The minimal negation operator
is a multigrade operator
where each
is a
-ary boolean function
defined in such a way that
in just those cases where exactly one of the arguments
is 0
.
In contexts where the initial letter
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,
=
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.
To express the general case of
in terms of familiar operations, it helps to introduce an intermediary concept:
Definition. Let the function
be defined for each integer
in the interval
by the following equation:
Then
is defined by the following equation:
If we think of the point
as indicated by the boolean product
or the logical conjunction
then the minimal negation
indicates the set of points in
that differ from
in exactly one coordinate. This makes
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, least action operator, or hedge operator, to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.
The remainder of this discussion proceeds on the algebraic boolean convention that the plus sign
and the summation symbol
both refer to addition modulo 2. Unless otherwise noted, the boolean domain
is interpreted so that
and
This has the following consequences:
The following properties of the minimal negation operators
may be noted:
Table 1 is a truth table for the sixteen boolean functions of type
, each of which is either a boundary of a point in
or the complement of such a boundary.
Table 1. Logical Boundaries and Their Complements | ||||
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|
Decimal | Binary | Sequential | Parenthetical | |
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1 1 1 1 0 0 0 0 | |||
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1 1 0 0 1 1 0 0 | |||
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1 0 1 0 1 0 1 0 | |||
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0 1 1 0 1 0 0 0 |
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|
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1 0 0 1 0 1 0 0 |
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|
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1 0 0 1 0 0 1 0 |
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|
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0 1 1 0 0 0 0 1 |
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|
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1 0 0 0 0 1 1 0 |
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|
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0 1 0 0 1 0 0 1 |
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|
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0 0 1 0 1 0 0 1 |
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|
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0 0 0 1 0 1 1 0 |
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|
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1 1 1 0 1 0 0 1 |
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|
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1 1 0 1 0 1 1 0 |
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|
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1 0 1 1 0 1 1 0 |
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|
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0 1 1 1 1 0 0 1 |
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|
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1 0 0 1 1 1 1 0 |
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|
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0 1 1 0 1 1 0 1 |
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|
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0 1 1 0 1 0 1 1 |
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|
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1 0 0 1 0 1 1 1 |
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This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing 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
of
points are the hypercube picture and the venn diagram picture. The hypercube picture associates each point of
with a unique point of the
-dimensional hypercube. The venn diagram picture associates each point of
with a unique "cell" of the venn diagram on
"circles".
In addition, each point of
is the unique point in the fiber of truth
of a singular proposition
, and thus it is the unique point where a singular conjunction of
literals is equal to 1.
For example, consider two cases at opposite vertices of the
-cube:
To pass from these limiting examples to the general case, observe that a singular proposition
can be given canonical expression as a conjunction of literals,
. Then the proposition
is 1 on the points adjacent to the point where
is 1, and 0 everywhere else on the cube.
For example, consider the case where
. Then the minimal negation operation
-- written more simply as
-- has the following venn diagram:
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Figure 2.
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For a contrasting example, the boolean function expressed by the form
has the following venn diagram:
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Figure 3.
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The fiber of 0 under, defined as
, is the set of points where
is 0.
The fiber of 1 under, defined as
, is the set of points where
is 1.
As of this snapshot date, this entry was owned by Jon Awbrey.