Conjunctive Queries

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BCQ = HOM View query Q as a relational structure Universe: Variables of the query Relations: sets of query-atoms for each database relation. Non-Boolean CQs Q:r(x) r 1 (x 1 ) r 2 (x 2 )… r m (x m ) The database D is itself a relational structure. By the definition of the semantics of FO queries, the Boolean conjunctive query evaluates to true iff there is a homomorphism h: QD. R1 R1 DatabaseD ….. R1 D=QiffHOM(Q,D) HowdoweformallydefineQ(D) Vive-versa, every HOM instance can be reformulated as a Boolean conjunctive query. 14 Non-Boolean CQs head(Q) body(Q) r(x) r 1 (x 1 ) r 2 (x 2 )… r m (x m ) h h homomorphismh h R1 R1 ….. R1 DatabaseD Problems Equivalent to BCQ • <strong>Conjunctive</strong> Query Containment Q ( 1 Q2 db Q1 db) Q2 ( db) • Query of Tuple Problem t Q(db) • Constraint Satisfaction in AI CSP BCQ • Clause Subsumption in Theorem Proving Q(D)={h(head(Q))|h HOM(body(Q),D)} s.t. C D 15
Combinatorial Crossword Puzzle All known general solution algorithms rely on backtracking Combinatorial Crossword Puzzle All known general solution algorithms rely on backtracking 1h: P A R I S 1v: L I M B O P A N D A L I N G O L A U R A P E T R A A N I T A P A M P A P E T E R We are looking for a homomorphism f:[1..26][A..Z] that translates 1H ={} into 1h, and so on. and so on 1h: P A R I S 1v: L I M B O P A N D A L I N G O L A U R A P E T R A A N I T A P A M P A P E T E R We are looking for a homomorphism f:[1..26][A..Z] that translates 1H ={} into 1h, and so on. and so on ContainmentofCQ’s • GivenconjunctivequeriesQ1andQ2,wesay thatQ1iscontainedinQ2,denotedasQ1 Q2, iff forall databasesD,Q1(D) Q2(D). • Example: –Q1:p(X,Y) arc(X,Z),arc(Z,Y) –Q2:p(X,Y) arc(X,Z),arc(W,Y) •DBisagraph;Q1producespathsoflength2, Q2producespairsofnodeswithanarcoutand in,respectively. 19 WhyCQContainment Containmenttestsusedforequivalencetests. Q1=Q2iffQ1 Q2andQ2 Q1 Equivalenceisfundamentaltoqueryoptimization Answeringqueriesusingviews. Dataintegration:collectingdatafromdifferentsources (e.g.onInternet),answeringyourspecificquery need toknowwhetherthesourcescan(fully,orpartially) answeryourquery. 20
Page 1 and 2: Three Problems: CQs and Structural
Page 3: Boolean Conjunctive Queries The pro
Page 7 and 8: ExampleConcluded Q1: p(X,Y) r(X,Z)
Page 9 and 10: Example Continued Q1: p(X,Y) r(X,Y)
Page 11 and 12: Cores endomorphism h: {Y b} B = {
Page 13 and 14: CQminimization Ans(X,Y,Z) R(X 2 ,Y
Page 15 and 16: Acyclic CSP Acyclic queries or CSPs
Page 17 and 18: GYO’ algorithm- removingears Algo
Page 19 and 20: d: 3 8 3 7 5 7 6 7 d(Y,P) r(Y,Z,U)
Page 21 and 22: s: 3 8 9 9 3 8 8 3 8 3 8 4 3 8 3 8
Page 23 and 24: … d: 3 8 3 7 5 7 6 7 d(Y,P) r(Y
Page 25 and 26: p(x,y) q q(x,z) … ABCQ is in LO
Page 27 and 28: NearlyAcyclicGraphs y Example: a cy
Page 29 and 30: Playing the game Playing the game q
Page 31 and 32: Beyond treewidth Bounded Degree of
Page 33 and 34: Hypertree Decompositions Grouping a
Page 35 and 36: Hypergraph and the tree decompositi
Page 37 and 38: QUESTION: Hypertree Decomposition =
Page 39 and 40: Hypertree decomposition Hypertree d
Page 41 and 42: Relationship GHW vs. HW: Observatio

Conjunctive Queries

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