On CLearnability in Description Logics
Ali Rezaei Divroodi
1
QuangThuy 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,2830 November 2012
Outline
Description Logics (DLs)
Concept Learning in DLs
Clearnability in DLs
A.R.Divroodi,Q.T.Ha,L.A.Nguyen,H.S.Nguyen
On CLearnability in Description Logics 2/29
What are Description Logics?
a family of logicbased Knowledge Representation formalisms
decidable fragments of classical rstorder 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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.NienhuysCheng,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 CLearnability 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 CLearnability 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):
Logicbased 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 Logicbased 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 Bisimulationbased Method of Concept Learning
for Knowledge Bases in Description Logics
A.R.Divroodi,Q.T.Ha,L.A.Nguyen,H.S.Nguyen
On CLearnability in Description Logics 12/29
PAClearning (Probably Approximately Correct Learning)
PAClearning 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.
PAClearnability is hard to investigate for DLs.
With respect to DLs,PAClearnability 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 CLearnability in Description Logics 13/29
PAClearning (Probably Approximately Correct Learning)
PAClearning 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.
PAClearnability is hard to investigate for DLs.
With respect to DLs,PAClearnability 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 CLearnability in Description Logics 14/29
Clearnability (Possibility of Correct Learning)
We study Clearnability instead of PAClearnability.
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 CLearnability in Description Logics 15/29
Clearnability (Possibility of Correct Learning)
We study Clearnability instead of PAClearnability.
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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability 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 CLearnability in Description Logics 25/29
Our Concept Learning Algorithm MiMoD
Use the bisimulationbased 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 CLearnability 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 CLearnability in Description Logics 27/29
Concluding Remarks
Clearnability is somehow weaker than PAClearnability,but
still worth investigating for the learning theory in DLs.
Our theorem shows a good property of the bisimulationbased
concept learning approach.
As future work,we intend to study Clearnability 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 CLearnability in Description Logics 28/29
Thank you for your attention!
A.R.Divroodi,Q.T.Ha,L.A.Nguyen,H.S.Nguyen
On CLearnability in Description Logics 29/29
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