minimal negation operator
primes \PMlinkescapephrasePrimes \PMlinkescapephrasesingular \PMlinkescapephraseSingular
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, (x, y, z) =
To express the general case of in terms of familiar operations, it helps to introduce an intermediary concept:
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 operation maps the bit sequence to its parity.
The following properties of the minimal negation operators may be noted:
The function (x, y) is the same as that associated with the operation and the relation
In contrast, (x, y, z) is not identical to
More generally, the function for is not identical to the boolean sum
1 Truth tables
|Table 1. Logical Boundaries and Their Complements|
|1 1 1 1 0 0 0 0|
|1 1 0 0 1 1 0 0|
|1 0 1 0 1 0 1 0|
|0 1 1 0 1 0 0 0||( p , q , r )|
|1 0 0 1 0 1 0 0||( p , q ,(r))|
|1 0 0 1 0 0 1 0||( p ,(q), r )|
|0 1 1 0 0 0 0 1||( p ,(q),(r))|
|1 0 0 0 0 1 1 0||((p), q , r )|
|0 1 0 0 1 0 0 1||((p), q ,(r))|
|0 0 1 0 1 0 0 1||((p),(q), r )|
|0 0 0 1 0 1 1 0||((p),(q),(r))|
|1 1 1 0 1 0 0 1||(((p),(q),(r)))|
|1 1 0 1 0 1 1 0||(((p),(q), r ))|
|1 0 1 1 0 1 1 0||(((p), q ,(r)))|
|0 1 1 1 1 0 0 1||(((p), q , r ))|
|1 0 0 1 1 1 1 0||(( p ,(q),(r)))|
|0 1 1 0 1 1 0 1||(( p ,(q), r ))|
|0 1 1 0 1 0 1 1||(( p , q ,(r)))|
|1 0 0 1 0 1 1 1||(( p , q , r ))|
2 Charts and graphs
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”.
For example, consider two cases at opposite vertices of the -cube:
The point with all 1’s as coordinates is the point where the conjunction of all posited variables evaluates to 1, namely, the point where:
The point with all 0’s as coordinates is the point where the conjunction of all negated variables evaluates to 1, namely, the point where:
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 (p, q, r) — has the following venn diagram:
|Figure 2. (p, q, r)|
For a contrasting example, the boolean function expressed by the form ((p),(q),(r)) has the following venn diagram:
|Figure 3. ((p),(q),(r))|
3 Glossary of basic terms
A boolean variable is a variable that takes its value from a boolean domain, as .
In situations where boolean values are interpreted as logical values, a boolean-valued function or a boolean function is frequently called a proposition.
Given a sequence of boolean variables, , each variable may be treated either as a basis element of the space or as a coordinate projection .
A literal is one of the propositions , in other words, either a posited basic proposition or a negated basic proposition , for some = to .
In mathematics generally, the fiber of a point under a function is defined as the inverse image .
In the case of a boolean-valued function , there are just two fibers:
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.
When 1 is interpreted as the logical value true, then is called the fiber of truth in the proposition . 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 for the fiber of truth in the proposition .
A singular boolean function is a boolean function whose fiber of 1 is a single point of .
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 .
A singular conjunction in is a conjunction of literals that includes just one conjunct of the pair for each = to .
A singular proposition can be expressed as a singular conjunction:
|Title||minimal negation operator|
|Date of creation||2013-03-22 17:48:13|
|Last modified on||2013-03-22 17:48:13|
|Owner||Jon Awbrey (15246)|
|Last modified by||Jon Awbrey (15246)|
|Author||Jon Awbrey (15246)|