Part 6: Description Logics

Languages for Ontologies

• In early days of Artificial Intelligence, ontologies were represented resorting to non-logic-based formalisms– Frames systems and semantic networks

• Graphical representation– arguably ease to design

– but difficult to manage with complex pictures

– formal semantics, allowing for reasoning was missing

Semantic Networks

• Nodes representing concepts (i.e. sets of classes of individual objects)

• Links representing relationships– IS_A relationship– More complex relationships may have nodes

Person

Female

ParentWoman

Mother

hasChild(1,NIL)

Logics for Semantic Networks

• Logics was used to describe the semantics of core features of these networks– Relying on unary predicates for describing sets of

individuals and binary predicates for relationship between individuals

• Typical reasoning used in structure-based representation does not require the full power of 1st order theorem provers– Specialized reasoning techniques can be applied

From Frames to Description Logics

• Logical specialized languages for describing ontologies

• The name changed over time– Terminological systems emphasizing that the language

is used to define a terminology– Concept languages emphasizing the concept-forming

constructs of the languages– Description Logics moving attention to the properties,

including decidability, complexity, expressivity, of the languages

Description Logic ALC• ALC is the smallest propositionally closed

Description Logics. Syntax:– Atomic type:

• Concept names, which are unary predicates• Role names, which are binary predicates

– Constructs• ¬C (negation)• C1 ⊓ C2 (conjunction)• C1 ⊔ C2 (disjunction)R.C (existential restriction)R.C (universal restriction)

Semantics of ALC

• Semantics is based on interpretations (I,.I) where .I maps:– Each concept name A to AI ⊆ I

• I.e. a concept denotes set of individuals from the domain (unary predicates)

– Each role name R to AI ⊆ I x I

• I.e. a role denotes pairs of (binary relationships among) individuals

• An interpretation is a model for concept C iffCI ≠ {}

• Semantics can also be given by translating to 1st order logics

Negation, conjunction, disjunction

• ¬C denotes the set of all individuals in the domain that do not belong to C. Formally– (¬C)I = I – CI

– {x: ¬C(x)}

• C1 ⊔ C2 (resp. C1 ⊓ C2) is the set of all individual that either belong to C1 or (resp. and) to C2– (C1 ⊔ C2)I = C1

I ⋃ C2I resp. (C1 ⊓ C2)I = C1

I ⋂ C2I

– {x: C1(x) ⌵ C2(x)} resp. {x: C1(x) C2(x)}• Persons that are not female

– Person ⊓ ¬Female• Male or Female individuals

– Male ⊔ Female

Quantified role restrictions

• Quantifiers are meant to characterize relationship between concepts

R.C denotes the set of all individual which relate via R with at least one individual in concept C– (R.C)I = {d ∈ I | (d,e) ∈ RI and e ∈ CI}

– {x | y R(x,y) C(Y)}

• Persons that have a female child– Person ⊓ hasChild.Female

Quantified role restrictions (cont)

R.C denotes the set of all individual for which all individual to which it relates via R belong to concept C– (R.C)I = {d ∈ I | (d,e) ∈ RI implies e ∈ CI}– {x | y R(x,y) C(Y)}

• Persons whose all children are Female– Person ⊓ hasChild.Female

• The link in the network above– Parents have at least one child that is a person, and

there is no upper limit for children hasChild.Person ⊓ hasChild.Person

Elephant example

• Elephants that are grey mammal which have a trunck– Mammal ⊓ bodyPart.Trunk ⊓ color.Grey

• Elephants that are heavy mammals, except for Dumbo elephants that are light– Mammal ⊓

(weight.heavy ⊔ (Dumbo ⊓ weight.Light)

Reasoning tasks in DL

• What can we do with an ontology? What does the logical formalism brings more?

• Reasoning tasks– Concept satisfiability (is there any model for C?)– Concept subsumption (does C1

I ⊆ C2I for all I?)

C1 ⊑ C2

• Subsumption is important because from it one can compute a concept hierarchy

• Specialized (decidable and efficient) proof techniques exist for ALC, that do not employ the whole power needed for 1st order logics– Based on tableau algorithms

Representing Knowledge with DL

• A DL Knowledge base is made of– A TBox: Terminological (background) knowledge

• Defines concepts.• Eg. Elephant ≐ Mammal ⊓ bodyPart.Trunk

– A ABox: Knowledge about individuals, be it concepts or roles

• E.g. dumbo: Elephant or (lisa,dumbo):haschild

• Similar to eg. Databases, where there exists a schema and an instance of a database.

General TBoxes

• T is finite set of equation of the form

C1 ≐ C2

• I is a model of T if for all C1 ≐ C2 ∈ T, C1I = C2

I

• Reasoning:– Satisfiability: Given C and T find whether there is a

model both of C and of T?

– Subsumption (C1 ⊑T C2): does C1I ⊆ C2

I holds for all models of T?

Acyclic TBoxes

• For decidability, TBoxes are often restricted to equations

A ≐ Cwhere A is a concept name (rather than expression)

• Moreover, concept A does not appear in the expression C, nor at the definition of any of the concepts there (i.e. the definition is acyclic)

ABoxes

• Define a set of individuals, as instances of concepts and roles

• It is a finite set of expressions of the form:– a:C– (a,b):Rwhere both a and b are names of individuals, C is a

concept and R a role• I is a model of an ABox if it satisfies all its

expressions. It satisfies– a:C iff aI ∈ CI

– (a,b):R iff (aI,bI) ∈ RI

Reasoning with TBoxes and ABoxes

• Given a TBox T (defining concepts) and an ABox A defining individuals– Find whether there is a common model (i.e.

find out about consistency)– Find whether a concept is subsumed by another

concept C1 ⊑T C2

– Find whether an individual belongs to a concept (A,T |= a:C), i.e. whether aI ∈ CI for all models of A and T

Inference under ALC

• Since the semantics of ALC can be defined in terms of 1st order logics, clearly 1st order theorem provers can be used for inference

• However, ALC only uses a small subset of 1st order logics– Only unary and binary predicates, with a very limited

use of quantifiers and connectives• Inference and algorithms can be much simpler

– Tableau Algorithms are used for ALC and mostly other description logics

• ALC is also decidable, unlike 1st order logics

More expressive DLs

• The limited use of 1st order logics has its advantages, but some obvious drawbacks: Expressivity is also limited

• Some concept definitions are not possible to define in ALC. E.g.– An elephant has exactly 4 legs

• (expressing qualified number restrictions)– Every mother has (at least) a child, and every son is the

child of a mother• (inverse role definition)

– Elephant are animal• (define concepts without giving necessary and sufficient

conditions)

Extensions of ALC

• ALCN extends ALC with unqualified number restrictions≤n R and ≥n R and =n R

– Denotes the individuals which relate via R to at least (resp. at most, exactly) n individuals

– Eg. Person ⊓ (≥ 2 hasChild)• Persons with at least two children

• The precise meaning is defined by (resp. for ≥ and =)– (≤n R)I = {d ∈ I | #{(d,e) ∈ RI} ≤ n }

• It is possible to define the meaning in terms of 1st order logics, with recourse to equality. E.g.– ≥2 R is {x: yz, y ≠ z R(x,y) R(x,z)}– ≤2 R is

{x: y,z,w, (R(x,y) R(x,z) R(x,w)) (y=z ⌵ y=w ⌵ z=w)}

Qualified number restriction

• ALCN can be further extended to include the more expressive qualified number restrictions

(≤n R C) and (≥n R C)and (=n R C)– Denotes the individuals which relate via R to at least (resp. at

most, exactly) n individuals of concept C– Eg. Person ⊓ (≥ 2 hasChild Female)

• Persons with at least two female children– E.g. Mammal ⊓ (=4 bodypart Leg)

• Mammals with 4 legs

• The precise meaning is defined by (resp. for ≥ and =)– (≤n R)I = {d ∈ I | #{(d,e) ∈ RI} ≤ n }

• Again, it is possible to define the meaning in terms of 1st order logics, with recourse to equality. E.g.– (≥2 R C) is {x: yz, y ≠ z C(y) C(z) R(x,y) R(x,z)}

Further extensions• Inverse relations

– R- denotes the inverse of R: R- (x,y) = R(y,x)• One of constructs (nominals)

– {a1, …, an}, where as are individuals, denotes one of a1, …, an

• Statements of subsumption in TBoxes (rather than only definition)

• Role transitivity– Trans(R) denotes the transitivity closure of R

• SHOIN is the DL resulting from extending ALC with all the above described extensions– It is the underlying logics for the Semantic Web language OWL-

DL– The less expressive language SHIF, without nominal is the basis

for OWL-Lite

Example

• From the w3c wine ontology– Wine ⊑

PotableLiquid ⊓ (=1 hasMaker) hasMaker.Winery)

• Wine is a potable liquid with exactly one maker, and the maker must be a winery

hasColor-.Wine ⊑ {“white”, “rose”, “red”}• Wines can be either white, rose or red.

– WhiteWine ≐ Wine ⊓ hasColor.{“white”} • White wines are exactly the wines with color white.