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Optimizing Query Answering under Ontological Constraints
Giorgio Orsi1,2 and Andreas Pieris2
1Institute for the Future of Computing
Oxford Martin School University of Oxford
2Department of Computer Science
Ontological Constraints Giorgio Orsi 1,2 and Andreas Pieris 2 1 - - PowerPoint PPT Presentation
Ontological Constraints Giorgio Orsi 1,2 and Andreas Pieris 2 1 - - PowerPoint PPT Presentation
Optimizing Query Answering under Ontological Constraints Giorgio Orsi 1,2 and Andreas Pieris 2 1 Institute for the Future of Computing Oxford Martin School University of Oxford 2 Department of Computer Science University of Oxford VLDB 2011
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Ontological Databases
D D D
ABox TBox
Ontological Reasoning DB Constraints Ontological DB
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Ontological Databases
D D D Q(X) 9Y (X,Y)
ABox TBox
Ontological Reasoning DB Constraints Ontological DB
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Ontological Databases
D D D
,
ABox TBox
{ t | D [ ² 9u (t,u) } Ontological Reasoning DB Constraints Ontological DB Q(X) 9Y (X,Y)
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Ontological Constraints (examples)
Concept Inclusions: 8X emp(X) person(X) (Inverse) Relation Inclusion: Relation Transitivity: 8X8Y8Z mgs(X,Y),mgs(Y,Z) mgs(X,Z) 8X8Y manages(X,Y) isManaged(Y,X) Participation: 8X emp(X) 9Y report(X,Y) Disjointness: 8X emp(X), customer(X) ? Functionality: 8X8Y8Z reports(X,Y),reports(X,Z) Y = Z
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Datalog§
¡ Datalog variant allowing in the head:
- 9-variables ! TGDs 8X8Y (X,Y) 9Z (X,Z)
- Equality atoms ! EGDs 8X (X) Xi=Xj
- Constant false (?) ! NCs 8X (X) ?
Datalog+
[Cali’ et Al, PODS 09]
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Datalog+
Datalog§ [Cali’ et Al, PODS 09]
¡ Datalog variant allowing in the head:
- 9-variables ! TGDs 8X8Y (X,Y) 9Z (X,Z)
- Equality atoms ! EGDs 8X (X) Xi=Xj
- Constant false (?) ! NCs 8X (X) ?
¡ But, query answering under Datalog+ is undecidable
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Datalog+
Datalog§ [Cali’ et Al, PODS 09]
¡ Datalog variant allowing in the head:
- 9-variables ! TGDs 8X8Y (X,Y) 9Z (X,Z)
- Equality atoms ! EGDs 8X (X) Xi=Xj
- Constant false (?) ! NCs 8X (X) ?
¡ Datalog+ is syntactically restricted ! Datalog§ ¡ But, query answering under Datalog+ is undecidable
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Datalog+
Datalog§ [Cali’ et Al, PODS 09]
¡ Datalog variant allowing in the head:
- 9-variables ! TGDs 8X8Y (X,Y) 9Z (X,Z)
- Equality atoms ! EGDs 8X (X) Xi=Xj
- Constant false (?) ! NCs 8X (X) ?
¡ Datalog+ is syntactically restricted ! Datalog§ ¡ But, query answering under Datalog+ is undecidable ¡ TGDs more expressive than inclusion dependencies 8D8P8A runs(D,P),area(P,A) 9E employee(E,D,A)
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The Chase Procedure
Input: Database D, set of TGDs Output: A model of D [ person(john) 8X person(X) 9Y father(Y,X) 8X8Y father(X,Y) person(X) D chase(D,) = D [ ?
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The Chase Procedure
Input: Database D, set of TGDs Output: A model of D [ person(john) D chase(D,) = D [ {father(z1,john) 8X person(X) 9Y father(Y,X) 8X8Y father(X,Y) person(X)
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The Chase Procedure
Input: Database D, set of TGDs Output: A model of D [ person(john) D chase(D,) = D [ {father(z1,john), person(z1) 8X person(X) 9Y father(Y,X) 8X8Y father(X,Y) person(X)
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The Chase Procedure
Input: Database D, set of TGDs Output: A model of D [ person(john) D chase(D,) = D [ {father(z1,john), person(z1), father(z2,z1) 8X person(X) 9Y father(Y,X) 8X8Y father(X,Y) person(X)
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The Chase Procedure
Input: Database D, set of TGDs Output: A model of D [ person(john) D chase(D,) = D [ {father(z1,john), person(z1), father(z2,z1), …} 8X person(X) 9Y father(Y,X) 8X8Y father(X,Y) person(X)
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Query Answering via Chase
[see, e.g., Deutsch, Nash & Remmel, PODS 08]
D [ ² Q , chase(D,) ² Q D
. . .
C = chase(D,) M1 M2 h1 h2
h1(C) h2(C)
Q h
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Q
Query Answering via Rewriting
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Q Q
compilation
Query Answering via Rewriting
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Q
evaluation
Q Q
compilation
D
Query Answering via Rewriting
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Chase vs Rewriting
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Linear TGDs
8X8Y r(X,Y) 9Z (X,Z)
single body atom ¡ Properly generalize inclusion dependencies. ¡ Enjoy the bounded-derivation depth property. ¡ FO-rewritable Query Answering in AC0 (data complexity).
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Q q promotesTo(A,B), customer(B) (original query) promoter(X) Y promotesTo(X,Y) promotesTo(X,Y) customer(Y) q promotesTo(A,B), customer(B)
Q
FO-rewritability: example [Gottlob et Al., ICDE 11]
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q promotesTo(A,B), customer(B) q promotesTo(A,B), customer(V0,B) { Y = B } ( V0 is fresh ) promoter(X) Y promotesTo(X,Y) promotesTo(X,Y) customer(Y)
Q
FO-rewritability: example [Gottlob et Al., ICDE 11]
Q q promotesTo(A,B), customer(B)
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q promotesTo(A,B), customer(B) q promotesTo(A,B), promotesTo(V0,B) ans(A) promotesTo(A,B) factorization { A = V0 } promoter(X) Y promotesTo(X,Y) promotesTo(X,Y) customer(Y)
Q
FO-rewritability: Example [Gottlob et Al., ICDE 11]
Q q promotesTo(A,B), customer(B)
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q promoter(A) promoter(X) Y promotesTo(X,Y) promotesTo(X,Y) customer(Y)
Q
FO-rewritability: example [Gottlob et Al., ICDE 11]
Q q promotesTo(A,B), customer(B) q promotesTo(A,B) {X = A, Y = B} q promotesTo(A,B), customer(B)
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UCQ rewriting (first-order)
promoter(X) Y promotesTo(X,Y) promotesTo(X,Y) customer(Y)
Q
FO-rewritability: example [Gottlob et Al., ICDE 11]
Q q promoter(A) q promotesTo(A,B), customer(B) q promotesTo(A,B) q promotesTo(A,B), customer(B)
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FO-rewritability
¡ Desirable properties of a FO-rewriting: independent on the DB executable by any DBMS easy to compute (e.g., polynomial time) small size (e.g., polynomial size)
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FO-rewritability
¡ Unions of Conjunctive Queries (UCQs) executable by any DBMS DB independent easy to optimize and distribute worst-case exponential size in Q and
Calvanese et Al, JAR 07 Perez Urbina et Al, JAL 09 Cali’ et Al, PODS 09 Gottlob et Al, ICDE 11 and others…
¡ Desirable properties of a FO-rewriting: independent on the DB executable by any DBMS easy to compute (e.g., polynomial time) small size (e.g., polynomial size)
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¡ Combined and hybrid FO-rewriting good computational properties (e.g., polynomial in size) requires access to the DB
Perez Urbina et Al, JAL 09 Kontchakov et Al., KR 10 Gottlob and Schwentick, DL 11
FO-rewritability
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¡ Purely intensional Datalog rewriting very compressed representation purely intensional requires view-creation or Datalog engine ¡ Combined and hybrid FO-rewriting good computational properties (e.g., polynomial in size) requires access to the DB
Perez Urbina et Al, JAL 09 Kontchakov et Al., KR 10 Gottlob and Schwentick, DL 11 Perez Urbina et Al, JAL 09 Rosati and Almatelli., KR 10
FO-rewritability
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Datalog Rewriting: Keep it First-Order!
¡ A Datalog query is (in general) not a first-order query a non-recursive Datalog query is a first-order query a bounded Datalog query is a first-order query
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¡ A Datalog query is (in general) not a first-order query a non-recursive Datalog query is a first-order query a bounded Datalog query is a first-order query ¡ Input: a (w.l.o.g. boolean) conjunctive query Q = <q,ρ> Q : q(X) p(X), s(X,Y) <q, q(X) p(X),s(X,Y) > a set of linear TGDs ¡ Output: a bounded Datalog query Q = <q,π >
Datalog Rewriting: Keep it First-Order!
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Datalog Rewriting: skolemization (and renaming)
r(X,Y) Z s(Y,Z) s(X,Y) Z p(Y,Y,Z) p(X,Y,Z) t(Z)
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r(X,Y) Z s(Y,Z) s(X,Y) Z p(Y,Y,Z) p(X,Y,Z) t(Z) r(X1,Y1) s(Y1,f1(Y1)) s(X2,Y2) p(Y2,Y2,f2(Y2)) p(X3,Y3,Z3) t(Z3) f
Datalog Rewriting: skolemization (and renaming)
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Datalog Rewriting: Skolemization (and renaming)
r(X,Y) Z s(Y,Z) s(X,Y) Z p(Y,Y,Z) p(X,Y,Z) t(Z) ¡ f and are equisatisfiable (not equivalent) ¡ Introduce one Skolem function for each existential variable r(X1,Y1) s(Y1,f1(Y1)) s(X2,Y2) p(Y2,Y2,f2(Y2)) p(X3,Y3,Z3) t(Z3) f
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Datalog Rewriting: Rule Saturation
¡ Apply resolution inference rule to rules in f at least one of the rules contains Skolem terms δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3) f
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Datalog Rewriting: Rule Saturation
¡ Apply resolution inference rule to rules in f at least one of the rules contains Skolem terms f [f] … r(X1,Y1) p(f1(Y1) ,f1(Y1), f2(f1(Y1))) … δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3)
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Datalog Rewriting: Properties of Rule Saturation
¡ [f] mimics the chase derivations.
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Datalog Rewriting: Properties of Rule Saturation
¡ [f] mimics the chase derivations. δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3)
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Datalog Rewriting: Properties of Rule Saturation
¡ [f] mimics the chase derivations. ¡ [f] depends only on . ¡ [f] is possibly infinite linear TGDs have BDDP: suffices to construct it up to k steps [f]k. δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3)
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Datalog Rewriting: Query Saturation
¡ resolve [f] with the query Q. use only rules with Skolem terms.
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Datalog Rewriting: Query Saturation
¡ resolve [f] with the query Q. use only rules with Skolem terms. … δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3) … [δ12]] : r (X1,Y1) p(f1(Y1) ,f1(Y1), f2(f1(Y1))) … Q s(A,B), p(B,B,C) [f] Q … Q r(X1,Y1), p(f1(Y1), f1(Y1),C) … [Q,f]
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Datalog Rewriting: Query Saturation
¡ bypasses chase derivations with function symbols … δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3) … [δ12]] : r (X1,Y1) p(f1(Y1) ,f1(Y1), f2(f1(Y1))) … Q s(A,B), p(B,B,C) [f] Q
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Datalog Rewriting: Finalization
¡ keep only the function-free rules from [f] [ [Q,f] ¡ derivations producing certain answers are captured by function-symbol-free rules.
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¡ use the predicate graph to reduce the number of rules in f δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3) f Q s(A,B), p(B,B,C) Q
Optimizations: Pruning
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Optimizations: Pruning
¡ use the predicate graph to reduce the number of rules in f δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3) f Q s(A,B), p(B,B,C) Q
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Optimizations: Pruning
¡ use the predicate graph to reduce the number of rules in f δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) δ3 : p(X3,Y3,Z3) t(Z3) f Q s(A,B), p(B,B,C) Q ¡ we are no longer independent on Q!
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Optimizations: Query Elimination
¡ eliminate implied atoms during query saturation δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2)) f Q Q s(A,B), p(B,B,C)
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Optimizations: Query Elimination
¡ eliminate implied atoms during query saturation f Q s(A,B) ² p(B,B,C) Q s(A,B), p(B,B,C) atom coverage δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2))
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Optimizations: Query Elimination
¡ eliminate implied atoms during query saturation f Q s(A,B), p(B,B,C) ≡ Q s(A,B) Q s(A,B) ² p(B,B,C) Q s(A,B), p(B,B,C) atom coverage δ1 : r (X1,Y1) s(Y1,f1(Y1)) δ2 : s(X2,Y2) p(Y2,Y2,f2(Y2))
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Optimizations: Query Elimination
¡ unique elimination strategy (w.r.t. the final size of the rewriting) see paper. ¡ given m = |body(ρ)| and n = || worst-case size of [f] [ [Q,f] is O((n∙m)m) worst-case size of Q = <q,π > is O(n+m)m ¡ atom coverage under linear TGDs can be checked in polynomial time see paper.
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Experimental Results
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Discussion
¡ Datalog rewriting is substantially more compact than UCQ rewriting. ¡ Unclear whether this always leads to increase in performance. ¡ Extend the procedure to larger classes of TGDs guarded TGDs [Cali’ et Al, PODS 09] non FO-rewritable sticky-join TGDs [Cali’ et Al, VLDB 10]
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