Translatable Updates of Selection Views under Constant Complement

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1 Translatable Updates of Selection Views under Constant Complement Enrico Franconi and Paolo Guagliardo Free University of Bozen-Bolzano, Italy 4 th September 2014 DEXA 2014, Munich (Germany) KRDB Research Centre for Knowledge and Data

2 Outline 1. The View Update Problem 2. Updatable Selection Views 3. Propagation of Updates 4. Mending Non-Updatable Selection Views 5. Summary and Future Work 1 / 18

3 The View Update Problem view definitions 2 / 18

4 The View Update Problem view definitions view update 2 / 18

5 The View Update Problem view definitions database update view update 2 / 18

6 The View Update Problem view definitions database update view update view definitions 2 / 18

7 Unambiguous Progagation of View Updates S source schema (finite set of relation symbols) Σ S intergrity contraints over S (source constraints) T target schema disjoint with S 3 / 18

8 Unambiguous Progagation of View Updates S source schema (finite set of relation symbols) Σ S intergrity contraints over S (source constraints) T target schema disjoint with S f view: maps instances of S satisfying Σ S to instances of T I f J 3 / 18

9 Unambiguous Progagation of View Updates S source schema (finite set of relation symbols) Σ S intergrity contraints over S (source constraints) T target schema disjoint with S f view: maps instances of S satisfying Σ S to instances of T I f J u J 3 / 18

10 Unambiguous Progagation of View Updates S source schema (finite set of relation symbols) Σ S intergrity contraints over S (source constraints) T target schema disjoint with S f view: maps instances of S satisfying Σ S to instances of T I f J u I f J 3 / 18

11 Unambiguous Progagation of View Updates S source schema (finite set of relation symbols) Σ S intergrity contraints over S (source constraints) T target schema disjoint with S f view: maps instances of S satisfying Σ S to instances of T I f J u I f J Requirement: I must exist and be unique (in such a case we say that u is uniquely translatable on J) 3 / 18

12 Views Specified by Constraints Σ ST one formula of the form x. T (x) ϕ(x) for each T T where ϕ is a (safe) query over S 4 / 18

13 Views Specified by Constraints Σ ST one formula of the form x. T (x) ϕ(x) for each T T where ϕ is a (safe) query over S A view f under Σ S and specified by Σ ST is updatable if each S(x) has an exact rewriting in terms of T under Σ S Σ ST 4 / 18

14 Views Specified by Constraints Σ ST one formula of the form x. T (x) ϕ(x) for each T T where ϕ is a (safe) query over S A view f under Σ S and specified by Σ ST is updatable if each S(x) has an exact rewriting in terms of T under Σ S Σ ST General Translatability Criterion For each S S, replace every occurrence of S(x) in Σ S Σ ST with its exact rewriting in terms of T The resulting set Σ T mentions only symbols in T An update u is uniquely translatable on J img(f) precisely if u(j) = Σ T (AC 0 data complexity) 4 / 18

15 Global Translatability Drawback: The general translatability criterion is applicable only when the target instances are materialized 5 / 18

16 Global Translatability Drawback: The general translatability criterion is applicable only when the target instances are materialized Solution: Given a view update u, check whether it is globally translatable, that is, translatable on every J img(f) 5 / 18

17 Global Translatability Drawback: The general translatability criterion is applicable only when the target instances are materialized Solution: Given a view update u, check whether it is globally translatable, that is, translatable on every J img(f) We show how this can be done for insertions / deletions sequences of insertions and deletions replacements w.r.t. selection views 5 / 18

18 Selection Views S 6 / 18

19 Selection Views T 1 selection S 6 / 18

20 Selection Views T 2 T 1 selection selection S 6 / 18

21 Selection Views T 2 T 1 T 3 selection selection selection S 6 / 18

22 Selection Views T 2 T 1 T 3 S 6 / 18

23 Selection Views T 2 T 1 T 3 S 6 / 18

24 Interpreted Attributes Data values from special domains (e.g., the integers or the reals) on which a set of predicates (e.g., smaller/greater than) and functions (e.g., addition and subtraction) are defined, according to a first-order language C 7 / 18

25 Interpreted Attributes Data values from special domains (e.g., the integers or the reals) on which a set of predicates (e.g., smaller/greater than) and functions (e.g., addition and subtraction) are defined, according to a first-order language C Notation S( x 1,..., x k, y 1,..., y n ) x 1,..., x k non-interpreted (values from dom) y 1,..., y n interpreted over idom associated with C 7 / 18

26 Interpreted Attributes Data values from special domains (e.g., the integers or the reals) on which a set of predicates (e.g., smaller/greater than) and functions (e.g., addition and subtraction) are defined, according to a first-order language C Notation S( x 1,..., x k, y 1,..., y n ) x 1,..., x k non-interpreted (values from dom) y 1,..., y n interpreted over idom associated with C We assume C to be closed under negation 7 / 18

27 Unit Two-Variable Per Inequality (UTVPI) Constraints A UTVPI constraint is a formula of the form ax + by d where x, y integer variables a, b { 1, 0, 1} (unit coefficients) d Z (an integer) 8 / 18

28 Unit Two-Variable Per Inequality (UTVPI) Constraints A UTVPI constraint is a formula of the form ax + by d where x, y integer variables a, b { 1, 0, 1} (unit coefficients) d Z (an integer) Fragment of linear arithmetic constraints over the integers Satisfiability of UTVPIs is decidable in polynomial time 8 / 18

29 Unit Two-Variable Per Inequality (UTVPI) Constraints A UTVPI constraint is a formula of the form ax + by d where x, y integer variables a, b { 1, 0, 1} (unit coefficients) d Z (an integer) Fragment of linear arithmetic constraints over the integers Satisfiability of UTVPIs is decidable in polynomial time BUTVPI = a Boolean combination of UTVPIs Satisfiability of BUTVPIs is NP-complete 8 / 18

30 Conditional Domain Constraints (CDCs) A conditional domain constraint is a formula of the form x, y. S(x, y) λ(x) δ(y) where λ(x) Boolean combination of equalities x = a, with x from x and a from dom δ(y) a formula in C For short S λ δ 9 / 18

31 Specification of Selection Views Each target symbol T T is defined by a formula of the form x, y. T (x, y) ( S(x, y) λ(x) σ(y) ) where λ(x) Boolean combination of equalities x = a, with x from x and a from dom σ(y) a formula in C For short T λ σ 10 / 18

32 Example S( x 1, x 2, x 3, y 1, y 2 ) Name Department Position Salary Bonus non-interpreted interpreted over Z 11 / 18

33 Example S( x 1, x 2, x 3, y 1, y 2 ) Name Department Position Salary Bonus non-interpreted interpreted over Z Let y 1, y 2 be expressed in thousands of euros per month Let a = ICT and b = Manager Source Constraints Employees in ICT gain at most 5 x 2 = a y 1 + y 2 5 Σ S x 3 = b y 2 2 Managers receive a bonus of at least 2 y 1 y 2 0 Bonus is never greater than salary 11 / 18

34 Example S( x 1, x 2, x 3, y 1, y 2 ) Name Department Position Salary Bonus non-interpreted interpreted over Z Let y 1, y 2 be expressed in thousands of euros per month Let a = ICT and b = Manager View Specification Σ ST T 1 x 2 a x 3 = b T 2 y 2 < 4 T 3 x 3 b Managers in departments other than ICT Employees getting strictly less than 4 as bonus Employees not working as managers 11 / 18

35 Derived Target Constraints When a selection view under Σ S and specified by Σ ST is updatable we have that Σ S Σ ST = x, y. S(x, y) T 1 (x, y) T n (x, y) 12 / 18

36 Derived Target Constraints When a selection view under Σ S and specified by Σ ST is updatable we have that Σ S Σ ST = x, y. S(x, y) T 1 (x, y) T n (x, y) Replacing S(x, y) by T 1 (x, y) T n (x, y) in Σ S Σ ST we get the following set Σ T of derived target constraints: T i λ δ every tuple in T i must satisfy the source CDCs 12 / 18

37 Derived Target Constraints When a selection view under Σ S and specified by Σ ST is updatable we have that Σ S Σ ST = x, y. S(x, y) T 1 (x, y) T n (x, y) Replacing S(x, y) by T 1 (x, y) T n (x, y) in Σ S Σ ST we get the following set Σ T of derived target constraints: T i λ δ every tuple in T i must satisfy the source CDCs T i λ i σ i every tuple in T i must satisfy the selection conditions of T i 12 / 18

38 Derived Target Constraints When a selection view under Σ S and specified by Σ ST is updatable we have that Σ S Σ ST = x, y. S(x, y) T 1 (x, y) T n (x, y) Replacing S(x, y) by T 1 (x, y) T n (x, y) in Σ S Σ ST we get the following set Σ T of derived target constraints: T i λ δ every tuple in T i must satisfy the source CDCs T i λ i σ i every tuple in T i must satisfy the selection conditions of T i T i λ j σ j T j every tuple in T i satisfying the selection conditions of T j must also be in T j 12 / 18

39 Translatability of Insertions and Deletions Theorem Let u be the insertion +J. Then, u is translatable on J if and only if J \ J = Σ T (look only at the newly inserted facts) 13 / 18

40 Translatability of Insertions and Deletions Theorem Let u be the insertion +J. Then, u is translatable on J if and only if J \ J = Σ T (look only at the newly inserted facts) u is globally translatable if and only if J = Σ T 13 / 18

41 Translatability of Insertions and Deletions Theorem Let u be the insertion +J. Then, u is translatable on J if and only if J \ J = Σ T (look only at the newly inserted facts) u is globally translatable if and only if J = Σ T Theorem Let u be the deletion J. Then, u is translatable on J if and only if J J = Σ T (look only at the facts that are effectively deleted) 13 / 18

42 Translatability of Insertions and Deletions Theorem Let u be the insertion +J. Then, u is translatable on J if and only if J \ J = Σ T (look only at the newly inserted facts) u is globally translatable if and only if J = Σ T Theorem Let u be the deletion J. Then, u is translatable on J if and only if J J = Σ T (look only at the facts that are effectively deleted) u is globally translatable if and only if J \ J = Σ T, where J is the maximal subset of J that does not satisfy cdc(σ T ) 13 / 18

43 Translatability of Insertions and Deletions Theorem Let u be the insertion +J. Then, u is translatable on J if and only if J \ J = Σ T (look only at the newly inserted facts) u is globally translatable if and only if J = Σ T Theorem Let u be the deletion J. Then, u is translatable on J if and only if J J = Σ T (look only at the facts that are effectively deleted) u is globally translatable if and only if J \ J = Σ T, where J is the maximal subset of J that does not satisfy cdc(σ T ) Theorem The global translatability problem for insertions and deletions is in AC 0 13 / 18

44 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn 14 / 18

45 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn +J 1 14 / 18

46 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn +J 1 J 2 14 / 18

47 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn +J 1 J 2 +J n 1 14 / 18

48 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn +J 1 J 2 +J n 1 J n 14 / 18

49 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn +J 1 J 2 +J n 1 J n equivalent to inserting A and simultaneously deleting D, where: a fact is in A iff it is inserted at some point in the sequence and it is not deleted afterwards a fact is in D iff it is deleted at some point in the sequence and it is not re-inserted afterwards 14 / 18

50 Translatability of Sequences of Insertions and Deletions +J 1,1 +J 1,m1 J 2,1 J 2,m2 +J n 1,1 +J n 1,mn 1 J n,1 J n,mn +J 1 J 2 +J n 1 J n equivalent to inserting A and simultaneously deleting D, where: a fact is in A iff it is inserted at some point in the sequence and it is not deleted afterwards a fact is in D iff it is deleted at some point in the sequence and it is not re-inserted afterwards Theorem The global translatability problem for sequences consisting of insertions and deletions is in AC 0 14 / 18

51 Translatability of Replacements A replacement u is specified by means of a surjective mapping r : D A where D and A are disjoint sets of facts 15 / 18

52 Translatability of Replacements A replacement u is specified by means of a surjective mapping r : D A where D and A are disjoint sets of facts Theorem Let u be the replacement specified by r : D A. Then, u is translatable on J if and only if D+r(J D) is 15 / 18

53 Translatability of Replacements A replacement u is specified by means of a surjective mapping r : D A where D and A are disjoint sets of facts Theorem Let u be the replacement specified by r : D A. Then, u is translatable on J if and only if D+r(J D) is Let D be the maximal subset of D that satisfies cdc(σ T ); u is globally translatable iff D = Σ T and, for each D D, r(d ) satisfies Σ T whenever D does 15 / 18

54 Translatability of Replacements A replacement u is specified by means of a surjective mapping r : D A where D and A are disjoint sets of facts Theorem Let u be the replacement specified by r : D A. Then, u is translatable on J if and only if D+r(J D) is Let D be the maximal subset of D that satisfies cdc(σ T ); u is globally translatable iff D = Σ T and, for each D D, r(d ) satisfies Σ T whenever D does Theorem The global translatability of a replacement specified by r : D A can be decided in linear time in the size of D 15 / 18

55 Non-Updatable Views Skip Problem: What if a view f is not updatable? 16 / 18

56 Non-Updatable Views Skip Problem: What if a view f is not updatable? Solution: Use another view g such that f g is updatable g is called a complement of f g provides information missing from f 16 / 18

57 Non-Updatable Views Skip Problem: What if a view f is not updatable? Solution: Use another view g such that f g is updatable g is called a complement of f g provides information missing from f Constant complement principle Update propagation must not affect, directly or indirectly, the contents of the view complement 16 / 18

58 Non-Updatable Views Skip Problem: What if a view f is not updatable? Solution: Use another view g such that f g is updatable g is called a complement of f g provides information missing from f Constant complement principle Update propagation must not affect, directly or indirectly, the contents of the view complement Complements must be as small as possible 16 / 18

59 Complements of Selection Views We provide necessary and sufficient conditions for the existence of a unique minimal complement 17 / 18

60 Complements of Selection Views We provide necessary and sufficient conditions for the existence of a unique minimal complement Selection conditions expressed by BUTVPIs There exists a unique minimal complement We provide an algorithm for finding the minimal complement 17 / 18

61 Complements of Selection Views We provide necessary and sufficient conditions for the existence of a unique minimal complement Selection conditions expressed by BUTVPIs There exists a unique minimal complement We provide an algorithm for finding the minimal complement Selection conditions expressed by UTVPIs There might not exist a unique minimal complement We provide an algorithm for finding one minimal complement 17 / 18

62 Complements of Selection Views We provide necessary and sufficient conditions for the existence of a unique minimal complement Selection conditions expressed by BUTVPIs There exists a unique minimal complement We provide an algorithm for finding the minimal complement Selection conditions expressed by UTVPIs There might not exist a unique minimal complement We provide an algorithm for finding one minimal complement Both algorithms have exponential worst-case running time in the size of the original view 17 / 18

63 Conclusion Summary The global translatability problem for sequences consisting of insertions and deletions is in AC 0 The global translatability problem for replacements can be decided in linear time 18 / 18

64 Conclusion Summary The global translatability problem for sequences consisting of insertions and deletions is in AC 0 The global translatability problem for replacements can be decided in linear time Future Work Update independence update a target relation without affecting the others Additional source constraints FDs and UINDs in addition to (suitably restricted) CDCs 18 / 18

Lossless Horizontal Decomposition with Domain Constraints on Interpreted Attributes

Lossless Horizontal Decomposition with Domain Constraints on Interpreted Attributes Ingo Feinerer 1 and Enrico Franconi 2 and Paolo Guagliardo 2 1 Vienna University of Technology 2 Free University of Bozen-Bolzano