Conjunctive Queries

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ConjunctiveQueryevaluation CQminimization THEOREM:BCQisNPComplete(combinedcomplexity) Proof:FollowsfromtheNPcompletenessofHOM Exercise: AsanalternativeproofofNPhardness,givea reductionfrom3SAT. COROLLARY: ConjunctiveQueryContainmentisNPcomplete COROLLARY: ConjunctiveQueryEquivalenceisNPcomplete •AconjunctivequeryQisminimal,ifitcontains minimalnumberofsubgoals – (subgoal=bodyatom) Thismeans:whenewerwedropanatom fromthebodyofQ,wegetaqueryQ’Q •ThetaskoftheCQminimization:minimizethe numberofsubgoals ofagivenCQ. Note:ThelatterproblemsareundecidableforFO 37 38 Cores Cores endomorphism h: {Y b} BCQ: Pp(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) B= { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d)} { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } Logical meaning X, Y, U, V: p(X,Y) & p(X,b) & p(a,b) & p(U,c) & p(U,V) & q(a,c,d) 39 40
Cores endomorphism h: {Y b} B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } Cores endomorphism h: {Y b} B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } REDUNDANT! X,Y p(X,Y) & P(X,b) X p(X,b) 41 REDUNDANT! X,Y p(X,Y) X p(X,b) 42 Cores endomorphism h: {Y b} B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } Cores endomorphism h: {Y b} B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } h(B) = { p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } h(B) = { p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) } X a V c h(B) can be further reduced by endomorphism g: {X a, V c} 43 44
Page 1 and 2: Three Problems: CQs and Structural
Page 3 and 4: Boolean Conjunctive Queries The pro
Page 5 and 6: Combinatorial Crossword Puzzle All
Page 7 and 8: ExampleConcluded Q1: p(X,Y) r(X,Z)
Page 9: Example Continued Q1: p(X,Y) r(X,Y)
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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