A Generalized Next-Closure Algorithm – Enumerating Semilattice Elements from a Generating Set

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A Generalized Next-Closure Algorithm – Enumerating Semilattice Elements from a Generating Set

Daniel Borchmann

TU Dresden, Institute of Algebra daniel.borchmann@mailbox.tu-dresden.de

Málaga, 14. October 2012

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Motivation

Reminder

The Next-Closure algorithm successively computes all closed sets of a closure operatorcon a finite setM.

Definition (Closure Operators on Sets)

LetM be a set. Thenc: P

(

M

)

ÝÑP

(

M

)

is called aclosure operatoronMif and only if

cisextensive, i. e.AĎ^c

(

A

)

for allAĎ^M

cismonotone, i. e.AĎBĎMimpliesc

(

A

)

Ďc

(

B

)

cisidempotent, i. e.c

(

c

(

A

)) =

Afor allAĎM.

Daniel Borchmann Enumerating Semilattice Elements Málaga, 14. October 2012 2 / 13

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Motivation

Reminder

The Next-Closure algorithm successively computes all closed sets of a closure operatorcon a finite setM.

Definition (Closure Operators on Sets)

LetM be a set. Thenc: P

(

M

)

_ÝÑP

(

M

)

is called aclosure operatoronMif and only if

cisextensive, i. e.AĎ^c

(

A

)

for allAĎ^M

cismonotone, i. e.AĎBĎMimpliesc

(

A

)

Ďc

(

B

)

cisidempotent, i. e.c

(

c

(

A

)) =

Afor allAĎM.

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Goal

Problem

How to enumerate things that are not closure operators on sets?

closure operators in a fuzzy setting? closure operators on ordered sets? Goal

Generalize Next-Closure to arbitrary closure operatorsonce and for all.

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Goal

Problem

closure operators in a fuzzy setting?

closure operators on ordered sets? Goal

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Goal

Problem

closure operators on ordered sets?

Goal

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Goal

Problem

closure operators on ordered sets?

Goal

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Next-Closure

Definition (Lectic Order)

Letăbe a strict linear order onM,iPMandA,BĎM.

ThenAăi Bif and only if

i

=

min_ď

(

Az^BY^Bz^A

)

andi P^B. Furthermore,Aă^Bif and only ifAăi Bfor somei P^M. Definition

Leti P^M,^AĎ^M^{such that}^c

(

A

) =

A. Then define A‘ⁱ :

=

c

(

_tj P^A|^j ăⁱu Y tⁱu

)

.

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Next-Closure

Letăbe a strict linear order onM,iPMandA,BĎM. ThenAăi Bif and only if

i

=

min_ď

(

Az^BY^Bz^A

)

andi P^B.

Furthermore,Aă^Bif and only ifAăi Bfor somei P^M. Definition

(

A

) =

=

c

(

)

.

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Next-Closure

i

=

min_ď

(

Az^BY^Bz^A

)

andi P^B. Furthermore,Aă^Bif and only ifAăi Bfor somei P^M.

Definition

(

A

) =

=

c

(

)

.

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Next-Closure

i

=

min_ď

(

Az^BY^Bz^A

)

andi P^B. Furthermore,Aă^Bif and only ifAăi Bfor somei P^M.

Definition

(

A

) =

=

c

(

)

.

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Next-Closure

Theorem (Next-Closure)

Let A‰M be such that c

(

A

) =

A. Define

A⁺ :

=

min_ăt^BĎ^M|^c

(

B

) =

B,Aă^Bu.

Then

A⁺

=

A‘ⁱ where i is maximal with Aăi A‘^i.

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Next-Closure

Theorem (Next-Closure)

Let A‰M be such that c

(

A

) =

A. Define

A⁺ :

=

min_ăt^BĎ^M|^c

(

B

) =

B,Aă^Bu.

Then

A⁺

=

A‘ⁱ where i is maximal with Aăi A‘^i.

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Generalizing Next-Closure

Now consider the expressionA‘ⁱ in more detail:

A‘ⁱ

=

c

(

t^j P^A|^j ăⁱu Y tⁱu

)

=

c

(

c

(

_tj P^A|^j ăⁱu

)

_Yc

(

_tiu

))

=

:c

(

tj PA|j ăiu

)

_c

(

tiu

)

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

Observation

We can solely work in thesemilattice

(

im

(

c

)

,_

)

of all closed sets ofc! Goal

Generalize Next-Closure to work onarbitrary, abstractly given semilattices.

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Generalizing Next-Closure

A‘ⁱ

=

c

(

)

=

c

(

c

(

_tj P^A|^j ăⁱu

)

_Yc

(

_tiu

))

=

:c

(

tj PA|j ăiu

)

_c

(

tiu

)

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

Observation

(

im

(

c

)

,_

)

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Generalizing Next-Closure

A‘ⁱ

=

c

(

)

=

c

(

c

(

_tj P^A|^j ăⁱu

)

_Yc

(

_tiu

))

=

:c

(

tj PA|jăiu

)

_c

(

tiu

)

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

Observation

(

im

(

c

)

,_

)

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Generalizing Next-Closure

A‘ⁱ

=

c

(

)

=

c

(

c

(

_tj P^A|^j ăⁱu

)

_Yc

(

_tiu

))

=

:c

(

tj PA|jăiu

)

_c

(

tiu

)

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

Observation

(

im

(

c

)

,_

)

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Generalizing Next-Closure

A‘ⁱ

=

c

(

)

=

c

(

c

(

_tj P^A|^j ăⁱu

)

_Yc

(

_tiu

))

=

:c

(

tj PA|jăiu

)

_c

(

tiu

)

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

Observation

(

im

(

c

)

,_

)

of all closed sets ofc!

Goal

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Generalizing Next-Closure

A‘ⁱ

=

c

(

)

=

c

(

c

(

_tj P^A|^j ăⁱu

)

_Yc

(

_tiu

))

=

:c

(

tj PA|jăiu

)

_c

(

tiu

)

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

Observation

(

im

(

c

)

,_

)

of all closed sets ofc!

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Generalizing Next-Closure

Plan

Things we have to generalize:

prerequisites (setting) lectic orders

A‘i

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Generalizing Next-Closure

Plan

prerequisites (setting)

lectic orders A‘i

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Generalizing Next-Closure

Plan

A‘i

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Generalizing Next-Closure

Plan

A‘i

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Generalizing Next-Closure – The Setting

Let

(

L,ďL

)

be a finitesemilattice, i. e.

(

L,ďL

)

is a finite ordered set and

for allx,yP^Lexists a least upper boundx_^y^of^x^and^y.

Lett^x1,. . .,x_nu Ď^Lbe a generating set of

(

L,ďL

)

, i. e. for eachyP^L^{it is} true that

y

=

^ł

xiďLy

x_i.

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Generalizing Next-Closure – The Setting

Let

(

L,ďL

)

(

L,ďL

)

for allx,yP^Lexists a least upper boundx_^yofxandy.

(

L,ďL

)

y

=

^ł

xiďLy

x_i.

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Generalizing Next-Closure – The Setting

Let

(

L,ďL

)

(

L,ďL

)

(

L,ďL

)

,

i. e. for eachyP^L^{it is} true that

y

=

^ł

xiďLy

x_i.

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Generalizing Next-Closure – The Setting

Let

(

L,ďL

)

(

L,ďL

)

(

L,ďL

)

y

=

^ł

xiďLy

x_i.

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Generalizing Next-Closure – Lectic Orders

Recall

Aăi B ðñ i

=

min_ă

(

AzBYBzA

)

andtiu ĎB.

Definition

Let 1ďⁱ ďⁿ^and^a,bPL. Then define

aăi b :ðñ i

=

min∆a,b andx_i ďLb, where

∆a,b:

=

tⁱ |

(

x_i ďLaandx_i ęLb

)

or

(

x_i ęL aandx_i ďL b

)

u.

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Generalizing Next-Closure – Lectic Orders

Recall

Aăi B ðñ i

=

min_ă

(

AzBYBzA

)

andtiu ĎB. Definition

aăi b :ðñ i

=

min∆a,b andx_i ďLb,

where

∆a,b:

=

tⁱ |

(

x_i ďLaandx_i ęLb

)

or

(

)

u.

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Generalizing Next-Closure – Lectic Orders

Recall

Aăi B ðñ i

=

min_ă

(

AzBYBzA

)

andtiu ĎB. Definition

aăi b :ðñ i

=

min∆a,b andx_i ďLb, where

∆a,b:

=

tⁱ |

(

x_i ďLaandx_i ęLb

)

or

(

)

u.

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Generalizing Next-Closure – A ‘ i

Recall

A‘ⁱ

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

.

Definition

LetaP^L^{and 1}ďⁱ ď^{n. Define}

a‘i :

=

^ł

jăi,xjďa

x_j_x_i.

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Generalizing Next-Closure – A ‘ i

Recall

A‘ⁱ

=

^ł

jăi c(tju)ĎA

c

(

t^ju

)

_^c

(

tⁱu

)

.

Definition

LetaP^L^{and 1}ďⁱ ď^{n. Define}

a‘i :

=

^ł

jăi,xjďa

x_j _x_i.

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Generalizing Next-Closure

Theorem

Let aP^{L. Define}

a⁺ :

=

min_ăt^bP^L|^aă^bu.

Then, if this minimum exists,

a⁺

=

a‘i with i being maximal with aăi a‘i.

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Generalizing Next-Closure

Theorem

Let aP^{L. Define}

a⁺ :

=

min_ăt^bP^L|^aă^bu.

Then, if this minimum exists,

a⁺

=

a‘i with i being maximal with aăi a‘i.

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Instances of the Generalized Algorithm

Consider the case of computing intents of a formal contextK

= (

G,M,I

)

.

Original Next-Closure is Generalized Next-Closure for the semilattice

(

Int

(

_K

)

,_

)

whereA_B:

= (

AYB

)

²and generating set

tm² |mPMu Y t H²u.

One can also consider the semilattice

(

Int

(

_K

)

,X

)

with generating set t^g¹ |^gP^Gu Y t^Mu.

Yields an algorithm with a lot of similarities toClose-by-One!

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Instances of the Generalized Algorithm

= (

G,M,I

)

. Original Next-Closure is Generalized Next-Closure for the semilattice

(

Int

(

_K

)

,_

)

whereA_^B:

= (

AY^B

)

(

Int

(

_K

)

,X

)

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Instances of the Generalized Algorithm

= (

G,M,I

)

(

Int

(

_K

)

,_

)

whereA_^B:

= (

AY^B

)

(

Int

(

_K

)

,X

)

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Instances of the Generalized Algorithm

= (

G,M,I

)

(

Int

(

_K

)

,_

)

whereA_^B:

= (

AY^B

)

(

Int

(

_K

)

,X

)

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Thank You for Your Attention!

Questions Welcome!