Download - Description Logics in RTE
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Description Logics in RTE
Kilian Evang
2009-07-20
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Description Logics
I a family of logics
I origins in research on knowledge representation systems
I widely used in practice, notably in Semantic Webtechnology
I address expressivity-tractability tradeoff: adequateknowledge representation, useful inferencing
I basic standard DL called ALI degree of expressivity of a DL can be expressed in terms
of additional constructs added to AL
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Individuals, Concepts, Roles
[Horridge et al., 2007], p. 13
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
SHOIN (D)
I chosen here because the XML description languageOWL DL is based on it
I OWL DL and its subset OWL Lite widely used inSemantic Web technology
I extends ALC of [Bedaride, 2003] by several constructs
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Expressions in SHOIN (D)
I individual namesI example: paulI denote individuals aka objects
I concepts (aka classes)I example: PersonI denote sets of individuals
I roles (aka properties)I example: hasChildI denote binary relations between individuals, i.e. sets of
ordered pairs of individuals
I formulasI terminological axiomsI assertions
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Interpretations
An interpretation I consists of
I a domain ∆I of individuals andI an interpretation function ·I that maps
I individual names to elements of ∆I
I concept descriptions to subsets of ∆I
I role descriptions to subsets of ∆I ×∆I
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Individual Names
Syntax: aSemantics: aI ∈ ∆I
Example: paulUnderstand: “the individual named paul”
Unique name assumption: an interpretation assigns eachindividual name a different individual.
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Atomic Roles
Syntax: RSemantics: RI ⊆ ∆I ×∆I
Example: hasChildUnderstand: “the set of all parent-child pairs”
Example: isChildOfUnderstand: “the set of all child-parent pairs”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Inverse Roles
Syntax: R−
Semantics: {(x, y) | (y, x) ∈ RI}
Example: hasChild−
Understand: “the set of all child-parent pairs”
Example: isChildOf−
Understand: “the set of all parent-child pairs”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Atomic Concepts
Syntax: ASemantics: AI ⊆ ∆I
Example: PersonUnderstand: “the set of all persons”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Conjunction
Syntax: C u DSemantics: (C u D)I = CI ∩ DI
Example: Person u FemaleUnderstand: “the set of all female persons”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Disjunction
Syntax: C t DSemantics: (C t D)I = CI ∪ DI
Example: Doctor tGardenerUnderstand: “the set of all doctors and gardeners”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Negation
Syntax: ¬CSemantics: (¬C )I∆I \ CI
Example: ¬FlowerUnderstand: “the set of all individuals that aren’t
flowers”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Exists Restriction
Syntax: ∃R.CSemantics: (∃R.C )I = {x | ∃y((x , y) ∈ RI ∧ y ∈ CI)}
Example: ∃hasChild.PersonUnderstand: “the set of all individulals that have a
child which is a person”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Number Restrictions
Syntax: > nP, 6 nPSemantics: (> nP)I = {x | |{y | (x , y) ∈ PI}| > n}
(6 nP)I = {x | |{y | (x , y) ∈ PI}| 6 n}
Example: > 3hasChildUnderstand: “the set of all individuals with at least
three children”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Value Restriction
Syntax: ∀R.CSemantics: (∀R.C )I =
{x | ∀y((x , y) ∈ RI → y ∈ CI)}
Example: ∀hasChild.FemaleUnderstand: “the set of all individuals all of whose
children are female (including allindividuals without any children)”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Nominals
Syntax: {o1, . . . , on}where o1, . . . , on are individual names
Semantics: {o1, . . . , on}I = {oI1 , . . . , oIn }
Example: {china, france,russia,uk,usa}Understand: “the set of the permanent members of
the UN security council”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
The Universal Concept and the Bottom Concept
Syntax: >Semantics: >I = ∆I
Syntax: ⊥Semantics: ⊥I = ∅
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Inclusions
Syntax: C v D (R v S)Semantics: An interpretation I
satisfies C v D (R v S)iff CI ⊆ DI (RI ⊆ SI).
Example: Apple v FruitUnderstand: “Every apple is a fruit.”
Example: hasTopping v hasIngredientUnderstand: “Having something as a topping also
means having it as an ingredient.”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Equalities
Syntax: C ≡ D (R ≡ S)Semantics: An interpretation I
satisfies C v D (R v S)iff CI = DI (RI = SI).
Example: SpicyPizza ≡Pizza u ∃hasTopping.SpicyTopping
Understand: “A SpicyPizza is defined to be a pizzawith a spicy topping.”
Example: isChildOf ≡ hasChild−
Understand: “isChildOf is defined to be the inverserole of hasChild.”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Transitive Roles
Syntax: R ∈ R+
Semantics: RI = (RI)+
Example: isPartOf ∈ R+
Understand: “If A is a part of B and B is a partof C, then A is also a part of C.”
I important for part-whole descriptions
I allows for defining concepts that have no finite model[Sattler, 1996]
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Concept Assertions
Syntax: C (a)Semantics: An interpretation I satisfies C (a) iff
aI ∈ CI .
Example: Father(peter)Understand: “Peter is a father.”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Role Assertions
Syntax: R(a, b)Semantics: (a, b)I ∈ RI
Example: hasChild(mary,paul)Understand: “Paul is a child of Mary.”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Concrete Domains
Rouhgly and intuitively, concrete domains are a languageextension that allows for “importing”
I “individuals” such as 18,√
2, "Zwolf Boxkampfer",or "Zwo"
I “roles” such as greaterThan or startsWithfrom worlds such as arithmetic or string manipulation intothe logic. OWL DL uses this to assignnumeric/string/date/... properties to individuals.
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Comparison of Four DLs
construct AL ALC S SHOIN (D)
atomic negation X X X Xconjunction X X X Xuniversal quantification X X X Xexistential quantification limited X X Xdisjunction X X Xtransitive roles X Xnumber restrictions Xrole hierarchies Xinverse roles X
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Knowledge Bases
I a knowledge base is a set of formulas (explicitknowledge)
I sometimes divided up into two subsets:I TBox
I contains only terminological axiomsI provides a general terminology
I ABoxI contains only assertionsI provides a specific world description
I also contains implicit knowledge
I implicit knowledge can be made explicit by reasoning
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
An Example Knowledge Base
TBox
Woman ≡ Person u Female
Man ≡ Person u ¬Woman
Mother ≡ Woman u ∃hasChild.Person
Father ≡ Man u ∃hasChild.Person
Parent ≡ Father tMother
Grandmother ≡ Mother u ∃hasChild.Parent
MotherWithManyChildren ≡ Motheru > 3hasChild
MotherWithoutDaughter ≡ Mother u ∀hasChild.¬Woman
Wife ≡ Woman u ∃hasHusband.Man
ABoxhasChild(mary, paul), Father(paul)
An example piece of implicit knowledge
Grandmother(mary)
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Modelhood
An interpretation I is a model of (satisifies)
I a formula φ iff it satisfies φ.
I a TBox T iff it is a model of every terminological axiomin T .
I an ABox A iff it is a model of every assertion in A.
I an ABox A with respect to a TBox T iff it is a modelof both A and T .
I a concept C iff CI is nonempty.
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Reasoning Tasks for Concepts
Let C ,D concepts and T a TBox (e.g. see above).I C is satisfiable wrt. T iff C and T have a common
model.I e.g. not satisfiable: Man uWoman
I C is subsumed by D wrt. T iff CI ⊆ DI for everymodel I of T .
I e.g. Mother is subsumed by WomanI C and D are equivalent wrt. T iff CI = DI for every
model I of T .I e.g. ∃hasChild.Person is equivalent to FathertMother
I C and D are disjoint wrt. T iff CI ∩ DI = ∅ for everymodel I of T .
I e.g. Man and Woman are disjoint
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Reasoning Tasks for Knowledge Bases
Let K a knowledge base.I consistency checking: K is consistent iff it has a
model.I e.g. above KB is consistent, adding Mother(paul)
would make it inconsistent
I instance checking: Given a concept C and anindividual name a, K entails C (a) iff K ∪ {¬C (a)} isinconsistent.
I e.g. Grandmother(mary) is entailed by above KB
I retrieval problem: Given a concept C , find allindividual names a such that K entails C (a).
I e.g. the result for ∃hasChild.Person would be {mary}I realization problem: Given an individual name a, find
the most specific concepts C such that K entails C (a).
I ...
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
[Bedaride, 2003]: RTE in Four Steps
I RTE in four steps:
1. represent T and H as two ABoxes2. make a TBox with background knowledge3. saturate ABoxes with TBox4. subgraph-detect ABox H in ABox T
I Example T/H pair:I T: “John buys a cat at the pet shop for 50 euros.”I H: “A shop sells an animal to John.”
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Step 1: Represent T and H as Two ABoxes
I ABox T = {CommercialTransaction(ct1), John(j1),PetShop(ps1),Cat(c1), 50Euros(p1), buyer(ct1, j1),seller(ct1,ps1), goods(ct1,c1),money(ct1,p1)}
I ABox H = {CommercialTransaction(ct2), John(j2),Shop(s2),Animal(a2),buyer(ct2, j2),seller(ct2, s2), goods(ct2,a2)}
I Note:I FrameNet frames and frame elements represented as
individuals, characterized by concept assertionsI connected via frame-specific rolesI no difference made between common/proper,
definite/indefinite, singular/plural NPI each ABox has its own set of individual names
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Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Step 2: TBox with Background Knowledge
I ABox T = {CommercialTransaction(ct1), John(j1),PetShop(ps1),Cat(c1), 50Euros(p1), buyer(ct1, j1),seller(ct1,ps1), goods(ct1,c1),money(ct1,p1)}
I ABox H = {CommercialTransaction(ct2), John(j2),Shop(s2),Animal(a2),buyer(ct2, j2),seller(ct2, s2), goods(ct2,a2)}
I TBox BK = {PetShop v Shop,Cat v Animal}I Note:
I atomic concepts mapped to WordNet synsets (how –WSD?)
I for each pair (Sh,St) of synsets from H and T, check ifthere is a relation and if so,
I add the appropriate axiom(s) to the TBox: Sh v St forhyponymy, St v Sh for hypernymy, Sh v St andSt v Sh for synonymy, Sh v ¬St and St v ¬Sh forantonymy
![Page 34: Description Logics in RTE](https://reader033.vdocument.in/reader033/viewer/2022052504/5495f834ac7959292e8b4f97/html5/thumbnails/34.jpg)
Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Step 3: Saturate ABoxes with TBox
I TBox BK = {PetShop v Shop,Cat v Animal}I ABox T ′ = {CommercialTransaction(ct1), John(j1),
PetShop(ps1),Cat(c1), 50Euros(p1), buyer(ct1, j1),seller(ct1,ps1), goods(ct1,c1),money(ct1,p1),Shop(ps1),Animal(c1)}
I ABox H ′ = {CommercialTransaction(ct2), John(j2),Shop(s2),Animal(a2),buyer(ct2, j2),seller(ct2, s2), goods(ct2,a2)}
I Note:I T ′ (H ′) is T (H) saturated with BK , i.e. containing
every assertion entailed by BK ∪ T (BK ∪ H)
![Page 35: Description Logics in RTE](https://reader033.vdocument.in/reader033/viewer/2022052504/5495f834ac7959292e8b4f97/html5/thumbnails/35.jpg)
Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
Step 4: Subgraph-Detect H ′ in T ′
I Let σ = {ct2/ct1, j2/j1,a2/c1, s2/ps1}I ABox T ′ = {CommercialTransaction(ct1), John(j1),
PetShop(ps1),Cat(c1), 50Euros(p1), buyer(ct1, j1),seller(ct1,ps1), goods(ct1,c1),money(ct1,p1),Shop(ps1),Animal(c1)}
I ABox H ′σ = {CommercialTransaction(ct1),John(j1),Shop(ps1),Animal(c1),buyer(ct1, j1),seller(ct1,ps1), goods(ct1,c1)}
I Note:I We detect entailment iff we can find a individual name
substitution σ such that H ′σ ⊆ T ′, i.e. all informationin H ′ is also in T ′.
![Page 36: Description Logics in RTE](https://reader033.vdocument.in/reader033/viewer/2022052504/5495f834ac7959292e8b4f97/html5/thumbnails/36.jpg)
Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
References
Franz Baader, Diego Calvanese, Deborah L. McGuiness,Daniele Nardi and Peter F. Patel-Schneider (2003)The description logic handbook: theory, implementation,and applicationsCambride University Press
Paul Bedaride (2003)Using Description Logics for Recognising TextualEntailmentIn: Proceedings of the Twelfth ESSLLI Student Session
![Page 37: Description Logics in RTE](https://reader033.vdocument.in/reader033/viewer/2022052504/5495f834ac7959292e8b4f97/html5/thumbnails/37.jpg)
Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
References
Matthew Horridge, Simon Jupp, Georgina Moulton,Alan Rector, Robert Stevens and Chris Wroe (2007)A Practical Guide to Building OWL Ontologies UsingProtege 4 and CO-ODE Tools, Edition 1.1
Ulrike Sattler (1996)A concept language extended with different kinds oftransitive rolesSpringer
![Page 38: Description Logics in RTE](https://reader033.vdocument.in/reader033/viewer/2022052504/5495f834ac7959292e8b4f97/html5/thumbnails/38.jpg)
Description Logicsin RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
TerminologicalAxioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
BackgroundKnowledge
ABox Saturation
Subgraph Detection
Back Matter
RteClassMember v ∃thanks−.{kilian}