General Concept Inclusions with High Confidence in Finite Interpretations

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General Concept Inclusions with High Confidence in Finite Interpretations

Daniel Borchmann

QuantLA Summer Workshop 2013

2013-09-12

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Motivation The Ultimate Goal

Goal

Use description logic ontologies to represent knowledge of certain domains

Problem

How to obtain these ontologies? Approach

Learn ontologies from domain data

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Goal

Use description logic ontologies to represent knowledge of certain domains Problem

How to obtain these ontologies?

Approach

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Goal

Approach

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Goal

Approach

Learn first versions of ontologies from domain data

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Goal

Extract terminological knowledge from factual knowledge.

Person Artist

Person

Person Writer child

child

Dchild.WriterĎArtist

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Goal

Person Artist

Person

Person Writer child

child

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Goal

Person Artist

Person

Person Writer child

child

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Goal

Extract terminological knowledge frominterpretations.

Person Artist

Person

Person Writer child

child

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Goal

Extract finite bases of GCIs frominterpretations.

Person Artist

Person

Person Writer child

child

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Motivation Errors in the Data

Experiment

§ Use information from Wikipedia about famous persons and their children (via DBpedia)

§ interpretation ℐ_DBpedia with 5626 individuals, 60 concept names

§ extract base of 1252 GCIs Some Results

CollegeCoach[MilitaryPersonĎK President[ Dchild.ArtistĎDchild.Actor

Dchild.CollegeCoach[ Dchild.Philosopher[PersonĎK

Observation

Dchild.JĎPerson does not hold in ℐ_DBpedia, because of 4

erroneous

counterexamples: Teresa_Carpio,Charles_Heung,Adam_Cheng,Lydia_Shum.

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Experiment

Observation

erroneous

(13)

Experiment

Observation

erroneous

(14)

Experiment

§ extract base of 1252 GCIs

Some Results

Observation

erroneous

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Experiment

Dchild.CollegeCoach[ Dchild.Philosopher[PersonĎK Observation

erroneous

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Experiment

CollegeCoach[MilitaryPersonĎK

President[ Dchild.ArtistĎDchild.Actor

erroneous

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Experiment

erroneous

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Experiment

Observation

erroneous

(19)

Experiment

Dchild.JĎPerson

does not hold in ℐ_DBpedia, because of 4

erroneous

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Experiment

Dchild.JĎPerson

, because of 4

erroneous

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Experiment

Dchild.JĎPerson

erroneous

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Experiment

Dchild.JĎPerson

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Outline

1 The Original Approach

2 Adding Confidence

3 Experiments withℐ_DBpedia

4 Exploring Confident GCIs

5 Conclusions

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The Original Approach

Outline

2 Adding Confidence

5 Conclusions

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The Original Approach Description Logics

Description Logic

Useℰℒ andℰℒ^K with usual semantics.

General Concept Inclusions

§ General Concept Inclusions (GCIs) are of the form C ĎD

whereC,D are ℰℒ^K-concept descriptions.

§ C ĎD holdsin ℐ if and only if

C^ℐ ĎD^ℐ

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Description Logic

where C,D are ℰℒ^K-concept descriptions.

C^ℐ ĎD^ℐ

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Description Logic

C^ℐ ĎD^ℐ

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Description Logic

C^ℐ ĎD^ℐ

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The Original Approach Formal Concept Analysis

Example (Formal Context)

small medium large inner outer moon no moon

Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Example (Contextual Derivation)

touter,smallu¹ “ tPlutou tNeptune,Jupiteru¹ “ touter,moonu

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Example (Contextual Derivation) touter,smallu¹

“ tPlutou tNeptune,Jupiteru¹ “ touter,moonu

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

touter,smallu¹ “ tPlutou

tNeptune,Jupiteru¹ “ touter,moonu

(33)

Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

1

“ touter,moonu

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

1

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Observation

touter,moonu¹ “ tJupiter,Saturn,Uranus,Neptune,Plutou

“ touteru¹

Theimplicationtouteru Ñ tmoonu holds.

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Observation

touter,moonu¹

“ tJupiter,Saturn,Uranus,Neptune,Plutou

“ touteru¹

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Observation

“ touteru¹

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Observation

“ touteru¹

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Mercury ˆ ˆ ˆ

Venus ˆ ˆ ˆ

Earth ˆ ˆ ˆ

Mars ˆ ˆ ˆ

Jupiter ˆ ˆ ˆ

Saturn ˆ ˆ ˆ

Uranus ˆ ˆ ˆ

Neptune ˆ ˆ ˆ

Pluto ˆ ˆ ˆ

Observation

“ touteru¹

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Definition

§ Aformal context is a triple K“ pG,M,Iq, whereG,M are sets and I ĎGˆM.

§ If AĎG, then its derivation inK is defined as A¹ :“ tmPM | @g PA:pg,mq PIu

(A¹ is the set of common attributes).

§ Analogously define B¹ forB ĎM (set of described objects).

§ If X,Y ĎM, then the pairpX,Yqis called animplication ofpG,M,Iq, writtenX ÑY.

§ X ÑY holdsin Kif and only ifX¹ĎY¹.

§ For each X ĎM, it is true thatX ĎX² and thatX ÑX² is valid in K.

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Definition

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Definition

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Definition

§ If X,Y ĎM, then the pairpX,Yqis called animplication ofpG,M,Iq,

writtenX ÑY.

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Definition

writtenX ÑY.

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Definition

written X ÑY.

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Definition

§ If X,Y ĎM, then the pairpX,Yqis called animplication ofpG,M,Iq, written X ÑY.

§ X ÑY holdsin Kif and only ifX¹ ĎY¹.

§ For eachX ĎM, it is true thatX ĎX² and thatX ÑX² is valid in K.

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Definition

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Definition

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The Original Approach The Result by Baader and Distel

Person Artist

Person

Person Writer child

child

t. . . ,Dchild.WriterĎArtist, . . .u Axiomatization

(Base of valid GCIs) ℐ

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Axiomatization (Base of valid

Implications)

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Person Artist

Person

Person Writer child

child

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

(52)

Person Artist

Person

Person Writer child

child

(Base of valid GCIs)

ℐ

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

(53)

Person Artist

Person

Person Writer child

child

(Base of valid GCIs)

ℐ

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

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Person Artist

Person

Person Writer child

child

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

(55)

Person Artist

Person

Person Writer child

child

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

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Person Artist

Person

Person Writer child

child

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

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Advantage

Formal Concept Analysis provides methods to extract implicational knowledgefrom formal contexts

Idea

Use these methods to learn GCIs from data! Parallels between FCA and DL

Formal Concept Analysis Description Logics objectsG individuals ∆^ℐ attributes M concept descriptions formal contexts K interpretationsℐ

implications GCIs

A¹,AĎM pd Aq^ℐ B¹,B ĎG ?

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Advantage

Idea

Use these methods to learn GCIs from data!

Parallels between FCA and DL

implications GCIs

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Advantage

Idea

Formal Concept Analysis Description Logics

objectsG individuals ∆^ℐ attributes M concept descriptions formal contexts K interpretationsℐ

implications GCIs

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Advantage

Idea

Formal Concept Analysis Description Logics objectsG individuals ∆^ℐ

attributes M concept descriptions formal contexts K interpretationsℐ

implications GCIs

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Advantage

Idea

Formal Concept Analysis Description Logics objectsG individuals ∆^ℐ attributes M concept descriptions

formal contexts K interpretationsℐ

implications GCIs

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Advantage

Idea

implications GCIs

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Advantage

Idea

implications GCIs

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Advantage

Idea

implications GCIs

A¹,AĎM pd Aq^ℐ

B¹,B ĎG ?

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Advantage

Idea

implications GCIs

A¹,AĎM pd Aq^ℐ

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Observation

Need to describe sets X Ď∆^ℐ as good as possible.

Definition

A concept descriptionC is a model-based most-specific concept description of X if and only if

§ X ĎC^ℐ and

§ for each D with X ĎD^ℐ, it is true that C is more specificthan D

Difficulty

Existence can not be guaranteed in ℰℒ^K ;ℰℒ^K_gfp Example (ℰℒ^K_gfp Concept Description)

C :“ pA,tA”Writer[ Dchild.B,B ”Writer[ Dchild.Auq

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Observation

Definition

A concept descriptionC is amodel-based most-specific concept description of X if and only if

§ X ĎC^ℐ and

§ for each D with X ĎD^ℐ, it is true thatC is more specificthan D Difficulty

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Observation

Definition

§ X ĎC^ℐ and

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Observation

Definition

§ X ĎC^ℐ and

§ for each D with X ĎD^ℐ, it is true thatC is more specificthan D

Difficulty

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Observation

Definition

§ X ĎC^ℐ and

Existence can not be guaranteed in ℰℒ^K

;ℰℒ^K_gfp Example (ℰℒ^K_gfp Concept Description)

(71)

Observation

Definition

§ X ĎC^ℐ and

Existence can not be guaranteed in ℰℒ^K ;ℰℒ^K_gfp

Example (ℰℒ^K_gfp Concept Description)

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Observation

Definition

§ X ĎC^ℐ and

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Definition Set

Mℐ :“ t K u YN_C Y t Dr.X^ℐ |X Ď∆^ℐ,X ‰ H,r PN_Ru.

and defineinduced formal context Kℐ of ℐ asKℐ “ p∆^ℐ,Mℐ,∇q, where px,Cq P∇ ðñ xPC^ℐ.

Theorem

Let ℐ be a finite interpretation. Ifℒ is a base of all confident implications of Kℐ, then the set

tl

U Ñ ppl

Uq^ℐq^ℐ | pU ÑU²q Pℒu

is a base of Thpℐq.

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Definition Set

and defineinduced formal context Kℐ ofℐ asKℐ “ p∆^ℐ,Mℐ,∇q, where px,Cq P∇ ðñ x PC^ℐ.

Theorem

tl

U Ñ ppl

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Definition Set

and defineinduced formal context Kℐ ofℐ asKℐ “ p∆^ℐ,Mℐ,∇q, where px,Cq P∇ ðñ x PC^ℐ.

Theorem

tl

U Ñ ppl

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Adding Confidence

Outline

2 Adding Confidence

5 Conclusions

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Adding Confidence An Observation

Recall

Dchild.JĎPerson

does not hold in ℐ_DBpedia, but there are only 4 erroneous counterexamples.

Observation

From 2551 individuals satisfying Dchild.J, 2547 also satisfy Person, i. e. conf_ℐ_DBpediapDchild.JĎPersonq “ 2547

2551

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Recall

Dchild.JĎPerson

Observation

From 2551 individuals satisfying Dchild.J, 2547 also satisfy Person, i. e.

conf_ℐ_DBpediapDchild.JĎPersonq “ 2547 2551

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Recall

Dchild.JĎPerson

Observation

From 2551 individuals satisfying Dchild.J, 2547 also satisfy Person, i. e.

confℐ_DBpediapDchild.JĎPersonq “ 2547 2551

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Adding Confidence Definition

Definition

Theconfidenceof C ĎD in ℐ is defined as confℐpC ĎDq:“

#1 if C^ℐ “ H,

|pC[Dq^ℐ|

|C^ℐ| otherwise.

Let c P r0,1s. Define

Th_cpℐq:“ tC ĎD |confℐpC ĎDq ěcu. New Goal

Axiomatize Th_cpℐq, i. e. find afinite baseof Th_cpℐq.

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Definition

#1 if C^ℐ “ H,

|pC[Dq^ℐ|

|C^ℐ| otherwise.

Th_cpℐq:“ tC ĎD |confℐpC ĎDq ěcu.

New Goal

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Definition

#1 if C^ℐ “ H,

|pC[Dq^ℐ|

|C^ℐ| otherwise.

Th_cpℐq:“ tC ĎD |confℐpC ĎDq ěcu.

New Goal

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Adding Confidence Bases of Confident GCIs

Question

How to find bases for Th_cpℐq?

Observation

Related work by M. Luxenburger on partial implications Approach (Luxenburger)

§ Separately axiomatizevalid GCIs andproperly confident GCIs

§ Consider concept descriptions of the form pC^ℐq^ℐ only

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Question

Observation

Related work by M. Luxenburger on partial implications

Approach (Luxenburger)

(85)

Question

Observation

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Question

Observation

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Question

Observation

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The Idea in FCA Let ℬbase of K,

𝒞“ tX² ÑY² |1ąconf_KpX² ÑY²q ěcu and 1ąconfpAÑBq ěc.

Then conf_KpAÑBq “conf_KpA² ÑB²q and ℬY𝒞|ùAÑA² ÑB² ÑB

(89)

§ Consider concept descriptions of the form pC^ℐq^ℐ only The Idea in FCA

Let ℬbase of K,

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Let ℬbase of K,

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Let ℬbase of K,

Then conf_KpAÑBq “conf_KpA² ÑB²q

and ℬY𝒞|ùAÑA² ÑB² ÑB

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Let ℬbase of K,

Then conf_KpAÑBq “conf_KpA² ÑB²q and ℬY𝒞|ùAÑA²

ÑB² ÑB

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Let ℬbase of K,

Then conf_KpAÑBq “conf_KpA² ÑB²q and ℬY𝒞|ùAÑA² ÑB²

ÑB

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Let ℬbase of K,

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Definition

Confpℐ,cq:“ tX^ℐ ĎY^ℐ |Y,X Ď∆ℐ,1ąconfℐpX^ℐ ĎY^ℐq ěcu.

Theorem

Let ℬbe a finite base of ℐ, and c P r0,1s. ThenℬYConfpℐ,cq is a finite base of Th_cpℐq.

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Definition

Confpℐ,cq:“ tX^ℐ ĎY^ℐ |Y,X Ď∆ℐ,1ąconfℐpX^ℐ ĎY^ℐq ěcu.

Theorem

Let ℬbe a finite base of ℐ, and c P r0,1s. ThenℬYConfpℐ,cq is a finite base of Th_cpℐq.

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Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Implications)

(98)

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Axiomatization (Base of confident

Implications)

(99)

(Base ofconfident GCIs)

ℐ

Kℐ Mℐ

∆^ℐ

ˆ ˆ . ˆ . ˆ

. . ˆ

tU ÑV |. . .u

Axiomatization (Base of confident

Implications)

(100)

Definition

If Kis a formal context, X ÑY an implication ofK, then conf_KpX ÑYq:“

#1 X¹ “ H

|pXYYq¹|

|X¹| otherwise is the confidenceof X ÑY inK.

Ifc P r0,1s, then Th_cpKq denotes the set of all implications of Kwith confidence at leastc in K.

Goal

Transform bases of Th_cpKqto bases of Th_cpℐq.

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Definition

#1 X¹ “ H

|pXYYq¹|

|X¹| otherwise

is the confidenceof X ÑY inK. Ifc P r0,1s, then Th_cpKq denotes the set of all implications of Kwith confidence at leastc in K.

Goal

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Definition

#1 X¹ “ H

|pXYYq¹|

|X¹| otherwise

is the confidenceof X ÑY inK. Ifc P r0,1s, then Th_cpKq denotes the set of all implications of Kwith confidence at leastc in K.

Goal

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Lemma

If X,Y ĎMℐ, then

conf_ℐpl

X Ďl

Yq “conf_K_ℐpX ÑYq.

Definition

If ℒ is a set of implications ofKℐ, then lℒ:“ tl

X Ďl

Y | pX ÑYq Pℒu.

Lemma

Let ℒY tX ÑY u be a set of implications ofKℐ. Then

ℒ|ù pX ÑYq ùñ l

ℒ|ùl

X Ďl Y.

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Lemma

If X,Y ĎMℐ, then

conf_ℐpl

X Ďl

Definition

X Ďl

Y | pX ÑYq Pℒu.

Lemma

ℒ|ùl

X Ďl Y.

(105)

Lemma

If X,Y ĎMℐ, then

conf_ℐpl

X Ďl

Definition

X Ďl

Y | pX ÑYq Pℒu.

Lemma

ℒ|ùl

X Ďl Y.

(106)

Theorem

Letℒ be a (confident) base of Th_cpKℐq. Thend

ℒis a (confident) base of Th_cpℐq.

“Corollary”

We can use data mining techniques to extract complete sets of confident implications from Kℐ, and thus to learn confident GCIs from interpretations!

(107)

Theorem

Letℒ be a (confident) base of Th_cpKℐq. Thend

ℒis a (confident) base of Th_cpℐq.

“Corollary”

We can use data mining techniques to extract complete sets of confident implications from Kℐ, and thus to learn confident GCIs from interpretations!

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Proof (Sketch).

Show

§ dℒ|ù pd

U Ďpd

Uq^ℐℐq for all U ĎMℐ

§ dℒ|ùConfpℐ,cq First claim: LetU ĎMℐ.

§ ℒ|ù pU ÑU²q

§ d

ℒ|ù pd U Ďq

Second Claim: LetpX^ℐ ĎY^ℐq PConfpℐ,cq.

§ X^ℐ ”d

X¹,Y^ℐ ”d Y¹

§ conf_K_ℐpX¹ ÑY¹q “conf_ℐpd

X¹ Ďd

Y¹q ěc

§ ℒ|ù pX¹ ÑY¹q

§ d

ℒ|ù pd

X¹ Ďd

Y¹q ” pX^ℐ ĎY^ℐq

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Proof (Sketch).

Show

§ dℒ|ù pd

U Ďpd

§ ℒ|ù pU ÑU²q

§ d

ℒ|ù pd U Ďq

§ X^ℐ ”d

X¹,Y^ℐ ”d Y¹

X¹ Ďd

Y¹q ěc

§ d

ℒ|ù pd

X¹ Ďd

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Proof (Sketch).

Show

§ dℒ|ù pd

U Ďpd

§ dℒ|ùConfpℐ,cq

First claim: LetU ĎMℐ.

§ ℒ|ù pU ÑU²q

§ d

ℒ|ù pd U Ďq

§ X^ℐ ”d

X¹,Y^ℐ ”d Y¹

X¹ Ďd

Y¹q ěc

§ d

ℒ|ù pd

X¹ Ďd

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Proof (Sketch).

Show

§ dℒ|ù pd

U Ďpd

§ ℒ|ù pU ÑU²q

§ d

ℒ|ù pd U Ďq

§ X^ℐ ”d

X¹,Y^ℐ ”d Y¹

X¹ Ďd

Y¹q ěc

§ d

ℒ|ù pd

X¹ Ďd