description logics dr. alexandra i. cristea. description logics description logics allow formal...
DESCRIPTION
Why Description Logics in OWL? OWL exploits results of 20+ years of DL research –Well defined (model theoretic) semantics –Formal properties well understood, e.g., computational complexity –Known reasoning algorithms –Implemented systems (highly optimised): FaCT++, Racer, Pellet, etc. Foundational research was crucial to the design of OWL –“Why not extend the language with feature x, which is clearly harmless?” –“Adding x would lead to undecidability — see proof in [... ]” 3 “I can’t find an efficient algorithm, but neither can all these famous people.” Garey & Johnson. Computers and Intractability: A Guide to the Theory of NP-CompletenessTRANSCRIPT
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Description LogicsDr. Alexandra I. Cristea
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Description LogicsDescription Logics allow formal concept definitions that can be reasoned about to be expressedNot a single logic, but a family of KR logicsSubsets of first-order logic (FOL)Advantages:– Well-defined model theory– Known computational complexity
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Why Description Logics in OWL?OWL exploits results of 20+ years of DL research– Well defined (model theoretic) semantics– Formal properties well understood, e.g., computational complexity
– Known reasoning algorithms– Implemented systems (highly optimised): FaCT++, Racer, Pellet, etc.
Foundational research was crucial to the design of OWL– “Why not extend the language with feature x, which is clearly harmless?”– “Adding x would lead to undecidability — see proof in [. . . ]”
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“I can’t find an efficient algorithm,but neither can all these famous people.”Garey & Johnson. Computers and Intractability:A Guide to the Theory of NP-Completeness. 1979.
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Why ontology reasoning?Given key role of ontologies in many applications, it is essential to provide tools and services to help users:– Design and maintain high quality ontologies, e.g.:– Meaningful — all named classes can have instances (remember ‘mad cows’?)– Correct — captures intuitions of domain experts– Minimally redundant — no unintended synonyms
– Answer queries, e.g.:– Find more general/specific classes– Retrieve individuals/tuples matching a given query
– But what is (formal) reasoning?The laws of correct reasoning are the subject of mathematical logic 4
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Exercise: give examples of (principles of) ‘correct reasoning’Aristotle:
Hamlet: “To be or not to be”David Hume: “The sun has risen in the east every morning up until now. Ergo the sun will also rise in the east tomorrow.”
If a drink is made with boiling water, it will be hot.This drink was not made with boiling water.Ergo this drink is not hot. 5
All humans are mortal.Socrates is a human.Ergo Socrates is mortal.
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Example concept definitions in DL
Woman = Person ⊓ FemaleMan = Person ⊓ ¬Woman
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Reasoning in DLA classifier (a reasoning engine) can construct class hierarchy from ontology concepts defsConcept definitions composed from primitives >> ontology is more maintainable
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Example Reasoning
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Example Reasoning
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Example Reasoning
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DL TerminologyDL separates assertions and concept definitions:A Box: Assertions– e.g. hasChild(john, mary)– This is the knowledge base
• T Box: Terminology– The definitions of concepts in the ontology– Example axioms for definitions
• C D [C is a subclass of D, D subsumes C]⊑• C ≡ D [C is defined by the expression D] 11
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DL architecture
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DL Terminology (2)Concept: class, category or type
Role: binary relation– Attributes are functional roles
Subsumption:– D subsumes C if C is a subclass of D : i.e. all Cs are Ds
Unfoldable terminologies:– The defined concept does not occur in the defining expression:– C ≡ D where C does not occur in the expression D
Language families– AL: Attributive Language– ALC: adds full negation to AL
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Language elements for concept expressions
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Universal restrictionUniversal (value) restriction: ∀R.CThe set {x| y, R(x, y) → y C}∀ ∈The set of things x such that for all y where x and y are related by R, y is in C.
e.g., hasChild.Parent∀Set of things x so that for all y where x and y are related by hasChild, y will be in class Parent; everything in set x is a child, everything in set y is a parent.That is, anything that is the object of the relation hasChild must be in class Parent, regardless of what the subject is.
This is a local statement: this is true for every statement in your dataset. 15
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Existential restrictionExistential restriction - also called exists restriction: ∃R.CThe set {x| y, R(x, y) y C}∃ ⋀ ∈The set of things x such that there exists a y where x and y are related via R and y is in class C.
e.g., hasChild.Doctor∃The set of all x’s such that x is related to y via hasChild and y is in class Doctor.the set of all children which have at least one parent who is a doctor
This is a local statement: this is true for at least one statement in your dataset.
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DL namingAttributive language - basic language which allows:
– atomic negation– concept intersection– universal restrictions– limited existential quantification
Frame based description language, allows:– concept intersection– universal restrictions– limited existential quantification– role restriction
allows:– concept intersection– existential restrictions (of full existential quantification)
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DL naming: Extensions
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DL naming: Exceptions
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Common DLs
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HomeworkExplain, based on the naming conventions you’ve just seen, what do the DL types for OWL2, Protégé, OWL-Lite, and OWL-DL stand for.
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Example concept expressionsParent ≡ “Persons who have (amongst other things) some children”
Person hasChild.⨅ ∃ ⊤ParentOfBoys ≡ “Persons who have some children, and only have children that are male”
Person ( hasChild. ) ( hasChild.Male)⨅ ∃ ⊤ ⨅ ∀ScottishParent ≡ “Persons who only have children that drink (amongst other things) some IrnBru”
Person ( hasChild. ( drink.IrnBru))⨅ ∀ ∃22
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Value and exists restrictions{a,b,c,d,e,f} are instances; Plant and Animal are classes
⊤⊑ Plant Animal(partition)
Plant Animal ⊓ ⊑⊥(disjointness) 23
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Value and exists restrictions{a,b,c,d,e,f} are instances; Plant and Animal are classes
∀eats.Animal = {a,b,c,e,f}∃eats.Animal = {c,d,e} ∃eats.Animal eats.Animal = {c,e}⊓ ∀ 24
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Model theoryΔ1 universal domain of individuals, let Δ1 ={a,b,c,d,e,f}eats1 set of pairs for the relation eats, leteats1 = {<d,a>,<d,e>,<e,d>,<e,f>,<c,f>}For all concepts C:
i) C1 ⊆ Δ 1ii) C1≠ ∅
Let Animal1 = {d,e,f} ∴ (¬Animal)1 = {a,b,c} ∴ ( eats.Animal)1={a,b,c,e,f}∀ ∴ ( eats. Animal)1 = {c,d,e}∃
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InferenceMeatEater ≡ eats. Animal = {a,b,c,e,f}∀Vegetarian ≡ eats. ¬Animal = {a,b,f}∀Omnivore ≡ eats. Animal = {c,d,e}∃
Inference:From the above classes we can see that:• MeatEater subsumes Vegetarian• Vegetarian is disjoint from Omnivorein this model, with these definitions.
The problem is to prove this for ALL models. 26
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DL InferenceInference can be expressed in terms of the model– Satisfiability of C: C1 is non-empty– Subsumption C D iff C1 D1 (“C is subsumed ⊑ ⊆
by D”)– Equivalence C ≡ D iff C1 = D1– Disjointness(C D) ⊑ iff C1 ∩ D1 ≡ ⊑ ∅• Tractable/terminating inference algorithms exist27
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Last time:– Started DL– DL terminology, some reasoning/ inference,
models, architecture
Next:– More on DL inference 28
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Value and exists restrictions{a,b,c,d,e,f} are instances; Plant and Animal are classes
Vegetarian = {a,b,f}Omnivore = {c,d,e}
disjoint?MeatEater= {a,b,c,e,f} 29
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DL Inference (2)
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DL Inference (2)
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DL Inference (2)
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Yes
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DL Inference (2)
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Yes
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DL Inference (2)
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Yes
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DL Inference (2)
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Yes
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DL Inference (2)
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Yes
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DL Inference (3)Inference has 2 equivalent notions - so implementing one lets us prove all 4 propertiesReduction to subsumption :⊑– Unsatisfiability of C: C ⊑⊥– Equivalence C≡D iff C D and D C⊑ ⊑– Disjointness (C D) ⊑⊑ ⊥Reduction to unsatisfiability CI = :∅– Subsumption C D iff (C ¬D) is unsatisfiable⊑ ⨅– Equivalence C≡D iff (C ¬D) and (D ¬C) are unsatisfiable⨅ ⨅– Disjointness (C D) is unsatisfiable⨅ 37
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DL SummaryDLs: family of languages based on subsets of FOL.Expressivity depends on language attributes.Attributes: indicated by letters; DL language names: series of letters => expressivity of DL language in its name.DLs allow complex expressions of how concepts relate to one another.Many algorithms (e.g., Tableaux Algorithms) allowing efficient reasoning over DLs.
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Additional Slides DL
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