Comparison and Combination of the Expressive Power of Description Logics and Logic Programs

# Comparison and Combination of the Expressive Power of Description Logics and Logic Programs

## Comparison and Combination of the Expressive Power of Description Logics and Logic Programs

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1. Comparison and Combination ofthe Expressive Power ofDescription Logics and Logic Programs Jidi (Judy) Zhao October 9, 2014

2. Motivation for Extending Description Logics with Horn Logic Rules By Benjamin Grosof, May, 2003 2

3. Examples of LP not representable in DL • DL cannot represent “more than one free variable at a time”. • FriendshipBetween(?X,?Y) • ← Man(?X) ∧ Woman(?Y). • DLs cannot directly support n-ary predicates • Traditional expressive DLs support transitive role axioms but they cannot derive values of properties • uncleOf (?X,?Z) ←brotherOf(?X,?Y) ∧ parentOf(?Y,?Z). • HomeWorker(?X) ← Work(?X, ?Y) ∧ Live(?X, ?Z) ∧ Loc(?Y,?W) ∧ Loc(?Z,?W)

4. Examples of DL not representable in LP • Horn Logic cannot represent a (1) disjunction or (2) existential in the head. • (1) State a subclass of a complex class expression which is a disjunction. E.g., • (Human u Adult) v (Man t Woman) • (2) State a subclass of a complex class expression which is an existential. E.g., • Radio v9hasPart.Tuner 4

5. Differences between DLs and LPs Description Logics Open World Assumption (OWA) May exist many models Generally no Unique Name Assumption (UNA) Classical negation Logic Programs Closed World Assumption (CWA) Only one model Unique Name Assumption (UNA) Negation As Failure (NAF) 5

6. Semantic Web Layer Cake Proof User Interface & Applications Trust Crypto Unifying Logic Rules: RIF Ontology: OWL Query: SPARQL RDFS Data interchange: RDF XML URI/IRI

7. Different approaches • approaches reducing description logics to logic programs • DLP • OWL-R DL and OWL 2 RL • Homogeneous approaches • OWL Rules • SWRL • hybrid approaches accessing description logics through queries in logic programs • AL-Log

8. Expressiveness of Description Logic Programs (DLP)

9. DLP comprises basic RDFS & more by Benjamin Grosof et al. • RDFS subset of DL permits the following statements: • Subclass, Domain, Range, Subproperty (also SameClass, SameProperty) • instance of class, instance of property • more DL statements beyond RDFS: • Using Intersection connective (conjunction) in class descriptions • Stating that a property (or inverse) is Transitive or Symmetric • Using Disjunction or Existential in a subclass expression • Using Universal in a superclass expression

10. DLP • Figure 1. Relationship between the fragments (profiles) of OWL 1.1 • http://www.webont.org/owl/1.1/tractable.html

11. DLP mappings

12. OWL 2 RL based on Description Logic Programs [DLP] is a syntactic profile of OWL 2 DL. allows for scalable reasoning using rule-based technologies. trades the full expressivity of the language for efficiency http://www.w3.org/2007/OWL/wiki/Profiles#OWL_2_RL 12

13. OWL 2 RL • achieved by restricting the use of OWL 2 constructs to certain syntactic positions. • Table 1. Syntactic Restriction on Class Expressions in SubClassOf Axioms

14. SWRL A Semantic Web Rule Language Combining OWL and RuleML SWRL is undecidable SWRL with the restriction of DL Safe rules is decidable Variables in DL Safe rules bind only to explicitly named individuals in the ontology. 14

15. AL-log [Donini et al., 1998] Provides hybrid reasoning with representational adequacy and deductive power An AL-log knowledge base K = (Σ, π) Σ is an ALCknowledge base, expressing knowledge about concepts, roles and individuals. π is a constrained Datalog program Defines an interface between DL and datalog by allowing Datalog program to “query” DL KB 15

16. Example 1 FP=Full Professor, FM=Faculty Member, NFP=Nonteaching Full Professor, AC=Advanced Course, BC=Basic Course, TC=Teaching, CO=Course, ST=Student, TP=Topic.

17. Conclusion of AL-Log Defines an interface between DL and datalog by allowing datalog program to “query” DL KB Results of DL satisfiability check used for checking constraints in query answering AL-log does not allow relational subsystem to deduce knowledge about the structural subsystem No roles allowed in rule bodies AL-log extended with roles in rule body by [Rosatti, 1999] [Eiter et al., 2004] extend the approach for more expressive DLs and more expressive LP language

18. Uncertainty extension of DL

19. Motivation for Extending Description Logics with Uncertainty “Everything is vague to a degree you do not realize till you have tried to make it precise.” -------Bertrand Russell British author, mathematician, & philosopher (1872 - 1970) Nobel Prize in Literature,1950 20

20. Motivation for Extending Description Logics with Uncertainty (Cont.) Uncertainty is an intrinsic feature of real-world knowledge and refers to a form of deficiency or imperfection in the information. The truth of such information is not precisely established. People work and make decisions with imprecise data in an uncertain world. 21

21. Probability, Possibility and Fuzzy logic Probabilistic Description Logic: Statistical information e.g. John is a student with the probability 0.6 and a teacher with the probability 0.4 Fuzzy Description Logic: Express vagueness and imprecision e.g. John is tall with the degree of truth 0.9 Possibilistic Description Logic: Particular rankings and preferences e.g. John prefers an ice cream to a beer 23

22. Probability, Possibility and Fuzzy logic (Cont.) Previous work on uncertainty extension to DL can be classified based on (a) the generalization of classical description logics (b) the supported forms of uncertain knowledge (c) the underlying semantics (d) their inference problems and reasoning algorithms. 24

23. A norm-parameterized fuzzy description logic [Zhao, Boley, Du, 2009]

24. Fuzzy Sets Fuzzy sets and set membership is the key to decision making when faced with uncertainty (Zadeh, 1965). Fuzzy Logic is particularly good at handling vagueness and imprecision. Generalize crisp sets to Fuzzy Sets (concepts). 26

25. Fuzzy values • Cheetahs run very fast. • John is young. • Mary is old. • John is tall. 27

26. Fuzzy Operations fuzzy intersection (t-norm) fuzzy union (s-norm) fuzzy set complement (negation) 29

27. A Knowledge Base (KB) <T,A>= a Tbox + an Abox A TBox (terminology) is a finite set of fuzzy concept inclusion axioms in FOC fuzzy concept equivalence axioms fuzzy DL Knowledge Bases(I) 30

28. fuzzy role inclusion axioms fuzzy role equivalence axioms An ABox (Assertion) is a set of fuzzy assertions about individuals fuzzy concept assertions fuzzy role assertions individual inequality fuzzy DL Knowledge Bases (II) 31

29. Semantics (I) • Semantics given by standard FO model theory and Fuzzy Logic • A fuzzy interpretation I is a tuple (I, •I) • Iis the domain (a set) • •Iis a mapping that maps: • Each object (individual/constant) to an element ofI • Each unary predicate (classe/concept) C to a membership function of CI: I →[0,1] • Each binary predicate (propertie/role) R to a membership function of RI: I ×I →[0,1] 32

30. Semantics (II) • Concept Negation E.g. • Concept Conjunction E.g. 33

31. Semantics (III) • Concept Disjunction E.g. • Role Exists Restriction in FOC existential quantier: supremum or least upper bound 34

32. Semantics (IV) • Role Exists Restriction • E.g.

33. Semantics (V) • At-least Number Restriction in FOC • Inverse Role

34. Reasoning Procedure

35. Probabilistic reasoning in terminological logics(Jaeger,1994) Propositional concept language (PCL) Syntax: Terminological axioms Probabilistic terminological axioms Probabilistic assertions Semantics: The probability measure that interprets an individual will be defined by Jeffrey’s rule.

36. Probabilistic reasoning in terminological logics(Jaeger,1994) Reasoning Tasks: (1)derive additional conditional probabilities. (2) derive additional probabilistic assertions. The former codifies statistical information that will be gained generally by observing a large number of individual objects and checking their membership of the various concepts. The latter expresses a degree of belief in a specific proposition. Its value most often will be justified only by a subjective assessment of likelihood.

37. Probabilistic reasoning in terminological logics(Jaeger,1994) Example: TBox PTBox PABox

38. Probabilistic reasoning in terminological logics(Jaeger,1994) Reasoning on TBox and PTBox:

39. Probabilistic reasoning in terminological logics(Jaeger,1994) Reasoning on KB: According to Jeffrey’ rule, Present a naive method for computing the probability of new knowledge

40. Research Challenges in DL Extensions Syntax and Semantics Decidability Reasoning algorithms for possible extensions Soundness and completeness Complexity/efficiency Effective methods for reasoning under uncertainty 44

41. Questions?