Changes

If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate.  This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''.  In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.  It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''threshold operator'', or ''least action operator'', to name but a few.

If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate.  This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''.  In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.  It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''threshold operator'', or ''least action operator'', to name but a few.

In what follows, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math> In this context, the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2. This has the following consequences:

The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2.  Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math>  This has the following consequences:

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