1. DL Overview Second 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 ALU ε are 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.  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