt]
 
\\[6pt]
\texttt{(} x, y \texttt{)}
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\texttt{(x, y)}
& = &
+
& = & \tilde{x}y \lor x\tilde{y}
\tilde{x}y \lor x\tilde{y}
+
& = & x^\prime y \lor x y^\prime
& = &
 
x^\prime y \lor x y^\prime
 
 
\\[6pt]
 
\\[6pt]
\texttt{(} x, y, z \texttt{)}
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\texttt{(x, y, z)}
& = &
+
& = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z}
\tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z}
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& = & x^\prime y z \lor x y^\prime z \lor x y z^\prime
& = &
 
x^\prime y z \lor x y^\prime z \lor x y z^\prime
 
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
  
It may also be noted that <math>\texttt{(} x, y \texttt{)}</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>\texttt{(} x, y \texttt{)}</math> and for <math>\texttt{(} x, y, z \texttt{)}</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint.  However, the function <math>\texttt{(} x, y, z \texttt{)}</math> is not the same thing as the function <math>x + y + z\!</math>.
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It may also be noted that <math>\texttt{(x, y)}</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>\texttt{(x, y)}</math> and for <math>\texttt{(x, y, z)}</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint.  However, the function <math>\texttt{(x, y, z)}</math> is not the same thing as the function <math>x + y + z\!</math>.
  
 
The minimal negation operator ('''mno''') has a legion of aliases:  ''logical boundary operator'', ''[[limen|limen operator]]'', ''threshold operator'', or ''least action operator'', to name but a few.  The rationale for these names is visible in the [[venn diagram]]s of the corresponding operations on [[set]]s.
 
The minimal negation operator ('''mno''') has a legion of aliases:  ''logical boundary operator'', ''[[limen|limen operator]]'', ''threshold operator'', or ''least action operator'', to name but a few.  The rationale for these names is visible in the [[venn diagram]]s of the corresponding operations on [[set]]s.
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To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube.
 
To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube.
  
For example, consider the case where <math>k = 3.\!</math>  Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}</math> &mdash; has the following venn diagram:
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For example, consider the case where <math>k = 3.\!</math>  Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram:
  
 
{| align="center" cellpadding="10" style="text-align:center"
 
{| align="center" cellpadding="10" style="text-align:center"
 
|
 
|
 
<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
 
<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
<p><math>\text{Figure 2.} ~~ \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}</math>
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<p><math>\text{Figure 2.}~~\texttt{(p, q, r)}</math>
 
|}
 
|}
  
For a contrasting example, the boolean function expressed by the form <math>\texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}</math> has the following venn diagram:
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For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram:
  
 
{| align="center" cellpadding="10" style="text-align:center"
 
{| align="center" cellpadding="10" style="text-align:center"
 
|
 
|
 
<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>
 
<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>
<p><math>\text{Figure 3.} ~~ \texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}</math>
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<p><math>\text{Figure 3.}~~\texttt{((p),(q),(r))}</math>
 
|}
 
|}
  

Revision as of 17:18, 23 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 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 may 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{(~)} & = & 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:

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

To pass from these limiting examples to the general case, observe that a singular proposition \(s : \mathbb{B}^k \to \mathbb{B}\) 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 \(x_1, \ldots, x_k, (x_1), \ldots, (x_k),\) in other words, either a posited basic proposition \(x_j\!\) or a negated basic proposition \((x_j),\!\) 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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