minimal negation operator

minimal negation operator

\PMlinkescapephrase

primes

\PMlinkescapephrase

Primes

\PMlinkescapephrase

singular

\PMlinkescapephrase

Singular

The

minimal negation operator

ν

is a

multigrade operator

(

ν

k

)

k

∈

ℕ

where each

ν

k

is a

k

-ary

boolean function

defined in such a way that

ν

k

⁢

(

x

1

,

…

,

x

k

)

=

1

in just those cases where exactly one of the

arguments

x

j

is

0

.

In

contexts

where the initial letter

ν

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)

=

ν

⁢

(

x

,

y

,

z

)

.

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

.

()

=

ν

0

=

0

=

false

(x)

=

ν

1

⁢

(

x

)

=

x

~

=

x

′

(x, y)

=

ν

2

⁢

(

x

,

y

)

=

x

~

⁢

y

∨

x

⁢

y

~

=

x

′

⁢

y

∨

x

⁢

y

′

(x, y, z)

=

ν

3

⁢

(

x

,

y

,

z

)

=

x

~

⁢

y

⁢

z

∨

x

⁢

y

~

⁢

z

∨

x

⁢

y

⁢

z

~

=

x

′

⁢

y

⁢

z

∨

x

⁢

y

′

⁢

z

∨

x

⁢

y

⁢

z

′

To express the general case of

ν

k

in terms of familiar

operations

, it helps to introduce an intermediary concept:

Definition

.

Let the

function

¬

j

:

𝔹

k

→

𝔹

be defined for each integer

j

in the interval

[

1

,

k

]

by the following

equation

:

¬

j

⁡

(

x

1

,

…

,

x

j

,

…

,

x

k

)

=

x

1

∧

…

∧

x

j

-

1

∧

¬

⁢

x

j

∧

x

j

+

1

∧

…

∧

x

k

.

Then

ν

k

:

𝔹

k

→

𝔹

is defined by the following equation:

ν

k

⁢

(

x

1

,

…

,

x

k

)

=

¬

1

⁡

(

x

1

,

…

,

x

k

)

∨

…

∨

¬

j

⁡

(

x

1

,

…

,

x

k

)

∨

…

∨

¬

k

⁡

(

x

1

,

…

,

x

k

)

.

If we think of the point

x

=

(

x

1

,

…

,

x

k

)

∈

𝔹

k

as indicated by the

boolean

product

x

1

⋅

…

⋅

x

k

or the logical conjunction

x

1

∧

…

∧

x

k

,

then the

minimal

negation

(

⁢

x

1

,

…

,

x

k

⁢

)

indicates the set of points in

𝔹

k

that differ from

x

in exactly one coordinate. This makes

(

⁢

x

1

,

…

,

x

k

⁢

)

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

𝔹

=

{

0

,

1

}

is interpreted so that

0

=

false

and

1

=

true

.

This has the following

consequences

:

•

The operation

x

+

y

is a function

equivalent

to the

exclusive disjunction

of

x

and

y

,

while its fiber of

1

is the

relation

of inequality between

x

and

y

.

•

The operation

∑

j

=

1

k

x

j

maps the bit

sequence

(

x

1

,

…

,

x

k

)

to its

parity

.

The following

properties

of the minimal negation operators

ν

k

:

𝔹

k

→

𝔹

may be noted:

•

The function

(x, y)

is the same as that associated with the operation

x

+

y

and the relation

x

≠

y

.

•

In contrast,

(x, y, z)

is not identical to

x

+

y

+

z

.

•

More generally, the function

ν

k

⁢

(

x

1

,

…

,

x

k

)

for

k

>

2

is not identical to the boolean sum

∑

j

=

1

k

x

j

.

•

The

inclusive disjunctions

indicated for the

ν

k

of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already

disjoint

.

1

Truth tables

Table 1 is a

truth table

for the sixteen boolean functions of type

f

:

𝔹

3

→

𝔹

, each of which is either a boundary of a point in

𝔹

3

or the

complement

of such a boundary.

Table 1. Logical Boundaries and Their Complements

L

1

L

2

L

3

L

4

Decimal

Binary

Sequential

Parenthetical

p

=

1 1 1 1 0 0 0 0

q

=

1 1 0 0 1 1 0 0

r

=

1 0 1 0 1 0 1 0

f

104

f

01101000

0 1 1 0 1 0 0 0

( p , q , r )

f

148

f

10010100

1 0 0 1 0 1 0 0

( p , q ,(r))

f

146

f

10010010

1 0 0 1 0 0 1 0

( p ,(q), r )

f

97

f

01100001

0 1 1 0 0 0 0 1

( p ,(q),(r))

f

134

f

10000110

1 0 0 0 0 1 1 0

((p), q , r )

f

73

f

01001001

0 1 0 0 1 0 0 1

((p), q ,(r))

f

41

f

00101001

0 0 1 0 1 0 0 1

((p),(q), r )

f

22

f

00010110

0 0 0 1 0 1 1 0

((p),(q),(r))

f

233

f

11101001

1 1 1 0 1 0 0 1

(((p),(q),(r)))

f

214

f

11010110

1 1 0 1 0 1 1 0

(((p),(q), r ))

f

182

f

10110110

1 0 1 1 0 1 1 0

(((p), q ,(r)))

f

121

f

01111001

0 1 1 1 1 0 0 1

(((p), q , r ))

f

158

f

10011110

1 0 0 1 1 1 1 0

(( p ,(q),(r)))

f

109

f

01101101

0 1 1 0 1 1 0 1

(( p ,(q), r ))

f

107

f

01101011

0 1 1 0 1 0 1 1

(( p , q ,(r)))

f

151

f

10010111

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

𝔹

k

of

2

k

points are the

hypercube

picture and the

venn diagram

picture. The hypercube picture associates each point of

𝔹

k

with a unique point of the

k

-dimensional hypercube. The venn diagram picture associates each point of

𝔹

k

with a unique ”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 equal to 1.

For example, consider two cases at opposite vertices of the

k

-cube:

•

The point

(

1

,

1

,

…

,

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

⁢

…

⁢

x

k

-

1

⁢

x

k

=

1

•

The point

(

0

,

0

,

…

,

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

⁢

)

⁢

…

⁢

(

⁢

x

k

-

1

⁢

)(

⁢

x

k

⁢

)

=

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

⁢

…

⁢

e

k

-

1

⁢

e

k

. Then the proposition

ν

⁢

(

e

1

,

e

2

,

…

,

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

ν

⁢

(

p

,

q

,

r

)

— 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 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-valued function

f

:

X

→

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

,

ν

⁢

(

x

j

)

}

for each

j

=

1

to

k

.

•

A singular proposition

s

:

𝔹

k

→

𝔹

can be expressed as a singular conjunction:

s

=

e

1

⁢

e

2

⁢

…

⁢

e

k

-

1

⁢

e

k

where

e

j

=

x

j

or

e

j

=

ν

⁢

(

x

j

)

for

j

=

1

⁢

to

⁢

k

Title

minimal negation operator

Canonical name

MinimalNegationOperator

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)

Numerical id

42

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 08A70

Classification

msc 08A40

Classification

msc 39A70

Classification

msc 39A12

Classification

msc 03G05

Classification

msc 03E20

Classification

msc 03C05

Classification

msc 03B05

Related topic

DifferentialLogic

Related topic

DifferentialPropositionalCalculus

Related topic

DifferentialPropositionalCalculusAppendices

Related topic

DifferentialPropositionalCalculusAppendix2

Related topic

DifferentialPropositionalCalculusAppendix3

Related topic

DifferentialPropositionalCalculusAppendix4

Related topic

PropositionalCalculus

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