Minimal negation operator

In logic and mathematics, the minimal negation operator $u!$ is a multigrade operator $(\; u\_\{k\})\_\{k\; in\; mathbb\{N$ where each $u\_\{k\}!$ is a "k"-ary boolean function defined in such a way that $u\_\{k\}(x\_1,\; ldots\; ,\; x\_k)\; =\; 1$ if and only if exactly one of the arguments $x\_\{j\}$ is 0.

In contexts where the initial letter $u!$ 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.

:

It may also be noted that $(x,\; y)!$ is the same function as $x\; +\; y!$ and $x\; e\; y$, and that the inclusive disjunctions indicated for $(x,\; y)!$ and for $(x,\; y,\; z)!$ may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function $(x,\; y,\; z)!$ is not the same thing as the function $x\; +\; y\; +\; z!$.

The minimal negation operator (mno) has a legion of aliases: "logical boundary operator", "limen operator", "threshold operator", or "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.

The next section discusses 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 already be familiar to many. As a result, the full definitions of the terms marked in bold are relegated to a Glossary at the end of the article.

Truth tables

Table 1 is a truth table for the sixteen boolean functions of type f : B³ → B, each of which is either a boundary of a point in B³ or the complement of such a boundary.

Charts and graphs

Two common ways of visualizing the space B^"k" of 2^"k" points are the hypercube picture and the venn diagram picture. Depending on how literally or figuratively one regards these pictures, each point of B^k is either identified with or represented by a point of the "k"-cube and also by a cell of the venn diagram on "k" "circles".

In addition, each point of B^k is the unique point in the fiber of truth $[|s|]$ of a singular proposition "s" : B^"k" → B, and thus it is the unique point where a singular conjunction of "k" literals is 1.

For example, consider two cases at opposite vertices of the cube:

* The point whose coordinates are all 1 is the unique point where the conjunction of all posited variables $x\_1\; x\_2\; ldots\; x\_\{n-1\}\; x\_n$ is 1.

* The point whose coordinates are all 0 is the unique point where the conjunction of all negated variables $(x\_1)(x\_2)ldots(x\_\{n-1\})(x\_n)$ is 1.

To pass from these limiting examples to the general case, observe that a singular proposition "s" : B^"k" → 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 $u\; (e\_1,\; e\_2,\; ldots,\; e\_\{k-1\},\; e\_k)$ 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 $u\; (p,\; q,\; r)!$, when there is no risk of confusion written more simply as $(p,\; q,\; r)!$, has the following Venn diagram:

:

For a contrasting example, the boolean function expressed by the form $((p),(q),(r))!$ has the following Venn diagram:

:

Glossary of basic terms

* A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, usually but not invariably with 0 = "false" and "1" = "true".

* A boolean variable "x" is a variable that takes its value from a boolean domain, as "x" ∈ B.

* In situations where boolean values are interpreted as logical values, a boolean-valued function "f" : "X" → B or a boolean function "g" : B^"k" → B is frequently called a proposition.

* Given a sequence of "k" boolean variables, "x"₁, …, "x"_"k", each variable "x"_"j" may be treated either as a basis element of the space B^"k" or as a coordinate projection "x"_"j" : B^"k" → B.

* This means that the "k" objects "x"_"j" for "j" = 1 to "k" are just so many boolean functions "x"_"j" : B^"k" → B , subject to logical interpretation as a set of "basic propositions" that generate the complete set of $2^\{2^k\}$ propositions over B^"k".

* A literal is one of the 2"k" propositions "x"₁, …, "x"_"k", ("x"₁), …, ("x"_"k"), in other words, either a "posited" basic proposition "x"_"j" or a "negated" basic proposition ("x"_"j"), for some "j" = 1 to "k".

* In mathematics generally, the fiber of a point "y" under a function "f" : "X" → "Y" is defined as the inverse image $f^\{-1\}(y)$.

* In the case of a boolean function "f" : B^"k" → B, there are just two fibers:
** 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.

* When 1 is interpreted as the logical value "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".

* A singular boolean function "s" : B^"k" → B is a boolean function whose fiber of 1 is a single point of B^"k".

* 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 B^"k".

* A singular conjunction in B^"k" → B is a conjunction of "k" literals that includes just one conjunct of the pair $\{\; x\_j,\; u\; (x\_j)\; \}$ for each "j" = 1 to "k".

* A singular proposition "s" : B^"k" → B can be expressed as a singular conjunction:
:

References

ee also

* Ampheck
* Anamnesis
* Boolean algebras canonically defined
* Boole, George
* Boolean algebra (logic)
* Boolean function
* Boolean logic
* Boolean-valued function
* Continuous predicate
* Differentiable manifold
* Differential topology
* Dynamical system
* Exclusive disjunction
* Leibniz, G.W.
* Logical connective
* Logical graph
* Meno
* Multigrade operator
* Parametric operator
* Peirce, C.S.
* Sole sufficient operator
* Truth table
* Universal algebra

External links

* [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs]
** [http://atlas.wolfram.com/01/01/ ECARs (Elementary Cellular Automata Rules)]
** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index]
** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Rule Icons]
** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples]
** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Boolean Forms]
* [http://www.flickr.com/photos/hexadecimal_time/2875084959/ Minimal negation operations in Venn and Hasse diagrams]