minimal negation operator MinimalNegationOperator 2009-05-16 11:17:20 2009-05-24 12:37:55 Definition % This is the default PlanetPhysics preamble. % as your knowledge of TeX increases, you % will probably want to edit this, but it % should be fine as is for beginners. % Almost certainly you want these: \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % Used for TeXing text within EPS files: \usepackage{psfrag} % Need this for including graphics (\includegraphics): \usepackage{graphicx} % For neatly defining theorems and propositions: %\usepackage{amsthm} % Making logically defined graphics: %\usepackage{xypic} % There are many more packages, add them here as you need them. % define commands here \PMlinkescapephrase{primes} \PMlinkescapephrase{Primes} \PMlinkescapephrase{singular} \PMlinkescapephrase{Singular} The \textbf{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{center}$\begin{array}{ccccc} (~) & = & 0 & = & \operatorname{false} \\[6pt] (x) & = & \tilde{x} & = & x' \\[6pt] (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\[6pt] (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{array}$\end{center} In what follows, $\mathbb{B} = \{ 0, 1 \}$ and ``$+$" denotes addition modulo 2. The following properties of the minimal negation operators $\nu_{k} : \mathbb{B}^k \to \mathbb{B}$ may be noted: \begin{itemize} \item The function $(x, y)$ is the same as that associated with the operation $x + y$ and the relation $x \ne y$. \item On the other hand, $(x, y, z)$ is not identical to $x + y + z$. \item More generally, none of the functions $\nu_{k}$ for $k \ge 3$ are identical to the boolean sums $\sum_{j=1}^k x_j$. \item The inclusive disjunctions indicated for the $\nu_{k}$ of more than one argument may be replaced with exclusive disjunctions without affecting the meaning of the expressions, since the terms disjoined are already disjoint. \end{itemize} The minimal negation operator (\textbf{mno}) has a legion of aliases: \textit{logical boundary operator}, \textit{limen operator}, \textit{threshold operator}, or \textit{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. \section{Truth tables} Table 1 is a truth table for the sixteen boolean functions of type $f : \mathbb{B}^3 \to \mathbb{B}$, each of which is either a boundary of a point in $\mathbb{B}^3$ or the complement of such a boundary. \begin{center}\begin{tabular}{|c|c|c|c|c|} \multicolumn{5}{c}{\textbf{Table 1. Logical Boundaries and Their Complements}} \\ \hline $L_1$ & $L_2$ & & $L_3$ & $L_4$ \\ \hline Decimal & Binary & & Sequential & Parenthetical \\ \hline & & $p =$ & 1 1 1 1 0 0 0 0 & \\ \hline & & $q =$ & 1 1 0 0 1 1 0 0 & \\ \hline & & $r =$ & 1 0 1 0 1 0 1 0 & \\ \hline\hline $f_{104}$ & $f_{01101000}$ & & 0 1 1 0 1 0 0 0 & $( p , q , r )$ \\ \hline $f_{148}$ & $f_{10010100}$ & & 1 0 0 1 0 1 0 0 & $( p , q , (r))$ \\ \hline $f_{146}$ & $f_{10010010}$ & & 1 0 0 1 0 0 1 0 & $( p , (q), r )$ \\ \hline $f_{97}$ & $f_{01100001}$ & & 0 1 1 0 0 0 0 1 & $( p , (q), (r))$ \\ \hline $f_{134}$ & $f_{10000110}$ & & 1 0 0 0 0 1 1 0 & $((p), q , r )$ \\ \hline $f_{73}$ & $f_{01001001}$ & & 0 1 0 0 1 0 0 1 & $((p), q , (r))$ \\ \hline $f_{41}$ & $f_{00101001}$ & & 0 0 1 0 1 0 0 1 & $((p), (q), r )$ \\ \hline $f_{22}$ & $f_{00010110}$ & & 0 0 0 1 0 1 1 0 & $((p), (q), (r))$ \\ \hline\hline $f_{233}$ & $f_{11101001}$ & & 1 1 1 0 1 0 0 1 & $(((p), (q), (r)))$ \\ \hline $f_{214}$ & $f_{11010110}$ & & 1 1 0 1 0 1 1 0 & $(((p), (q), r ))$ \\ \hline $f_{182}$ & $f_{10110110}$ & & 1 0 1 1 0 1 1 0 & $(((p), q , (r)))$ \\ \hline $f_{121}$ & $f_{01111001}$ & & 0 1 1 1 1 0 0 1 & $(((p), q , r ))$ \\ \hline $f_{158}$ & $f_{10011110}$ & & 1 0 0 1 1 1 1 0 & $(( p , (q), (r)))$ \\ \hline $f_{109}$ & $f_{01101101}$ & & 0 1 1 0 1 1 0 1 & $(( p , (q), r ))$ \\ \hline $f_{107}$ & $f_{01101011}$ & & 0 1 1 0 1 0 1 1 & $(( p , q , (r)))$ \\ \hline $f_{151}$ & $f_{10010111}$ & & 1 0 0 1 0 1 1 1 & $(( p , q , r ))$ \\ \hline \end{tabular}\end{center} \section{Charts and graphs} This section presents 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 be familiar to many readers, so the full definitions of the terms marked in bold are relegated to a Glossary at the end of the article. Two ways of visualizing the space $\mathbb{B}^k$ of $2^k$ points are the \textit{hypercube} picture and the \textit{venn diagram} picture. Depending on how literally or figuratively one regards these pictures, each point of $\mathbb{B}^k$ is either identified with or represented by a point of the $k$-cube and also by a cell of the $k$- ``circle" venn diagram. In addition, each point of $\mathbb{B}^k$ is the unique point in the \textbf{fiber of truth} $[|s|]$ of a \textbf{singular proposition} $s : \mathbb{B}^k \to \mathbb{B}$, and thus it is the unique point where a \textbf{singular conjunction} of $k$ \textbf{literals} is equal to 1. For example, consider two cases at opposite vertices of the $k$-cube: \begin{itemize} \item 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: \[ x_1\ x_2\ \ldots\ x_{k-1}\ x_k = 1 \] \item 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: \[ (x_1)(x_2) \ldots (x_{k-1})(x_k) = 1 \] \end{itemize} 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: \begin{center}\begin{tabular}{c} \includegraphics[scale=0.8]{MinimalNegationOperator1} \\ Figure 2.\ $(p, q, r)$ \end{tabular}\end{center} For a contrasting example, the boolean function expressed by the form $((p),(q),(r))$ has the following venn diagram: \begin{center}\begin{tabular}{c} \includegraphics[scale=0.8]{MinimalNegationOperator2} \\ Figure 3.\ $((p),(q),(r))$ \end{tabular}\end{center} \section{Glossary of basic terms} \begin{itemize} \item A \textbf{boolean domain} $\mathbb{B}$ is a generic 2-element set, say, $\mathbb{B} = \{ 0, 1 \}$, whose elements are interpreted as logical values, usually but not invariably with 0 = \textit{false} and 1 = \textit{true}. \item A \textbf{boolean variable} $x$ is a variable that takes its value from a boolean domain, as $x \in \mathbb{B}$. \item 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 \textbf{proposition}. \item 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 $x_j : \mathbb{B}^k \to \mathbb{B}$. \item This means that the $k$ objects $x_j$ for $j$ = $1$ to $k$ are just so many boolean functions $x_j : \mathbb{B}^k \to \mathbb{B}$, subject to logical interpretation as a set of \textit{basic propositions} that generate the complete set of $2^{2^k}$ propositions over $\mathbb{B}^k$. \item A \textbf{literal} is one of the $2k$ propositions $x_1, \ldots, x_k, (x_1), \ldots, (x_k)$, in other words, either a \textit{posited} basic proposition $x_j$ or a \textit{negated} basic proposition $(x_j)$, for some $j$ = $1$ to $k$. \item In mathematics generally, the \textbf{fiber} of a point $y$ under a function $f : X \to Y$ is defined as the inverse image $f^{-1}(y)$. \item In the case of a boolean-valued function $f : X \to \mathbb{B}$, there are just two fibers: \begin{quote} 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. \end{quote} \item When 1 is interpreted as the logical value \textit{true}, then $f^{-1}(1)$ is called the \textbf{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 $f$. \item A \textbf{singular boolean function} $s : \mathbb{B}^k \to \mathbb{B}$ is a boolean function whose fiber of 1 is a single point of $\mathbb{B}^k$. \item In the interpretation where 1 equals \textit{true}, a singular boolean function is called a \textbf{singular proposition}. \item Singular boolean functions and singular propositions serve as functional or logical representatives of the points in $\mathbb{B}^k$. \item A \textbf{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 $j$ = $1$ to $k$. \item A singular proposition $s : \mathbb{B}^k \to \mathbb{B}$ can be expressed as a singular conjunction: \begin{quote}$\begin{array}{lll} s & = & e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k \\ \text{where} & e_j & = x_j \\ \text{or} & e_j & = \nu(x_j) \\ \text{for} & j & = 1 ~\text{to}~ k \end{array}$\end{quote} \end{itemize}