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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) - PowerPoint PPT Presentation
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DL OverviewSecond Pass
Ming Fang06/19/2009
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) FOL constructors: intersection, union,
negation, universal quantifier, etc.
Description Language: A Simple Example The basic description language: AL
A,B: atomic concepts R: atomic roles C,D: concept descriptions
Semantics of Concepts Interpretation I consists of:
1) a non-empty set : the domain of interpretation2) an interpretation function: assigns A a set ; assigns R a binary relation
Extensions of AL Union( ) :
Full existential quantification( ):
Number restrictions( ):
Negation( ):
AL-family Because union and full existential
quantification can be expressed using negation, and vice versa, ALC and ALUε are interchangeable.
Knowledge Base Architecture of DL knowledge representation
system
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)
A Family Relationships Example
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:
Semantics Definitorial: descriptive semantics Non-definitorial: fixpoint semantics Example:
Momo: a man having only male offspringLeast fixpoints: all James are MomosGreatest fixpoints: all James and all Charles are Momos
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.
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.
Assertions(ABox) Introduce individuals by giving them names Assert properties of these individuals Have the form: C(a), R(b, c) “open-world semantics”
Inferences TBox
Inferences cont’
Inferences cont’ Eliminate acyclic Tbox by expansion: easier
for developing reasoning procedures.
Expansion could be computationally costly. Source of complexity in TBox reasoning.
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.
Inferences cont’ An interesting example
Open-world reasoning may require to make case analyses.
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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