1. Description Logics
in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Description Logics in RTE Terminological
Axioms
Assertions
Concrete Domains
Comparison
Reasoning
Kilian Evang for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
Background
2009-07-20 Knowledge
ABox Saturation
Subgraph Detection
Back Matter

2. Description Logics
Description Logics in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
a family of logics Roles
Concepts
Terminological
origins in research on knowledge representation systems Axioms
Assertions
widely used in practice, notably in Semantic Web Concrete Domains
Comparison
technology Reasoning
for Concepts
address expressivity-tractability tradeoff: adequate for Knowledge Bases
[Bedaride, 2003]
knowledge representation, useful inferencing T and H
Background
basic standard DL called AL Knowledge
ABox Saturation
degree of expressivity of a DL can be expressed in terms Subgraph Detection
Back Matter
of additional constructs added to AL

3. Description Logics
Individuals, Concepts, Roles in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Axioms
Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter
[Horridge et al., 2007], p. 13

4. Description Logics
SHOIN (D) in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Axioms
chosen here because the XML description language Assertions
Concrete Domains
OWL DL is based on it Comparison
Reasoning
OWL DL and its subset OWL Lite widely used in for Concepts
for Knowledge Bases
Semantic Web technology [Bedaride, 2003]
extends ALC of [Bedaride, 2003] by several constructs T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

5. Description Logics
Expressions in SHOIN (D) in RTE
Kilian Evang
Introduction
individual names SHOIN (D)
Individual Names
example: paul Roles
Concepts
denote individuals aka objects Terminological
Axioms
concepts (aka classes) Assertions
Concrete Domains
example: Person Comparison
Reasoning
denote sets of individuals for Concepts
roles (aka properties) for Knowledge Bases
[Bedaride, 2003]
example: hasChild T and H
Background
denote binary relations between individuals, i.e. sets of Knowledge
ABox Saturation
ordered pairs of individuals Subgraph Detection
formulas Back Matter
terminological axioms
assertions

6. Description Logics
Interpretations in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
An interpretation I consists of Axioms
Assertions
a domain ∆I of individuals and
Concrete Domains
Comparison
an interpretation function ·I that maps Reasoning
for Concepts
I
individual names to elements of ∆ for Knowledge Bases
concept descriptions to subsets of ∆I [Bedaride, 2003]
T and H
role descriptions to subsets of ∆I × ∆I Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

7. Description Logics
Individual Names in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Syntax: a Terminological
Axioms
Semantics: a I ∈ ∆I Assertions
Concrete Domains
Comparison
Example: paul Reasoning
for Concepts
Understand: “the individual named paul” for Knowledge Bases
[Bedaride, 2003]
Unique name assumption: an interpretation assigns each T and H
Background
Knowledge
individual name a different individual. ABox Saturation
Subgraph Detection
Back Matter

8. Description Logics
Atomic Roles in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Syntax: R Terminological
Axioms
Semantics: R I ⊆ ∆I × ∆I Assertions
Concrete Domains
Comparison
Example: hasChild Reasoning
for Concepts
Understand: “the set of all parent-child pairs” for Knowledge Bases
[Bedaride, 2003]
Example: isChildOf T and H
Background
Knowledge
Understand: “the set of all child-parent pairs” ABox Saturation
Subgraph Detection
Back Matter

9. Description Logics
Inverse Roles in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Syntax: R− Concepts
Terminological
Axioms
Semantics: {(x, y) | (y, x) ∈ R I } Assertions
Concrete Domains
Comparison
Example: hasChild− Reasoning
for Concepts
Understand: “the set of all child-parent pairs” for Knowledge Bases
[Bedaride, 2003]
Example: isChildOf − T and H
Background
Knowledge
Understand: “the set of all parent-child pairs” ABox Saturation
Subgraph Detection
Back Matter

10. Description Logics
Atomic Concepts in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Axioms
Syntax: A Assertions
Concrete Domains
Semantics: AI ⊆ ∆I Comparison
Reasoning
for Concepts
Example: Person for Knowledge Bases
Understand: “the set of all persons” [Bedaride, 2003]
T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

11. Description Logics
Conjunction in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Syntax: C D Axioms
Assertions
Semantics: (C D)I = C I ∩ D I Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
Example: Person Female [Bedaride, 2003]
Understand: “the set of all female persons” T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

12. Description Logics
Disjunction in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Syntax: C D Axioms
Assertions
Semantics: (C D)I = C I ∪ D I Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
Example: Doctor Gardener [Bedaride, 2003]
Understand: “the set of all doctors and gardeners” T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

13. Description Logics
Negation in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Syntax: ¬C Terminological
Axioms
Semantics: (¬C )I ∆I C I Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
Example: ¬Flower for Knowledge Bases
[Bedaride, 2003]
Understand: “the set of all individuals that aren’t T and H
flowers” Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

14. Description Logics
Exists Restriction in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Syntax: ∃R.C Terminological
Axioms
Semantics: (∃R.C )I = {x | ∃y ((x, y ) ∈ R I ∧ y ∈ C I )} Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
Example: ∃hasChild.Person for Knowledge Bases
[Bedaride, 2003]
Understand: “the set of all individulals that have a T and H
child which is a person” Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

15. Description Logics
Number Restrictions in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Syntax: nP, nP Concepts
Terminological
nP)I = {x | |{y | (x, y ) ∈ P I }|
Axioms
Semantics: ( n} Assertions
nP)I = {x | |{y | (x, y ) ∈ P I }|
Concrete Domains
( n} Comparison
Reasoning
for Concepts
for Knowledge Bases
Example: 3hasChild [Bedaride, 2003]
T and H
Understand: “the set of all individuals with at least Background
Knowledge
three children” ABox Saturation
Subgraph Detection
Back Matter

16. Description Logics
Value Restriction in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Syntax: ∀R.C Concepts
Semantics: (∀R.C )I = Terminological
Axioms
Assertions
{x | ∀y ((x, y ) ∈ R I → y ∈ C I )} Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
Example: ∀hasChild.Female
[Bedaride, 2003]
Understand: “the set of all individuals all of whose T and H
Background
children are female (including all Knowledge
ABox Saturation
individuals without any children)” Subgraph Detection
Back Matter

17. Description Logics
Nominals in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Syntax: {o1 , . . . , on } Concepts
Terminological
Axioms
where o1 , . . . , on are individual names Assertions
{o1 , . . . , on }I = {o1 , . . . , on }
I I Concrete Domains
Semantics: Comparison
Reasoning
for Concepts
for Knowledge Bases
Example: {china, france, russia, uk, usa} [Bedaride, 2003]
T and H
Understand: “the set of the permanent members of Background
Knowledge
the UN security council” ABox Saturation
Subgraph Detection
Back Matter

18. Description Logics
The Universal Concept and the Bottom Concept in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Axioms
Syntax: Assertions
Concrete Domains
Semantics: I = ∆I Comparison
Reasoning
for Concepts
Syntax: ⊥ for Knowledge Bases
Semantics: ⊥I = ∅ [Bedaride, 2003]
T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

19. Description Logics
Inclusions in RTE
Kilian Evang
Introduction
Syntax: C D (R S) SHOIN (D)
Individual Names
Semantics: An interpretation I Roles
Concepts
satisfies C D (R S) Terminological
Axioms
iff C I ⊆ D I (R I ⊆ S I ). Assertions
Concrete Domains
Comparison
Reasoning
for Concepts
Example: Apple Fruit for Knowledge Bases
Understand: “Every apple is a fruit.” [Bedaride, 2003]
T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Example: hasTopping hasIngredient
Back Matter
Understand: “Having something as a topping also
means having it as an ingredient.”

20. Description Logics
Equalities in RTE
Kilian Evang
Syntax: C ≡ D (R ≡ S) Introduction
Semantics: An interpretation I SHOIN (D)
Individual Names
satisfies C D (R S) Roles
Concepts
iff C I = D I (R I = S I ). Terminological
Axioms
Assertions
Concrete Domains
Comparison
Example: SpicyPizza ≡ Reasoning
for Concepts
Pizza ∃hasTopping.SpicyTopping for Knowledge Bases
Understand: “A SpicyPizza is defined to be a pizza [Bedaride, 2003]
T and H
with a spicy topping.” Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter
Example: isChildOf ≡ hasChild−
Understand: “isChildOf is defined to be the inverse
role of hasChild.”

21. Description Logics
Transitive Roles in RTE
Kilian Evang
Introduction
SHOIN (D)
Syntax: R ∈ R+ Individual Names
Semantics: R I = (R I )+ Roles
Concepts
Terminological
Axioms
Assertions
Concrete Domains
Example: isPartOf ∈ R+ Comparison
Reasoning
Understand: “If A is a part of B and B is a part for Concepts
for Knowledge Bases
of C, then A is also a part of C.”
[Bedaride, 2003]
T and H
Background
important for part-whole descriptions Knowledge
ABox Saturation
Subgraph Detection
allows for defining concepts that have no finite model Back Matter
[Sattler, 1996]

22. Description Logics
Concept Assertions in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Syntax: C (a) Terminological
Axioms
Semantics: An interpretation I satisfies C (a) iff Assertions
Concrete Domains
aI ∈ C I . Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
Example: Father(peter) T and H
Understand: “Peter is a father.” Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

23. Description Logics
Role Assertions in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Syntax: R(a, b) Axioms
Assertions
Semantics: (a, b)I ∈ R I Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
Example: hasChild(mary, paul) [Bedaride, 2003]
Understand: “Paul is a child of Mary.” T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter

24. Description Logics
Concrete Domains in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Rouhgly and intuitively, concrete domains are a language Concepts
Terminological
extension that allows for “importing” Axioms
√ Assertions
Concrete Domains
“individuals” such as 18, 2, "Zw¨lf Boxk¨mpfer",
o a Comparison
or "Zw¨"
o Reasoning
for Concepts
“roles” such as greaterThan or startsWith for Knowledge Bases
[Bedaride, 2003]
from worlds such as arithmetic or string manipulation into T and H
Background
the logic. OWL DL uses this to assign Knowledge
ABox Saturation
numeric/string/date/... properties to individuals. Subgraph Detection
Back Matter

25. Description Logics
Comparison of Four DLs in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
construct AL ALC S SHOIN (D) Concepts
Terminological
atomic negation Axioms
Assertions
conjunction Concrete Domains
universal quantification Comparison
existential quantification limited Reasoning
for Concepts
disjunction for Knowledge Bases
transitive roles [Bedaride, 2003]
number restrictions T and H
Background
role hierarchies Knowledge
ABox Saturation
inverse roles Subgraph Detection
Back Matter

26. Description Logics
Knowledge Bases in RTE
Kilian Evang
Introduction
SHOIN (D)
a knowledge base is a set of formulas (explicit Individual Names
Roles
knowledge) Concepts
Terminological
Axioms
sometimes divided up into two subsets: Assertions
Concrete Domains
TBox Comparison
contains only terminological axioms Reasoning
for Concepts
provides a general terminology for Knowledge Bases
ABox [Bedaride, 2003]
T and H
contains only assertions Background
Knowledge
provides a specific world description ABox Saturation
Subgraph Detection
also contains implicit knowledge Back Matter
implicit knowledge can be made explicit by reasoning

27. Description Logics
An Example Knowledge Base in RTE
Kilian Evang
TBox
Introduction
SHOIN (D)
Woman ≡ Person Female Individual Names
Roles
Man ≡ Person ¬Woman Concepts
Terminological
Mother ≡ Woman ∃hasChild.Person Axioms
Assertions
Father ≡ Man ∃hasChild.Person Concrete Domains
Comparison
Parent ≡ Father Mother Reasoning
for Concepts
Grandmother ≡ Mother ∃hasChild.Parent for Knowledge Bases
MotherWithManyChildren ≡ Mother 3hasChild [Bedaride, 2003]
T and H
MotherWithoutDaughter ≡ Mother ∀hasChild.¬Woman Background
Knowledge
Wife ≡ Woman ∃hasHusband.Man ABox Saturation
Subgraph Detection
Back Matter
ABox
hasChild(mary, paul), Father(paul)
An example piece of implicit knowledge
Grandmother(mary)

28. Description Logics
Modelhood in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
An interpretation I is a model of (satisifies) Roles
Concepts
Terminological
a formula φ iff it satisfies φ. Axioms
Assertions
Concrete Domains
a TBox T iff it is a model of every terminological axiom Comparison
in T . Reasoning
for Concepts
an ABox A iff it is a model of every assertion in A. for Knowledge Bases
[Bedaride, 2003]
an ABox A with respect to a TBox T iff it is a model T and H
Background
of both A and T . Knowledge
ABox Saturation
a concept C iff C I is nonempty.
Subgraph Detection
Back Matter

29. Description Logics
Reasoning Tasks for Concepts in RTE
Kilian Evang
Introduction
Let C , D concepts and T a TBox (e.g. see above). SHOIN (D)
C is satisfiable wrt. T iff C and T have a common Individual Names
Roles
model. Concepts
Terminological
Axioms
e.g. not satisfiable: Man Woman Assertions
Concrete Domains
C is subsumed by D wrt. T iff C I ⊆ D I for every Comparison
model I of T . Reasoning
for Concepts
e.g. Mother is subsumed by Woman for Knowledge Bases
C and D are equivalent wrt. T iff C I = D I for every [Bedaride, 2003]
T and H
model I of T . Background
Knowledge
e.g. ∃hasChild.Person is equivalent to Father Mother ABox Saturation
Subgraph Detection
C and D are disjoint wrt. T iff C I ∩ D I = ∅ for every Back Matter
model I of T .
e.g. Man and Woman are disjoint

30. Description Logics
Reasoning Tasks for Knowledge Bases in RTE
Kilian Evang
Let K a knowledge base. Introduction
consistency checking: K is consistent iff it has a SHOIN (D)
Individual Names
model. Roles
Concepts
e.g. above KB is consistent, adding Mother(paul) Terminological
Axioms
would make it inconsistent Assertions
Concrete Domains
instance checking: Given a concept C and an Comparison
individual name a, K entails C (a) iff K ∪ {¬C (a)} is Reasoning
for Concepts
inconsistent. for Knowledge Bases
e.g. Grandmother(mary) is entailed by above KB [Bedaride, 2003]
T and H
retrieval problem: Given a concept C , find all Background
Knowledge
individual names a such that K entails C (a). ABox Saturation
Subgraph Detection
e.g. the result for ∃hasChild.Person would be {mary} Back Matter
realization problem: Given an individual name a, find
the most specific concepts C such that K entails C (a).
...

31. Description Logics
[Bedaride, 2003]: RTE in Four Steps in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
RTE in four steps: Terminological
Axioms
1. represent T and H as two ABoxes Assertions
Concrete Domains
2. make a TBox with background knowledge Comparison
3. saturate ABoxes with TBox Reasoning
for Concepts
4. subgraph-detect ABox H in ABox T for Knowledge Bases
Example T/H pair: [Bedaride, 2003]
T and H
T: “John buys a cat at the pet shop for 50 euros.” Background
Knowledge
H: “A shop sells an animal to John.” ABox Saturation
Subgraph Detection
Back Matter

32. Description Logics
Step 1: Represent T and H as Two ABoxes in RTE
Kilian Evang
Introduction
ABox T = {CommercialTransaction(ct1), John(j1), SHOIN (D)
Individual Names
PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), Roles
Concepts
seller(ct1, ps1), goods(ct1, c1), money(ct1, p1)} Terminological
Axioms
Assertions
ABox H = {CommercialTransaction(ct2), John(j2), Concrete Domains
Comparison
Shop(s2), Animal(a2), buyer(ct2, j2), Reasoning
seller(ct2, s2), goods(ct2, a2)} for Concepts
for Knowledge Bases
Note: [Bedaride, 2003]
T and H
FrameNet frames and frame elements represented as Background
Knowledge
individuals, characterized by concept assertions ABox Saturation
connected via frame-specific roles Subgraph Detection
Back Matter
no difference made between common/proper,
definite/indefinite, singular/plural NP
each ABox has its own set of individual names

33. Description Logics
Step 2: TBox with Background Knowledge in RTE
Kilian Evang
ABox T = {CommercialTransaction(ct1), John(j1), Introduction
PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), SHOIN (D)
Individual Names
seller(ct1, ps1), goods(ct1, c1), money(ct1, p1)} Roles
Concepts
ABox H = {CommercialTransaction(ct2), John(j2), Terminological
Axioms
Shop(s2), Animal(a2), buyer(ct2, j2), Assertions
Concrete Domains
seller(ct2, s2), goods(ct2, a2)} Comparison
Reasoning
TBox BK = {PetShop Shop, Cat Animal} for Concepts
for Knowledge Bases
Note: [Bedaride, 2003]
T and H
atomic concepts mapped to WordNet synsets (how – Background
Knowledge
WSD?) ABox Saturation
for each pair (Sh , St ) of synsets from H and T, check if Subgraph Detection
Back Matter
there is a relation and if so,
add the appropriate axiom(s) to the TBox: Sh St for
hyponymy, St Sh for hypernymy, Sh St and
St Sh for synonymy, Sh ¬St and St ¬Sh for
antonymy

34. Description Logics
Step 3: Saturate ABoxes with TBox in RTE
Kilian Evang
Introduction
SHOIN (D)
TBox BK = {PetShop Shop, Cat Animal} Individual Names
Roles
ABox T = {CommercialTransaction(ct1), John(j1), Concepts
Terminological
Axioms
PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), Assertions
Concrete Domains
seller(ct1, ps1), goods(ct1, c1), money(ct1, p1), Comparison
Shop(ps1), Animal(c1)} Reasoning
for Concepts
ABox H = {CommercialTransaction(ct2), John(j2), for Knowledge Bases
[Bedaride, 2003]
Shop(s2), Animal(a2), buyer(ct2, j2), T and H
Background
seller(ct2, s2), goods(ct2, a2)} Knowledge
ABox Saturation
Note: Subgraph Detection
Back Matter
T (H ) is T (H) saturated with BK , i.e. containing
every assertion entailed by BK ∪ T (BK ∪ H)

35. Description Logics
Step 4: Subgraph-Detect H in T in RTE
Kilian Evang
Introduction
SHOIN (D)
Let σ = {ct2/ct1, j2/j1, a2/c1, s2/ps1} Individual Names
Roles
ABox T = {CommercialTransaction(ct1), John(j1), Concepts
Terminological
PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), Axioms
Assertions
seller(ct1, ps1), goods(ct1, c1), money(ct1, p1), Concrete Domains
Comparison
Shop(ps1), Animal(c1)} Reasoning
for Concepts
ABox H σ = {CommercialTransaction(ct1), for Knowledge Bases
John(j1), Shop(ps1), Animal(c1), buyer(ct1, j1), [Bedaride, 2003]
T and H
seller(ct1, ps1), goods(ct1, c1)} Background
Knowledge
ABox Saturation
Note: Subgraph Detection
We detect entailment iff we can find a individual name Back Matter
substitution σ such that H σ ⊆ T , i.e. all information
in H is also in T .

36. Description Logics
References in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Franz Baader, Diego Calvanese, Deborah L. McGuiness, Concepts
Terminological
Daniele Nardi and Peter F. Patel-Schneider (2003) Axioms
Assertions
The description logic handbook: theory, implementation, Concrete Domains
Comparison
and applications Reasoning
Cambride University Press for Concepts
for Knowledge Bases
[Bedaride, 2003]
Paul Bedaride (2003) T and H
Using Description Logics for Recognising Textual Background
Knowledge
ABox Saturation
Entailment Subgraph Detection
In: Proceedings of the Twelfth ESSLLI Student Session Back Matter

37. Description Logics
References in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Matthew Horridge, Simon Jupp, Georgina Moulton, Terminological
Axioms
Alan Rector, Robert Stevens and Chris Wroe (2007) Assertions
Concrete Domains
A Practical Guide to Building OWL Ontologies Using Comparison
Prot´g´ 4 and CO-ODE Tools, Edition 1.1
e e Reasoning
for Concepts
for Knowledge Bases
Ulrike Sattler (1996) [Bedaride, 2003]
A concept language extended with different kinds of T and H
Background
transitive roles Knowledge
ABox Saturation
Springer Subgraph Detection
Back Matter

38. Description Logics
in RTE
Kilian Evang
Introduction
SHOIN (D)
Individual Names
Roles
Concepts
Terminological
Axioms
Assertions
RteClassMember ∃thanks− .{kilian} Concrete Domains
Comparison
Reasoning
for Concepts
for Knowledge Bases
[Bedaride, 2003]
T and H
Background
Knowledge
ABox Saturation
Subgraph Detection
Back Matter