1 Model Theory and Calculus for DL-Lite Evgeny Kharlamov Diego Calvanese, Werner Nutt Free University of Bozen-Bolzano Dresden University of Technology October 2006

2 Motivation

3 Problem: Data Integration Information Sources User Interface Query: q

4 Ontology Information Sources Solution: q Data Integration System Motivation

5 Solution: Ontology Information Sources q Motivation Data Warehouse

6 Motivation Pre-process (data from the sources): Incompleteness of the sources wrt the ontology 23Golf 7 …  VW is a Car VW Car … 7Golf... …

7 Solution: Ontology Information Sources q Motivation Data Warehouse DL-Lite Size??

8 Ontology Information Sources Solution: q Data Integration System Motivation q 1,..., q n

9 Motivation Evaluation of Mediators:  Response time  Correctness of answers q L1L1 q 1,..., q n L3L3 L2L2

10 Motivation Evaluation of Mediators:  Response time ~ LogSpace  Correctness of answers ~ correct q DL-Lite q 1,..., q n UCQs CQs

11 Ontology Information Sources QuOnto: q Data Integration System QuOnto q 1,..., q n CQ DL-Lite UCQ

12 Aim of this Thesis Better 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-Lite Vocabulary (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-Lite Ontology:  Inclusion dependency: VW IsA Car VW IsA Has_engine  Disjointness: VW IsA ¬ Mercedes Has_engine IsA ¬ Animal

16 DL-Lite Ontology:  Functional dependency func (Has_id) func (Has_engine)

17 DL-Lite Data (sources): Car(vw_golf) Has_engine(vw_golf, td)

18 Universal Models

19 Universal Models VW  Car Mercedes  Car VW  ¬Mercedes Car  ¬Animal func (Has_id) func (Has_engine)...

20 Universal Models Properties:  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 BobPerson Bill Father Person Sam Father Person …

22 Chase of Polynomial Size weakly-acyclic ontology VW  Car Mercedes  Car VW  ¬Mercedes Car  ¬Animal func (Has_id) func (Has_engine)... pol(n+m) m n

23 Chase of polynomial size: Chase as Data Warehouse Information Sources q User Interface weakly-acyclic Ontology =

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 Query Ontology 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 Conclusion We 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