A Survey of Temporal Knowledge Representations

Transcription

1 A Survey of Temporal Knowledge Representations Advisor: Professor Abdullah Tansel Second Exam Presentation

2 Knowledge Representations logic-based logic-base formalisms formalisms more complex and difficult to understand all based on formal logic Syntax Semantics E.g.: Description Logics non logic-based formalisms closer to human intuition therefore easier to understand usually don t have consistent semantics E.g.: Semantic Networks Frame-based representations Rule-based representations

3 FOL as Semantic Web Language? Why not simply take FOL for Ontologies? FOL can do everything.. compare higher programming Languages to assemblers FOL has high expressivity too bulky for modelling not appropriate to find consensus in modelling proof theoretically very complex (semidecidable) FOL is also not a markup language Look for an appropriate fragment of FOL

4 Description Logics (DLs) DLs are Fragments of FOL In DL from simple descriptions more complex descriptions are created with the help of Constructors. DLs differ in the applied constructors (Expressivity) DLs have been developed from semantic Networks DLs are decidable (most times) DLs possess sufficient expressivity (most times) DLs are related to modal logics e.g., W3C Standard OWL 2 DL is based on description logics SHROIQ(D)

5 Attributive Language with complements ALC Building Blocks: Concepts/ Classes Roles / Properties Individuals Student(John) Individual John is of class Student Lecture(Database) Individual Database is of class lecture attendslecture(john,database) John attends the lecture Database

6 ALC - Building Blocks Atomic Types Concept names A,B,... Special concepts - Top (universal concept) - Bottom concept Role names R,S,... Constructors Negation: C Conjunction: C Disjunction: C Existential quantifier: R.C Universal quantifier: R.C D D

7 ALC - Building Blocks Class Inclusion Professor FacultyMember every Professor isa Faculty Member In FOL equals ( x)(professor(x) FacultyMember(x)) Class Equivalence Professor FacultyMember the Faculty Members are exactly the Professors In FOL equals ( x)(professor(x) FacultyMember(x))

8 ALC - Complex Class Relations Conjunction Disjunction Negation Professor (Person UniversityEmployee) (Person Student) ( x)(professor(x) ((Person(x) Λ UniversityEmployee(x)) V (Person(x) Λ Student(x)))

9 ALC - Quantifiers on Roles Strict Binding of the Range of a Role to a Class Examination hassupervisor.professor An Examination must be supervised by a Professor ( x)(examination(x) ( y)(hassupervisor(x,y) Professor(y))) Open Binding of the Range of a Role to a Class Examination hassupervisor.person Every Examinationhas at least one supervisor (who is a person) ( x)(examination(x) ( y)(hassupervisor(x,y) Λ Person(y)))

10 ALC - Formal Syntax Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) C,D := A C C D C D R.C R.C

11 ALC - Formal Syntax Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) C,D := A C C D C D R.C R.C An ALC TBox contains assertions of the form C and C D, where C,D are complex classes. D

12 ALC - Formal Syntax Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) C,D := A C C D C D R.C R.C An ALC TBox contains assertions of the form C and C D, where C,D are complex classes. An ALC ABox contains assertions of the form C(a) and R(a,b), where C is a complex Class, R a Role and a,b Individuals. D

13 ALC - Formal Syntax Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) C,D := A C C D C D R.C R.C An ALC TBox contains assertions of the form C and C D, where C,D are complex classes. An ALC ABox contains assertions of the form C(a) and R(a,b), where C is a complex Class, R a Role and a,b Individuals. An ALC-Knowledge Base contains an ABox and a TBox. D

14 ALC - Semantic (Interpretation) we define a model-theoretic semantic for ALC (i.e. Entailment will be defined via Interpretations) an Interpretation I=(Δ I,. I ) contains a set Δ I (Domain)of Individuals and an interpretation function Individual names a to domain elements a I Δ I. I that maps Class names C to a set of domain elements C I Δ I Role names R to a set of pairs of domain elements R I Δ I Δ I

15 ALC - Semantic (Interpretation) Individual Names Class Names Role Names

16 ALC - Semantic (Interpretation) Extension for complex classes: I = Δ I and I =

17 ALC - Semantic (Interpretation) Extension for complex classes: I = Δ I and I = (C (C D) I D) I = = C I C I D I D I and

18 ALC - Semantic (Interpretation) Extension for complex classes: I = Δ I and I = (C (C D) I D) I = = C I C I D I D I and ( C) I = Δ I \ C I

19 ALC - Semantic (Interpretation) Extension for complex classes: I = Δ I and I = (C (C D) I D) I = = C I C I D I D I and ( C) I = Δ I \ C I R.C={a Δ I ( b Δ I )((a,b) R I b C I )}

20 ALC - Semantic (Interpretation) Extension for complex classes: I = Δ I and I = (C (C D) I D) I = = C I C I D I D I and ( C) I = Δ I \ C I R.C={a Δ I ( b Δ I )((a,b) R I b C I )} R.C={a Δ I ( b Δ I )((a,b) R I b C I )}

21 ALC - Semantic (Interpretation)...and Axioms: C(a) holds, iff a I C I R(a,b) holds, iff (a I,b I ) R I C D holds, iff C I D I C D holds, iff C I = D I

22 ALC - Knowledgebase Terminological Knowledge (TBox) Axioms that describe the structure ofthe modeled domain (conceptional schema): Human parentof.human Orphan Human hasparent.alive Assertional Knowledge (ABox) Axioms that describe specific situations (data): Orphan(harrypotter) hasparent(harrypotter,jamespotter)

23 Tableaux Algorithm: Proof algorithm to check the consistency of a logical formula by inferring that its negation is a contradiction (proof by refutation). Construct tree, where each node is marked with a logical formula. A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path; a branch of the path represents a disjunction. The tree is created by successive application of the Tableaux Extension Rules. (q r) (p r) r (q r) (p r) r q (p r) r r p r

24 Tableaux Algorithm for Description Logics Transformation to Negation normalform necessary Let W be a knowledge base, Substitute C D by C D and D C Substitute C D by C D. Apply the NNF Transformations from the next page Resulting knowledge base NNF(W) Negation normalform of W. Negation is placed directly in front of atomic classes.

25 Tableaux Transformation in Negation Normalform NNF Transformations NNF(C) = C, if C NNF( C) = C, if NNF( C) = NNF(C D) = NNF(C D) = NNF( (C D)) NNF( (C D)) NNF( R.C) = NNF( R.C) = NNF( R.C) NNF( R.C) NNF(C) NNF(C) NNF(C) = = is atomic C is atomic NNF(D) NNF(D) = NNF( C) = NNF( C) R.NNF(C) R.NNF(C) R.NNF( C) R.NNF( C) NNF( D) NNF( D) W and NNF(W) are logically equivalent.

26 Tableaux Transformation in Negation Normalform Example: P (E U) ( E D) In NNF: P (E U) (E D) C D (C D)= = C C D D

27 Tableaux Extension Rules for DL Selection C(a) W (ABox) R(a,b) W (ABox) C W (TBox) (C D)(a) A (C D)(a) A ( R.C)(a) A ( R.C)(a) A Action Add C(a) Add R(a,b) Add C(a) for a known Individual a Add C(a) and D(a) Split the path. Add (1) C(a) and (2) D(a) Add R(a,b) and C(b) for a new Individual b if R(a,b) A, then add C(b) Ifthe resulting tableaux is closed, the original knowledge base is unsatisfiable. Only select elements that lead to new elements within the tableaux. If this is not possible, then the algorithm terminates and the original knowledge base is satisfiable.

28 Tableaux Algorithm (DL) Example: P E U D Professor Person University Employee Student Knowledge Base: P (E U) (E D) Is P E a logical consequence? Knowledge Base (with [negated] query)in NNF: { P (E U) (E D),(P E)(a)}

29 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)}

30 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A:

31 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base)

32 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a)

33 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a)

34 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base)

35 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base) (5 β from 4)) P(a) (6)((E U) (E D))(a)

36 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base) (5 β from 4)) P(a) (6)((E U) (E D))(a) (7 β from 6) (E U)(a) (8) (E D)(a)

37 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base) (5 β from 4)) P(a) (6)((E U) (E D))(a) (7 β from 6) (E U)(a) (8) (E D)(a) (9 α from 7) E(a)

38 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base) (5 β from 4)) P(a) (6)((E U) (E D))(a) (7 β from 6) (E U)(a) (8) (E D)(a) (9 α from 7) E(a) (11 α from 7)U(a)

39 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base) (5 β from 4)) P(a) (6)((E U) (E D))(a) (7 β from 6) (E U)(a) (8) (E D)(a) (9 α from 7) E(a) (10 α from 8) E(a) (11 α from 7)U(a)

40 Tableaux Algorithm (DL) Example: Knowledge Base W:{ P (E U) (E D),(P E)(a)} Tableaux A: (1)(P E)(a)(from knowledge base) (2 α from 1) P(a) (3 α from 1) E(a) (4)( P (E U) (E D))(a)(from knowledge base) (5 β from 4)) P(a) (6)((E U) (E D))(a) (7 β from 6) (E U)(a) (8) (E D)(a) (9 α from 7) E(a) (11 α from 7)U(a) (10 α from 8) E(a) (12 α from 8) D(a)

41 Tableaux Algorithm (DL) with Blocking The Selection of ( R.C)(a) in the tableaux path A is blocked, if there is already an individualb with {C C(a) A} {C C(b) A}. Two possibilities of termination: 1. Closing the Tableaux. Knowledge Base is unsatisfiable. 2. All non blocked selections from thetableaux don t lead to an extension. Knowledge Base is satisfiable.

42 Temporal extensions of DLs How can time be modelled?

43 Temporal extensions of DLs How can time be modelled? Point-based notion of time Interval-based notion of time

44 Temporal extensions of DLs How can time be modelled? Point-based notion of time Interval-based notion of time How can the temporal dimension be handled?

45 Temporal extensions of DLs How can time be modelled? Point-based notion of time Interval-based notion of time How can the temporal dimension be handled? Implicit notion of time: sequences of events through state-change representations Explicit notion of time: definition of temporal operators and new formulae

46 Temporal extensions of DLs How can time be modelled? Point-based notion of time Interval-based notion of time How can the temporal dimension be handled? Implicit notion of time: sequences of events through state-change representations Explicit notion of time: definition of temporal operators and new formulae Internal point of view: different states of an individual are modelled as different individual components External point of view: an individual has different states in different moments

47 An Example How to become a doctor: Be a PhD student for some years (5-6) Defend a thesis Become a doctor Let s try to model it with different temporal Description Logics!

48 Combining Description Logics and Tense Operators: ALCT [Sch93] Combination of ALC with point-based modal temporal connectives Time part of the semantic structure Recall: an individual has different states at different moments Temporal connectives can be applied only to concepts

49 An Example Every doctor has been a PhD student in the past Every thesis defenderis a PhD student that has been a PhD student in the past

50 An Example Every doctor has been a PhD student in the past Every thesis defenderis a PhD student that has been a PhD student in the past

51 Example Every doctor has been a PhD student in the past Every thesis defenderis a PhD student that has been a PhD student in the past

52 Interval-based Temporal DLs Schmiedel s Formalism Idea Examples Bettini,The Undecidable Realm Idea Examples Artale and Franconi, Towards Decidable Logic Idea Examples

53 Interval-based Temporal DLs: Characteristics Interval-based - Explicit, eg: alltime, TE - Follows the external approach, eg: C<i,a> for temporalconcept assertions and R<i, a,b> for temporal role assertions

54 Schmiedel s Formalism: Idea The first of such an interval-based temporal DL (proposed in 1990) Underlying DL = FLEN R (no T,,, but with cardinality restrictions on roles, and role conjunction) - Temporal operators = at, alltime, sometime - Subsumption is argued to be undecidable

55 Schmiedel s Formalism: Examples Concept: PhD students during 1993 (at 1993 PhDStudent) Terminological Axiom: Doctors were PhD students

56 The Undecidable Realm: Idea Proposed by Bettini Variable-free extension with existential and universal temporal quantification - Arbitrary relationships between more than two intervals can t be represented Satisfiability and subsumptionare undecidable Starting from the DL ALCN Two concept constructors: TE.C and TE.C

57 The Undecidable Realm: Examples Concept: Persons who become a doctor sometime after.doctor, eg: <1990, michael>belongs to this, if <1992, michael>belongs to Doctor Terminological Axiom: Doctors were PhD students Doctor (metby ).PhDStudent

58 Towards Decidable Logics: Idea Developed by Artale and Franconi Decidable: Expressivity is reduced, universal quantificationon temporal variables has been eliminated Underlying DL (most expressive): T L-ALCF Temporal variables are introduced by the temporalexistential quantifier, excluding the predefined temporal var #

59 Towards Decidable Logics: Examples Terminological Axiom: Doctors were PhD students Terminological Axiom: PhD thesis defendersfinish their PhDs

60 The Semantic Web The Semantic Web is an extension of the current web in which information is given well-defined meaning, better enabling computers and people to work in cooperation. [Berners-Lee et al, 2001]

61 The Semantic Web Web pages are written in HTML HTML describes the structure of information HTML describes the syntax not the semantics If computers machines can understand the meaning behind the information on the web, not just simply display it. They can learn what we are interested in. They can help us better find what we want. This is really what the semantic web is all about.

62 The Semantic Web The goal of the semantic web is that it would be possible for machines to understand the information on the web rather than simply display. The major obstacle to this goal is the fact that most information on the web is designed solely for human consumption. This information should be structured in a way that machines can understand and process that information. The key technological threads that are currently employed in the development of Semantic Web are: extensible Markup Language (XML), Resource Description Framework (RDF), and Web Ontology Language (OWL)

63 RDF Triple RDF is a W3C standard for describing resources in the web An RDF Triple (s,p,o) is such that: s, p are URI-s, ie, resources on the Web; o is a URI or a literal s, p, and o stand for subject, property, and object RDF is a general model for such triples with machine readable formats like RDF/XML.

64 RDF Graph Representation

65 RDF Schema: Basic Ideas RDF is a universal language that enables users to describe their own vocabularies. But, RDF does not make assumption about any particular domain. It is up to user to define this in RDF schema.

66 What does RDF Schema add? Defines vocabulary for RDF Organizes this vocabulary in a typed hierarchy Class, subclassof, type Property, subpropertyof domain, range Staff subclassof subclassof Schema(RDFS) Lecturer domain supervisedby range Research Assistant type Tom supervisedby Alan type Data(RDF)

67 Temporal RDF

68 OWL Web Ontology Language (OWL) extends RDF by adding several other constructs such as owl:class (in addition to the rdfs:class), relationships between class and individuals, and property characteristics Follows Description Logic in having class, property, and individuals Allows setting the cardinality constraints

69 Reification WorksFor(Employee, Company, TimeInterval) representing the fact that an employee works for a company during a specific time interval.

70 4D fluent WorksFor(Employee, Company, TimeInterval) representing the fact that an employee works for a company during a specific time interval.

71 Thank You