61
Description Logics Dr. Alexandra I. Cristea

Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Embed Size (px)

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-Completeness

Citation preview

Page 1: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Description LogicsDr. Alexandra I. Cristea

Page 2: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

2

Page 3: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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-Completeness. 1979.

Page 4: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 5: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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.

Page 6: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Example concept definitions in DL

Woman = Person ⊓ FemaleMan = Person ⊓ ¬Woman

6

..

Page 7: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Reasoning in DLA classifier (a reasoning engine) can construct class hierarchy from ontology concepts defsConcept definitions composed from primitives >> ontology is more maintainable

7

Page 8: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Example Reasoning

8

Page 9: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Example Reasoning

9

Page 10: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Example Reasoning

10

Page 11: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 12: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL architecture

12

Page 13: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

13

Page 14: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Language elements for concept expressions

14

Page 15: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 16: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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.

16

Page 17: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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)

17

Page 18: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL naming: Extensions

18

Page 19: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL naming: Exceptions

19

Page 20: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Common DLs

20

Page 21: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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.

21

Page 22: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 23: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Value and exists restrictions{a,b,c,d,e,f} are instances; Plant and Animal are classes

⊤⊑ Plant Animal(partition)

Plant Animal ⊓ ⊑⊥(disjointness) 23

Page 24: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 25: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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}∃

25

Page 26: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 27: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 28: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Last time:– Started DL– DL terminology, some reasoning/ inference,

models, architecture

Next:– More on DL inference 28

Page 29: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 30: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

30

Page 31: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

31

Page 32: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

32

Yes

Page 33: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

33

Yes

Page 34: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

34

Yes

Page 35: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

35

Yes

Page 36: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

DL Inference (2)

36

Yes

Page 37: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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

Page 38: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

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.

38

Page 39: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

Additional Slides DL

39

Page 40: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

40

Page 41: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

41

Page 42: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

42

Page 43: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

43

Page 44: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

44

Page 45: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

45

Page 46: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

46

Page 47: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

47

Page 48: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

48

Page 49: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

49

Page 50: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

50

Page 51: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

51

Page 52: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

52

Page 53: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

53

Page 54: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

54

Page 55: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

55

Page 56: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

56

Page 57: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

57

Page 58: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

58

Page 59: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

59

Page 60: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

60

Page 61: Description Logics Dr. Alexandra I. Cristea. Description Logics Description Logics allow formal concept definitions that can be reasoned about to be expressed

61