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Model Theory and Calculus for DL-Lite

Evgeny KharlamovDiego Calvanese, Werner Nutt

Free University of Bozen-BolzanoDresden University of Technology

October 2006

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Motivation

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MotivationProblem: Data Integration

Information Sources

User Interface

Query: q

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Ontology

Information Sources

Solution:

qData Integration

System

Motivation

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Solution:Ontology

Information Sources

q

Motivation

Data Warehouse

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MotivationPre-process (data from the sources):

Incompleteness of the sources wrt the ontology

23 Golf 7

…

VW is a Car VW Car

…

7 Golf ...

…

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Solution:Ontology

Information Sources

q

Motivation

Data Warehouse

DL-Lite

Size??

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Ontology

Information Sources

Solution:

qData Integration

System

Motivation

q1, . . . , qn

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MotivationEvaluation of Mediators:

Response time Correctness of answers

q

L1

q1, . . . , qn

L3L2

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MotivationEvaluation of Mediators:

Response time ~ LogSpace Correctness of answers ~ correct

q

DL-Lite

q1, . . . , qn

UCQs

CQs

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Ontology

Information Sources

QuOnto:

qData Integration

System

QuOnto

q1, . . . , qn

CQ

DL-Lite

UCQ

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Aim of this ThesisBetter understanding of properties of

DL-Lite

Relationship: ontology - size of the Warehouse Relationship: ontology - query answering

Response time Correctness of answers

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DL-Lite

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DL-LiteVocabulary (of the ontology):

Classes: Car Elements that participate in a relation: A = {x | there is y s.t. Has_engine(x,y)} B = {y | there is x s.t. Has_engine(x,y)}

Relations: Has_engine

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DL-LiteOntology:

Inclusion dependency:VW IsA CarVW IsA Has_engine

Disjointness:VW IsA ¬ MercedesHas_engine IsA ¬ Animal

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DL-LiteOntology:

Functional dependency func (Has_id)func (Has_engine)

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DL-LiteData (sources):

Car(vw_golf)Has_engine(vw_golf, td)

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Universal Models

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Universal Models

VW Car Mercedes CarVW ¬Mercedes Car ¬Animalfunc (Has_id) func (Has_engine) . . .

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Universal ModelsProperties:

If there is a completion UM If there is a UM there is a class of Ums Chase of a DB with an Ontology is a UM

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Universal Models

Infinite universal models:

Bob is a Person Every person has a father Every father is a person No one can be an ancestor

of him/herself

Bob Person

BillFatherPerson

SamFatherPerson

…

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Chase of Polynomial Size

weakly-acyclicontology

VW Car Mercedes CarVW ¬Mercedes Car ¬Animalfunc (Has_id) func (Has_engine) . . .

pol(n+m) m

n

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Chase of polynomial size:Chase as Data Warehouse

Information Sources

q

User Interfaceweakly-

acyclicOntology

=

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Results

Introduced the notion of UM Shown that any chase is a UM Proposed weakly-acyclic ontologies for

which chase is finite and of polynomial size

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Deduction as Query Answering

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Deduction as Query Answering

Information Sources QueryOntology

T(Information Sources)

T(Query)

T(Ontology)

Calculus

All Answers

Derivation

Extended Horn Logic (EHL)

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Extended Horn Logic

HL:X Y Z bro(X,Z):- bro(X,Y), bro(Y,Z)

EHL: X Y Z bro(bob,Z):- bro(X,Y), bro(Y,bob)

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Calculus

Extends Resolution-based calculus with

Extended resolution Query homorphisms

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Results Introduced EHL Defined reduction from DL-Lite to EHL Introduced a calculus for EHL Shown soundness and completeness of the

calculus wrt query answering

query answering in DL-Lite is reducible to reasoning in EHL

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ConclusionWe investigated properties of DL-Lite logic:

Model theory: Universal models other properties

Proof Theory Calculus as a tool for query answering

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Further work

Extend query language (in QuOnto)

Find good algorithms and optimisations

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Thank you