relation reduction

relation reduction

Relation

reduction

and

relational reducibility

have to do with the extent to which a given relation is determined by an indexed family or a

sequence

of other relations, called the

relation dataset

. The relation under examination is called the

reductand

. The relation dataset typically consists of a specified set of other relations, simpler in some measure than the reductand, called the

reduction base

, plus a specified

operation on relations

, called the

reduction method

or the

reduction step

.

A question of relation reduction or relational reducibility is sometimes posed as a question of

relation reconstruction

or

relational reconstructibility

, since a useful way of stating the question is to ask whether the reductand can be reconstructed from the relational dataset.

A relation that is not uniquely determined by a particular relation dataset is said to be irreducible in just that respect. A relation that is not uniquely determined by any relation dataset in a particular class of relation datasets is said to be irreducible in respect of that class.

Contents:

1

Discussion

2

Projective reducibility of relations

3

Projective reducibility of triadic relations

3.1

Examples of projectively irreducible relations

3.2

Examples of projectively reducible relations

3.3

Summary

1

Discussion

The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of “getting new relations from old” in order to say precisely what is meant by the claim that the relation

L

is reducible to the set of relations

{

L

j

:

j

∈

J

}

. This is tantamount to claiming that if one is given a set of “simpler” relations

L

j

, for indices

j

in some set

J

, that this

collection

of data would somehow or other fix the original relation

L

that one is seeking to analyze, to determine, to specify, or to synthesize.

In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely:

•

Reduction under

composition

(

http://planetmath.org/RelationComposition

).

•

Reduction under

projections

.

As it happens, there is an interesting relationship between these two notions of reducibility, the

implications

of which may be taken up partly in

parallel

with the discussion of the basic

concepts

.

2

Projective reducibility of relations

It is convenient to begin with the

projective reduction

of relations, partly because this type of reduction is simpler and more intuitive (in the visual sense), but also because a number of conceptual tools that are needed in any case arise quite naturally in the projective setting.

The work of intuiting how projections operate on multidimensional relations is often facilitated by keeping in mind the following sort of geometric image:

•

Picture a

k

-adic relation

L

as a body that resides in a

k

-dimensional space

X

. If the domains of the relation

L

are

X

1

,

…

,

X

k

, then the

extension

of the relation

L

is a subset of the

cartesian product

X

=

X

1

×

…

×

X

k

.

In this setting, the

interval

K

=

[

1

,

k

]

=

{

1

,

…

,

k

}

is called the

index set

of the

indexed family

of sets

X

1

,

…

,

X

k

.

For any subset

F

of the index set

K

, there is the corresponding subfamily of sets,

{

X

j

:

j

∈

F

}

, and there is the corresponding cartesian product over this subfamily, notated and defined as

X

F

=

∏

j

∈

F

X

j

.

For any point

x

in

X

, the

projection

of

x

on the

subspace

X

F

is notated as

proj

F

⁡

(

x

)

, or still more simply as

proj

F

⁡

x

.

More generally, for any relation

L

⊆

X

, the projection of

L

on the subspace

X

F

is written as

proj

F

⁡

(

L

)

, or still more simply as

proj

F

⁡

L

.

The question of

projective reduction

for

k

-adic relations can be stated with moderate generality in the following way:

•

Given a set of

k

-place relations in the same space

X

and a set of projections from

X

to the associated subspaces, do the projections afford

sufficient

data to tell the different relations apart?

3

Projective reducibility of triadic relations

See main entry (

http://planetmath.org/TriadicRelation

) on

triadic relations

.

By way of illustrating the different sorts of things that can

occur in

considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main entry on that subject.

3.1

Examples of projectively irreducible relations

The 3-adic relations

L

0

and

L

1

are shown in the next two Tables:

L

0

=

{

(

x

,

y

,

z

)

∈

𝔹

3

:

x

+

y

+

z

=

0

}

X

Y

Z

0

0

0

0

1

1

1

0

1

1

1

0

L

1

=

{

(

x

,

y

,

z

)

∈

𝔹

3

:

x

+

y

+

z

=

1

}

X

Y

Z

0

0

1

0

1

0

1

0

0

1

1

1

Viewed in terms of

operations

on relational data tables, a 2-adic projection of a 3-adic relation

L

is the 2-adic relation that results from deleting one column of the table for

L

and then deleting all but one row of any resulting rows that happen to be identical in

content

. In other words, the

multiplicity

of any repeated row is ignored.

In the case of the above two relations,

L

0

,

L

1

⊆

X

×

Y

×

Z

≅

𝔹

3

, the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.

proj

X

⁢

Y

⁡

L

0

proj

X

⁢

Z

⁡

L

0

proj

Y

⁢

Z

⁡

L

0

X

Y

X

Z

Y

Z

0

0

0

0

0

0

0

1

0

1

1

1

1

0

1

1

0

1

1

1

1

0

1

0

proj

X

⁢

Y

⁡

L

1

proj

X

⁢

Z

⁡

L

1

proj

Y

⁢

Z

⁡

L

1

X

Y

X

Z

Y

Z

0

0

0

1

0

1

0

1

0

0

1

0

1

0

1

0

0

0

1

1

1

1

1

1

It is clear on inspection that the following three equations hold:

proj

X

⁢

Y

⁡

L

0

=

proj

X

⁢

Y

⁡

L

1

,

proj

X

⁢

Z

⁡

L

0

=

proj

X

⁢

Z

⁡

L

1

,

proj

Y

⁢

Z

⁡

L

0

=

proj

Y

⁢

Z

⁡

L

1

.

These equations say that

L

0

and

L

1

cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, either relation is said to be

irreducible with respect to 2-adic projections

. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is

projectively irreducible

, the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations.

3.2

Examples of projectively reducible relations

The 3-adic

sign relations

L

A

and

L

B

are shown in the next two Tables:

L

A

= Sign Relation of Interpreter A

Object

Sign

Interpretant

A

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

i

⁢

"

A

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

i

⁢

"

B

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

B

⁢

"

B

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

u

⁢

"

B

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

B

⁢

"

B

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

u

⁢

"

L

B

= Sign Relation of Interpreter B

Object

Sign

Interpretant

A

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

u

⁢

"

A

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

u

⁢

"

B

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

B

⁢

"

B

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

i

⁢

"

B

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

B

⁢

"

B

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

i

⁢

"

In the case of the two sign relations,

L

A

,

L

B

⊆

X

×

Y

×

Z

≅

O

×

S

×

I

, the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.

proj

O

⁢

S

⁡

L

A

proj

O

⁢

I

⁡

L

A

proj

S

⁢

I

⁡

L

A

Object

Sign

Object

Interpretant

Sign

Interpretant

A

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

i

⁢

"

A

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

i

⁢

"

B

`

⁢

`

⁢

B

⁢

"

B

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

A

⁢

"

B

`

⁢

`

⁢

u

⁢

"

B

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

u

⁢

"

proj

O

⁢

S

⁡

L

B

proj

O

⁢

I

⁡

L

B

proj

S

⁢

I

⁡

L

B

Object

Sign

Object

Interpretant

Sign

Interpretant

A

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

A

⁢

"

A

`

⁢

`

⁢

u

⁢

"

A

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

A

⁢

"

`

⁢

`

⁢

u

⁢

"

B

`

⁢

`

⁢

B

⁢

"

B

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

A

⁢

"

B

`

⁢

`

⁢

i

⁢

"

B

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

B

⁢

"

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

i

⁢

"

It is clear on inspection that the following three inequations hold:

proj

O

⁢

S

⁡

L

A

≠

proj

O

⁢

S

⁡

L

B

,

proj

O

⁢

I

⁡

L

A

≠

proj

O

⁢

I

⁡

L

B

,

proj

S

⁢

I

⁡

L

A

≠

proj

S

⁢

I

⁡

L

B

.

These inequations say that

L

A

and

L

B

can be distinguished from each other solely on the basis of their 2-adic projection data. But this is not enough to say that either one of them is projectively reducible to their 2-adic projection data. To say that a 3-adic relation is projectively reducible in that respect, one has to show that it can be distinguished from

every

other 3-adic relation on the basis of the 2-adic projection data alone.

In other words, to show that a 3-adic relation

L

on

O

×

S

×

I

is

reducible

or

reconstructible

in the 2-adic projective sense, it is necessary to show that no distinct

L

′

on

O

×

S

×

I

exists such that

L

and

L

′

have the same set of projections. Proving this takes a much more comprehensive or exhaustive investigation of the space of possible relations on

O

×

S

×

I

than looking merely at one or two relations at a time.

Fact.

As it happens, each of the relations

L

A

and

L

B

is uniquely determined by its 2-adic projections. This can be seen by following the proof that is given below.

Before tackling the proof, however, it will speed things along to recall a few ideas and notations from other articles.

•

If

L

is a relation over a set of domains that includes the domains

U

and

V

, then the abbreviated notation

L

U

⁢

V

can be used for the projection

proj

U

⁢

V

⁡

L

.

•

The operation of reversing a projection asks what elements of a bigger space

project

onto given elements of a smaller space. The set of

elements

that project onto

x

under a given projection

f

is called the

fiber

of

x

under

f

and is written

f

-

1

⁢

(

x

)

or

f

-

1

⁢

x

.

•

If

X

is a

finite set

, the

cardinality

of

X

, written

card

⁡

(

X

)

or

|

X

|

, means the number of elements in

X

.

Proof.

Let

L

be either one of the relations

L

A

or

L

B

. Consider any

coordinate

position

(

s

,

i

)

in the

S

⁢

I

-plane

S

×

I

. If

(

s

,

i

)

is not in

L

S

⁢

I

then there can be no element

(

o

,

s

,

i

)

in

L

, therefore we may restrict our attention to positions

(

s

,

i

)

in

L

S

⁢

I

, knowing that there exist at least

|

L

S

⁢

I

|

=

8

elements in

L

, and seeking only to determine what objects

o

exist such that

(

o

,

s

,

i

)

is an element in the fiber of

(

s

,

i

)

. In other words, for what

o

in

O

is

(

o

,

s

,

i

)

in the fiber

proj

S

⁢

I

-

1

⁡

(

s

,

i

)

? Now, the circumstance that

L

O

⁢

S

has exactly one element

(

o

,

s

)

for each coordinate

s

in

S

and that

L

O

⁢

I

has exactly one element

(

o

,

i

)

for each coordinate

i

in

I

, plus the “coincidence” of it being the same

o

at any one choice for

(

s

,

i

)

, tells us that

L

has just the one element

(

o

,

s

,

i

)

over each point of

S

×

I

. All together, this proves that both

L

A

and

L

B

are reducible in an informative sense to 3-tuples of 2-adic relations, that is, they are

projectively 2-adically reducible

.

3.3

Summary

The

projective

analysis

of 3-adic relations, illustrated by means of concrete examples, has been pursued just far enough at this point to state this clearly demonstrated result:

•

Some 3-adic relations are, and other 3-adic relations are not, reducible to, or reconstructible from, their 2-adic projection data. In short, some 3-adic relations are projectively reducible and some 3-adic relations are projectively irreducible.

Title

relation reduction

Canonical name

RelationReduction

Date of creation

2013-10-24 16:56:16

Last modified on

2013-10-24 16:56:16

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

36

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 68R01

Classification

msc 68P15

Classification

msc 08A02

Classification

msc 05C65

Classification

msc 05B30

Classification

msc 05B20

Classification

msc 03B10

Classification

msc 03E20

Synonym

relation identification

Synonym

relation reconstruction

Related topic

RelationTheory

Related topic

RelationComposition2

Related topic

RelationConstruction

Related topic

TacitExtension

Defines

relational reducibility

Defines

relational irreducibility

Defines

reductand

Defines

reduction base

Defines

reduction method

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