Minimal negation operator

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 \(\nu_k : \mathbb{B}^k \to \mathbb{B}\) defined in such a way that \(\nu_k (x_1, \ldots , x_k) = 1\) 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: \(\text{Figure 2.}~~\texttt{(p, q, r)}\) For a contrasting example, the boolean function expressed by the form \(\texttt{((p),(q),(r))}\) has the following venn diagram: \(\text{Figure 3.}~~\texttt{((p),(q),(r))}\) 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 Template:Col-break Ampheck Anamnesis BACD Boole, George Boolean algebra Boolean function Boolean logic Boolean-valued function Template:Col-break Continuous predicate Differentiable manifold Differential topology Dynamical system Exclusive disjunction Leibniz, G.W. Logical connective Logical graph Template:Col-break Meno Multigrade operator Parametric operator Peirce, C.S. Sole sufficient operator Truth table Universal algebra Zeroth order logic Template:Col-end External links Minimal Negation Operator @ PlanetMath Minimal Negation Operator @ PlanetPhysics Wolfram Atlas of Simple Programs Elementary Cellular Automata Rules (ECARs) ECAR Index ECAR Icons ECAR Examples ECAR Formulas <sharethis />