Processing Unions of Conjunctive Queries with Negation under Limited Access Patterns Alan Nash Department of Mathematics Bertram Lud¨ascher San Diego Supercomputer Center University of California, San Diego
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Data Integration •
Problem: Integrate sources with limited query capabilities
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Example: Global-as-View (GAV) integration: • Given: query Q against global view GV over source schemas Si : Q(¯ x) ← QGV (¯ x, y¯) •
Find: query plan P with subqueries Qi against sources Si with limited access patterns: Q(¯ x) ← P (Q1 , . . . , Qn )(¯ x, y¯)
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Web Service Composition •
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Problem: Declarative composition of web services into larger web service “workflows” or ”dataflows.” Idea: Specify composite web service plans as declarative queries, then do query planning over sources with limited access patterns. Example: Given web service “relations” WS1 (in y, out z) and WS2 (in x, out y), the declarative query Q(x, z) ← WS2 (x, y), WS1 (y, z) WS2
WS1
becomes a web service plan x −→ y −→ z.
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Access Patterns Assume you have the following interface, which requires you enter an author, a title, or a subject. author title subject publisher We model this as a relation B(a, t, s, p) with three access patterns: iooo, oioo, ooio.
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Queries Under Access Patterns What kinds of queries can we answer? • List all books • Title and publisher for non-‘Springer’ books by ‘Knuth’ • Books by ‘Knuth’ or ‘Aho’
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Classes of Queries • • • •
Q1 (a, t, s, p) ← B(a, t, s, p) Q2 (t, p) ← ¬B( ‘Knuth’ , t, s, ‘Springer’ ), B( ‘Knuth’ , t, s, p) Q3 (a, t, p) ← B( ‘Knuth’ , t, s, p) Q3 (a, t, p) ← B( ‘Aho’ , t, s, p) Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t)
These are CQ, CQ¬ ,UCQ, and UCQ¬ respectively.
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Executable Queries A UCQ¬ query is executable if every variable appears first, positively, in an output slot in the body. Q3 can be annotated as follows Q3 (a, t, p) ← B iooo ( ‘Knuth’ , t, s, p) Q3 (a, t, p) ← B iooo ( ‘Aho’ , t, s, p) Q3 is executable; the rest are not. •
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Orderable Queries A UCQ¬ query is orderable if its subgoals can be reordered to get an executable query. Q2 can be reordered to get Q02 (t, p) ← B iooo ( ‘Knuth’ , t, s, p), ¬B iooo ( ‘Knuth’ , t, s, ‘Springer’ ) Q2 and Q3 are orderable; Q1 and Q4 are not. •
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Feasible Queries A UCQ¬ query is feasible if it is equivalent to an ¬ executable UCQ query. Q4 is equivalent to Q04 (a) ← Lo (t), B oioo (a, t, s, p) Q2 , Q3 , and Q4 are feasible; Q1 is not. •
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: None • ans(Q4 )(a)← Bindings: None • ans(Q4 )(a)←
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: None • ans(Q4 )(a) ← Lo (t) Bindings: None • ans(Q4 )(a) ← Lo (t)
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: t • ans(Q4 )(a) ← Lo (t) Bindings: t • ans(Q4 )(a) ← Lo (t)
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: t ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p) Bindings: t •
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ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p)
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: t, a, s, p ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p) Bindings: t, a, s, p •
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ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p)
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: t, a, s, p ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p) Bindings: t, a, s, p •
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ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p), ¬C ii (a, t)
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The Answerable Part Access Patterns: B iooo , B oioo , B ooio , Lo , C ii • Q4 (a) ← B(a, t, s, p), L(t), C(b, t) Q4 (a) ← B(a, t, s, p), L(t), ¬C(a, t) Bindings: t, a, s, p ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p) Bindings: t, a, s, p •
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ans(Q4 )(a) ← Lo (t), B oioo (a, t, s, p), ¬C ii (a, t)
ans(Q) can be computed efficiently (in quadratic time).
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The Answerable Part For Q ∈ CQ¬ : • A subgoal is Q-answerable if there is an executable query E including that subgoal and subgoals from Q. • If Q is unsatisfiable, then ans(Q) := false. Otherwise, ans(Q) is the query given by the Q-answerable subgoals in Q. • Head of ans(Q): variables in the head of Q and in the body of ans(Q). If Q ∈ UCQ¬ with Q = Q1 ∪ . . . ∪ Qk , then ans(Q) := ans(Q1 ) ∪ . . . ∪ ans(Qk ). •
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Query Containment •
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Q1 is contained in Q2 (Q1 v Q2 ) if, for every database D, Q1 (D) ⊆ Q2 (D). Q1 is equivalent to Q2 (Q1 ≡ Q2 ) if Q1 v Q2 and Q2 v Q1 . Checking containment of CQ or UCQ is NP-complete. Checking containment of CQ¬ or UCQ¬ is ΠP2 -complete. ΠP2 := coNPNP . That is, ΠP2 is what can be computed in coNP with access to an NP oracle.
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Outline • • •
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We introduced access patterns. We defined executable, orderable, and feasible. Executable and orderable are syntactic notions. Feasible is a semantic notion that depends on equivalence. We have shown how to compute ans(Q), the “answerable part” of Q. We present our main results. We show how to compute the underestimate query Qu and the overestimate query Qo . We show how to use Qu and Qo at runtime.
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Summary of Definitions •
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A UCQ¬ query is safe if every variable appears positively in the body. A UCQ¬ query is executable if every variable appears first, positively, in an output slot in the body. A UCQ¬ query is orderable if its subgoals can be reordered to get an executable query. A UCQ¬ query is feasible if it is equivalent to an executable UCQ¬ query. ans(Q) is the answerable part of Q.
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The Feasibility Problem Given Q ∈ UCQ¬ , determine whether Q is feasible. • Known: NP-complete for CQ and UCQ. •
We show: ΠP2 -complete for CQ¬ and UCQ¬ .
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Results Assume Q, E ∈ UCQ¬ . • ans(Q) is executable.
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Results Assume Q, E ∈ UCQ¬ . • ans(Q) is executable. •
Q is feasible iff ans(Q) ≡ Q.
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Results Assume Q, E ∈ UCQ¬ . • ans(Q) is executable. • •
Q is feasible iff ans(Q) ≡ Q. That is, UCQ¬ feasibility reduces to containment.
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Results Assume Q, E ∈ UCQ¬ . • ans(Q) is executable. • • •
Q is feasible iff ans(Q) ≡ Q. That is, UCQ¬ feasibility reduces to containment. ¬
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Feasibility of CQ and UCQ : ΠP2 -complete.
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Results Assume Q, E ∈ UCQ¬ . • ans(Q) is executable. • • • •
Q is feasible iff ans(Q) ≡ Q. That is, UCQ¬ feasibility reduces to containment. ¬
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Feasibility of CQ and UCQ : ΠP2 -complete. If Q v E and E is executable, then Q v ans(Q) v E. That is, if there is a minimal executable query containing Q, it is equivalent to ans(Q).
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Compile-time vs. Runtime •
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At compile time (we have the query Q but no database D) we can check whether Q is feasible. At runtime (we have the query Q and a database D) we can check whether we have Q(D), regardless of whether Q is feasible. If we do not have exactly Q(D), we can often quantify how close we are.
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Under- and Overestimates • •
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Access patterns: S o , Roo , B ii , T oo Q(x, y) ← ¬S(y), R(x, y), B(x, z) Q(x, y) ← T (x, y) Underestimate: Qu (x, y) ← T (x, y) Overestimate: Qo (x, y) ← R(x, y), ¬S(y) Qo (x, y) ← T (x, y)
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Example •
Q(x, y) ← ¬S(y), R(x, y), B(x, z) Q(x, y) ← T (x, y)
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Qu (x, y) ← T (x, y)
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Qo (x, y) ← R(x, y), ¬S(y) Qo (x, y) ← T (x, y)
T
R 1 2 3 4
Qu
S 1 2 5 6
Qo 1 2 3 4 ∆
1 2 3 4 5 6
B 5 7 Q
1 8 3 9 1 2 3 4
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Algorithm Answer* ?
procedure A NSWER (Q) (Qu , Qo ) := P LAN ? (Q) ansu := A NSWER (Qu , D) anso := A NSWER (Qo , D) ∆ := anso \ ansu output ansu if ∆ = ∅ then output “answer is complete” else output “answer not known complete;” output “possibly part of the answer:” output ∆ if ∆ has no null values then u| output “answer at least” |ans |anso | “complete”
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Algorithm Plan* ?
procedure P LAN (Q) for i := 1 to n do Ai := A NSWERABLE (Qi , P) Ui := Q i \ Ai Ai if Ui = ∅ u Qi := ⊥ otherwise v¯ := x¯ \ vars(Ai ) Qoi := Ai and ( v¯ = null ) Qu := Qu1 ∨ · · · ∨ Qun Qo := Qo1 ∨ · · · ∨ Qon output Qu , Qo
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Algorithm Feasible* procedure F EASIBLE (Q) ? u o (Q , Q ) := P LAN (Q) if Qu = Qo then return true else if Qo contains null then return false else return Qo v Q
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Summary •
Feasibility of UCQ¬ : ΠP2 -complete.
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Feasibility of CQ¬ : ΠP2 -complete.
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ans(Q): minimal executable query containing Q. Unified algorithm for CQ, UCQ, CQ¬ , UCQ¬ . Runtime approximations.
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Future Work • • • •
Beyond UCQ¬ : FO (PODS 2004) Integrity constraints Views with access patterns Recursion
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