Knowledge Representation and Description Logic Part 2

Transcription

1 Knowledge Representation and Description Logic Part 2 Renata Wassermann renata@ime.usp.br Computer Science Department University of São Paulo September 2014 IAOA School Vitória Renata Wassermann Knowledge Representation and Description Logic Part 2 1 / 40

2 Interpretations - Blocks world A Green B C Non green Is there a green block directly on top of a non-green one? Renata Wassermann Knowledge Representation and Description Logic Part 2 2 / 40

3 Interpretations - Blocks world A Green B C Non green S = {On(a,b), On(b,c), Green(a), Green(c)} Renata Wassermann Knowledge Representation and Description Logic Part 2 3 / 40

4 Interpretations - Blocks world A Green B C Non green S = {On(a,b), On(b,c), Green(a), Green(c)} α = x y(green(x) Green(y) On(x,y)) Renata Wassermann Knowledge Representation and Description Logic Part 2 3 / 40

5 Interpretations - Blocks world A Green B C Non green S = {On(a,b), On(b,c), Green(a), Green(c)} α = x y(green(x) Green(y) On(x,y)) S =α? Renata Wassermann Knowledge Representation and Description Logic Part 2 3 / 40

6 Exercise on interpretations (Brachman & Levesque) For each of the following sentences, give a logical interpretation that makes that sentence false and the other two sentences true: (a) x y z((p(x, y) P(y, z)) P(x, z)) (b) x y((p(x, y) P(y, x)) x = y) (c) x y(p(a, y) P(x, b)) Renata Wassermann Knowledge Representation and Description Logic Part 2 4 / 40

7 (a) x y z((p(x, y) P(y, z)) P(x, z)) (b) x y((p(x, y) P(y, x)) x = y) (c) x y(p(a, y) P(x, b)) I = D, I D = {d 1, d 2, d 3 } I [P] = {(d 1, d 2 ), (d 2, d 3 )} I [a] = d 3 I [b] = d 2 Renata Wassermann Knowledge Representation and Description Logic Part 2 5 / 40

8 (a) x y z((p(x, y) P(y, z)) P(x, z)) (b) x y((p(x, y) P(y, x)) x = y) (c) x y(p(a, y) P(x, b)) I = D, I D = {d 1, d 2 } I [P] = {(d 1, d 2 ), (d 2, d 1 ), (d 1, d 1 ), (d 2, d 2 )} = DÖD I [a] = d 1 I [b] = d 2 Renata Wassermann Knowledge Representation and Description Logic Part 2 6 / 40

9 (a) x y z((p(x, y) P(y, z)) P(x, z)) (b) x y((p(x, y) P(y, x)) x = y) (c) x y(p(a, y) P(x, b)) I = D, I D = {d 1, d 2 } I [P] = {(d 1, d 2 )} I [a] = d 1 I [b] = d 1 Renata Wassermann Knowledge Representation and Description Logic Part 2 7 / 40

10 Deductive Inference Process of finding entailments of KB. Renata Wassermann Knowledge Representation and Description Logic Part 2 8 / 40

11 Deductive Inference Process of finding entailments of KB. Process is sound if whenever it produces α, then KB = α Renata Wassermann Knowledge Representation and Description Logic Part 2 8 / 40

12 Deductive Inference Process of finding entailments of KB. Process is sound if whenever it produces α, then KB = α Process is complete if whenever KB = α, it produces α Renata Wassermann Knowledge Representation and Description Logic Part 2 8 / 40

13 Satisfiability KB = α iff KB { α} has no model Renata Wassermann Knowledge Representation and Description Logic Part 2 9 / 40

14 Satisfiability KB = α iff KB { α} has no model Undecidable for first-order logic! Renata Wassermann Knowledge Representation and Description Logic Part 2 9 / 40

15 Satisfiability KB = α iff KB { α} has no model Undecidable for first-order logic! If we want to have inference procedures that always give an answer, tradeoff expressivity for tractability. Renata Wassermann Knowledge Representation and Description Logic Part 2 9 / 40

16 What are DLs? (Horrocks) Family of logic-based knowledge representation formalisms well-suited for the representation of and reasoning about terminological knowledge descendants of semantics networks, frame-based systems, and KL-ONE aka terminological KR systems, concept languages, etc. Renata Wassermann Knowledge Representation and Description Logic Part 2 10 / 40

17 Brief history of Description Logics Semantic Networks Peirce 1909: Logic of the future Quillian s thesis 60s: Semantic memory The meaning of a concept comes from its relationship to other concepts The information is stored by interconnecting nodes with labelled arcs. Cat Fish is a lives in Mammal Water Renata Wassermann Knowledge Representation and Description Logic Part 2 11 / 40

18 Brief history of Description Logics Semantic Networks Vertebra Cat has Fur has is a has is a is a Animal Mammal Bear is a Fish is a Whale Water lives in lives in Renata Wassermann Knowledge Representation and Description Logic Part 2 12 / 40

19 Brief history of Description Logics Woods What s in a link? (1975) cat colour black Renata Wassermann Knowledge Representation and Description Logic Part 2 13 / 40

20 Brief history of Description Logics Woods What s in a link? (1975) cat colour black All cats are all black. Renata Wassermann Knowledge Representation and Description Logic Part 2 13 / 40

21 Brief history of Description Logics Woods What s in a link? (1975) cat colour black All cats are all black. All cats have some colour black. Renata Wassermann Knowledge Representation and Description Logic Part 2 13 / 40

22 Brief history of Description Logics Woods What s in a link? (1975) cat colour black All cats are all black. All cats have some colour black. There is a cat which is all black. Renata Wassermann Knowledge Representation and Description Logic Part 2 13 / 40

23 Brief history of Description Logics Woods What s in a link? (1975) cat colour black All cats are all black. All cats have some colour black. There is a cat which is all black. There is a cat which has some colour black. Renata Wassermann Knowledge Representation and Description Logic Part 2 13 / 40

24 Brief history of Description Logics KL-ONE Brachman s thesis, Harvard, 1977 William A. Woods, James G. Schmolze, The KL-ONE family, 1992 Used by companies and research institutes. Focus on NLP. Semantics must be clear. Concept structuring primitives. Basis for what comes next... Renata Wassermann Knowledge Representation and Description Logic Part 2 14 / 40

25 Brief history of Description Logics KL-ONE Separate terminological and assertional knowledge External meaning of is a (sets and instances) Multiple inheritance Allows for automatic classification Renata Wassermann Knowledge Representation and Description Logic Part 2 15 / 40

26 Brief history of Description Logics KL-ONE Separate terminological and assertional knowledge External meaning of is a (sets and instances) Multiple inheritance Allows for automatic classification Example: infer that Woman with sons is more specific than Woman with children Renata Wassermann Knowledge Representation and Description Logic Part 2 15 / 40

27 Brief history of Description Logics The beginning of the modern era The Tractability of Subsumption in Frame-Based Description Languages, Brachman and Levesque, Here we present evidence as to how the cost of computing one kind of inference is directly related to the expressiveness of the representation language. As it turns out, this cost is perilously sensitive to small changes in the representation language. Even a seemingly simple frame-based description language can pose intractable computational obstacles. Renata Wassermann Knowledge Representation and Description Logic Part 2 16 / 40

28 Main ideas DL Knowledge Base Description of the concepts and their properties Description of concrete situations. Renata Wassermann Knowledge Representation and Description Logic Part 2 17 / 40

29 Main ideas Concepts Represent classes (sets) May be atomic or constructed from other concepts: Female Female Human Renata Wassermann Knowledge Representation and Description Logic Part 2 18 / 40

30 Main ideas Roles Represent relations (properties) May be used to restrict concepts: Female Human haschild.female Female haschild.human Renata Wassermann Knowledge Representation and Description Logic Part 2 19 / 40

31 Main ideas TBox A TBox contains the terminological knowledge: Concept definitions (introduce names for concepts): Father Man haschild. Axioms (restrict models): Mother Woman BlackCat Cat hascolour.black Renata Wassermann Knowledge Representation and Description Logic Part 2 20 / 40

32 Main ideas ABox The ABox contains assertions about individuals: Concept assertions: BlackCat(mimi) Mother haschild.woman(mary) Role assertions: haschild(mary,betty) Renata Wassermann Knowledge Representation and Description Logic Part 2 21 / 40

33 The typical language Attributive Concept Language with Complements (Schmidt-Schauß and Smolka, 1991). Concept construction closed under boolean operations. Basis for more expressive languages. Renata Wassermann Knowledge Representation and Description Logic Part 2 22 / 40

34 Syntax Concepts C, D A C C D C D R.C R.C Renata Wassermann Knowledge Representation and Description Logic Part 2 23 / 40

35 Semantics Interpretations I: I (domain of interpretation) function that assigns to each: A, a set A I I r, a binary relation R I I I. a, an element a I I Renata Wassermann Knowledge Representation and Description Logic Part 2 24 / 40

36 Semantics Extending the interpretation function I = I I = ( C) I = I \C I (C D) I = C I D I (C D) I = C I D I ( R.C) I = {a I b.(a, b) R I b C I } ( R.C) I = {a I b.(a, b) R I b C I } Renata Wassermann Knowledge Representation and Description Logic Part 2 25 / 40

37 Semantics Semantics of the TBox An interpretation I satisfies C D iff C I = D I. C D iff C I D I. a TBox T iff it satisfies all elements of T Renata Wassermann Knowledge Representation and Description Logic Part 2 26 / 40

38 Semantics Semantics of the ABox An interpretation I satisfies C(a) iff a I C I. r(a, b) iff (a I, b I ) r I. an ABox A iff it satisfies all elements of A Renata Wassermann Knowledge Representation and Description Logic Part 2 27 / 40

39 Semantics DL Knowledge Base An ALC Knowledge Base is a pair Σ = T, A, where T is a TBox A is an ABox An interpretation I is a model of Σ if it satisfies T and A. A knowledge base Σ is said to be satisfiable if it admits a model. Renata Wassermann Knowledge Representation and Description Logic Part 2 28 / 40

40 Semantics Logical Consequence Σ = ϕ iff every model of Σ is a model of ϕ Renata Wassermann Knowledge Representation and Description Logic Part 2 29 / 40

41 Semantics Logical Consequence Σ = ϕ iff every model of Σ is a model of ϕ teaches.course GraduateStudent Professor teaches(john, cs101) Course(cs101) Professor(john) Renata Wassermann Knowledge Representation and Description Logic Part 2 29 / 40

42 Semantics Logical Consequence Σ = ϕ iff every model of Σ is a model of ϕ teaches.course GraduateStudent Professor teaches(john, cs101) Course(cs101) Professor(john) Σ =GraduateStudent(john) Renata Wassermann Knowledge Representation and Description Logic Part 2 29 / 40

43 Semantics Example (Horrocks) - TBox HogwartsStudent Student attendsschool.hogwarts HogwartsStudent haspet.(owl Cat Toad) haspet.phoenix Wizard Phoenix (Owl Cat Toad) Renata Wassermann Knowledge Representation and Description Logic Part 2 30 / 40

44 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) Inferences: Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

45 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) Inferences: attendsschool.hogwarts(harrypotter) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

46 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) haspet(harrypotter, hedwig) Inferences: attendsschool.hogwarts(harrypotter) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

47 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) haspet(harrypotter, hedwig) Inferences: attendsschool.hogwarts(harrypotter) (Owl Cat Toad)(hedwig) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

48 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) haspet(harrypotter, hedwig) Phoenix(fawks) Inferences: attendsschool.hogwarts(harrypotter) (Owl Cat Toad)(hedwig) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

49 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) haspet(harrypotter, hedwig) Phoenix(fawks) Inferences: attendsschool.hogwarts(harrypotter) (Owl Cat Toad)(hedwig) haspet(harrypotter,fawks) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

50 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) haspet(harrypotter, hedwig) Phoenix(fawks) haspet(dumbledore,fawks) Inferences: attendsschool.hogwarts(harrypotter) (Owl Cat Toad)(hedwig) haspet(harrypotter,fawks) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

51 Semantics Example (Horrocks) - ABox Facts: HogwartsStudent(harryPotter) haspet(harrypotter, hedwig) Phoenix(fawks) haspet(dumbledore,fawks) Inferences: attendsschool.hogwarts(harrypotter) (Owl Cat Toad)(hedwig) haspet(harrypotter,fawks) Wizard(dumbledore) and HogwartsStudent(dumbledore) Renata Wassermann Knowledge Representation and Description Logic Part 2 31 / 40

52 Semantics Reasoning Services Satisfiability: Checking whether Σ has a model Concept Satisfiability (w.r.t. Σ): Checking whether there is a model I of Σ such that C I Σ = C Student Person Renata Wassermann Knowledge Representation and Description Logic Part 2 32 / 40

53 Semantics Reasoning Services Subsumption (w.r.t. Σ): Checking whether for all models I of Σ, C I D I. Σ = C D Student Person This is the first problem that was studied (classification). For simple logics, solved by structural methods. Renata Wassermann Knowledge Representation and Description Logic Part 2 33 / 40

54 Semantics Reasoning Services Instance Checking (w.r.t. Σ): Checking whether for all models I of Σ, a I C I. Σ = C(a) Renata Wassermann Knowledge Representation and Description Logic Part 2 34 / 40

55 Semantics Reasoning Services Instance Checking (w.r.t. Σ): Checking whether for all models I of Σ, a I C I. Σ = C(a) Retrieval Finding the individuals a s.t. for all models I of Σ, a I C I {a Σ = C(a)} Renata Wassermann Knowledge Representation and Description Logic Part 2 34 / 40

56 Semantics Reasoning Services Instance Checking (w.r.t. Σ): Checking whether for all models I of Σ, a I C I. Σ = C(a) Retrieval Finding the individuals a s.t. for all models I of Σ, a I C I {a Σ = C(a)} Realization Finding the concepts C s.t. for all models I of Σ, a I C I {C Σ = C(a)} Renata Wassermann Knowledge Representation and Description Logic Part 2 34 / 40

57 Semantics Reasoning services All boils down to satisfiability. Renata Wassermann Knowledge Representation and Description Logic Part 2 35 / 40

58 Semantics Reasoning services All boils down to satisfiability. Concept satisfiability: Σ = C Renata Wassermann Knowledge Representation and Description Logic Part 2 35 / 40

59 Semantics Reasoning services All boils down to satisfiability. Concept satisfiability: Σ = C iff Σ {(C)(x)} is satisfiable. Renata Wassermann Knowledge Representation and Description Logic Part 2 35 / 40

60 Semantics Reasoning services All boils down to satisfiability. Concept satisfiability: Σ = C iff Σ {(C)(x)} is satisfiable. Subsumption: Σ = C D Renata Wassermann Knowledge Representation and Description Logic Part 2 35 / 40

61 Semantics Reasoning services All boils down to satisfiability. Concept satisfiability: Σ = C iff Σ {(C)(x)} is satisfiable. Subsumption: Σ = C D iff Σ {(C D)(x)} is not satisfiable. Renata Wassermann Knowledge Representation and Description Logic Part 2 35 / 40

62 Translation into FOL Concepts are translated into formulas with one free variable: t(c D) = t(c) t(d) t( r.c) = yr(x, y) t y (C) t( r.c) = yr(x, y) t y (C) And then axioms C D correspond to x(t(c) t(d)) Renata Wassermann Knowledge Representation and Description Logic Part 2 36 / 40

63 Translation into FOL Cat Mammal xcat(x) Mammal(x) Student haspet.(owl Cat Toad) x(student(x) y (haspet(x,y) (Owl(y) Cat(y) Toad(y)))) Renata Wassermann Knowledge Representation and Description Logic Part 2 37 / 40

64 Below ALC EL C, D A C D R.C Looks rather inexpressive, but: SNOMED, Galen, etc. Renata Wassermann Knowledge Representation and Description Logic Part 2 38 / 40

65 Above ALC SHOIQ S = ALC+ transitive roles H = role hierarchies O = nominals I = inverse of roles Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 2 39 / 40

66 Above ALC SHOIQ S = ALC+ transitive roles (ancestor) H = role hierarchies O = nominals I = inverse of roles Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 2 39 / 40

67 Above ALC SHOIQ S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) O = nominals I = inverse of roles Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 2 39 / 40

68 Above ALC SHOIQ S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) O = nominals ({hogwarts}) I = inverse of roles Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 2 39 / 40

69 Above ALC SHOIQ S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) O = nominals ({hogwarts}) I = inverse of roles (ispetof = haspet 1 ) Q = qualified number restriction Renata Wassermann Knowledge Representation and Description Logic Part 2 39 / 40

70 Above ALC SHOIQ S = ALC+ transitive roles (ancestor) H = role hierarchies (hasson haschild) O = nominals ({hogwarts}) I = inverse of roles (ispetof = haspet 1 ) Q = qualified number restriction (> 3hasChild.Male) Renata Wassermann Knowledge Representation and Description Logic Part 2 39 / 40

71 Above ALC OWL Web Ontology Language W3C standard for ontologies since 2004 Based on SHOIQ XML syntax Renata Wassermann Knowledge Representation and Description Logic Part 2 40 / 40