Introduction to the Semantic Web - Description...
Transcript of Introduction to the Semantic Web - Description...
Description Logic Basics Beyond ALC
ITS 489 ◦ ICT ◦ SIIT
Introduction to the Semantic WebDescription Logics
Boontawee [email protected]
August 17, 2009
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Description Logic Basics Beyond ALC
Outline
1 Description Logic Basics
2 Beyond ALC
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Description Logic Basics Beyond ALC
Syntax...of the Description Logic ALC
Definition
Let CN and RN be two disjoint sets of concept and role names,respectively. Then the following are concept descriptions:
>, ⊥, A,
¬C ,
C u D,
C t D,
∃r .C ,
∀r .C ,
where A ∈ CN, r ∈ RN, and C , D are concept descriptions.
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Examples...of ALC concept descriptions
Example
1 Person uMaleMale persons
2 Person u ∃child.>Persons that have a child
3 Father tMotherCollection of fathers and mothers
4 Mother u ∀child.HappyMothers whose all children are happy
5 Woman u ∀child.HappyWomen whose all children, if any, are happy
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Semantics...of the Description Logic ALC
The semantics in DL is defined by means of interpretations,which is based on the set theory
Definition
An interpretation I is a pair (∆I , ·I) of
the interpretation domain ∆I and
the interpretation function ·IMapping each concept to a subset of ∆I
Mapping each role to a subset of ∆I ×∆I
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Interpretations...of ALC concept descriptions
Definition
>I := ∆I
⊥I := ∅AI ⊆ ∆I
(¬C )I := ∆I\C I(C u D)I := C I ∩ DI
(C t D)I := C I ∪ DI
(∃r .C )I := {d | there exists e s.t. (d , e) ∈ rI and e ∈ C I}(∀r .C )I := {d | for all e, if (d , e) ∈ rI then e ∈ C I}
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Examples...of an interpretation
Example
Let ∆I be {d0, d1, d2, d3, d4, d5},and ·I defined as follows:
PersonI ={d0, d1, d2, d3, d4}
MaleI = {d1, d3}
FemaleI = {d0, d2, d4}
HappyI = {d3, d4, d5}
1 (Person uMale)I
=
2 (Person u ∃child.>)I
=
3 (Father tMother)I
=
4 (Mother u ∀child.Happy)I
=
5 (Woman u ∀child.Happy)I
=
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Concept AxiomsConcept definitions
TBox defines the terminology by means of concept definitions
Concept definition
A ≡ C(Complete) concept definitionConcept name A is defined to be precisely the same asconcept description C , specifying both necessary andsufficient conditions
A v CPrimitive concept definitionConcept name A is defined to satisfy the necessaryconditions given in concept description C
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Examples...of ALC concept definitions
Example
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.PersonFather ≡ Man u ∃child.Person
Mother ≡ Woman u ∃child.PersonFatherWODaughter ≡ Father u ∀child.Male
ManWODaughter ≡ Man u ∀child.MaleFamily v SocialUnit u ∃part.Person
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Concept AxiomsConcept definitions
The semantics is again based on interpretations
Semantics of concept definition
Let I be an interpretation.
I satisfies A ≡ C if AI = C I
I.e. A and C are interpreted to be the same set of objects
I satisfies A v C if AI ⊆ C I
I.e. A is interpreted to be a subset of C ’s interpretation
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Interpretations...of ALC concept descriptions
Example
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.PersonFather ≡ Man u ∃child.Person
Mother ≡ Woman u ∃child.PersonFatherWODaughter ≡ Father u ∀child.Male
ManWODaughter ≡ Man u ∀child.MaleFatherWOChild ≡ Father u ∀child.⊥
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Concept AxiomsGeneral concept inclusions
TBox can constrain on concepts readily defined somewhere
General concept inclusion (GCI)
C v DConcept description C is more specific than (or, mustimply) concept description D
Example
Man uMarried u ∃hasPet.Dog v HappyParent v ∃hasResponsibility.ChildRaise
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Description Logic-based OntologyTBox & ABox
Definition
An ontology based on Description Logic, or knowledge based,comprises:
TBox—terminological partBackground knowledge about the domain in general
ABox—assertional partFacts about individual objects from the domain
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TBoxesVarious kinds of TBoxes abound
Unfoldable TBox
Only concept definitions can be used, and the followingconditions must be respected:
Definitional—there can be at most one definition foreach concept nameI.e. Impossible to have A ≡ C and A v D in TAcyclic—there is no cyclic dependency in TI.e. Impossible to have A ≡ B u C and B ≡ ∃r .A in T
General TBox
Both GCIs and concept definitions can be used!
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Examples of TBoxes
Is this an unfoldable TBox?
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.Person
Is this an unfoldable TBox?
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.PersonPerson ≡ Man tWoman
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General TBox vs Unfoldable TBox
Every unfoldable TBox is a general TBox
Using GCIs to simulate concept definitions
A v C is a special kind of GCI
A ≡ C can be simulated by two GCIs to formulateimplications in both directions
A v C — the necessary conditionC v A — the sufficient condition
Is this a general TBox?
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.PersonPerson ≡ Man tWoman
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Semantics of GCIs and TBoxes
The semantics is again based on interpretations
Semantics of GCIs
Let I be an interpretation. Then I satisfies C v D ifC I ⊆ DI
I.e. C is interpreted to be a subset of D’s interpretation
Semantics of TBox
Let T be a (general or unfoldable) TBox. An interpretation Iis a model of T if I satisfies all axioms in T
If A ≡ D, then AI = DI
If C v D, then C I ⊆ DI
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Reasoning ProblemsConcept satisfiability
Concepts are intended to represent classes of objects
They should not be interpreted as the empty set ∅
Concept satisfiability
Concept C is satisfiable if there is an interpretation I suchthat CI 6= ∅
Concept C is satisfiable w.r.t. TBox T if there is a model Iof T such that CI 6= ∅
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ExamplesConcept satisfiability
This is an unsatisfiable concept
(Man u ∃child.Female) u ∀child.⊥
Are these satisfiable?
⊥, Female uMale, A t ¬A, A u ¬A, ∀r .⊥, ∃r .⊥
Now consider satisfiability w.r.t. TBox
Man ≡ Person uMaleWoman ≡ Person u Female
Male ≡ ¬Female
Is Female uMale satisfiable w.r.t. the TBox?
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Reasoning ProblemsConcept subsumption
Concepts in DL correspond to classes in OWL
Concept subsumption in DL correspond to subclassrelation in OWL
Concept subsumption
Concept D subsumes C (C v D) if there, for allinterpretations I, CI ⊆ DI
Concept D subsumes C w.r.t. TBox T (C vT D) if there, forall models I of T , CI ⊆ DI
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ExamplesConcept subsumption
Without TBox
∃child.Doctor t ∃child.Lawyer v ∃child.(Doctor t Lawyer)∀child.(Rich u Happy) v ∀child.Rich u ∀child.Happy
Which subsumptions hold true?
⊥ v A, A v >, A u B v A, A v A t B
Now consider subsumption w.r.t. TBox
Parent ≡ Person u ∃child.PersonFather ≡ Man u ∃child.Person
Man ≡ Person uMale
Is it the case that Father vT Parent?
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Syntax...of the Description Logic ALC
Definition
Let CN and RN be two disjoint sets of concept and role names,respectively. Then the following are concept descriptions:
Atomic concepts >, ⊥, A,
Negation ¬C ,
Conjunction C u D,
Disjunction C t D,
Existential restriction ∃r .C ,
Value restriction ∀r .C ,
where A ∈ CN, r ∈ RN, and C , D are concept descriptions.
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Syntax...of the Description Logic SHOIN & SHOIQ
Definition
Apart from CN and RN, we have a third disjoint set ofindividuals Ind. We can use all the concept constructs inALC, and MORE!
Nominal concept {a}Unqualified number restriction (≤ n r), (≥ n r)
Qualified number restriction (≤ n r C ), (≥ n r C )
where a ∈ Ind, r ∈ RN, and C is concept description.
SHOIQ is more than just a concept language
S stands for ALC plus role transitivity: trans(r)
H stands for role hierarchy: r v s
I stands for inverse role: r−B Suntisrivaraporn ◦ Introduction to the Semantic Web 25
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Semantics...of Description Logics
The semantics in DL is defined by means of interpretations,which is based on the set theory
Definition
An interpretation I is a pair (∆I , ·I) of
the interpretation domain ∆I and
the interpretation function ·IMapping each concept to a subset of ∆I
Mapping each role to a subset of ∆I ×∆I
Mapping each individual a to aI ∈ ∆I
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Interpretations...of ALC concept descriptions
Definition
>I := ∆I
⊥I := ∅AI ⊆ ∆I
(¬C )I := ∆I\C I(C u D)I := C I ∩ DI
(C t D)I := C I ∪ DI
(∃r .C )I := {d | there exists e s.t. (d , e) ∈ rI and e ∈ C I}(∀r .C )I := {d | for all e, if (d , e) ∈ rI then e ∈ C I}
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Interpretations...of additional constructors in SHOIN & SHOIQ
Concept constructors
{a}I := {aI}(≤ n r)I := {d | ]{e : (d , e) ∈ rI} ≤ n}(≥ n r)I := {d | ]{e : (d , e) ∈ rI} ≥ n}
(≤ n r C )I := {d | ]{e : (d , e) ∈ rI and e ∈ C I} ≤ n}(≥ n r C )I := {d | ]{e : (d , e) ∈ rI and e ∈ C I} ≥ n}
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Interpretations...of additional constructors in SHOIN & SHOIQ
Role constructors
(r−)I := {(b, a) | (a, b) ∈ rI}
Role axiom constructors
Interpretation I satisfies
trans(r)if, for all d , e, f ∈ ∆I , (d , e), (e, f ) ∈ rI ⇒ (d , f ) ∈ rI
r v sif rI ⊆ sI
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Examples...of SHOIQ concept definitions
Example
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.PersonPerson v ∃child−.Person
Chinese v Person u (≤ 1 child Chinese)
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Examples...of SHOIQ concept definitions
Example
Man ≡ Person uMaleWoman ≡ Person u Female
Parent ≡ Person u ∃child.PersonPerson v ∃child−.Person
Chinese v Person u (≤ 1 child Chinese)ParentA ≡ Parent u ∃child.Rich u ∃child.HappyParentB ≡ Parent u ∃child.Male u ∃child.FemaleParentC ≡ Parent u (≥ 2 child)
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