1 Comparison and Combination of the Expressive Power of Description Logics and Logic Programs Jidi (Judy) Zhao December 7, 2015

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

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 v 9 hasPart.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 URI/IRI Data interchange: Rules: RIF Unifying Logic Trust Proof Ontology: OWL Crypto RDFS User Interface & Applications XML Query: SPARQL RDF

7 Different approaches 1.approaches reducing description logics to logic programs A.DLP B.OWL-R DL and OWL 2 RL 2.Homogeneous approaches A.OWL Rules B.SWRL 3.hybrid approaches accessing description logics through queries in logic programs A.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

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 

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 Subclass ExpressionsSuperclass Expressions a class a nominal class (OneOf) intersection of class expressions (ObjectIntersectionOf) union of class expressions (ObjectUnionOf) existential quantification to a class expressions (ObjectSomeValuesFrom) existential quantification to an individual (ObjectHasValue) a class intersection of classes (ObjectIntersectionOf) universal quantification to a class expressions (ObjectAllValuesFrom) at-most 1 cardinality restrictions (ObjectMaxCardinality 1) existential quantification to an individual (ObjectHasValue)

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 ALC knowledge 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.

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18 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

19 Uncertainty extension of DL

20 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 ( ) Nobel Prize in Literature,

21 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

22 URW3 Situation Report: uncertainty ontology  URW3 22

23 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

24 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

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

26 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

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

28 Membership Functions 28

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

30 A Knowledge Base (KB) = 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

31 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

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

33 Semantics (II) 33  Concept Negation E.g.  Concept Conjunction E.g.

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

35 Semantics (IV)  Role Exists Restriction  E.g.

36 Semantics (V)  At-least Number Restriction in FOC  Inverse Role

37 Semantics (VI) 37

38 Reasoning Procedure

39 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.

40 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.

41 Probabilistic reasoning in terminological logics(Jaeger,1994)  Example:  TBox  PTBox  PABox

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

43 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

44 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

45 Questions?