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Exchanging OWL 2 QL Knowledge Bases

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Exchanging OWL 2 QL Knowledge Bases

  1. 1. Exchanging OWL 2 QL Knowledge Bases Vlad Ryzhikov joint work with E. Botoeva, D. Calvanese and M. Arenas KRDB Research Centre, Free University of Bozen-Bolzano, Italy ryzhikov@inf.unibz.it Talk at University of KwaZulu-Natal, Durban, South Africa Vlad Ryzhikov Free University of Bozen-Bolzano 1/16
  2. 2. Knowledge Base Exchange Problem given a mapping M between the disjoint signatures Σ and Σ and a source knowledge base (KB) K, find a target KB K that is a solution for K under M. M Σ Σ1 Σ2 target signature source signature A T D B T A C B C solution A source KB K Vlad Ryzhikov A target KB K Free University of Bozen-Bolzano 2/16
  3. 3. Data Exchange vs. Knowledge Base Exchange • In Data Exchange (DE) only mappings M (in some scenarios, also solutions K ) use constraints – in Knowledge Base Exchange (KBE) source KB K, M, and K use constraints. Vlad Ryzhikov Free University of Bozen-Bolzano 3/16
  4. 4. Data Exchange vs. Knowledge Base Exchange • In Data Exchange (DE) only mappings M (in some scenarios, also solutions K ) use constraints – in Knowledge Base Exchange (KBE) source KB K, M, and K use constraints. • We consider DL-LiteR as the language for the constraints; it is a formal counterpart of OWL 2 QL standard. Vlad Ryzhikov Free University of Bozen-Bolzano 3/16
  5. 5. Data Exchange vs. Knowledge Base Exchange • In Data Exchange (DE) only mappings M (in some scenarios, also solutions K ) use constraints – in Knowledge Base Exchange (KBE) source KB K, M, and K use constraints. • We consider DL-LiteR as the language for the constraints; it is a formal counterpart of OWL 2 QL standard. • Some definitions of solutions in DE apply to KBE, however, KBE allows for other natural definitions, which are easier to compute. Vlad Ryzhikov Free University of Bozen-Bolzano 3/16
  6. 6. Solutions Solution is a target KB K that “at best” preserves the meaning of a source KB K w.r.t. a mapping M. Vlad Ryzhikov Free University of Bozen-Bolzano 4/16
  7. 7. Solutions Solution is a target KB K that “at best” preserves the meaning of a source KB K w.r.t. a mapping M. Example: K = {A(a)}, M = {A Vlad Ryzhikov A }, Free University of Bozen-Bolzano K = {A (a)} − solution? 4/16
  8. 8. Solutions Solution is a target KB K that “at best” preserves the meaning of a source KB K w.r.t. a mapping M. Example: K = {A(a)}, M = {A K = {A(a), A B}, M = {A A }, A ,B K = {A (a)} − solution? B }, K = {A (a), B (a)} − solution? K = {A (a), A Vlad Ryzhikov Free University of Bozen-Bolzano B } − solution? 4/16
  9. 9. Solutions Solution is a target KB K that “at best” preserves the meaning of a source KB K w.r.t. a mapping M. Example: K = {A(a)}, M = {A K = {A(a), A B}, M = {A A }, A ,B K = {A (a)} − solution? B }, K = {A (a), B (a)} − solution? K = {A (a), A K = {A Vlad Ryzhikov B}, M = {A A ,B B }, K = {A Free University of Bozen-Bolzano B } − solution? B } − solution? 4/16
  10. 10. Solutions Solution is a target KB K that “at best” preserves the meaning of a source KB K w.r.t. a mapping M. Example: K = {A(a)}, M = {A K = {A(a), A B}, M = {A A }, A ,B K = {A (a)} − solution? B }, K = {A (a), B (a)} − solution? K = {A (a), A K = {A B}, M = {A A ,B B }, K = {A B } − solution? B } − solution? Different definitions make different K above solutions! Vlad Ryzhikov Free University of Bozen-Bolzano 4/16
  11. 11. Definitions DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and nulls; Σ be a set of DL-LiteR concept names A and role names P, then define basic concepts B ::= A | ∃P | ∃P − concept inclusions B1 concept disjointness B1 B2 B2 basic roles R ::= P | P − role inclusions R1 ⊥ role disjointness R1 R2 R2 ⊥ concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . , Vlad Ryzhikov Free University of Bozen-Bolzano 5/16
  12. 12. Definitions DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and nulls; Σ be a set of DL-LiteR concept names A and role names P, then define basic concepts B ::= A | ∃P | ∃P − concept inclusions B1 concept disjointness B1 B2 B2 basic roles R ::= P | P − role inclusions R1 ⊥ role disjointness R1 R2 R2 ⊥ concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . , DL-LiteR knowledge base is a set of concept/role inclusions/disjointness (called TBox) and concept/role membership assertions (called ABox if without nulls, otherwise extended ABox). Vlad Ryzhikov Free University of Bozen-Bolzano 5/16
  13. 13. Definitions DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and nulls; Σ be a set of DL-LiteR concept names A and role names P, then define basic concepts B ::= A | ∃P | ∃P − concept inclusions B1 concept disjointness B1 B2 B2 basic roles R ::= P | P − role inclusions R1 ⊥ role disjointness R1 R2 R2 ⊥ concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . , DL-LiteR knowledge base is a set of concept/role inclusions/disjointness (called TBox) and concept/role membership assertions (called ABox if without nulls, otherwise extended ABox). Mapping: defined over a pair of disjoint signatures Σ, Σ as the set of concept inclusions/disjointness, where B is over Σ and B over Σ . Vlad Ryzhikov Free University of Bozen-Bolzano 5/16
  14. 14. Definitions DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and nulls; Σ be a set of DL-LiteR concept names A and role names P, then define basic concepts B ::= A | ∃P | ∃P − concept inclusions B1 concept disjointness B1 B2 B2 basic roles R ::= P | P − role inclusions R1 ⊥ role disjointness R1 R2 R2 ⊥ concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . , DL-LiteR knowledge base is a set of concept/role inclusions/disjointness (called TBox) and concept/role membership assertions (called ABox if without nulls, otherwise extended ABox). Mapping: defined over a pair of disjoint signatures Σ, Σ as the set of concept inclusions/disjointness, where B is over Σ and B over Σ . Semantics: standard, no unique name assumption. Vlad Ryzhikov Free University of Bozen-Bolzano 5/16
  15. 15. Solutions We consider three notions: • Universal solution Inherited from incomplete data exchage; analogious to model conservative extentions or Σ-model inseparability Vlad Ryzhikov Free University of Bozen-Bolzano 6/16
  16. 16. Solutions We consider three notions: • Universal solution Inherited from incomplete data exchage; analogious to model conservative extentions or Σ-model inseparability • Universal UCQ-solution (UCQ = Union of Conjunctive Queries) Based on what can be extracted from source and target with unions of conjunctive queries; analogious to query conservative extentions or Σ-query inseparability Vlad Ryzhikov Free University of Bozen-Bolzano 6/16
  17. 17. Solutions We consider three notions: • Universal solution Inherited from incomplete data exchage; analogious to model conservative extentions or Σ-model inseparability • Universal UCQ-solution (UCQ = Union of Conjunctive Queries) Based on what can be extracted from source and target with unions of conjunctive queries; analogious to query conservative extentions or Σ-query inseparability • Representation Like Universal UCQ-solution, but defined w.r.t. K and K containing only TBox; uses universal quantification over possible the source and target ABoxes Vlad Ryzhikov Free University of Bozen-Bolzano 6/16
  18. 18. Universal Solutions • Let Mod(K) be the set of models of K. • Let I, J be a pair of DL-LiteR interpretations over signatures, respectively Σ and Γ, s.t. Σ ⊆ Γ, then I is said to agree with J on Σ, if aI = aJ for all constants a; AI = AJ and P I = P J for all concept and role names A and P from Σ. Given a Γ interpretation J , denote by agrΣ (J ) the set of all Σ interpretations that agree with J on Σ; we also use agrΣ (J ), where J is a set of Σ interpretations. Vlad Ryzhikov Free University of Bozen-Bolzano 7/16
  19. 19. Universal Solutions • Let Mod(K) be the set of models of K. • Let I, J be a pair of DL-LiteR interpretations over signatures, respectively Σ and Γ, s.t. Σ ⊆ Γ, then I is said to agree with J on Σ, if aI = aJ for all constants a; AI = AJ and P I = P J for all concept and role names A and P from Σ. Given a Γ interpretation J , denote by agrΣ (J ) the set of all Σ interpretations that agree with J on Σ; we also use agrΣ (J ), where J is a set of Σ interpretations. • Let the mapping M be between the signatures Σ and Σ ; a KB K over Σ is said to be a universal solution (US) for a KB K over Σ under M if Mod(K ) = agrΣ (Mod(K ∪ M)). Vlad Ryzhikov Free University of Bozen-Bolzano 7/16
  20. 20. Universal Solutions contd. Mod(K ) = agrΣ (Mod(K ∪ M)) Examples: K = {A(a)}, M = {A Vlad Ryzhikov A }, Free University of Bozen-Bolzano K = {A (a)} − US 8/16
  21. 21. Universal Solutions contd. Mod(K ) = agrΣ (Mod(K ∪ M)) Examples: K = {A(a)}, M = {A K = {A(a), A Vlad Ryzhikov B}, M = {A A }, A ,B K = {A (a)} − US B }, K = {A (a), B (a)} − US K = {A (a), A B } − not US Free University of Bozen-Bolzano 8/16
  22. 22. Universal Solutions contd. Mod(K ) = agrΣ (Mod(K ∪ M)) Examples: K = {A(a)}, M = {A K = {A(a), A B}, M = {A K = {∃R(a)}, M = {R Vlad Ryzhikov A }, A ,B R , ∃R − K = {A (a)} − US B }, K = {A (a), B (a)} − US K = {A (a), A B } − not US B }, K = {R (a, n), B (n)} − US Free University of Bozen-Bolzano 8/16
  23. 23. Universal Solutions contd. Mod(K ) = agrΣ (Mod(K ∪ M)) Examples: K = {A(a)}, M = {A K = {A(a), A B}, M = {A K = {∃R(a)}, M = {R K = {A B ⊥, A(a), B(b)}, M = {A Vlad Ryzhikov A }, A ,B R , ∃R − A ,B K = {A (a)} − US B }, K = {A (a), B (a)} − US K = {A (a), A B } − not US B }, K = {R (a, n), B (n)} − US B} Free University of Bozen-Bolzano − no US exists 8/16
  24. 24. Universal Solutions contd. US is a fundamental and well-behaved notion in DE, however, in KBE it has a number of issues: • Nulls required in ABox, which are not part of OWL 2 QL standard. Vlad Ryzhikov Free University of Bozen-Bolzano 9/16
  25. 25. Universal Solutions contd. US is a fundamental and well-behaved notion in DE, however, in KBE it has a number of issues: • Nulls required in ABox, which are not part of OWL 2 QL standard. • USs cannot contain any non-trivial TBox, i.e., only (extended) ABoxes can be materialized as the target. Vlad Ryzhikov Free University of Bozen-Bolzano 9/16
  26. 26. Universal Solutions contd. US is a fundamental and well-behaved notion in DE, however, in KBE it has a number of issues: • Nulls required in ABox, which are not part of OWL 2 QL standard. • USs cannot contain any non-trivial TBox, i.e., only (extended) ABoxes can be materialized as the target. • USs “very often” do not exists, when the source KB K1 contains disjointness assertions. Reason: no unique name assumption, as it is the case in OWL 2 QL. Vlad Ryzhikov Free University of Bozen-Bolzano 9/16
  27. 27. Universal UCQ-solutions Universal UCQ-solution is a “softer” notion of solution, that avoids the above mentioned issues. It uses UCQs q and certains answers cert(q, K): Vlad Ryzhikov Free University of Bozen-Bolzano 10/16
  28. 28. Universal UCQ-solutions Universal UCQ-solution is a “softer” notion of solution, that avoids the above mentioned issues. It uses UCQs q and certains answers cert(q, K): • Let the mapping M be between the signatures Σ and Σ ; a KB K over Σ is said to be a universal UCQ-solution (UUCQS) for a KB K over Σ under M if cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ . Vlad Ryzhikov Free University of Bozen-Bolzano 10/16
  29. 29. Universal UCQ-solutions Universal UCQ-solution is a “softer” notion of solution, that avoids the above mentioned issues. It uses UCQs q and certains answers cert(q, K): • Let the mapping M be between the signatures Σ and Σ ; a KB K over Σ is said to be a universal UCQ-solution (UUCQS) for a KB K over Σ under M if cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ . • if only the inclusion ⊇ in the equation above satisfied, K is called a UCQ-solution Vlad Ryzhikov Free University of Bozen-Bolzano 10/16
  30. 30. Universal UCQ-solutions contd. cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ Examples: K = {A(a)}, M = {A Vlad Ryzhikov A }, Free University of Bozen-Bolzano K = {A (a)} − UUCQS 11/16
  31. 31. Universal UCQ-solutions contd. cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ Examples: K = {A(a)}, M = {A K = {A(a), A Vlad Ryzhikov B}, M = {A A }, A ,B K = {A (a)} − UUCQS B }, K = {A (a), B (a)} − UUCQS K = {A (a), A B } − UUCQS Free University of Bozen-Bolzano 11/16
  32. 32. Universal UCQ-solutions contd. cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ Examples: K = {A(a)}, M = {A K = {A(a), A B}, M = {A K = {∃R(a)}, M = {R Vlad Ryzhikov A }, A ,B R , ∃R − K = {A (a)} − UUCQS B }, K = {A (a), B (a)} − UUCQS K = {A (a), A B } − UUCQS B }, Free University of Bozen-Bolzano K = {∃R (a), ∃R − B } − UUCQS 11/16
  33. 33. Universal UCQ-solutions contd. cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ Examples: K = {A(a)}, M = {A K = {A(a), A B}, M = {A K = {∃R(a)}, M = {R K = {A B ⊥, A(a), B(b)}, M = {A Vlad Ryzhikov A }, A ,B R , ∃R − A ,B K = {A (a)} − UUCQS B }, K = {A (a), B (a)} − UUCQS K = {A (a), A B } − UUCQS B }, B} Free University of Bozen-Bolzano K = {∃R (a), ∃R − B } − UUCQS K = {A (a), B (b)} −UUCQS 11/16
  34. 34. Universal UCQ-solutions contd. • UUCQS is a notion of the solution, that is better suited for KBE. Vlad Ryzhikov Free University of Bozen-Bolzano 12/16
  35. 35. Universal UCQ-solutions contd. • UUCQS is a notion of the solution, that is better suited for KBE. • Still, this notion is dependent on data, i.e., ABox; computing UUCQS requires processing big amounts of frequently changing data. Vlad Ryzhikov Free University of Bozen-Bolzano 12/16
  36. 36. Universal UCQ-solutions contd. • UUCQS is a notion of the solution, that is better suited for KBE. • Still, this notion is dependent on data, i.e., ABox; computing UUCQS requires processing big amounts of frequently changing data. • UCQ-representation is a notion of the solution, that is not dependent on data. Vlad Ryzhikov Free University of Bozen-Bolzano 12/16
  37. 37. UCQ-representation For the definition, we need to consider UCQ-solutions over KBs consisting of only ABoxes. Cosider A = {A(a)} and M = {A A }, then • A = {A (a), A (b)} - UCQ-solution; • A = {A (b)} - not UCQ-solution. Vlad Ryzhikov Free University of Bozen-Bolzano 13/16
  38. 38. UCQ-representation For the definition, we need to consider UCQ-solutions over KBs consisting of only ABoxes. Cosider A = {A(a)} and M = {A A }, then • A = {A (a), A (b)} - UCQ-solution; • A = {A (b)} - not UCQ-solution. Let the mapping M be between the signatures Σ and Σ ; a TBox T over Σ is said to be UCQ-representaton (UCQR) for a TBox T over Σ under M if cert(q, T ∪ A ∪ M) = cert(q, T ∪ A ). A : A is an ABox over Σ that is a UCQ-solution for A under M for • each UCQ q over Σ , • ABox A over Σ such that T ∪ A is consistent. Vlad Ryzhikov Free University of Bozen-Bolzano 13/16
  39. 39. UCQ-representation cont. cert(q, T ∪ A ∪ M) = cert(q, T ∪ A ) A : A is an ABox over Σ that is a UCQ-solution for A under M for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent. Examples: T = {A Vlad Ryzhikov A}, M = {A A }, T = {A Free University of Bozen-Bolzano A } − UCQR 14/16
  40. 40. UCQ-representation cont. cert(q, T ∪ A ∪ M) = cert(q, T ∪ A ) A : A is an ABox over Σ that is a UCQ-solution for A under M for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent. Examples: T = {A A }, T = {A Vlad Ryzhikov A}, M = {A B}, M = {A A ,B T = {A B }, A } − UCQR T = {A A } − not UCQR T = {A B } − UCQR Free University of Bozen-Bolzano 14/16
  41. 41. UCQ-representation cont. cert(q, T ∪ A ∪ M) = cert(q, T ∪ A ) A : A is an ABox over Σ that is a UCQ-solution for A under M for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent. Examples: T = {A A }, T = {A B}, M = {A A ,B T = {A Vlad Ryzhikov A}, M = {A B}, M = {B T = {A B }, B }, A } − UCQR T = {A A } − not UCQR T = {A B } − UCQR Free University of Bozen-Bolzano − no UCQR exists 14/16
  42. 42. UCQ-representation cont. cert(q, T ∪ A ∪ M) = cert(q, T ∪ A ) A : A is an ABox over Σ that is a UCQ-solution for A under M for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent. Examples: T = {A A}, M = {A A }, T = {A B}, M = {A A ,B T = {A B}, M = {B B }, ⊥}, M = {A A ,B T = {A Vlad Ryzhikov B T = {A B }, A } − UCQR T = {A A } − not UCQR T = {A B } − UCQR − no UCQR exists B }, T = {A Free University of Bozen-Bolzano B ⊥} − UCQR T = ∅ − UCQR 14/16
  43. 43. Summary of Complexity Results Membership Universal solutions UCQ-representations Non-emptiness Universal solutions UCQ-representations ABoxes extended ABoxes in NP NP-complete NLogSpace-complete ABoxes extended ABoxes in NP PSpace-hard, in ExpTime NLogSpace-complete • Membership problem: given source KB K1 , target KB K2 , and the mapping M, decide, if K2 is correct. • Non-emptyness problem: given source KB K1 and the mapping M, decide, if there exists a target KB K2 , such that it is correct. Vlad Ryzhikov Free University of Bozen-Bolzano 15/16
  44. 44. Summary of Complexity Results Membership Universal solutions UCQ-representations Non-emptiness Universal solutions UCQ-representations ABoxes extended ABoxes in NP NP-complete NLogSpace-complete ABoxes extended ABoxes in NP PSpace-hard, in ExpTime NLogSpace-complete • Membership problem: given source KB K1 , target KB K2 , and the mapping M, decide, if K2 is correct. • Non-emptyness problem: given source KB K1 and the mapping M, decide, if there exists a target KB K2 , such that it is correct. • Universal UCQ-solution: membership is PSpace-hard, no other results yet - future work. Vlad Ryzhikov Free University of Bozen-Bolzano 15/16
  45. 45. Thank you for your attention! Vlad Ryzhikov Free University of Bozen-Bolzano 16/16

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