introduction to the semantic web - description...
TRANSCRIPT
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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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:
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
Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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Description Logic Basics Beyond ALC
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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