On C-Learnability in Description Logics

Ali Rezaei Divroodi

1

Quang-Thuy Ha

2

Linh Anh Nguyen

1

Hung Son Nguyen

1

1:University of Warsaw

2:College of Technology,Vietnam National University,Hanoi

ICCCI

Ho Chi Minh city,28-30 November 2012

Outline

Description Logics (DLs)

Concept Learning in DLs

C-learnability in DLs

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 2/29

What are Description Logics?

a family of logic-based Knowledge Representation formalisms

decidable fragments of classical rst-order logic

closely related to propositional modal logics

the logical base of OWL (Web Ontology Language) and

Semantic Web

describe domain in terms of:

concepts (classes of objects),

roles (binary relationships between objects),

individuals (objects)

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 3/29

Examples of Concepts

Person uFemale

Person u 9hasChild:Male

Person u 8hasChild:(Doctor tLawyer)

Person u 1 hasChild

Person u 2 hasChild:Female

Participant u 9attend:Talk

Participant u 8attend:(Talk u:Boring)

Speaker u 9gives:(Talk u 8topic:DL)

Speaker u 8gives:(Talk u 9topic:(DL tFuzzyLogic))

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 4/29

Examples of Terminological Axioms

rstYearCourse v 8isTaughtBy:Professor

mathCourse v 9isTaughtBy:f949352g

academicStaMember v 9teaches:undergraduateCourse

course v 1isTaughtBy

department v 10hasMember u 30hasMember

course v:staMember

peopleAtUni = staMember tstudent

facultyInCS = faculty u 9belongsTo:fCSDepartmentg

adminSta = staMember u:(faculty ttechSupportSta )

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 5/29

Concept Learning in DLs vs Trad.Binary Classication

Concept Learning in DLs is similar to binary classication in

traditional machine learning.

The dierence is that in DLs objects are described not only by

attributes but also by relationship between objects.

The description language is richer for expressing concepts.

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 6/29

Major Settings of Concept Learning in DLs

1

Given a knowledge base KB in a DL L and sets E

+

,E

of

individuals,learn a concept C in L such that:

a) KB j= C(a) for all a 2 E

+

,and

b) KB j=:C(a) for all a 2 E

.

E

+

contains positive examples of C,

E

contains negative examples of C.

2

The second setting diers from the previous one only in that

the condition b) is replaced by the weaker one:

KB 6j= C(a) for all a 2 E

.

3

Given an interpretation I and sets E

+

,E

of individuals,

learn a concept C in L such that:

a) I j= C(a) for all a 2 E

+

,and

b) I j=:C(a) for all a 2 E

.

Note that I 6j= C(a) is the same as I j=:C(a).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 7/29

Major Settings of Concept Learning in DLs

1

Given a knowledge base KB in a DL L and sets E

+

,E

of

individuals,learn a concept C in L such that:

a) KB j= C(a) for all a 2 E

+

,and

b) KB j=:C(a) for all a 2 E

.

E

+

contains positive examples of C,

E

contains negative examples of C.

2

The second setting diers from the previous one only in that

the condition b) is replaced by the weaker one:

KB 6j= C(a) for all a 2 E

.

3

Given an interpretation I and sets E

+

,E

of individuals,

learn a concept C in L such that:

a) I j= C(a) for all a 2 E

+

,and

b) I j=:C(a) for all a 2 E

.

Note that I 6j= C(a) is the same as I j=:C(a).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 8/29

Major Settings of Concept Learning in DLs

1

Given a knowledge base KB in a DL L and sets E

+

,E

of

individuals,learn a concept C in L such that:

a) KB j= C(a) for all a 2 E

+

,and

b) KB j=:C(a) for all a 2 E

.

E

+

contains positive examples of C,

E

contains negative examples of C.

2

The second setting diers from the previous one only in that

the condition b) is replaced by the weaker one:

KB 6j= C(a) for all a 2 E

.

3

Given an interpretation I and sets E

+

,E

of individuals,

learn a concept C in L such that:

a) I j= C(a) for all a 2 E

+

,and

b) I j=:C(a) for all a 2 E

.

Note that I 6j= C(a) is the same as I j=:C(a).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 9/29

Previous Work on Concept Learning in DLs:Before 2012

Based on renement operators as in inductive logic programming:

Using the rst setting:

L.Badea and S.-H.Nienhuys-Cheng,2000

L.Iannone,I.Palmisano,and N.Fanizzi,2007

Using the second setting:

N.Fanizzi,C.d'Amato,and F.Esposito,2008

J.Lehmann and P.Hitzler,2010

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 10/29

Bisimulation and Discernibility

In modal logics and state transition systems:

two states are indiscernible if they are bisimilar

(i.e.,if one state can simulates the other and vice versa)

DLs are variants of modal logics.

In DLs,two objects are indiscernible if they are bisimilar.

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 11/29

Previous Work on Concept Learning in DLs:In 2012

Based on bisimulation:

Using the third setting:

L.A.Nguyen and A.Sza las,2012 (a chapter in book):

Logic-based Roughication

T.-L.Tran,Q.-T.Ha,T.-L.-G.Hoang,L.A.Nguyen,

H.S.Nguyen and A.Sza las,KSE'2012:

Concept Learning for Description Logic-based Inf.Systems

Using the rst setting:

Q.-T.Ha,T.-L.-G.Hoang,L.A.Nguyen,H.S.Nguyen,

A.Sza las and T.-L.Tran,SoICT'2012:

A Bisimulation-based Method of Concept Learning

for Knowledge Bases in Description Logics

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 12/29

PAC-learning (Probably Approximately Correct Learning)

PAC-learning is a framework for mathematical analysis of

traditional machine learning proposed by Valiant in 1984.

The learner receives samples and must select from a certain

class a hypothesis that approximates the function to be learnt.

The goal is that,with high probability,the selected hypothesis

will have low generalization error.

The learner must be able to learn the concept in polynomial

time given any arbitrary approximation ratio,probability of

success,or distribution of the samples.

PAC-learnability is hard to investigate for DLs.

With respect to DLs,PAC-learnability was studied only for

a very restricted DL (by Cohen and Hirsh in 1994).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 13/29

PAC-learning (Probably Approximately Correct Learning)

PAC-learning is a framework for mathematical analysis of

traditional machine learning proposed by Valiant in 1984.

The learner receives samples and must select from a certain

class a hypothesis that approximates the function to be learnt.

The goal is that,with high probability,the selected hypothesis

will have low generalization error.

The learner must be able to learn the concept in polynomial

time given any arbitrary approximation ratio,probability of

success,or distribution of the samples.

PAC-learnability is hard to investigate for DLs.

With respect to DLs,PAC-learnability was studied only for

a very restricted DL (by Cohen and Hirsh in 1994).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 14/29

C-learnability (Possibility of Correct Learning)

We study C-learnability instead of PAC-learnability.

The starting point (which is easy to see)

There exists an algorithm for binary classication in traditional

machine learning such that,for every concept C being a boolean

function to be learnt,if the training set is good enough,the

algorithm returns a concept equivalent to C.

The studied problem

Can the above claim be extended for concept learning in DLs?

What DLs can that claim be extended for?

Assume that training information systems are

interpretations in DLs (i.e.,the third setting is used).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 15/29

C-learnability (Possibility of Correct Learning)

We study C-learnability instead of PAC-learnability.

The starting point (which is easy to see)

There exists an algorithm for binary classication in traditional

machine learning such that,for every concept C being a boolean

function to be learnt,if the training set is good enough,the

algorithm returns a concept equivalent to C.

The studied problem

Can the above claim be extended for concept learning in DLs?

What DLs can that claim be extended for?

Assume that training information systems are

interpretations in DLs (i.e.,the third setting is used).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 16/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 17/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 18/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 19/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 20/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 21/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 22/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 23/29

The Considered Description Logics

We consider DLs that extend the basic DL ALC with features

amongst I (inverse roles),Q

k

(quantied number restrictions with

numbers bounded by a constant k),Self (local re exivity of a role).

Roles and concepts are dened inductively as follows:

every role name is a role,every concept name is a concept,

if r is a role name then the inverse r

(if allowed) is a role,

if C and D are concepts,R is a role,r is a role name,and

h,k are natural numbers then:

>,?,:C,C uD,C tD,8R:C and 9R:C are concepts,

if the feature Q

k

is allowed and h k

then h R:C and <h R:C are concepts,

if the feature Self is allowed then 9r:Self is a concept

(standing for the set of objects x such that r(x;x) holds).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 24/29

Our Result

Theorem

Any concept C in any description logic that extends ALC with

some features amongst I,Q

k

,Self can be learnt if the training

information system is good enough.

That is,there exists a learning algorithm such that,for every

concept C of those logics,there exists a training information

system consistent with C such that applying the learning algorithm

to the system results in a concept equivalent to C.

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 25/29

Our Concept Learning Algorithm MiMoD

Use the bisimulation-based concept learning method proposed

by Nguyen and Sza las.

Partition the domain by using selectors.

Apart from traditional selectors of the form A (a concept

name),use also the following selectors (if they are allowed):

9r:Self;9r:C

i

;9r

:C

i

;h r:C

i

;h r

:C

i

;

where r is a role name and C

i

is the concept characterizing

the block number i in the current partition.

During the granulation process always choose a block and a

selector to partition it that result in new blocks characterized

by concepts with minimal modal depth,i.e.with minimal

nesting depth of 9R and h R.

The name MiMoD stands for\minimizing modal depth".

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 26/29

Techniques Used for Proving the Theorem

We introduced and used universal interpretations and

bounded bisimulation in DLs.

Universal interpretation

An interpretation I is universal with respect to the considered

DL language if,for every satisable concept C of that

language,C

I

6=;.

Lemma:In any DL that extends ALC with some features

amongst I,Q

k

,Self,there exists a nite universal

interpretation that can eectively be constructed.

(Here,the sets of concept names and role names are nite.)

A property of bounded bisimulation in DLs

Two objects are indiscernible by concepts with modal depth not

greater than n if they belong to the same abstract class of an

equivalence relation being a bisimulation with depth n.

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 27/29

Concluding Remarks

C-learnability is somehow weaker than PAC-learnability,but

still worth investigating for the learning theory in DLs.

Our theorem shows a good property of the bisimulation-based

concept learning approach.

As future work,we intend to study C-learnability in other DLs

and for the case when there is background knowledge (like a

TBox and/or an RBox).

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 28/29

Thank you for your attention!

A.R.Divroodi,Q.-T.Ha,L.A.Nguyen,H.-S.Nguyen

On C-Learnability in Description Logics 29/29

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