description logics in rte

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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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)”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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 = ∅


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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]


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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)


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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 ...


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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.”


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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)


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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 ′.


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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


Kilian Evang

Introduction

SHOIN (D)

Individual Names

Roles

Concepts


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}

description logics in rte

Technology