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DL Overview Second Pass

DL Overview Second Pass. Ming Fang 06/19/2009. Outlines. Description Languages Knowledge Representation in DL Logical Inference in DL. From last presentation. Unary predicates: denote concepts(sets of individuals ) Binary predicates: denote roles(binary relationships between individuals)

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DL Overview Second Pass

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  1. DL OverviewSecond Pass Ming Fang 06/19/2009

  2. Outlines • Description Languages • Knowledge Representation in DL • Logical Inference in DL

  3. From last presentation • Unary predicates: denote concepts(sets of individuals ) • Binary predicates: denote roles(binary relationships between individuals) • FOL constructors: intersection, union, negation, universal quantifier, etc.

  4. Description Language: A Simple Example • The basic description language: AL • A,B: atomic concepts • R: atomic roles • C,D: concept descriptions

  5. Semantics of Concepts • Interpretation I consists of: 1) a non-empty set : the domain of interpretation 2) an interpretation function: assigns A a set ; assigns R a binary relation

  6. Extensions of AL • Union( ) : • Full existential quantification( ): • Number restrictions( ): • Negation( ):

  7. AL-family • Because union and full existential quantification can be expressed using negation, and vice versa, ALC and ALUεare interchangeable.

  8. Knowledge Base • Architecture of DL knowledge representation system

  9. Terminologies(TBox) • Terminological axioms: statements about how concepts or roles are related to each other. • Inclusion VS. Equality • Definition: atomic concept on left-hand side of an equality • Base symbols (primitive concepts) VS. Name symbols (defined concepts)

  10. A Family Relationships Example

  11. Base Interpretation(J ): an interpretation that only interprets the base symbols. • Extension of J(I):an interpretation that also interprets the name symbols. • A terminology T is definitorial if every base interpretation has exactly one extension that is a model for T. • If T is acyclic, then it is definitorial. • There are cyclic T that are definitorial:

  12. Semantics • Definitorial: descriptive semantics • Non-definitorial: fixpoint semantics • Example: Momo: a man having only male offspring Least fixpoints: all James are Momos Greatest fixpoints: all James and all Charles are Momos

  13. Existence of Fixpoint Models • Least and greatest fixpoint models need not exist for every terminology. • Fixpoint models exist, but there is neither a least one or greatest one. • There exist a lfp-model and a gfp-model for a negation free terminology.

  14. Inclusion Axioms • Specialization: an inclusion whose left-hand side is atomic. • Become convenient when one is not able to define the concept in all details. • The terminology loses its definitorial impact, even if it is acyclic. • Normalization: convert into a regular T by • 1) choosing a new base symbol for every • 2) replacing with • stands for qualities that distinguish a women among persons.

  15. Assertions(ABox) • Introduce individuals by giving them names • Assert properties of these individuals • Have the form: C(a), R(b, c) • “open-world semantics”

  16. Inferences • TBox

  17. Inferences cont’

  18. Inferences cont’ • Eliminate acyclic Tbox by expansion: easier for developing reasoning procedures. • Expansion could be computationally costly. • Source of complexity in TBox reasoning.

  19. Inferences cont’ • ABox • 1) Consistency check: is there a model for A andT • 2) Instance check: • 3) Retrieval problem: given an ABox A and a concept C, find all individuals a such that • 4) Realization problem: find a most specific concepts C for an individual a such that All relevant inference problems can be reduced to the consistency problem for ABox if the DL allows for conjunction and negation.

  20. Inferences cont’ • An interesting example • Open-world reasoning may require to make case analyses.

  21. Some Leftovers • Nested quantifier? • L3? • The language consists of all formulae of FOL that can be built using three variables. • ALC can be translated into L2

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