Difference between revisions of "Minimal negation operator"

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: This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math>
 
: This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math>
  
* A '''literal''' is one of the 2''k'' propositions ''x''<sub>1</sub>,&nbsp;…,&nbsp;''x''<sub>''k''</sub>, (''x''<sub>1</sub>),&nbsp;…,&nbsp;(''x''<sub>''k''</sub>), in other words, either a ''posited'' basic proposition ''x''<sub>''j''</sub> or a ''negated'' basic proposition (''x''<sub>''j''</sub>), for some ''j'' = 1 to ''k''.
+
; Literal
 +
: A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, (x_1), \ldots, (x_k),</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>(x_j),\!</math> for some <math>j = 1 ~\text{to}~ k.</math>
  
 
* In mathematics generally, the '''[[fiber (mathematics)|fiber]]''' of a point ''y'' under a function ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'' is defined as the inverse image <math>f^{-1}(y)</math>.
 
* In mathematics generally, the '''[[fiber (mathematics)|fiber]]''' of a point ''y'' under a function ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'' is defined as the inverse image <math>f^{-1}(y)</math>.

Revision as of 02:01, 21 May 2009

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 & = & \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 \(\nu (p, q, r)\!\), when there is no risk of confusion written more simply as \((p, q, r)\!\), has the following venn diagram:

Minimal Negation Operator 1.jpg

\(\text{Figure 1.}\quad (p, q, r)\!\)

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

Minimal Negation Operator 2.jpg

\(\text{Figure 2.}\quad ((p),(q),(r))\!\)

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 \(j = 1 ~\text{to}~ k.\)
  • 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 : Bk → 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 : Bk → B is a boolean function whose fiber of 1 is a single point of Bk.
  • 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 Bk.
  • A singular conjunction in Bk → 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 : Bk → 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

Aficionados



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