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