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A Proof Theory for DL-Lite

A Proof Theory for DL-Lite. Diego Calvanese, Evgeny Kharlamov, Werner Nutt Free University of Bozen-Bolzano June 2007. DL-Lite is a “Nice” Logic. 1. Covers basic constructs of UML, ER. Concepts:. Assertions: . DL-Lite is a “Nice” Logic. 2. Answering Conjunctive Queries (CQs):

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A Proof Theory for DL-Lite

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  1. A Proof Theory for DL-Lite Diego Calvanese, Evgeny Kharlamov, Werner Nutt Free University of Bozen-Bolzano June 2007

  2. DL-Lite is a “Nice” Logic 1.Covers basic constructs of UML, ER Concepts: Assertions:

  3. DL-Lite is a “Nice” Logic 2.AnsweringConjunctive Queries (CQs): • Data complexity: LogSpace in • Combined complexity : PTime in • Query complexity: NP in Concepts: Assertions:

  4. DL-Lite is a “Nice” Logic 3.Rewritingtechniquesallow one to exploit current DBMS for CQ answering Concepts: Assertions:

  5. Conjunctive Queries over Knowledge Bases Knowledge Base : Conjunctive Query: Certain Answers:

  6. Perfect Query Reformulation[Calvanese et al. 05] Rewriting rules: Inclusion assertions: Procedure: apply a rewriting rule to a query and possibly factorize the result

  7. Perfect Query Reformulation[Calvanese et al. 05] Rewriting rules: Original CQ: ... ... ... ...

  8. Answering Conjunctive Queries • Store the ABox as RDB:RDB = { Person(Bob), HasFather(Bob,John) } • Query the RDB with the queries, obtained with the Perfect Query Reformulation:

  9. Questions • Which are the characteristics ofthat make it so nice for answering CQs? • Do these characteristics allow for alternative approaches to answering CQs?

  10. DL-Lite is a Fragment of an Extended Horn Logic (EHL) • Extended Horn Clause: Assertions: EHL Clauses:

  11. Properties of Extended Horn Logic • An EHL program has (at least one) Universal Model (UM), which can be homomorphically embedded into any other model • UMs are unique up to homomorphic equivalence • Answering CQs over all modelsis equivalent to answering CQs over any UM

  12. Resolution-Based Calculus forDL-Lite • Resolution (with facts and rules) • Factorization • -resolution (with rules containing )

  13. The Calculus at Work Goal: • Calculus Rules: • Resolution ( ), • Factorization ( ), • -resolution ( ) Extended Horn Program:

  14. Properties of the Calculus • Composition of substitutionsalong all successful derivations returns all certain answers • Soundness • Completeness

  15. Complete Strategies for Rule Application • Loop checking • Postponing resolution with facts until the very end • equivalent to the Perfect Query Reformulation • Selecting only one atom at a time for rule application ("Live-Only strategy") • SLD-Resolution is a special case of "Live-Only“for programs without  • early failure detection

  16. Conclusions • DL-Lite has nice characteristics because it is (essentially) Horn Logic • Existing query answering algorithms correspond to resolution strategies • Ideas from computational logic lead to • alternative approaches • optimization of existing techniques

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