dl overview second pass
DESCRIPTION
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 PresentationTRANSCRIPT
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DL OverviewSecond Pass
Ming Fang06/19/2009
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Outlines Description Languages Knowledge Representation in DL Logical Inference in DL
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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.
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Description Language: A Simple Example The basic description language: AL
A,B: atomic concepts R: atomic roles C,D: concept descriptions
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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
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Extensions of AL Union( ) :
Full existential quantification( ):
Number restrictions( ):
Negation( ):
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AL-family Because union and full existential
quantification can be expressed using negation, and vice versa, ALC and ALUε are interchangeable.
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Knowledge Base Architecture of DL knowledge representation
system
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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)
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A Family Relationships Example
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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:
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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
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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.
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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.
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Assertions(ABox) Introduce individuals by giving them names Assert properties of these individuals Have the form: C(a), R(b, c) “open-world semantics”
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Inferences TBox
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Inferences cont’
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Inferences cont’ Eliminate acyclic Tbox by expansion: easier
for developing reasoning procedures.
Expansion could be computationally costly. Source of complexity in TBox reasoning.
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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.
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Inferences cont’ An interesting example
Open-world reasoning may require to make case analyses.
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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