Datalog : A Family of Languages for Ontology Querying

Transcription

1 Datalog : A Family of Languages for Ontology Querying Georg Gottlob Department of Computer Science University of Oxford joint work with Andrea Calì,Thomas Lukasiewicz, Marco Manna, Andreas Pieris et al.

2 1 st Motivation: Ontological Query Answering Input: a knowledge base K = hd, i a Boolean conjunctive query Q Question: K ² Q (or, equivalently, D [ ² Q) Old database problem CQ answering over incomplete databases

3 2 nd Motivation: Decidable FO fragments (TGDs) Good data complexity! 8X8Y8Z R(X,Y,Z),S(Y),P(X,Z) 9W Q(X,W) 8X8Y8Z R(X,Y),S(Y,Z,Z)! 9W P(Y,W) Guarded Linear Sticky Sticky-join 8X8Y R(X,X,Y)! 9Z R(Y,Y,Z) 8X8Y8Z S(X,Y),R(Y,Z)! 9W P(Y,W)

4 2 nd Motivation: Decidable FO fragments (TGDs) Good data complexity! 8X8Y8Z R(X,Y,Z),S(Y),P(X,Z) 9W Q(X,W) Query: R(Y,Y,Z),P(Z,W) Add rule: 8Y8Z8W R(Y,Y,Z),P(Z,W) 8X8Y8Z R(X,Y),S(Y,Z,Z)! 9W P(Y,W) Guarded Linear Sticky Sticky-join 8X8Y R(X,X,Y)! 9Z R(Y,Y,Z) 8X8Y8Z S(X,Y),R(Y,Z)! 9W P(Y,W)

5 3 rd Motivation: Decidable FO fragments (current work) Tame FO Guarded FO Sticky-join FO

6 (Some) Known Results Data Complexity Combined Complexity DL-Lite in AC 0 [Calvanese et al., JAR 07] NP-complete [Calvanese et al., JAR 07] EL + ELH PTIME-complete [Rosati, DL 07] NP-complete [Rosati, DL 07] Horn-SHIQ PTIME-complete [Eiter et al., JELIA 08] EXPTIME-complete [Eiter et al., JELIA 08] IDs + FKDs in AC 0 [Calì et al., IJCAI 03] PSPACE-complete [Johnson & Klug, JCSS 84] Datalog PTIME-complete EXPTIME-complete

7 Our Goal Datalog-based Formalism Description Logics (DL-Lite, EL, ) Datalog Relational Constraints (IDs, FKDs, ) without losing tractable data complexity

8 Ontological Reasoning and Datalog DL Assertion Concept Inclusion emp v person Concept Product sen-emp emp v morethan (Inverse) Role Inclusion reports v mgr Role Transitivity trans(mgr) Datalog Rule emp(x) person(x) sen-emp(x),emp(y) morethan(x,y) reports(x,y) mgr(y,x) mgr(x,y),mgr(y,z) mgr(x,z)

9 Ontological Reasoning and Datalog DL Assertion Concept Inclusion emp v person Concept Product sen-emp emp v morethan (Inverse) Role Inclusion reports v mgr Role Transitivity trans(mgr) Datalog Rule emp(x) person(x) sen-emp(x),emp(y) morethan(x,y) reports(x,y) mgr(y,x) mgr(x,y),mgr(y,z) mgr(x,z) Participation emp v 9report emp(x) 9Y report(x,y) Disjointness emp u customer v? emp(x),customer(x)? Functionality funct(reports) reports(x,y),reports(x,z) Y = Z

10 Datalog Datalog: 8X8Y (X,Y) R(X) Extend Datalog by allowing in the head: - Existential quantification (9) - TGDs: 8X8Y (X,Y) 9Z (X,Z) - Equality atoms (=) - EGDs: 8X (X) X i = X j - Constant false (?) - Negative constraints: 8X (X)? But query answering under Datalog[9] is undecidable [see, e.g., Beeri & Vardi, ICALP 81] Datalog[9,=,?] is syntactically restricted! Datalog

11 The Chase Procedure Input: Database D, set of TGDs Output: A model of D [ D person(john) 8P person(p) 9F father(f,p) 8F8P father(f,p) person(f) chase(d, ) = D [?

12 The Chase Procedure Input: Database D, set of TGDs Output: A model of D [ D person(john) 8P person(p) 9F father(f,p) 8F8P father(f,p) person(f) chase(d, ) = D [ {father(z 1,john)

13 The Chase Procedure Input: Database D, set of TGDs Output: A model of D [ D person(john) 8P person(p) 9F father(f,p) 8F8P father(f,p) person(f) chase(d, ) = D [ {father(z 1,john), person(z 1 )

14 The Chase Procedure Input: Database D, set of TGDs Output: A model of D [ D person(john) 8P person(p) 9F father(f,p) 8F8P father(f,p) person(f) chase(d, ) = D [ {father(z 1,john), person(z 1 ), father(z 2,z 1 )

15 The Chase Procedure Input: Database D, set of TGDs Output: A model of D [ D person(john) 8P person(p) 9F father(f,p) 8F8P father(f,p) person(f) chase(d, ) = D [ {father(z 1,john), person(z 1 ), father(z 2,z 1 ), }

16 The Chase Procedure Input: Database D, set of TGDs Output: A model of D [ D person(john) 8P person(p) 9F father(f,p) 8F8P father(f,p) person(f) chase(d, ) = D [ {father(z 1,john), person(z 1 ), father(z 2,z 1 ), } infinite instance

17 Query Answering via Chase Q h C = chase(d, ) D h 1 h 2 h 1 (C)... h 2 (C) M 1 M 2 D [ ² Q, chase(d, ) ² Q [see, e.g., Fagin, Kolaitis, Miller & Popa, TCS 05]

18 Early Positive Result Query answering under IDs is decidable P(X,Y) 9Z R(X,Y,Z) - PSPACE-complete in combined complexity - NP-complete for bounded arity Q D chase(d, )... [Johnson & Klug, JCSS 84]

19 Guardedness All 8-variables occur in one body atom - guard atom 8X8Y8Z R(X,Y,Z),S(Y),P(X,Z) 9W Q(X,W) guard Chase has finite treewidth ) decidability of query answering [Calì, G. & Kifer, KR 08] Query answering is PTIME-complete in data complexity [Calì, G. & Lukasiewicz, PODS 09] Properly extends ELH (same data complexity)

20 Ontology Querying ELH: Popular DL (for biological applications) with PTIME data complexity [Baader, IJCAI 03 and Rosati, DL 07] ELH TBox A v B A u B v C Datalog Representation 8X A(X) B(X) 8X A(X),B(X) C(X) 9R.A v B A v 9R.B R v P 8X R(X,Y), A(Y) B(X) 8X A(X) 9Y R(X,Y), B(Y) 8X8Y R(X,Y) P(X,Y)

21 Linearity Just one atom in the body 8X8Y R(X,Y)! 9Z (X,Z) guard Linear TGDs are trivially guarded Query answering is in AC 0 in data complexity (first-order rewritability) [Calì, G. & Lukasiewicz, PODS 09]

22 First-Order Rewritable TGDs Q compilation Q FO

23 First-Order Rewritable TGDs Q compilation Q FO translation Q * SQL

24 First-Order Rewritable TGDs Q compilation Q FO translation Q * SQL evaluation D 8D (D [ ² Q, D ² Q * ) Query answering is in AC 0 in data complexity [Vardi, PODS 95]

25 Bounded Derivation-Depth Property (BDDP) Q D depth independent of D P chase(d, ) chase(d, ) ² Q ) P ² Q [Calì, G. & Lukasiewicz, PODS 09]

26 Bounded Derivation-Depth Property (BDDP) Q D depth independent of D P chase(d, ) BDDP ) First-Order Rewritability [Calì, G. & Lukasiewicz, PODS 09]

27 Linearity Just one atom in the body 8X8Y R(X,Y)! 9Z (X,Z) trivially guard Linear TGDs are trivially guarded Query answering is in AC 0 in data complexity (first-order rewritability) [Calì, Gottlob & Lukasiewicz, PODS 09] Properly extends DL-Lite (same data complexity)

28 Ontology Querying DL-Lite: Popular family of DLs with AC 0 data complexity (OWL 2 QL) [Calvanese, De Giacomo, Lembo, Lenzerini & Rosati, JAR 07] DL-Lite TBox Datalog Representation A v B 8X A(X) B(X) A v 9R 8X A(X) 9Y R(X,Y) 9R v A 8X8Y R(X,Y) A(X) R v P 8X8Y R(X,Y) P(X,Y)

29 But What about joins in rule bodies? 8A8D8P runs(d,p),area(p,a) 9E employee(e,d,p,a) What about the DL assertion concept product? 8E8M elephant(e),mouse(m) biggerthan(e,m)

30 But What about joins in rule bodies? 8A8D8P runs(d,p),area(p,a) 9E employee(e,d,p,a) What about the DL assertion concept product? 8E8M elephant(e),mouse(m) biggerthan(e,m) No tree-like models guaranteed 8X8Y R(X,Y) 9Z R(Y,Z) 8X8Y R(X,Y) S(X) Infinitely many symbols in S 8X8Y S(X),S(Y) P(X,Y) P forms an infinite clique

31 Stickiness 8A8D8P runs(d,p),area(p,a) 9E employee(e,d,p,a) 8E8M elephant(e),mouse(m) biggerthan(e,m) [Calì, G. & Pieris, VLDB 10]

32 Stickiness 8X8Y8Z R(X,Y),P(Y,Z) 9W T(X,Y,W) 8X8Y8Z T(X,Y,Z) 9W S(Y,W) [Calì, G. & Pieris, VLDB 10]

33 Stickiness 8X8Y8Z R(X,Y),P(Y,Z) 9W T(X,Y,W) 8X8Y8Z T(X,Y,Z) 9W S(Y,W) 8X8Y8Z R(X,Y),P(Y,Z) 9W T(X,Y,W) 8X8Y8Z T(X,Y,Z) 9W S(X,W) [Calì, G. & Pieris, VLDB 10]

34 Stickiness: Marking Procedure Initial Marking: mark all occurrences of body-variables that do not appear in all head-atoms 8V8W R 1 (V,W) 9X9Y9Z R 2 (W,X,Y,Z) 8V8W8X8Y R 2 (V,W,X,Y) 9Z R 1 (W,Z),R 3 (W,Y),R 4 (Y,X) 8W8X8Y R 3 (W,X),R 4 (X,Y) 9Z R 5 (Z,Y,X)

35 Stickiness: Marking Procedure Initial Marking: mark all occurrences of body-variables that do not appear in all head-atoms Propagation Step: propagate the marking to body-variables via the head-atoms 8V8W R 1 (V,W) 9X9Y9Z R 2 (W,X,Y,Z) 8V8W8X8Y R 2 (V,W,X,Y) 9Z R 1 (W,Z),R 3 (W,Y),R 4 (Y,X) 8W8X8Y R 3 (W,X),R 4 (X,Y) 9Z R 5 (Z,Y,X)

36 Stickiness Marked variables occur just once in the body of each TGD 8V8W R 1 (V,W) 9X9Y9Z R 2 (W,X,Y,Z) 8V8W8X8Y R 2 (V,W,X,Y) 9Z R 1 (W,Z),R 3 (W,Y),R 4 (Y,X) 8W8X8Y R 3 (W,X),R 4 (X,Y) 9Z R 5 (Z,Y,X) The chase has the sticky property (backward-resolution terminates)

37 Sticky Property D chase(d, ) 8X8Y8Z R(X,Y),P(Y,Z,Z) 9W Q(W,Y,Z) R(a,z 1 ) P(z 1,z 2,z 2 )

38 Sticky Property D chase(d, ) 8X8Y8Z R(X,Y),P(Y,Z,Z) 9W Q(W,Y,Z) h R(a,z 1 ) P(z 1,z 2,z 2 ) Q(z 3,z 1,z 2 )

39 Sticky Property D chase(d, ) 8X8Y8Z R(X,Y),P(Y,Z,Z) 9W Q(W,Y,Z) h R(a,z 1 ) P(z 1,z 2,z 2 ) Q(z 3,z 1,z 2 ) z 1 and z 2 occur in every atom

40 Stickiness Marked variables occur just once in the body of each TGD 8V8W R 1 (V,W) 9X9Y9Z R 2 (W,X,Y,Z) 8V8W8X8Y R 2 (V,W,X,Y) 9Z R 1 (W,Z),R 3 (W,Y),R 4 (Y,X) 8W8X8Y R 3 (W,X),R 4 (X,Y) 9Z R 5 (Z,Y,X) The chase has the sticky property (backward-resolution terminates) Query answering is in AC 0 in data complexity (first-order rewritability) [Calì, G. & Pieris, VLDB 10] Properly extends DL-Lite (same data complexity)

41 Polynomial Witness Property (PWP) chase(d, ) D polynomial size w.r.t Q and (for fixed arity) Q [G. & Schwentick, KR 12]

42 Polynomial Witness Property (PWP) chase(d, ) D polynomial size w.r.t Q and (for fixed arity) Q PWP ) non-recursive Datalog rewriting of polynomial size [G. & Schwentick, KR 12]

43 Polynomial Witness Property (PWP) chase(d, ) D polynomial size w.r.t Q and (for fixed arity) Q Holds for the first-order rewritable Datalog fragments (linear and sticky) [G. & Schwentick, KR 12]

44 Finite Controllability? D [ ² Q, D [ ² fin Q Holds for inclusion dependencies [Rosati, PODS 06] Holds for guarded TGDs (in fact, for the guarded fragment) [Bárány, Gottlob & Otto, LICS 10] Holds for sticky TGDs [Gogacz & Marcinkowski, next talk]

45 Additional Features EGDs, e.g., 8X8Y8Z reports(x,y),reports(x,z)! Y = Z Non-Conflicting EGDs: do not interact with TGDs Preliminary check without adding complexity Negative constraints, e.g., 8X emp(x),customer(x)!? Check without adding complexity

46 Additional Features EGDs, e.g., 8X8Y8Z reports(x,y),reports(x,z)! Y = Z Non-Conflicting EGDs: do not interact with TGDs Preliminary check without adding complexity Negative constraints, e.g., 8X emp(x),customer(x)!? Check without adding complexity Finite controllability does not hold D = {R(a,b)} 8X8Y R(X,Y) 9Z R(Y,Z) = 8X8Y8Z R(Y,X),R(Z,X) Y = Z D [ ² Q but D [ ² fin Q Q R(A,a)

47 Datalog : Overview Guarded Linear Sticky 8X8Y R(X,X,Y)! 9Z R(Y,Y,Z) 8X8Y8Z8W S(X,Y),R(Z,W)! 9V S(Y,V),R(W,V)

48 Datalog : Overview Guarded Linear Sticky? without losing first-order rewritability and PWP

49 Sticky-Join TGDs Marked variables can appear more than once in a single atom Providing that the following does not happen: 8X 1 8Y 1 8Z 1 8W 1 p(x 1,Y 1 ),p(w 1,Z 1 ) s(x 1,Y 1,W 1,Z 1 ) Y 1 and W 1 are in join 8X 2 8Y 2 8Z 2 s(x 2,Y 2,Y 2,Z 2 ) p(x 2,Z 2 )

50 Sticky-Join TGDs Marked variables can appear more than once in a single atom Providing that the following does not happen: 8X 1 8Y 1 8Z 1 8W 1 p(x 1,Y 1 ),p(w 1,Z 1 ) s(x 1,Y 1,W 1,Z 1 ) Y 1 and W 1 are in join 8X 2 8Y 2 8Z 2 s(x 2,Y 2,Y 2,Z 2 ) p(x 2,Z 2 ) Same complexity as sticky TGDs and enjoy PWP But harder to identify them - PSPACE-complete

51 Datalog : Overview 8X8Y8Z R(X,Y),S(Y,Z,Z)! 9W P(Y,W) Guarded Linear Sticky Sticky-join 8X8Y R(X,X,Y)! 9Z R(Y,Y,Z) 8X8Y8Z S(X,Y),R(Y,Z)! 9W P(Y,W)

52 Datalog : Overview Guarded Linear Sticky Sticky-join? without losing PTIME data complexity

53 Tame TGDs (TTGDs) Check that each rule is guarded* or sticky-join (and thus sticky) 8X8Y8Z R(X,Y) 9Z R(Y,Z) sticky-join and guarded* 8X8Y T(X),R(X,Y) T(Y) 8X8Y T(X),T(Y) 9Z P(X,Y,Z) Guarded* not sticky-join sticky-join not guarded Not first-order rewritable due to the second rule Chase has infinite treewidth since P forms an infinite clique *) Predicates in heads of non-guarded rules are unguarded. Atoms whose predicate is unguarded are not allowed to serve as guards!

54 Tame TGDs The following set of TGDs is not tame 8X8Y8Z R(X,Y) 9Z R(Y,Z) sticky-join and guarded* 8X8Y T(X),R(X,Y) T(Y) Guarded* not sticky-join 8X8Y T(X),T(Y) 9Z P(X,Y,Z) neither sticky-join nor guarded 8X8Y P(X,X,Y) Q(Y) sticky-join and guarded *) Predicates in heads of non-guarded rules are unguarded. Atoms whose predicate is unguarded are not allowed to serve as guards!

55 No bounded tw model, but bounded tw resolution proof scheme!

56 Datalog : Overview DL-Lite PTIME-complete PTIME-complete FO-rewritable Guarded Linear Sticky Sticky-join Tame ELH IDs + FKDs

57 Datalog : Summary of Complexity Results Data Fixed Combined Guarded PTIME NP 2EXPTIME Linear in AC 0 NP PSPACE Sticky in AC 0 NP EXPTIME Sticky-Join in AC 0 NP EXPTIME Tame PTIME NP 2EXPTIME Same complexity with negative constraints and non-conflicting EGDs

58 current work Finite controllability of Tamed TGDs: Reduction to Sticky TGDs

59 current work Tame FO Guarded FO Sticky-join FO

60 current work Tame FO + GNF Guarded FO Sticky-join FO

61 Thank you!