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 INFORMATION SYSTEMS MAJOR: COMPUTER SCIENCE CODE: 62.48.01.01 SUMMARY OF DOCTORAL THESIS OF COMPUTER HUE, 2015 This thesis was completed at: College of Sciences, Hue University Supervisors: Assoc Prof Dr.Sc Nguyen Anh Linh, Warsaw University, Poland Dr Hoang Thi Lan Giao, College of Sciences, Hue University Reviewer 1: Prof Dr.Sc Hoang Van Kiem University of Information Technology, VNU-HCM Reviewer 2: Assoc Prof Dr Doan Van Ban Institute of Information Technology, VAST Reviewer 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 PREFACE Concept learning in description logics (DLs) is similar to binary classification in traditional machine learning However, the problem in the context of DLs differs from the traditional setting in that objects are described not only by attributes but also by binary 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 − of individuals, learn a concept C in LΣ,Φ such that: KB |= C(a) for all a ∈ E + , and KB |= ¬C(a) for all a ∈ E − The sets E + and E − contain positive examples and negative ones of C , respectively • Setting (2): This setting differs from the previous one only in that the second condition is replaced by the weaker one: KB |= 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: I |= C(a) for all a ∈ E + , and I |= ¬C(a) for all a ∈ E − Note that I |= C(a) is the same as I |= ¬C(a) Concept learning in DLs was studied by a number of researchers The related work can be divided into three main groups The first group focuses on learnability in DLs [4, 8] Cohen and Hirsh studied PAC-learnability of description logic and proposed a concept learning algorithm called LCSLearn, which is based on “least common subsumers” [4] In [8] Franzier and Pitt provided some results on learnability in the CLASSIC description logic using some kinds 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 Hitzler introduced methods from inductive logic programing for concept learning in DL knowledge 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 and Szalas applied bisimulation in DLs to model indiscernibility of objects [14] They proposed a general bisimulation-based concept learning method for DL-based information systems Divroodi et al [5] studied C-learnability in DLs These works mentioned concept learning in DLs using Setting (3) Apart from the works of Nguyen and Szalas, Divroodi, which are base on bisimulation to guide the search for the result, all other works use refinement operators as in inductive logic programming and/or scoring functions-based search strategies These works focus on concept learning using Setting (1) and Setting (2) for the simple DLs such as ALER, ALN and ALC The works [14, 5] studied bisimulation-based concept learning in DLs using Setting (3) Both of works did not mentioned concept learning in DLs using Setting (1) and Setting (2) From the surveys as outlined above, we found that concept learning in DLs is a key problem It is used to build useful concepts for semantic systems and ontologies Therefore, it impacts on many practical applications which apply Semantic Web for systems This thesis studies bisimulation-based concept learning methods in DLs The main 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 for the 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 systems in DLs using Setting (3); • 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 (1) and Setting (2) Chương 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 atomic concepts 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 The description 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 ∃r.C and ∀r.C are also concepts of ALC , ⊥, ¬C , C D, C D, Definition 1.2 (Semantics) An interpretation in description logic ALC is a pair I = ∆I , ·I , where ∆I is a non-empty set called the domain of I and ·I is a mapping called 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 relation rI ⊆ ∆I × ∆I The interpretation of complex concepts as defined as follows: I = ∆I , ⊥I = ∅, (¬C)I = ∆I \ C I , (∃r.C)I = {x ∈ ∆I | ∃y ∈ ∆I [rI (x, y) ∧ C I (y)]}, (C D)I = C I ∩ DI , (∀r.C)I = {x ∈ ∆I | ∀y ∈ ∆I [rI (x, y) ⇒ C I (y)]}, (C D)I = C I ∪ DI 1.1.3 Knowledge representation From individuals, concepts and roles, one can build a system for representing and reasoning based on DLs include: box of role axioms (RBox), box of terminological axioms (TBox), box of individual assertions (ABox), reasoning system and user interface 1.1.4 Expressiveness Knowledge expressiveness of a DL depends on concept and role constructors which are 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 ALC reg Definition 1.3 (ALC reg 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, while the elements in ΣR are called atomic roles The description logic ALC reg allows concepts and roles defined recursively as follows: • if A ∈ ΣC then A is a concept of ALC reg , • if r ∈ ΣR then r is a role of ALC reg , • if C and D are concepts, R and S are roles then – ε, R ◦ S , R S , R∗ , C? are roles of ALC reg , – , ⊥, ¬C , C D, C D, ∃R.C and ∀R.C are concepts of ALC reg The interpretation of complex roles in ALC reg are defined as follows: (R ◦ S)I = RI ◦ S I , εI = { x, x | x ∈ ∆I }, (R S)I = RI ∪ S I , (C?)I = { x, x | C I (x)} (R∗ )I = (RI )∗ , 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 by I (inverse), O (nominal), F (functionality), N (unqualified number restriction), Q (qualified number restriction), U (universal role), Self (local reflexivity of a role) A set of DL-features is a set consisting of zero or some of these names Definition 1.4 (The LΣ,Φ Language) Let Σ be a DL-signature, Φ be a set of DLfeatures and L stand for ALC reg The DL language LΣ,Φ allows object roles and concepts defined recursively as follows: • 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 = d are concepts of LΣ,Φ , • if A ∈ ΣnA and d ∈ range(A) then A ≤ d, A < d, A ≥ d and A > d are concepts 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 S , R∗ and C? are object roles of LΣ,Φ , , ⊥, ¬C , C D, C 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 if if if if if if if F ∈ Φ then ≤ r is a concept of LΣ,Φ , {F, I} ⊆ Φ then ≤ r− is a concept of LΣ,Φ , N ∈ Φ then ≥ n r and ≤ n r are concepts of LΣ,Φ , {N , I} ⊆ Φ then ≥ n r− and ≤ n r− are concepts of LΣ,Φ , Q ∈ Φ then ≥ n r.C and ≤ n r.C are concepts of LΣ,Φ , {Q, I} ⊆ Φ then ≥ n r− C and ≤ n r− C are concepts of LΣ,Φ , U ∈ Φ then U is an object role of LΣ,Φ , Self ∈ Φ then ∃r.Self is a concept of LΣ,Φ Definition 1.5 (Semantics of LΣ,Φ ) An interpretation in LΣ,Φ is a pair I = ∆I , ·I , where ∆I is a non-empty set called the domain of I and ·I is a mapping called 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 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 σ I ⊆ ∆I × range(σ) The interpretation function ·I is extended to complex object roles and complex concepts as shown in Figure 1.1, where #Γ stands for the cardinality of the set Γ (C?)I = { x, x | C I (x)} (R∗ )I = (RI )∗ (R ◦ S)I = RI ◦ S I εI = { x, x | x ∈ ∆I } (R− )I = (RI )−1 (R S)I = RI ∪ S I {a}I = {aI } U I = ∆I × ∆I (C D)I = C I ∪ DI (C D)I = C I ∩ DI I (A ≤ d)I = {x ∈ ∆I | AI (x) xác định AI (x) ≤ d} = ∆I ⊥I = ∅ (A ≥ d)I = {x ∈ ∆I | AI (x) xác định AI (x) ≥ d} (¬C)I = ∆I \ C I (A = d)I = {x ∈ ∆I | AI (x) = d} (A = d)I = (¬(A = d))I (A < d)I = ((A ≤ d) (A = d))I (A > d)I = ((A ≥ d) (A = d))I (∀R.C)I = {x ∈ ∆I | ∀y [RI (x, y) ⇒ C I (y)]} (∃r.Self)I = {x ∈ ∆I | rI (x, x)} (∃R.C)I = {x ∈ ∆I | ∃y [RI (x, y) ∧ C I (y)]} (∃σ.{d})I = {x ∈ ∆I | σ I (x, d)} (≥ n R.C)I = {x ∈ ∆I | #{y | RI (x, y) ∧ C I (y)} ≥ n} (≥ n R)I = (≥ n R )I (≤ n R.C)I = {x ∈ ∆I | #{y | RI (x, y) ∧ C I (y)} ≤ n} (≤ n R)I = (≤ n R )I Hình 1.1: Interpretation of complex object roles and complex concepts 1.3 Normal forms 1.3.1 Negation normal form of concepts A concept C is in negation normal form if negation only occurs in front of concept names 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 role R is 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± | r ∈ ΣoR } A basic object role is an element in Σ±oR oR = ΣoR ∪ {r (respectively, ΣoR ) if the considered language allows inverse roles (respectively, does not 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 the form ε r or R1 ◦ .◦Rk r, where k ≥ 1, r ∈ ΣoR and R1 , , Rk are basic object roles 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 object roles of LΣ,Φ different from U A role axiom in LΣ,Φ is a role inclusion axiom or a role assertion in LΣ,Φ Definition 1.7 (Box of role axioms) A box of role axioms (RBox) in LΣ,Φ is a finite set of role axioms in LΣ,Φ 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 D, where C and D are concepts in LΣ,Φ A concept equivalent axiom in LΣ,Φ is an expression of the form C ≡ D, where C and D are concepts in LΣ,Φ A terminological axiom in LΣ,Φ is a general concept inclusion axiom or a concept equivalent axiom in LΣ,Φ 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 expression of the form C(a), R(a, b), ¬R(a, b), a = b and a = b, where C is a concept and R is an object role of LΣ,Φ 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 = R, T , A , 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, TBox T , ABox A), denoted by I |= R (respectively, I |= T , I |= A) if it validates all the role axioms of R (respectively, terminological axioms of T , individual assertions of A) An interpretation I is a model of knowledge base KB = R, T , A , denoted by I |= KB , if it is a model of R, T , A Example 1.1 This example concerns knowledge bases about publications Let Φ = {I, O, N , Q}, ΣI = {P1 , P2 , P3 , P4 , P5 , P6 }, ΣC = {Pub, Awarded , UsefulPub, Ad }, ΣdA = ΣC , ΣnA = {Year }, ΣoR = {cites, cited_by}, ΣdR = ∅, − R = {cites cited_by, cited_by − cites, Irr(cites)}, T ={ Pub, UsefulPub ≡ ∃cited_by }, A0 = {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 = A0 ∪ {(¬∃cited_by )(P1 ), (∀cited_by.{P2 , P3 , P4 })(P5 )} Then KB = R, T , A0 and KB = R, T , A0 are knowledge bases in LΣ,Φ The axiom P states that the domain of any model of KB or KB consists of only publications 1.5 Reasoning in description logic There are a number of reasoning problems in DL-based knowledge bases One can uses two type of algorithms to solve them include structural subsumption algorithms and tableau algorithms Structural subsumption algorithms have proved effective for simple DLs such as FL0 , FL⊥ , ALN , while tableau ones are usually used to solve reasoning problems for a lager class of DLs such as ALC [11], ALCI [12], ALCIQ [12], SHIQ [13], Summary of Chapter In this chapter, we have introduced a general of DLs, knowledge expressiveness of DLs Based on the syntax and semantics of DLs, we have presented about knowledge base, model of knowledge base and the keys of reasoning in DLs Apart from the general language based on the DLs ALC reg 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 basic elements of the language, include discrete and numeric attributes This approach is suitable for practical information systems based on description logic Chương BISIMUALTION IN DESCRIPTION LOGICS AND INVARIANCE 2.1 Introduction Bisimulations are studied in modal logics [2], [17] Bisimulation is viewed as a binary relation associating systems which describe the similar of two states in that one system as well as the similar of Kripke models Divroodi and Nguyen studied bisimulations for a number of DLs [6] 2.2 Bisimulation 2.2.1 Preliminary 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 I be interpretations in LΣ,Φ A binary relation Z ⊆ ∆I × ∆I is called an LΣ† ,Φ† -bisimulation between I and I 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 , x , y ∈ ∆I : Z(aI , aI ) Z(x, x ) ⇒ [AI (x) ⇔ AI (x )] Z(x, x ) ⇒ [B I (x) = B I (x ) or both are undefined] [Z(x, x ) ∧ rI (x, y)] ⇒ ∃y ∈ ∆I [Z(y, y ) ∧ rI (x , y )] [Z(x, x ) ∧ rI (x , y )] ⇒ ∃y ∈ ∆I [Z(y, y ) ∧ rI (x, y)] Z(x, x ) ⇒ [σ I (x, d) ⇔ σ I (x , d)], (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) if I ∈ Φ† then [Z(x, x ) ∧ rI (y, x)] ⇒ ∃y ∈ ∆I [Z(y, y ) ∧ rI (y , x )] [Z(x, x ) ∧ rI (y , x )] ⇒ ∃y ∈ ∆I [Z(y, y ) ∧ rI (y, x)], (2.7) (2.8) if O ∈ Φ† then Z(x, x ) ⇒ [x = aI ⇔ x = aI ], (2.9) Z(x, x ) ⇒ #{y | rI (x, y)} = #{y | rI (x , y )}, (2.10) if N ∈ Φ† then if {N , I} ⊆ Φ† then Z(x, x ) ⇒ #{y | rI (y, x)} = #{y | rI (y , x )}, (2.11) Z(x, x ) ⇒ [#{y | rI (x, y)} ≤ ⇔ #{y | rI (x , y )} ≤ 1], (2.12) if F ∈ Φ† then • ≤ r, if F ∈ Φ† and r ∈ Σ†oR , • ≤ r− , if {F, I} ⊆ Φ† and r ∈ Σ†oR , • ≥ l r and ≤ m r, if N ∈ Φ† , r ∈ Σ†oR , < l ≤ #∆I and ≤ m < #∆I , • ≥ l r− and ≤ m r− , if {N , I} ⊆ Φ† , r ∈ Σ†oR , < l ≤ #∆I and ≤ m < #∆I , • ≥ l r.Cit and ≤ m r.Cit , if Q ∈ Φ† , r ∈ Σ†oR , ≤ t ≤ k , < l ≤ #CiIt and ≤ m < #CiIt , • ≥ l r− Cit and ≤ m r− Cit , if {Q, I} ⊆ Φ† , r ∈ Σ†oR , ≤ t ≤ k , < l ≤ #CiIt and ≤ m < #CiIt , • ∃r.Self , if Self ∈ Φ† and r ∈ Σ†oR Theorem 3.1 Let I be an information system in LΣ,Φ , and let Σ† ⊆ Σ and Φ† ⊆ Φ To reach the partition corresponding to the equivalence relation ∼Σ† ,Φ† ,I it suffices to start from the partition {∆I } and repeatedly granulate it by using the basic selectors Definition 3.5 (Simple selectors) Let Y1 , , Yn be the blocks that have been created so far in the process of granulating {∆I }, where each block Yi is characterized by a concept Ci such that Yi = CiI A simple selector in LΣ† ,Φ† for splitting a block is either a basic selector or a concept of one of the following forms: • A ≤ d and A < d, where A ∈ Σ†nA , d ∈ range(A) and d is not a minimal element of range(A), • A ≥ d and A > d, where A ∈ Σ†nA , d ∈ range(A) and d is not a maximal element of range(A), • ∃r , ∃r.Ci and ∀r.Ci , where r ∈ Σ†oR and ≤ i ≤ n, • ∃r− , ∃r− Ci and ∀r− Ci , if I ∈ Φ† , r ∈ Σ†oR and ≤ i ≤ n, • ≥ l r.Ci and ≤ m r.Ci , if Q ∈ Φ† , r ∈ Σ†oR , ≤ i ≤ n, < l ≤ #CiI and ≤ m < #CiI , • ≥ l r− Ci and ≤ m r− Ci , if {Q, I} ⊆ Φ† , r ∈ Σ†oR , ≤ i ≤ n, < l ≤ #CiI and ≤ m < #CiI Let D be a set of available selectors Together with the current partition Y, we have D = {D1 , D2 , , Dh }, called the current set of selectors Extended selectors are defined by using the current set of selectors and object roles of the sublanguage as follows Definition 3.6 (Extended selectors) Let the current set of selectors be D = {D1 , D2 , , Dh } An extended selector in LΣ† ,Φ† for splitting a block is a concept of one of the following forms: • ∃r.Du and ∀r.Du , where r ∈ Σ†oR and Du ∈ D, • ∃r−.Du and ∀r−.Du , if I ∈ Φ† , r ∈ Σ†oR and Du ∈ D, 15 • ≥ l r.Du and ≤ m r.Du , if Q ∈ Φ† , r ∈ Σ†oR , Du ∈ D, < l ≤ #DuI and ≤ m < #DuI , • ≥ l r− Du and ≤ m r− Du , if {Q, I} ⊆ Φ† , r ∈ Σ†oR , Du ∈ D, < l ≤ #DuI and ≤ m < #DuI 3.2.3 Simplicity of concepts Definition 3.7 (Modal depth) Let C be a concept in the normal form The modal depth of C , denoted by mdepth(C), is defined to be: • if C is of the form , ⊥, A, A = d, A = d, A > d, A ≥ d, A < d or A ≤ d, • mdepth(D) if C is the normal form of ¬D, • if C is of the form ∃σ.{d}, ∃r.Self , ≥ n R or ≤ n R, • + mdepth(D) if C is of the form ∃R.D, ∀R.D, ≥ n R.D or ≤ n R.D, • max{mdepth(D1 ), mdepth(D2 ), , mdepth(Dn )} if C is of the form {D1 , D2 , , Dn } or {D1 , D2 , , Dn } Definition 3.8 (Length) Let C be a concept in the normal form The length of C , denoted by length(C), is defined to be: • if C is or ⊥, • if C is of the form A, A = d, A = d, A > d, A ≥ d, A < d or A ≤ d, • length(D) if C is the normal form of ơD, if C is of the form ∃σ.{d}, ∃r.Self , ≥ n R or ≤ n R, • + length(D) if C is of the form ∃R.D or ∀R.D, • + length(D) if C is of the form ≥ n R.D or ≤ n R.D, • 1+length(D1 )+length(D2 )+· · ·+length(Dn ) if C is of the form {D1 , D2 , , Dn } or {D1 , D2 , , Dn } A concept is said to be the simplest if its length and its modal depth is minimal 3.2.4 Entropy-based measures Let I is an information system, X and Y be subsets of ∆I , where X plays the role of a set of positive examples, Y plays a block in a partition Definition 3.9 (Entropy) Entropy of Y w.r.t X in ∆I of the information system I , denoted by E∆I (Y /X), is defined as follows: 0, if Y ∩ X = ∅ or Y ⊆ X (3.1) E∆I(Y/X)= #XY #XY #XY #XY log2 − log2 , otherwise − #Y #Y #Y #Y where XY stands for the set X ∩ Y and XY stands for the set X ∩ Y 16 Definition 3.10 (Information gain) Information gain for a selector D to split Y w.r.t X in ∆I of the information system I , denoted by IG∆I (Y /X, D), is defined as follows: #DI Y #DI Y I E∆I (D Y /X)+ E∆I (DI Y /X) (3.2) IG∆I (Y /X, D) = E∆I (Y /X) − #Y #Y where DI Y stands for the set DI ∩ Y and DI Y stands for the set DI ∩ Y In the case ∆I and X are clear from the context, we write E(Y ) and IG(Y, D) instead of E∆I (Y /X) and IG∆I (Y /X, D), respectively 3.2.5 Algorithm of concept learning in description logics using Setting (3) We now describe a bisimulation-based concept learning method for information systems in DLs via Algorithm 3.1 Function chooseBlockSelector is used to choose the best block and selector to split first Function chooseBlockSelector Input : Y, D Output: Yij , Sij such that IG(Yij , Sij ) is maximal, where Yij ∈ Y and Sij ∈ D BS := ∅; foreach Yij ∈ Y foreach Du ∈ D compute IG(Yij , Du ); S := arg max{IG(Yij , Du )}; let Sij be the simplest concept in S; BS := BS ∪ { Yij , Sij }; Du ∈D choose Yij , Sij ∈ BS such that IG(Yij , Sij ) is maximal; return Yij , Sij ; 3.3 Illustrative examples The following examples show the full picture of the effectiveness of basic and extended selectors as mentioned in Section 3.2.2 Firstly, we consider an example of knowledge base and an information system specified by that knowledge base Example 3.1 Given a knowledge base KB = R, T , A in LΣ,Φ with Σ = ΣI ∪ ΣdA ∪ ΣnA ∪ ΣoR ∪ ΣdR and Φ = {I}, where: ΣI = {Ava, Britt, Colin, Dave, Ella, F lor, Gigi, Harry}, ΣC = {Human, M ale, F emale, N ephew, N iece}, ΣoR = {hasChild, hasP arent, hasSibling}, ΣdA = ΣC , ΣnA = ∅, ΣdR = ∅ R = {hasP arent ≡ hasChild− , Sym(hasSibling)}, T = {Human ≡ , N iece ≡ F emale N ephew ≡ M ale ∃hasChild− (∃hasSibling ), ∃hasChild− (∃hasSibling )}, 17 Algorithm 3.1: bisimualtion-based concept learning for information systems in description logics Input : I, Σ† , Φ† , E = E − , E + Output: A concept C such that: • I |= C(a) for all a ∈ E + , and • I |= C(a) for all a ∈ E − I n := 1; Y1 := ∆ ; C1 := ; Y := {Y1 }; D = ∅; create and add selectors to D; /* Definition 3.4, 3.5 and/or 3.6 */ while (Y is not consistent with E) Yij , Sij := chooseBlockSelector(Y, D); if (Yij is not split by SiIj ) then break; 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 s := n + 1; t := n + 2; n := n + 2; Ys := Yij ∩ SiIj ; Cs := Cij Sij ; I Yt := Yij ∩ (¬Sij ) ; Ct := Cij ¬Sij ; if (Yij is not split by E) then LargestContainer [s] := LargestContainer [ij ]; LargestContainer [t] := LargestContainer [ij ]; else if (Ys is not split by E) then LargestContainer [s] := s; if (Yt is not split by E) then LargestContainer [t] := t; Y := Y ∪ {Ys , Yt } \ {Yij }; create and add new selectors to D; /* Definition 3.4, 3.5 and/or 3.6 */ J := ∅; C := ∅; if (Y is consistent with E) then foreach Yij ∈ Y if (Yij contains some aI with a ∈ E + ) then J := J ∪ {LargestContainer [ij ]}; foreach l ∈ J C := C ∪ {Cl }; C := C ; return Crs := Simplify (C); else return failure; A = {F emale(Ava), F emale(Britt), M ale(Colin), M ale(Dave), F emale(Ella), F emale(F lor), F emale(Gigi), hasChild(Ava, Dave), hasChild(Ava, Ella), M ale(Harry), hasChild(Britt, F lor), hasChild(Colin, Gigi), hasChild(Colin, Harry), hasSibling(Britt, Colin), hasSibling(Colin, Britt), hasSibling(Dave, Ella), hasSibling(Ella, Dave), hasSibling(Gigi, Harry), hasSibling(Harry, Gigi)} 18 An information system I of the knowledge base KB can be constructed as follows: ∆I = {Ava, Britt, Colin, Dave, Ella, F lor, Gigi, Harry}, HumanI = ∆I , AvaI = Ava, BrittI = Britt, , Harry I = Harry, F emaleI = {Ava, Britt, Ella, F lor, Gigi}, M aleI = {Colin, Dave, Harry}, hasChildI = { Ava, Dave , Ava, Ella , Britt, F lor , Colin, Gigi , Colin, Harry }, hasP arentI = { Dave, Ava , Ella, Ava , F lor, Britt , Gigi, Colin , Harry, Colin }, hasSibling I = { Britt, Colin , Colin, Britt , Dave, Ella , Ella, Dave , Gigi, Harry , Harry, Gigi }, (hasSibling − )I = hasSibling I , N ieceI = {F lor, Gigi}, N ephewI = {Harry} The next example demonstrates that using both simple selectors and extended selectors is better than using only simple selectors Example 3.2 Consider the information system I given in Example 3.1, and the sublanguage LΣ† ,Φ† with Σ† = {F emale, hasChild, hasSibling} and Φ† = {I}, and X = {F lor, Gigi} (i.e, E = E + , E − with E + = {F lor, Gigi} and E − = {Ava, Britt, Colin, Dave, Ella, Harry}) Learning a definition of X Once can think X as the set of instances of the concept N iece ≡ F emale ∃hasChild− (∃hasSibling ) in I Learning a definition of X in LΣ† ,Φ† by using only simple selectors The resulting concept is simplified to Crs as follows: Crs ≡ F emale ∀hasChild.⊥ (∀hasChild− (¬F emale) ∀hasSibling.⊥) Learning a definition of X in LΣ† ,Φ† by using both simple selectors and extended selectors The resulting concept is Crs ≡ F emale ∃hasChild− (∃hasSibling ) In the second case, the concept ∃hasChild− (∃hasSibling ) is an extended selector It is created by applying the second rule in Definition 3.6 to ∃hasSibling which is one of the available selectors D 3.4 Experimental results The datasets are used for our setting, including the WebKB [16], PokerHand [3] and Family We show the information details of experimental results is as follows: • the average (Avg.) modal depth (Dep.) of the origin concepts (Org.), • the average length (Len.) of the origin concepts, • the average modal depth of the resulting concepts (Res.), • the average length of the resulting concepts, • the average accuracy (Acc.), precision (Pre.), recall (Rec.) and F1 measures, • the standard variant, minimum (Min) and maximum (Max) values of accuracy, precision, recall and F1 measures 19 Bảng 3.1: Evaluation results on the WebKB, PokerHand and Family datasets using 100 random concepts in the DL ALCIQ Avg Dep Res./Org Avg Len Res./Org Avg Acc [Min;Max] Avg Pre [Min;Max] Avg Rec [Min;Max] Avg F1 [Min;Max] Simple Selectors 0.82/1.02 6.81/4.41 93.84±13.50 [33.69;100.0] 92.09±17.04 [32.08;100.0] 92.82±17.32 [23.08;100.0] 91.59±16.68 [27.69;100.0] Simple & Extended Selectors 0.84/1.02 3.40/4.41 94.60±12.20 [33.69;100.0] 92.81±15.93 [32.08;100.0] 93.14±17.17 [23.08;100.0] 92.33±16.17 [27.69;100.0] Simple Selectors 1.41/2.60 37.02/15.97 97.17±08.61 [50.57;100.0] 95.96±14.99 [01.67;100.0] 94.95±14.40 [01.67;100.0] 94.66±14.64 [01.67;100.0] Simple & Extended Selectors 1.23/2.60 3.47/15.97 99.44±02.15 [83.25;100.0] 98.68±09.08 [01.67;100.0] 98.06±09.58 [01.67;100.0] 98.18±09.14 [01.67;100.0] Simple Selectors 2.38/3.34 78.50/18.59 88.50±16.65 [27.91;100.0] 90.60±18.57 [04.55;100.0] 85.66±22.36 [07.69;100.0] 86.09±20.10 [08.70;100.0] Simple & Extended Selectors 2.29/3.34 10.20/18.59 92.79±14.35 [27.91;100.0] 91.99±18.40 [04.55;100.0] 91.75 ±19.82 [07.69;100.0] 90.39±19.89 [08.70;100.0] WebKB dataset PokerHand dataset Family dataset As can be seen in Table 3.1 and the other tables, it is clear that extended selectors are highly effective for reducing the length of the resulting concepts and for obtaining better classifiers We also tested the method using common concepts and sets of given objects on the Family and PokerHand datasets This demonstrates that extended selectors efficiently support the bisimulation-based concept learning method Summary of Chapter In this chapter, we have formulated and proved an important theorem on basic selectors We have generalized and extended the bisimulation-based concept learning algorithm using Setting (3) Together with the partition strategies, this algorithm is checked and tested on two aspects, including theory and experiment Apart from basic selectors, we introduced and used simple selectors as well as extended selectors for our method We used information gain to choose blocks and selectors for granulating partitions Our method is also implemented for a number of DLs We tested the method using simple selectors and extended selectors for different datasets Our experimental results show that the method is valuable and extended selectors support it significantly 20 Chương CONCEPT LEARNING FOR KNOWLEDGE BASE IN DESCRIPTION LOGISC 4.1 Introduction In this chapter the concept learning problems are considered as the two following settings: • Setting (1): Let LΣ,Φ be a decidable logic with the semi-finite model property, Ad ∈ ΣC be a special concept name standing for the “decision attribute”, and KB = R, T , A0 be a knowledge base in LΣ,Φ without using Ad Let E = E + , E − , where E + and E − be disjoint subsets of ΣI such that the knowledge base KB = R, T , A with A = A0 ∪ {Ad (a) | a ∈ E + } ∪ {¬Ad (a) | a ∈ E − } is satisfiable The problem is to learn a concept C as a definition of Ad in the logic LΣ,Φ restricted to a given sublanguage LΣ† ,Φ† with Σ† ⊆ Σ \ {Ad } and Φ† ⊆ Φ such that: KB |= C(a) for all a ∈ E + , and KB |= ¬C(a) for all a ∈ E − • Setting (2): This setting is similar to Setting (1) but the second condition is replaced by a weaker one: KB |= C(a) for all a ∈ E − Note that, two above problems is considered in the open world assumption 4.2 Partitioning the domain of an interpretation We propose an algorithm for granulating partition via Function partition It uses Function chooseBlockSelector for deciding the block and selector to split first Example 4.1 Consider the knowledge bases KB and KB given in Example 1.1 and the interpretation I is a model of KB and KB as follows: ∆I = {P1 , P2 , P3 , P4 , P5 , P6 }, xI = x với x ∈ {P1 , P2 , P3 , P4 , P5 , P6 }, Pub I = ∆I , Awarded I = {P1 , P4 , P6 }, UsefulPub I = {P2 , P3 , P4 , P5 , P6 }, cites I = { P1 , P2 , P1 , P3 , P1 , P4 , , P4 , P5 , P4 , P6 }, cited_by I = (cites I )−1 , the function Year I is specified as usual Let E = E + , E − with E + = {P4 , P6 } E − = {P1 , P2 , P3 , P5 }, the sublanguage LΣ† ,Φ† , where Σ† = {Awarded , cited_by} and Φ† = ∅ The steps of granulation process of ∆I in I via Function partition are described as follows: Y1 := ∆I , C1 := , Y := {Y1 } Dividing Y1 by Awarded we obtain: • Y2 := {P1 , P4 , P6 }, C2 := Awarded , • Y3 := {P2 , P3 , P5 }, C3 := ¬Awarded , ⇒ Y := {Y2 , Y3 } 21 Function partition - Granulating the domain of an interpretation in description logics 10 11 12 13 14 15 Input : I, Σ† , Φ† , E = E + , E − Output: Y = {Yi1 , Yi2 , , Yik } is a partition of ∆I such that Y consistent with E n := 1; Y1 := ∆I ; C1 := ; Y := {Y1 }; D := ∅; create and add selectors to D; /* Definition 3.4, 3.5 and/or 3.6 */ while (Y is not consistent with E) Yij , Sij := chooseBlockSelector(Y, D); if (Yij is not split by SiIj ) then break; s := n + 1; t := n + 2; n := n + 2; Ys := Yij ∩ SiIj ; Cs := Cij Sij ; Yt := Yij ∩ (¬Sij )I ; Ct := Cij ¬Sij ; Y := Y ∪ {Ys , Yt } \ {Yij }; create and add new selectors to D; /* Definition 3.4, 3.5 and/or 3.6 */ if (Y is consistent with E) then return Y; else return failure; We choose the best selector ∃cited_by to split Y2 : • Y4 := {P4 , P6 }, C4 := C2 ∃cited_by , • Y5 := {P1 }, C5 := C2 ¬∃cited_by , ⇒ Y := {Y3 , Y4 , Y5 } The obtained partition is Y = {Y3 , Y4 , Y5 } consistent with E 4.3 Concept learning in description logics using Setting (1) 4.3.1 Algorithm BBCL The main idea of Algorithm BBCL is to used models of KB , bisimulation in those models and decision tree to guide the search for the concept C This algorithm uses Function partition as mentioned in Section 4.2 for granulating the domain ∆I of I which is a model of KB 4.3.2 Algorithm dual-BBCL Algorithm dual-BBCL is used to learn a concept in DLs using Setting (1) One can exchange E + and E − , apply Algorithm BBCL to get a concept Crs and then return the resulting concept ¬Crs 4.3.3 Correctness of Algorithm BBCL Proposition 4.1 (Correctness of Algorithm BBCL) Algorithm BBCL is sound That is, if Algorithm BBCL returns a concept Crs then Crs is a solution of the problem of concept learning for knowledge base in DLs using Setting (1) 4.3.4 Illustrative examples Example 4.2 Let KB = R, T , A0 be the knowledge base given in Example 1.1 and E = E + , E − with E + = {P4 , P6 }, E − = {P1 , P2 , P3 , P5 }, Σ† = {Awarded , 22 Algorithm 4.1: BBCL - Concept learning in description logics using Setting (1) Input : KB, Σ† ⊆ Σ \ {Ad }, Φ† ⊆ Φ, E = E + , E − , k Output: A concept C of LΣ† ,Φ† such that: • KB |= C(a) for all a ∈ E + , and • KB |= ¬C(a) for all a ∈ E − 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 C := ∅; C0 := ∅; while not (too hard to extend C) construct a (next) finite interpretation I of KB or I = I|k , where I is an infinite model of KB; Y := partition (I, Σ† , Φ† , E); /* partitonting ∆I by Function partition */ foreach Yij ∈ Y such that ∃a ∈ E +: aI ∈ Yij and ∀a ∈ E −: aI ∈ Yij if (KB |= ¬Cij (a) for all a ∈ E − ) then if (KB |= (Cij C)) then C := C ∪ {Cij }; else C0 := C0 ∪ {Cij }; if (KB |= ( C)(a) for all a ∈ E + ) then go to 20; while not (too hard to extend C) D := D1 D2 · · · Dl , where D1 , D2 , , Dl randomly selected from C0 ; if (KB |= ¬D(a) for all a ∈ E − and KB |= (D C)) then C := C ∪ {D}; if (KB |= ( C)(a) for all a ∈ E + ) then go to 20; return failure; foreach (D ∈ C) if (KB |= (C \ {D})(a) for all a ∈ E + ) then C := C \ {D}; C := C; return Crs := simplify (C); /* simplying C */ cited_by} and Φ† = ∅ Learning a definition of Ad with KB = R, T , A , where A = A0 ∪ {Ad (a) | a ∈ E + } ∪ {¬Ad (a) | a ∈ E − } Steps of Algorithm BBCL are described as follows: C := ∅, C0 := ∅ KB has many models, but the most natural one is I specified as in Example 4.1 Applying Function partition to granulate the domain ∆I of I , we have a partition Y = {Y3 , Y4 , Y5 } consistent with E corresponding to concepts C3 , C4 , C5 , where Y3 = {P2 , P3 , P5 }, Y4 = {P4 , P6 }, Y5 = {P1 } C3 ≡ ¬Awarded , C4 ≡ Awarded ∃cited_by , C5 ≡ ¬Awarded ∃cited_by Since Y4 ⊆ E + we consider C4 ≡ Awarded 23 ∃cited_by Since KB |= ¬C4 (a) for all a ∈ E − we add C4 to C Therefore, we have C ≡ {C4 } and C ≡ C4 Since KB |= ( C)(a) for all a ∈ E + we have C ≡ C ≡ Awarded ∃cited_by C is already in the normal form and cannot be simplied, we return Crs ≡ Awarded ∃cited_by Example 4.3 We now consider Algorithm dual-BBCL for Example 4.2 but exchange E + and E − Algorithm dual-BBCL has the same first three steps as in Example 4.2 and then continuous as follow: Since Y3 ⊆ E + we consider C3 ≡ ¬Awarded Since KB |= ¬C3 (a) for all a ∈ E − we add C3 to C Therefore, we have C = {C3 } Since Y5 ⊆ E + we consider C5 ≡ Awarded ¬∃cited_by Since KB |= ¬C5 (a) for all a ∈ E − and C5 is not subsumed by C w.r.t KB , we add C5 to C Therefore, we have C = {C3 , C5 } and C ≡ C3 C5 C)(a) for all a ∈ E + we have C ≡ C ≡ ¬Awarded (Awarded ¬∃cited_by ) We normalize C to Crs ≡ ¬Awarded ¬∃cited_by and return it as the result Since KB |= ( If one wants to have a result for the dual learning problem as stated in Example 4.2 we negate to Crs The resulting concept is ¬Crs ≡ ¬(¬Awarded ¬∃cited_by ) ≡ Awarded ∃cited_by 4.4 Concept learning in description logics using Setting (2) 4.4.1 Algorithm BBCL2 Concept learning method for knowledge base in DLs using Setting (2) is described via Algorithm 4.2 4.4.2 Correctness of Algorithm BBCL2 Proposition 4.2 (Correctness of Algorithm BBCL2) Algorithm BBCL2 is sound That is, if Algorithm BBCL2 returns a concept Crs then Crs is a solution of the problem of concept learning for knowledge base in DLs using Setting (2) 4.4.3 Illustrative examples Example 4.4 Let KB = R, T , A0 be a knowledge base given as in Example 1.1 and E = E + , E − , Σ† , Φ† given as in Example 4.2 Let KB = R, T , A with A = A0 ∪ {Ad (a) | a ∈ E + } ∪ {¬Ad (a) | a ∈ E − } Algorithm BBCL2 has the same first three steps as Example 4.2 (adding E0− := ∅) and then continuous as follows: Since Y3 ⊆ E − we consider C3 ≡ ¬Awarded Since KB |= ¬C3 (a) for all a ∈ E + we add ¬C3 in C and add all elements of Y3 in E0− Therefore, we have C = {C3 } and E0− = {P2 , P3 , P5 } Since Y5 ⊆ E − we consider C5 ≡ Awarded ¬∃cited_by Since KB |= ¬C5 (a) for all a ∈ E + and C is not subsumed by C5 w.r.t KB we add ¬C5 in C Therefore, we have C = {¬C3 , ¬C5 } E0− = {P1 , P2 , P3 , P5 } Since E0− = E − we have C ≡ C ≡ ¬¬Awarded ¬(Awarded ¬∃cited_by ) We normalize C to Crs ≡ Awarded ∃cited_by and return it as the result 24 Algorithm 4.2: BBCL2 - Concept learning in description logics using Setting (2) Input : KB, Σ† , Φ† , E = E + , E − , k Output: A concept C such that: • KB |= C(a) for all a ∈ E + , and • KB |= C(a) for all a ∈ E − 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 C := ∅; C0 := ∅; E0− := ∅; while not (too hard to extend C) and (E0− = E − ) construct a (next) finite interpretation I of KB or I = I|k , where I is an infinite model of KB; Y :=partition (I, Σ† , Φ† , E); /* partitioning ∆I by Function partition */ foreach Yij ∈ Y such that ∃a ∈ E − : aI ∈ Yij and ∀a ∈ E + : aI ∈ Yij if (KB |= ¬Cij (a) for all a ∈ E + ) then if (KB |= ( C ¬Cij )) then C := C ∪ {¬Cij }; E0− := E0− ∪ {a ∈ E − | aI ∈ Yij }; else C0 := C0 ∪ {¬Cij }; while not (too hard to extend C) and (E0− = E − ) D := D1 D2 · · · Dl , where D1 , D2 , , Dl randomly selected from C0 ; if (KB |= D(a) for all a ∈ E + ) then if (KB |= ( C) D) and (∃a ∈ E − \ E0− : KB |= ( C D)(a)) then C := C ∪ {D}; E0− := E0− ∪ {a | a ∈ E − \ E0− , KB |= ( C)(a)}; if (E0− = E − ) then foreach D ∈ C if (KB |= (C \ {D})(a) for all a ∈ E − ) then C := C \ {D}; C := C; return Crs := simplify (C); else return failure; /* simplying C */ Summary of Chapter Based on Algorithm 3.1, this chapter presents a method for granulating partitions of the domain of an interpretation in DLs Next, we propose algorithms BBCL, dualBBCL and BBCL2 to solve two problems of concept learning for knowledge base in DLs using Setting (1) and Setting (2) The main idea of these algorithms is to use models of knowledge base, bisimulation in those models and decision tree to guide the search for the resulting concept The correctness of algorithms BBCL and BBCL2 is also proved via lemmas, respectively These algorithms can be applied for a large class of DLs which are the extension of ALC Σ,Φ with finite model property or semi-finite model property, where Φ ⊆ {I, F, N , Q, O, U, Self} 25 CONCLUSION Conclusion DLs play important role in knowledge base presentation and reasoning systems Since DLs are treated as the foundation of OWL (a language used to model the semantics and ontology systems as recommended by W3C), DLs were studied by a number of researchers In semantic systems, finding suitable concepts, building definitions for these concepts to system specification are very natural Concept learning in DLs is one of the solutions for finding and building the suitable concepts This thesis focuses on studying concept learning in DLs using different settings The results of the thesis can be summarized as follows: We have built description logic language LΣ,Φ based on ALC reg with the set of DL-features I , O, N , Q, F , U , Self In addition, we take attributes as basic elements of the language to describe practical information systems We have built bisimulation for the considered larger class of DLs The definitions, theorems, lemmas on bisimulation and the invariant for bisimulation have developed and proved for the mentioned class of DLs We have developed the bisimulation-based concept learning algorithm for DLbased information systems using Setting (3) We used basic, simple and extended selectors for granulating the partitions Search strategies were also proposed on information gain and the simplification of concepts We have built threes algorithms BBCL, dual-BBCL, BBCL2 to solve the concept learning problems for knowledge bases in DLs using Setting (1) and Setting (2) Future work From the results of this thesis, the problems are needed to study in the future as follows: Building efficient concept learning strategies by using different measures for deciding which block should be split first and comparing between strategies Building concept learning modules in DLs using different settings like an API integrated the other available systems Studying semi-supervised, unsupervised and statistic learning algorithms for knowledge bases in DLs Studying C-learnability for different DLs 26 REFERENCES [1] L Badea and S.-H Nienhuys-Cheng A refinement operator for description logics In Proceedings of the 10th International Conference on Inductive Logic Programming, 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Sen, G M Namata, M Bilgic, L Getoor, B Gallagher, and T Eliassi-Rad Collective classification in network data AI Magazine, 29(3):93–106, 2008 [17] J van Benthem Modal Logic for Open Minds Center for the Study of Language and Inf, 2010 28 LIST OF THE AUTHOR’S PUBLICATIONS T.-L Tran, Q.-T Ha, T.-L.-G Hoang, L A Nguyen, H S Nguyen, and A Szalas Concept learning for description logic-based information systems In Proceedings of the 2012 Fourth International Conference on Knowledge and Systems Engineering, KSE’2012, pages 65–73 IEEE Computer Society, 2012 (ISBN: 9781-4673-2171-6) Q.-T Ha, T.-L.-G Hoang, L A Nguyen, H S Nguyen, A Szalas, and T.-L Tran A bisimulation-based method of concept learning for knowledge bases in description logics In Proceedings of the Third Symposium on Information and Communication Technology, SoICT’2012, pages 241–249 ACM, 2012 (ISBN: 978-1-4503-1232-5) Trần Thanh Lương, Hoàng Thị Lan Giao Áp dụng độ đo entropy để phân hoạch khối cho hệ thống thông tin dựa logic mô tả Kỷ yếu Hội thảo quốc gia lần thứ XV: Một số vấn đề chọn lọc Công nghệ Thông tin Truyền thông, trang 11–18, Nhà xuất Khoa học Kỹ thuật, 2013 T.-L Tran, Q.-T Ha, T.-L.-G Hoang, L A Nguyen, and H S Nguyen Bisimulationbased concept learning in description logics In Proceedings of CS&P’2013, pages 421–433 CEURWS.org, 2013 (ISBN: 978-83-62582-42-6) T.-L Tran, L A Nguyen, and T.-L.-G Hoang A domain partitioning method for bisimulationbased concept learning in description logics In Proceedings of ICCSAMA’2014, volume 282 of Advances in Intelligent Systems and Computing, pages 297–312 Springer International Publishing, 2014 (ISBN: 978-3-319-065694) T.-L Tran, Q.-T Ha, T.-L.-G Hoang, L A Nguyen, and H S Nguyen Bisimulationbased concept learning in description logics Fundam Inform., Vol 133, No 2-3, pages 287–303, 2014 (ISSN: 0169-2968) T.-L Tran and T.-L.-G Hoang Entropy-based measures for partitioning the domain of an interpretation in description logics Journal of Science, Hue University, Vol 98, No 8, pages 97–101, 2014 (ISSN: 1859-1388) T.-L Tran, L A Nguyen, and T.-L.-G Hoang Bisimulation-based concept learning for information systems in description logics Vietnam Journal of Computer Science, Vol 2, pages 149–167, 2015 (ISSN: 2196-8888 (print version); ISSN: 2196-8896 (electronic version)) 29 ... for partitioning data Thus, we can use the sublanguages and bisimulation -based methods for learning machine problems in DLs 12 Chương CONCEPT LEARNING FOR INFORMATION SYSTEMS IN DESCRIPTION LOGICS... bisimualtion -based concept learning for information systems in description logics Input : I, Σ† , Φ† , E = E − , E + Output: A concept C such that: • I |= C(a) for all a ∈ E + , and • I |= C(a) for all... respectively 3.2.5 Algorithm of concept learning in description logics using Setting (3) We now describe a bisimulation -based concept learning method for information systems in DLs via Algorithm