Conjunctive Queries

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QueryHomomorphisms (akaContainmentMappings) Example Amappingfromthevariables ofCQQ2to thevariablesofCQQ1,suchthat: TheheadofQ2ismappedtotheheadofQ1. Q1:p(X,Y) r(X,Z),g(Z,Z),r(Z,Y) Q2:p(A,B) r(A,C),g(C,D),r(D,B) Eachsubgoal (=bodyatom)ofQ2ismappedto somesubgoal ofQ1withthesamepredicate. HomomorphismTheorem: ThereisahomomorphismfromQ2toQ1if andonlyifQ1 Q2. 21 Q1looksfor: X Q2looksfor: A Z C Y D SinceC=Dispossible, expectQ1 Q2. B 22 Example Continued • WheneverthereisapathfromX toY,it mustbethatX hasanarcout,andY an arcin. • Thus,everyfact(tuple)producedbyQ1is alsoproducedbyQ2. •Thatis,Q1 Q2. Example Continued Q1: p(X,Y) r(X,Z) , g(Z,Z) , r(Z,Y) Q2: p(A,B) r(A,C) , g(C,D) , r(D,B) Homomorphism:h(A)=X;h(B)=Y;h(C)=h(D)=Z. 23 24
ExampleConcluded Q1: p(X,Y) r(X,Z) , g(Z,Z) , r(Z,Y) Q2: p(A,B) r(A,C) , g(C,D) , r(D,B) •NocontainmentmappingfromQ1toQ2. – g(Z,Z)canonlybemappedtog(C,D). •Nootherg subgoals inQ2. –ButthenZmustmaptobothCandD impossible. • Thus,Q1properlycontainedinQ2. 25 ProofoftheHomomorphismTheorem Hom(Q2,Q1) iff Q1 Q2. a) hHOM(Q2,Q1) Q1 Q2(thisistheonlyif direction) LetDbeadatabaseandAQ1(D). Then gHOM(body(Q1),D),suchthatA=g(head(Q1)). Q2:head(Q2) body(Q2) h h Q1:head(Q1) body(Q1) A g g AD Buttheng hisahomomorphisminHOM(body(Q2),D), andg h(head(Q2))=g(h(head(Q2))=g(head(Q1))=A ThusA Q2(D). Insummary,AQ1(D),AQ2(D),thusQ1 Q2. 26 ProofoftheHomomorphismTheorem Hom(Q2,Q1) iff Q1 Q2. b)Q1 Q2 hHOM(Q2,Q1)(thisistheif direction) a) hHOM(Q2,Q1) Q1 Q2(thisistheonlyif direction) LetDbeadatabaseandAQ1(D). Then gHOM(body(Q1),D),suchthatA=g(head(Q1)). Q2:head(Q2) body(Q2) g h h h g h Q1:head(Q1) body(Q1) A g g AD Buttheng hisahomomorphisminHOM(body(Q2,D)), andg h(head(Q2))=g(h(head(Q2))=g(head(Q1))=A ThusA Q2(D). Insummary,AQ1(D),AQ2(D),thusQ1 Q2. 27 Q2: p(A,B) r(A,C) , g(C,D) , r(D,B) Q1: p(X,Y) r(X,Z) , g(Z,Z) , r(Z,Y) CanonicalDatabaseCan Q1 : Generateanewconstantf(U)=U foreachvariable U Can Q1 =f(atoms(body(Q)))={r(X,Z), g(Z,Z), r(Z,Y)} dom(Can Q1 )={X,Y,Z} Note:Themappingf:var(Q1) {newconstants}isanisomorphism where f(atoms(body(Q1)))=Can Q1 and f 1 (Can Q1 )=atoms(body(Q1) 28
Page 1 and 2: Three Problems: CQs and Structural
Page 3 and 4: Boolean Conjunctive Queries The pro
Page 5: Combinatorial Crossword Puzzle All
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

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