1 Model Theory and Calculus for DL-Lite Evgeny Kharlamov Diego Calvanese, Werner Nutt Free University of Bozen-Bolzano Dresden University of Technology.

1

Model Theory and Calculus for DL-Lite

Evgeny KharlamovDiego Calvanese, Werner Nutt

Free University of Bozen-BolzanoDresden University of Technology

October 2006

2

Motivation

3

MotivationProblem: Data Integration

Information Sources

User Interface

Query: q

4

Ontology

Information Sources

Solution:

qData Integration

System

Motivation

5

Solution:Ontology

Information Sources

q

Motivation

Data Warehouse

6

MotivationPre-process (data from the sources):

Incompleteness of the sources wrt the ontology

23 Golf 7

…

VW is a Car VW Car

…

7 Golf ...

…

7

Solution:Ontology

Information Sources

q

Motivation

Data Warehouse

DL-Lite

Size??

8

Ontology

Information Sources

Solution:

qData Integration

System

Motivation

q1, . . . , qn

9

MotivationEvaluation of Mediators:

Response time Correctness of answers

q

L1

q1, . . . , qn

L3L2

10

MotivationEvaluation of Mediators:

Response time ~ LogSpace Correctness of answers ~ correct

q

DL-Lite

q1, . . . , qn

UCQs

CQs

11

Ontology

Information Sources

QuOnto:

qData Integration

System

QuOnto

q1, . . . , qn

CQ

DL-Lite

UCQ

12

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

13

DL-Lite

14

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

15

DL-LiteOntology:

Inclusion dependency:VW IsA CarVW IsA Has_engine

Disjointness:VW IsA ¬ MercedesHas_engine IsA ¬ Animal

16

DL-LiteOntology:

Functional dependency func (Has_id)func (Has_engine)

17

DL-LiteData (sources):

Car(vw_golf)Has_engine(vw_golf, td)

18

Universal Models

19

Universal Models

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

20

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

21

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

…

22

Chase of Polynomial Size

weakly-acyclicontology

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

pol(n+m) m

n

23

Chase of polynomial size:Chase as Data Warehouse

Information Sources

q

User Interfaceweakly-

acyclicOntology

=

24

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

25

Deduction as Query Answering

26

Deduction as Query Answering

Information Sources QueryOntology

T(Information Sources)

T(Query)

T(Ontology)

Calculus

All Answers

Derivation

Extended Horn Logic (EHL)

27

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)

28

Calculus

Extends Resolution-based calculus with

Extended resolution Query homorphisms

29

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

30

ConclusionWe investigated properties of DL-Lite logic:

Model theory: Universal models other properties

Proof Theory Calculus as a tool for query answering

31

Further work

Extend query language (in QuOnto)

Find good algorithms and optimisations

32

Thank you