Doctoral thesis summary: Concept learning for description logic based information systems

Research objectives: Building a method of granulating partitions of domain of interpretation in DLs. This method is base on bisimulation and using suitable selectors as well as information gain measure; proposing bisimulation-based concept learning algorithms for knowledge bases in DLs using Setting.

HUE UNIVERSITY COLLEGE OF SCIENCES

TRAN THANH LUONG

CONCEPT LEARNING FOR DESCRIPTION LOGIC-BASED

This thesis was completed at:

College of Sciences, Hue University

Supervisors:

1 Assoc Prof Dr.Sc Nguyen Anh Linh, Warsaw University, Poland

2 Dr Hoang Thi Lan Giao, College of Sciences, Hue University

Reviewer 1: Prof Dr.Sc Hoang Van Kiem

University of Information Technology, VNU-HCMReviewer 2: Assoc Prof Dr Doan Van Ban

Institute of Information Technology, VASTReviewer 3: Assoc Prof Dr Nguyen Mau Han

College of Sciences, Hue University

The thesis will be presented at the Committee of Hue University,

to be held by Hue University at h , / /2015

The thesis can be found at the follow libraries:

• National Library of Vietnam

• Library Information Center College of Science, Hue University

Concept learning in description logics (DLs) is similar to binary classification intraditional machine learning However, the problem in the context of DLs differs fromthe traditional setting in that objects are described not only by attributes but also bybinary relations between objects The major settings of concept learning in DLs are

as follows:

• Setting (1): Given a knowledge base KB in a DL LΣ,Φ and sets E+, E− ofindividuals, learn a concept C in LΣ,Φ such that:

1 KB |= C(a) for all a ∈ E+, and

2 KB |= ¬C(a) for all a ∈ E−

The sets E+ and E− contain positive examples and negative ones of C, tively

respec-• Setting (2): This setting differs from the previous one only in that the secondcondition is replaced by the weaker one: KB 6|= C(a) for all a ∈ E−

• Setting (3): Given an interpretation I and sets E+, E− of individuals, learn

a concept C in a DL LΣ,Φ such that:

1 I |= C(a) for all a ∈ E+, and

2 I |= ¬C(a) for all a ∈ E−

Note that I 6|= C(a) is the same as I |= ¬C(a)

Concept learning in DLs was studied by a number of researchers The related workcan be divided into three main groups

The first group focuses on learnability in DLs [4, 8] Cohen and Hirsh studiedPAC-learnability of description logic and proposed a concept learning algorithm calledLCSLearn, which is based on “least common subsumers” [4] In [8] Franzier and Pittprovided some results on learnability in the CLASSIC description logic using somekinds of queries and either the exact learning model or the PAC model

The second group studies concept learning in DLs by using refinement operators.Badea and Nienhuys-Cheng [1] studied concept learning in the DL ALER, Iannone et

al [9] also investigated learning algorithms by using refinement operators for the richer

DL ALC The both of works studied concept learning in DLs using Setting (1) In [7]Fanizzi et al introduced the DL-FOIL system that is adapted to concept learning for

DL representations supporting the OWL-DL language In [10] Lehmann and Hitzlerintroduced methods from inductive logic programing for concept learning in DL knowl-edge bases Their algorithm, DL-Learner, exploits genetic programming techniques All

of works studied concept learning in DLs using Setting (2)

The last group exploits bisimulation for concept learning in DLs [6] Nguyen andSza las applied bisimulation in DLs to model indiscernibility of objects [14] They pro-posed a general bisimulation-based concept learning method for DL-based informationsystems Divroodi et al [5] studied C-learnability in DLs These works mentioned con-cept learning in DLs using Setting (3)

Apart from the works of Nguyen and Sza las, Divroodi, which are base on ulation to guide the search for the result, all other works use refinement operators

bisim-as in inductive logic programming and/or scoring functions-bbisim-ased search strategies.These works focus on concept learning using Setting (1) and Setting (2) for the simpleDLs such as ALER, ALN and ALC The works [14, 5] studied bisimulation-basedconcept learning in DLs using Setting (3) Both of works did not mentioned conceptlearning in DLs using Setting (1) and Setting (2)

From the surveys as outlined above, we found that concept learning in DLs is akey problem It is used to build useful concepts for semantic systems and ontologies.Therefore, it impacts on many practical applications which apply Semantic Web forsystems This thesis studies bisimulation-based concept learning methods in DLs Themain goals of the thesis are:

• Studying the syntax, semantic of a large richer DLs by allowing to use attributes

as basic elements of the language, data roles and DL-features as F, N This class

of DLs covers useful DLs, with well-known DLs like ALC, SHIQ, SHOIQ,

SROIQ, ;

• Formulating and extending the definitions, theorems, lemmas on bisimulation forthe mentioned class of DLs We use bisimulation notions to model indiscernibility

of objects as well as for concept learning in DLs;

• Developing bisimulation-based concept learning algorithms for information tems in DLs using Setting (3);

sys-• Building a method of granulating partitions of domain of interpretation in DLs.This method is base on bisimulation and using suitable selectors as well as infor-mation gain measure

• Proposing bisimulation-based concept learning algorithms for knowledge bases inDLs using Setting (1) and Setting (2)

Chương 1.

DESCRIPTION LOGIC AND KNOWLEDGE BASE

1.1 Overview of description logics

1.1.1 Introduction

DLs are built from three basic parts include a set of individuals, set of atomicconcepts and set of atomic roles

1.1.2 Description language ALC

Definition 1.1 (ALC Syntax) Let ΣC be a set of concept names and ΣR be a set

of role names (ΣC ∩ ΣR = ∅) The elements in ΣC are called atomic concepts Thedescription logic ALC allows concepts defined recursively as follows:

• if A ∈ ΣC then A is a concept of ALC,

• if C and D are concepts and r ∈ ΣR is a role then >, ⊥, ¬C, C u D, C t D,

Definition 1.2 (Semantics) An interpretation in description logic ALC is a pair

I = h∆I, ·Ii, where ∆I is a non-empty set called the domain of I and·I is a mappingcalled the interpretation function of I that associates each individual a ∈ ΣI with

an element aI ∈ ∆I, each concept name A ∈ ΣC with a set AI ⊆ ∆I, each role name

r ∈ ΣoR with a binary relationrI ⊆ ∆I× ∆I The interpretation of complex concepts

Knowledge expressiveness of a DL depends on concept and role constructors whichare allowed to use DLs mainly differ in their expressive power and syntactic structures.1.1.5 Description logics nomenclature

• ALC - is an abbreviation for attributive language with complements

• S - ALC + transitive roles • F - functional roles

• N - unqualified number restrictions • R - complex role inclusions

• H - subroles • I - inverse roles

• Q - qualified number restrictions • O - nominals

1.2 Syntax and semantics of description logics

1.2.1 Description language ALCreg

Definition 1.3 (ALCreg Syntax) LetΣC be a set of concept names andΣR be a set ofrole names (ΣC ∩ ΣR = ∅) The elements in ΣC are called atomic concepts, while theelements in ΣR are called atomic roles The description logic ALCreg allows conceptsand roles defined recursively as follows:

• if A ∈ ΣC then A is a concept of ALCreg,

• if r ∈ ΣR then r is a role of ALCreg,

• if C and D are concepts, R and S are roles then

– ε, R ◦ S, R t S, R∗, C? are roles of ALCreg,

– >, ⊥, ¬C, C u D, C t D, ∃R.C and ∀R.C are concepts of ALCreg 

The interpretation of complex roles in ALCreg are defined as follows:

(R ◦ S)I = RI ◦ SI, (R t S)I = RI ∪ SI, (R∗)I = (RI)∗,

εI = {hx, xi | x ∈ ∆I}, (C?)I = {hx, xi | CI(x)}

1.2.2 The LΣ,Φ language

A DL-signature is a finite set Σ = ΣI ∪ ΣdA∪ ΣnA∪ ΣoR∪ ΣdR, where ΣI is a set

of individuals, ΣdA is a set of discrete attributes, ΣnA is a set of numeric attributes,

ΣoR is a set of object role names, and ΣdR is a set of data roles All the sets ΣI, ΣdA,

ΣnA, ΣoR, ΣdR are pairwise disjoint

We consider some DL-features denoted byI (inverse),O(nominal), F ity), N (unqualified number restriction), Q(qualified number restriction),U (universalrole), Self (local reflexivity of a role) A set of DL-features is a set consisting of zero

(functional-or some of these names

Definition 1.4 (The LΣ,Φ Language) Let Σ be a DL-signature, Φ be a set of features and L stand for ALCreg The DL language LΣ,Φ allows object roles and con-cepts defined recursively as follows:

DL-• if r ∈ ΣoR then r is an object role of LΣ,Φ,

• if A ∈ ΣC then A is concept of LΣ,Φ,

• if A ∈ ΣA \ ΣC and d ∈ range(A) then A = d and A 6= d are concepts of LΣ,Φ,

• if A ∈ ΣnA and d ∈ range(A) then A ≤ d, A < d, A ≥ d and A > d areconcepts of LΣ,Φ,

• if R and S are object roles of LΣ,Φ, C and D are concepts of LΣ,Φ, r ∈ ΣoR,

σ ∈ ΣdR, a ∈ ΣI, and n is a natural number then

– ε, R ◦ S, R t S, R∗ and C? are object roles of LΣ,Φ,

– >, ⊥, ¬C, C u D, C t D, ∀R.C and ∃R.C are concepts of LΣ,Φ,

– if d ∈ range(σ) then ∃σ.{d} is a concept of LΣ,Φ,

– if I ∈ Φ then R− is an object role of LΣ,Φ,

– if O ∈ Φ then {a} is a concept of LΣ,Φ,

– if F ∈ Φ then ≤ 1 r is a concept of LΣ,Φ,

– if {F , I} ⊆ Φ then ≤ 1 r− is a concept of LΣ,Φ,

– if N ∈ Φ then ≥ n r and ≤ n r are concepts of LΣ,Φ,

– if {N , I} ⊆ Φ then ≥ n r− and ≤ n r− are concepts of LΣ,Φ,

– if Q ∈ Φ then ≥ n r.C and ≤ n r.C are concepts of LΣ,Φ,

– if {Q, I} ⊆ Φ then ≥ n r−.C and ≤ n r−.C are concepts of LΣ,Φ,

– if U ∈ Φ then U is an object role of LΣ,Φ,

Definition 1.5 (Semantics of LΣ,Φ) An interpretation in LΣ,Φ is a pair I = h∆I, ·Ii,where ∆I is a non-empty set called the domain of I and ·I is a mapping called theinterpretation function of I that associates each individual a ∈ ΣI with an element

aI ∈ ∆I, each concept nameA ∈ ΣC with a setAI ⊆ ∆I, each attribute A ∈ ΣA\ΣC

with a partial function AI : ∆I → range(A), each object role name r ∈ ΣoR with

a binary relation rI ⊆ ∆I × ∆I, and each data role σ ∈ ΣdR with a binary relation

σI ⊆ ∆I × range(σ) The interpretation function ·I is extended to complex objectroles and complex concepts as shown in Figure 1.1, where#Γstands for the cardinality

(C?)I = {hx, xi | CI(x)}

εI = {hx, xi | x ∈ ∆I} {a} I = {aI} UI = ∆I × ∆ I

1.3.1 Negation normal form of concepts

A concept C is in negation normal form if negation only occurs in front of conceptnames in C

1.3.2 Storage normal form of concepts

Storage normal form of concepts is built based on negation normal form and set.Concepts in the storage normal form are represented in a set of sub-concept

1.3.3 Converse normal form of roles

An object roleRis in the converse normal form if the inverse constructor is applied

in R only to role names in R, which are different from U

Let Σ±oR = ΣoR ∪ {r− | r ∈ ΣoR} A basic object role is an element in Σ±oR

(respectively, ΣoR) if the considered language allows inverse roles (respectively, doesnot allow inverse roles)

1.4 Knowledge base in description logics

1.4.1 Box of Role Axioms

Definition 1.6 (Role axiom) A role inclusion axiom in LΣ,Φ is an expression of theform ε v rorR1◦ .◦Rk v r, wherek ≥ 1, r ∈ ΣoR andR1, , Rk are basic objectroles of LΣ,Φ different from U A role assertion in LΣ,Φ is an expression of the form

Ref(r), Irr(r), Sym(r), Tra(r), or Dis(R, S), where r ∈ ΣoR and R, S are objectroles of LΣ,Φ different from U A role axiom in LΣ,Φ is a role inclusion axiom or a role

Definition 1.7 (Box of role axioms) A box of role axioms (RBox) in LΣ,Φ is a finite

1.4.2 Box of terminological axioms

Definition 1.8 (Terminological axiom) A general concept inclusion axiom in LΣ,Φ is

an expression of the form C v D, where C and D are concepts in LΣ,Φ A conceptequivalent axiom in LΣ,Φ is an expression of the form C ≡ D, where C and D areconcepts in LΣ,Φ A terminological axiom in LΣ,Φ is a general concept inclusion axiom

Definition 1.9 (Box of terminological axioms) A box of terminological (TBox) in

LΣ,Φ is a finite set of terminological axioms in LΣ,Φ 

1.4.3 Box of individual assertions

Definition 1.10 (Individual assertion) An individual assertion in LΣ,Φ is an sion of the form C(a), R(a, b), ¬R(a, b), a = b and a 6= b, where C is a concept and

Definition 1.11 (Box of individual assertions) A box of individual assertions (ABox)

in LΣ,Φ is a finite set of individual assertions in LΣ,Φ 

1.4.4 Knowledge base and model of knowledge base

Definition 1.12 (Knowledge base) A knowledge base in LΣ,Φ is a triple KB =

hR, T , Ai, where R is an RBox, T is a TBox and A is an ABox in LΣ,Φ 

Definition 1.13 (Model) An interpretation I is a model of RBox R (respectively,TBoxT , ABoxA), denoted byI |= R(respectively,I |= T, I |= A) if it validates allthe role axioms of R (respectively, terminological axioms of T, individual assertions

of A) An interpretation I is a model of knowledge baseKB = hR, T , Ai, denoted by

ΣoR= {cites, cited_by}, ΣdR = ∅,

R = {cites− v cited_by, cited_by− v cites, Irr(cites)},

T = {> v Pub, UsefulPub ≡ ∃cited_by.>},

A00= {Awarded (P1), ¬Awarded (P2), ¬Awarded (P3), Awarded (P4),

¬Awarded (P5), Awarded (P6), Year (P1) = 2010, Year (P2) = 2009,

Year (P3) = 2008, Year (P4) = 2007, Year (P5) = 2006, Year (P6) = 2006,cites(P1, P2), cites(P1, P3), cites(P1, P4), cites(P1, P6), cites(P2, P3),cites(P2, P4), cites(P2, P5), cites(P3, P4), cites(P3, P5), cites(P3, P6),cites(P4, P5), cites(P4, P6)},

A0= A00∪ {(¬∃cited_by.>)(P1), (∀cited_by.{P2, P3, P4})(P5)}

Then KB00 = hR, T , A00iandKB0 = hR, T , A0iare knowledge bases in LΣ,Φ Theaxiom > v P states that the domain of any model of KB00 or KB0 consists of only

1.5 Reasoning in description logic

There are a number of reasoning problems in DL-based knowledge bases One canuses two type of algorithms to solve them include structural subsumption algorithmsand tableau algorithms Structural subsumption algorithms have proved effective forsimple DLs such as F L0, F L⊥, ALN, while tableau ones are usually used to solvereasoning problems for a lager class of DLs such asALC [11],ALCI [12],ALCIQ[12],

SHIQ [13],

Summary of Chapter 1

In this chapter, we have introduced a general of DLs, knowledge expressiveness ofDLs Based on the syntax and semantics of DLs, we have presented about knowledgebase, model of knowledge base and the keys of reasoning in DLs Apart from the generallanguage based on the DLs ALCreg with the features I (inverse role), O (nominal),

F (functionally), N (unqualified number restrictions), Q (qualified restriction), U

(universal role),Self (local reflexivity of an object role), we also took attributes as basicelements of the language, include discrete and numeric attributes This approach issuitable for practical information systems based on description logic

Definition 2.1 (Bisimulation) Let Σ and Σ† be DL-signatures such that Σ† ⊆ Σ, Φ

and Φ† be sets of DL-features such that Φ† ⊆ Φ, I and I0 be interpretations in LΣ,Φ

A binary relation Z ⊆ ∆I × ∆I0 is called an LΣ† ,Φ†-bisimulation between I and I0

if the following conditions hold for every a ∈ Σ†I, A ∈ Σ†C, B ∈ Σ†A \ Σ†C, r ∈ Σ†oR,

σ ∈ Σ†dR, d ∈ range(σ), x, y ∈ ∆I, x0, y0 ∈ ∆I0 :

Z(x, x0) ⇒ [BI(x) = BI0(x0) or both are undefined] (2.3)

[Z(x, x0) ∧ rI(x, y)] ⇒ ∃y0 ∈ ∆I0[Z(y, y0) ∧ rI0(x0, y0)] (2.4)

[Z(x, x0) ∧ rI0(x0, y0)] ⇒ ∃y ∈ ∆I[Z(y, y0) ∧ rI(x, y)] (2.5)

Z(x, x0) ⇒ [σI(x, d) ⇔ σI0(x0, d)], (2.6)

if I ∈ Φ† then

[Z(x, x0) ∧ rI(y, x)] ⇒ ∃y0 ∈ ∆I0[Z(y, y0) ∧ rI0(y0, x0)] (2.7)

[Z(x, x0) ∧ rI0(y0, x0)] ⇒ ∃y ∈ ∆I[Z(y, y0) ∧ rI(y, x)], (2.8)

if {F , I} ⊆ Φ† then

Z(x, x0) ⇒ [#{y | rI(y, x)} ≤ 1 ⇔ #{y0 | rI0(y0, x0)} ≤ 1], (2.13)

if Q ∈ Φ† then

if Z(x, x0) holds then, for every r ∈ Σ†oR, there exists a bijection

h : {y | rI(x, y)} → {y0 | rI0(x0, y0)} such that h ⊆ Z, (2.14)

if {Q, I} ⊆ Φ† then

if Z(x, x0) holds then, for every r ∈ Σ†oR, there exists a bijection

h : {y | rI(y, x)} → {y0 | rI0(y0, x0)} such that h ⊆ Z, (2.15)

Lemma 2.1

1 The relation {hx, xi | x ∈ ∆I} is an LΣ† ,Φ †-bisimulation between I and I

2 IfZ is anLΣ† ,Φ †-bisimulation betweenI and I0thenZ−1 is anLΣ† ,Φ †-bisimulationbetween I0 and I

3 If Z1 is an LΣ† ,Φ †-bisimulation between I0 and I1, Z2 is an LΣ† ,Φ †-bisimulationbetween I1 and I2 then Z1◦ Z2 is an LΣ† ,Φ†-bisimulation between I0 and I2

4 If Z is a set of Lӆ ,ֆ-bisimulations between I and I0 then S

Z is an Lӆ ,ֆ

2.2.2 Bisimilarity and equivalence relation

Definition 2.2 Let I and I0 be interpretations in LΣ,Φ We say that I is LΣ† ,Φ †bisimilar to I0 if there exists an LΣ† ,Φ†-bisimulation between I and I0 

-Definition 2.3 Let I and I0 be interpretations in LΣ,Φ, x ∈ ∆I and x0 ∈ ∆I0 Wesay that x is LΣ† ,Φ †-bisimilar to x0 if there exists an LΣ† ,Φ †-bisimulation between I

Definition 2.4 Let I and I0 be interpretations in LΣ,Φ, x ∈ ∆I and x0 ∈ ∆I0

We say that x is LΣ† ,Φ†-equivalent to x0 if, for every concept C of LΣ† ,Φ†, x ∈ CI

2.3 Invariant for bisimulation

2.3.1 Relation between bisimulations, concepts and roles

Lemma 2.2 Let I and I0 be interpretations in LΣ,Φ, Z is an LΣ† ,Φ†-bisimulationbetween I and I0 Then, for every concept C in LΣ† ,Φ †, every object role R of LΣ† ,Φ †,every x, y ∈ ∆I, x0, y0 ∈ ∆I0 and every a ∈ Σ†I, the following properties hold:

[Z(x, x0) ∧ RI(x, y)] ⇒ ∃y0 ∈ ∆I0 | [Z(y, y0) ∧ RI0(x0, y0)] (2.20)

[Z(x, x0) ∧ RI0(x0, y0)] ⇒ ∃y ∈ ∆I | [Z(y, y0) ∧ RI(x, y)], (2.21)

if O ∈ Φ† then:

Z(x, x0) ⇒ [RI(x, aI) ⇔ RI0(x0, aI0)]  (2.22)2.3.2 Invariance of concept

Definition 2.5 (Invariant concept) A concept C of Lӆ ,ֆ is said to be invariant for

LΣ† ,Φ†-bisimulation if Z(x, x0) holds then x ∈ CI iif x0 ∈ CI0 for every interpretation

I, I0 in LΣ,Φ and every LΣ† ,Φ †-bisimulation Z between I and I0, where Σ† ⊆ Σ,

Theorem 2.1 All concepts of LΣ† ,Φ † are invariant for LΣ† ,Φ †-bisimulation 

This theorem allows to model indiscernibility of objects by the sublanguage Lӆ ,ֆ.Indiscernibility of objects is one of basic features for partitioning data

2.3.3 Invariance of knowledge base

Definition 2.6 A TBox T (respectively, ABoxA) inLΣ† ,Φ † is said to be invariant for

LΣ† ,Φ †-bisimulation if, for every interpretationsI andI0inLΣ,Φ, there exists anLΣ† ,Φ †bisimulation betweenI andI0 thenI is a model ofT (respectively,A) iifI0 is a model

and basic object roles R1, R2, , Rk of LΣ† ,Φ † (k ≥ 0) such that x0 = aI, xk = x

and RIi(xi−1, xi) holds for every 1 ≤ i ≤ k [6]

Theorem 2.2 Let T be a TBox in LΣ† ,Φ †, I and I0 be unreachable-objects-free pretations (w.r.t LΣ† ,Φ†) in LΣ,Φ such that there exists an LΣ† ,Φ†-bisimulation between

inter-I and I0 Then I is a model of T iif I0 is a model of T 

Theorem 2.3 Let A be an ABox in LΣ† ,Φ† If O ∈ Φ† or A contains only assertions

of the form C(a) then A is invariant for LΣ† ,Φ †-bisimulation 

Corollary 2.2 Let KB = hR, T , Aibe a knowledge base in LΣ† ,Φ † such that R = ∅

and either O ∈ Φ† or A contains only assertions of the form C(a), I and I0

be unreachable-objects-free interpretations (for LΣ† ,Φ †) in LΣ,Φ such that an LΣ† ,Φ †bisimulation between I and I0 Then I is a model KB iif I0 is a model of KB 

-2.4 The Hennessy-Milner property for bisimulation

Definition 2.7 An interpretationI in LΣ,Φ is said to be finitely branching (or finite) w.r.t LΣ† ,Φ† if, for every x ∈ ∆I and every role r ∈ Σ†oR then:

image-• the set {y ∈ ∆I | rI(x, y)} is finite,

• if I ∈ Φ† then the set {y ∈ ∆I | rI(y, x)} is finite 

Theorem 2.4 (The Hennessy-Milner property) Let Σ and Σ† be DL-signatures suchthat Σ† ⊆ Σ, Φ and Φ† be sets of DL-features such that Φ† ⊆ Φ Let I and I0 beinterpretations in LΣ,Φ, finitely branching w.r.t.LΣ† ,Φ † and such that for everya ∈ Σ†I,

aI is LΣ† ,Φ †-equivalent to aI0 Assume U /∈ Φ† or Σ†I 6= ∅ Then x ∈ ∆I is LΣ† ,Φ †equivalent to x0 ∈ ∆I0 iff there exists an LΣ† ,Φ†-bisimulation Z between I and I0 such

Corollary 2.3 Let Σ and Σ† be DL-signatures such that Σ† ⊆ Σ, Φ and Φ† be sets

of DL-features such that Φ† ⊆ Φ, I and I0 be finite interpretations in LΣ,Φ Assumethat Σ†I 6= ∅ and, for every a ∈ Σ†I, aI is LΣ† ,Φ †-equivalent to aI0 Then the relation

{hx, x0i ∈ ∆I × ∆I0 | x is LΣ† ,Φ †-equivalent to x0} is an LΣ† ,Φ †-bisimulation between

2.5 Auto-bisimulation

Definition 2.8 (Auto-bisimulation) Let I be an interpretation in LΣ,Φ An LΣ† ,Φ†auto-bisimulation of I is an LΣ† ,Φ †-bisimulation between I and itself An LΣ† ,Φ †-auto-bisimulation of I is said to be the largest if for every LΣ† ,Φ†-bisimulation Z0 of I then

Given an interpretation I in LΣ,Φ, by ∼Σ† ,Φ†,I we denote the largest LΣ† ,Φ†bisimulation of I, and by ≡Σ† ,Φ † ,I we denote the binary relation on ∆I with theproperty that x ≡Σ† ,Φ†,I x0 iff x is LΣ† ,Φ†-equivalent to x0

-auto-Theorem 2.5 Let Σ and Σ† be DL-signatures such that Σ† ⊆ Σ, Φ and Φ† be sets

of DL-features such that Φ† ⊆ Φ, and I be an interpretation in LΣ,Φ Then:

• the largest LΣ† ,Φ †-auto-bisimulation of I exists and is an equivalence relation,

• if I is finitely branching w.r.t LΣ† ,Φ† then the relation ≡Σ† ,Φ†,I is the largest

LΣ† ,Φ †-auto-bisimulation of I (i.e the relations ≡Σ† ,Φ † ,I and ∼Σ† ,Φ † ,I coincide).

We say that a set Y is divided by a set X if Y \ X 6= ∅ and Y ∩ X 6= ∅ Thus, Y

is not divided by X if either Y ⊆ X or Y ∩ X = ∅ A partition Y = {Y1, Y2, , Yn}

is consistent with a set X if, for every 1 ≤ i ≤ n, Yi is not divided by X

Theorem 2.6 Let Σ and Σ† be DL-signatures such that Σ† ⊆ Σ, Φ and Φ† be sets

of DL-features such that Φ† ⊆ Φ, I be a finite interpretation in LΣ,Φ, and X ⊆ ∆I.Let Y be a partition of ∆I by the relation ∼Σ† ,Φ†,I Then:

1 if there exists a concept C of LΣ† ,Φ † such that CI = X then the partition Y isconsistent with X,

2 if the partition Y is consistent with X then there exists a concept C of LΣ† ,Φ †

Summary of Chapter 2

By the language LΣ,Φ and the sublanguage LΣ† ,Φ †, this chapter presents tions and invariant for bisimulations in the class of DLs as mentioned in Chapter 1.The definitions, theorems, lemmas and corollaries are developed from results of theworks [6], [14] for a larger class of DLs We have proved the theorems, lemmas showed

bisimula-in this chapter Invariance is one of the techniques that allows to model bisimula-indiscernibility

of objects by the sublanguage Lӆ ,ֆ Indiscernibility of objects is the key features forpartitioning data Thus, we can use the sublanguages and bisimulation-based methodsfor learning machine problems in DLs

Chương 3.

CONCEPT LEARNING FOR INFORMATION SYSTEMS

IN DESCRIPTION LOGICS

3.1 Information systems

3.1.1 Traditional information systems

An information system is defined as follows [15]:

Definition 3.1 An information system is a tuple IS = hU, A, V, ρi, where:

• U is a non-empty finite set, called universal set of objects,

• A is a non-empty finite set, called set of attributes,

• V = S

a∈A

Va, where Va is a non-empty set of values of the attribute a ∈ A and Va

called range of values of a,

• ρ : U×A → V is an information function, such that ρ(u, a) ∈ Va for every u ∈ U

Limitation of traditional information systems does not describe relationships tween objects in those systems

be-3.1.2 Description logic-based information systems

Definition 3.2 (Acyclic knowledge base) An Acyclic knowledge base inLΣ,Φ is a pair

KB = hR, T , Ai, where:

• R is a finite list (ψ1, ψ2, , ψm) Each ψi is a role axiom of the form r ≡ R,where R is an objects role of LΣ,Φ and r ∈ ΣoR is an object role name notoccurring in R, A and ψ1, ψ2, , ψi−1,

• T is a finite list (ϕ1, ϕ2, , ϕn) Each ϕi is a terminological axiom of the form

A ≡ C, where C is a concept of LΣ,Φ and A ∈ ΣC is a concept name notoccurring in C, A and ϕ1, ϕ2, , ϕi−1,

• A is a finite set of individual assertions 

Given a acyclic knowledge base KB = hR, T , Ai A model I of KB in LΣ,Φ iscalled a standard model if I satisfies the following conditions:

• ∆I = ΣI (i.e, the domain of I consists all individual names in Σ),

• if A ∈ ΣC is a primitive concept of KB then AI = {a | A(a) ∈ A},

• if B ∈ ΣA \ ΣC then BI : ∆I → range(B) is a partial function such that

BI(aI) = c if (B(a) = c) ∈ A,

• if r ∈ ΣoR is a primitive object role of KB then rI= {ha, bi| r(a, b) ∈ A},

• if σ ∈ ΣdR then σI = {ha, di | σ(a, d) ∈ A},