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Containment

Containment. CSE 590 DB Rachel Pottinger. Outline. Introduction Motivation Formal definition Algorithms for different complexities An application: rewriting queries using views. Containment, what is it?.

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Containment

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  1. Containment CSE 590 DB Rachel Pottinger

  2. Outline • Introduction • Motivation • Formal definition • Algorithms for different complexities • An application: rewriting queries using views

  3. Containment, what is it? • For two queries, Q1 and Q2, if all of the answers to Q1 are a subset of those for Q2 for all databases, then Q1 is contained in Q2. • Denoted as Q1 Q2. • For general datalog, this is undecidable (by reduction from decision problems for context free languages)

  4. Why should I care? • Containment is useful in a number of situations, including: • Query minimization • Independence of queries using updates • Rewriting queries using views • Interesting logic problem

  5. More definitions • Equivalence of queries: Q1Q2 if they return the same answers for all databases. This is the same as Q1 Q2 and Q2 Q1 • Conjunctive query - a query that is formed only of conjunctions of predicates. • Q(X,Y):- e(X,Z),e(Z,Y)

  6. Containment Mapping • Let Q1 and Q2 be two conjunctive queries • Q1: I :- J1, …, Jl • Q2: H :- G1, …, Gk • A symbol mapping h is said to be a containment mapping if h turns Q2 into Q1; that is, h(H)= I, and for each i = 1,2,…,k, there is some j such that h(Gi)=Jj. There is no requirement that each Jj be the target of some Gi

  7. Proof Sketch • If there’s a containment mapping from Q2 to Q1, then Q1 Q2 • Suppose  maps Vars(Q2)Vars(Q1) • Let D be a database and  be an answer •  is a mapping from Vars(Q1) D • •  Vars(Q2) D • The rest of the proof follows later

  8. Example of homomorphism rules • Q1: fp(X,Y) :- e(Y,X), e(X,Z) • Q2: fp(A,B) :- e(B,A), e(C,A),e(A,D) • For Q1 Q2, map from Q2 to Q1

  9. Test for containment of a conjunctive query (Q1Q2) • Freeze the body of Q1, and put this into a canonical database • Apply Q2 to the canonical database • If Q1 can be derived from Q2 on the canonical database, then Q1 Q2, otherwise not

  10. A chilling example Q1: p(X,Z) :- a(X,Y), a(Y,Z) Q2: p(X,Z) :- a(X,U), a(V,Z) Canonical Database of Q1

  11. Proof continued • If Q1 Q2,then there is a containment mapping • Since Q1 Q2, we know that if we apply Q2 to the canonical database formed from Q1, we’ll get back the same fact we got from applying it to Q1, which makes a mapping from Q2 to Q1.

  12. Conjunctive queries with negation • Negation in the heads of the subgoals, ie: Q(X,Y):- e(X,Z),e(Z,Y) • The Levy and Sagiv test looks at an exponential number of canonical databases, thus is P2 complete • Consider all partitions of Q1; form canonical databases for all of them, D1, … Dk • For each database Di, see if the database makes all subgoals of Q1 true. • For all Di’s passing step 2, see if it the head of Q1 can be derived by applying Q2 • If so, then Q1 Q2, else not

  13. A negative example • Q1: p(X,Z):-a(X,Y), a(Y,Z), a(X,Z) • Q2: p(A,C):-a(A,B),a(B,C), a(A,D)

  14. Conjunctive Queries with Arithmetic Comparisons • Q(X,Y):-e(X,Z),e(Z,Y), Z < Y • Treat the same as the negated subgoals, only a check must be made for each ordering of each partition • Also P2 complete for dense domain such as reals

  15. Example with arithmetic comparisons • Q1:p(X,Z):-a(X,Y), a(Y,Z), X < Y • Q2:p(A,C):-A(A,B),A(B,C), A < C • false, see x = z = 0, y = 1

  16. Other complexity results •  queries restricted to queries Q1 and Q2 such that all database predicates have arity at most 2 and every database predicate occurs at most three times in the body of Q1 - P2 • Conjunctive queries where Q1 is fixed- NP complete • Conjunctive queries where Q2 is fixed - polynomial • Conjunctive query containment where Q2 is an acyclic query - polynomial time • Conjunctive queries where every database predicate occurs at most twice in the body of Q1 - linear time

  17. Rewriting Queries Using Views • Useful in query optimization • Good for query minimization • Needed to make the best use of cached information • Necessary in data integration

  18. Views • A view is a relation that is not part of the conceptual model, but is visible to the user. • Useful for common expressions, or protecting data • Example: If you had faculty(name, office, ssn) you may want students to access faculty_office(name, office)

  19. Views (con’t.) • Views can be either materialized or virtual • In data integration, data sources can be thought of as views

  20. An example of rewriting queries using views • Suppose you had two databases: • One has famous people and whether they are right or left handed • One has the birthdays of famous people • You want the birthdays of all of the lefties

  21. Containment in rewriting • Query of q(X):-e(X,Y), e(Y,X) • View of v(A,B):- e(A,C),e(C,B)

  22. A more complicated example • Q(x,u):-p(x,y),p0(y,z),p1(x,w),p2(w,u) • V1(a,b):-p(a,c),p0(c,b),p1(a,d) • V2(a,b):-p1(a,b) • V3(a,b):-p2(a,b)

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