Part 6: Description Logics. Languages for Ontologies In early days of Artificial Intelligence,...
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Transcript of Part 6: Description Logics. Languages for Ontologies In early days of Artificial Intelligence,...
![Page 1: Part 6: Description Logics. Languages for Ontologies In early days of Artificial Intelligence, ontologies were represented resorting to non-logic-based.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d7f5503460f94a63152/html5/thumbnails/1.jpg)
Part 6: Description Logics
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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
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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)
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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
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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
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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)
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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
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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
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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
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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
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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)
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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
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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.
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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?
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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)
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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
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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
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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
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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)
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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)}
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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)}
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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
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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.