Part 6: Description Logics

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 hasChild (1,NIL) Female Woman Parent Mother

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 (DI,.I) where .I maps: • Each concept name A to AI⊆DI • I.e. a concept denotes set of individuals from the domain (unary predicates) • Each role name R to AI⊆DI x DI • I.e. a role denotes pairs of (binary relationships among) individuals • An interpretation is a model for concept C iff CI≠ {} • 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 = DI – 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 = C1I⋃ C2I resp. (C1⊓ C2)I = C1I⋂ 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 ∈DI | (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 ∈DI | (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 C1I⊆ 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: Elephantor (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 = C2I • Reasoning: • Satisfiability: Given C and T find whether there is a model both of C and of T? • Subsumption (C1⊑T C2): does C1I⊆ C2I holds for all models of T?

Acyclic TBoxes • For decidability, TBoxes are often restricted to equations A ≐ C where 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):R where 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 ∈DI | #{(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: yz, 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 ∈DI | #{(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: yz, 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.

Part 6: Description Logics