On Answering Queries in the Presence of Limited Access Patterns

1
On Answering Queries in the Presence of Limited
Access Patterns

• Chen Li
• Stanford University
• joint work with Edward Chang, UC Santa Barbara

2
A movie database
Q(Award) - r(henry fonda,Movie),
s(Movie,Award)

• r(Star, Movie)

s(Movie, Award)
3
Limited access patterns
Should provide a star.
Should provide a movie.
s(Movie, Award)

• r(Star, Movie)

4
Answering Q given the restrictions
Q(Award) - r(henry fonda,Movie),
s(Movie,Award)
s(Movie, Award)

• r(Star, Movie)

5
The answer is complete

• We did not retrieve all the tuples from the
  relations.
• Still we computed all tuples in the answer to the
  query.

Q(Award) - r(henry fonda,Movie),
s(Movie,Award)

• r(Star, Movie)

s(Movie, Award)
6
Change the restriction

• We cannot compute the complete answer to Q.
• There can always be some tuples that are not
  retrievable.

Q(Award) - r(henry fonda,Movie),
s(Movie,Award)

• r(Star, Movie)

s(Movie, Award)
7
General questions

• Given a query on relations with limited access
  patterns, can we compute its complete answer by
  accessing the relations with legal patterns?
• Stable queries
• Different classes of queries
• Another problem studied testing query
  containment in the presence of binding patterns.

8
Rest of the talk

• Binding patterns, query stability
• Testing stability of queries
• Conjunctive queries
• Unions of conjunctive queries
• Conjunctive queries with arithmetic comparisons
• Datalog queries
• Dynamic computability of complete answer to
  conjunctive queries
• Conclusion and related work

9
(I) Binding patterns

• Attributes with adornments
• b bound
• f free
• Example
• r(Starb, Movief), s(Movieb, Awardf)
• A relation can have multiple binding patterns.

10

• Reasons of the restrictions
• Web search forms
• Legacy databases
• Security concerns
• Observations
• If a relation does not have an all-free
  binding pattern, then after certain queries are
  sent to this relation, there can always be some
  tuples that have not been retrieved.

11
Query stability

• A query Q on relations with binding patterns is
  stable if for any database, we can compute Qs
  complete answer by accessing the relations with
  legal patterns.
• The complete answer is the computable answer if
  we could retrieve all the tuples from the
  relations.
• Use partial tuples to derive the complete answer
  we need reasoning.

12
Assumptions about bindings

• Use values from Q and results from the relations
  as bindings
• The definition says for any database
• Relations not in the query can be assumed to be
  empty
• Not allowed try arbitrary strings as bindings to
  access the relations
• Does not terminate
• Impractical

13
(II) Testing stability of queries

• Conjunctive query
• q(X) - g1(X1),,gn(Xn)
• Feasible order of some subgoals of a CQ Q.
• Each subgoal in the order is executable
• That is, we have enough bound variables to
  satisfy one binding pattern of the relation
• Example
• Q(Award) - r(henry fonda,Movie),
• s(Movie,Award)

14
Feasible CQs

• A CQ is feasible if it has a feasible order of
  all its subgoals.
• Lemma A feasible CQ is stable.
• Testing feasibility of a CQ
• A greedy algorithm Inflationary

15
What if Q is not feasible?

• Q(Award) - r(henry fonda,Movie),
• s(Movie,Award),r(Star,Movie)
• Not feasible variable Star cannot be bound
• Equivalent to the old query
• Q(Award) - r(henry fonda,Movie),
• s(Movie,Award)
• The new query Q is stable!

16
Testing stability of a CQ

• Theorem
• A CQ Q is stable iff its minimal equivalent Qm
  is feasible.
• Minimal equivalent query Qm
• Qm is unique

17
Main idea of the proof

• Construct two databases of the relations
• They have the same observable tuples, but yield
  different answers to the query
• Thus, we cannot tell whether the computed answer
  is complete or not

Same observable tuples
Different answers to Q
18
Two algorithms for CQs

• Algorithm CQStable
• Minimize Q, get its minimal equivalent Qm
• Test feasibility of Qm by calling Inflationary
• Algorithm CQStable
• Compute all executable subgoals of Q
• If all subgoals become executable, then Q is
  stable
• Otherwise, test equivalence between Q and the new
  query with the executable subgoals
• CQStable is more efficient than CQStable
• Testing stability of a CQ is NP-complete.

19
Other classes of queries

• Unions of CQs two algorithms
• CQs with arithmetic comparisons
• An algorithm for the testing stability
• Datalog queries
• Undecidable
• Give a sufficient condition for stability of
  Datalog

20
(III) Dynamic computability of complete answer to
CQs

• For a nonstable CQ Q, for certain database,
  Qs complete answer might be computed.

21
An example

• Q1 ans(B) - r(a,B,C),s(C,D)
• Not stable
• For the following database, we can still compute
  Q1s complete answer b1,b2.

r(Ab, Bf, Cf)
s(Cf, Db)
p(Df)
d1
a
b1
c1
d1
c1
d2
a
b2
c2
d2
c2



a
b2
c3



22
Change the head argument

• Q2 ans(D) - r(a,B,C),s(C,D)
• Still not stable
• For the database, we cannot compute Q2s complete
  answer.

r(Ab, Bf, Cf)
s(Cf, Db)
p(Df)
d1
a
b1
c1
d1
c1
d2
a
b2
c2
d2
c2



a
b2
c3



23
Difference between Q1 and Q2

• b f f f b
• Q1 ans(B) - r(a,B,C),s(C,D)
• Q2 ans(D) - r(a,B,C),s(C,D)
• Q1s head argument B is bound by the executable
  subgoal r(a,B,C).
• Q2s head argument D is not bound by the
  executable subgoal r(a,B,C).

24
Generalization

• q(X) - g1(X1), , gk(Xk),
• gk1(Xk1), , gn(Xn)
• Executable subgoals E g1(X1),, gk(Xk)
• If all arguments in X are bound in E
• we might compute its complete answer.
• The computability is database dependent.
• If some arguments in X are not bound in E
• we can never compute its complete answer.
• Unless the relation after the subgoals in E is
  empty.

25
A decision tree

• It guides the planning process of computing the
  complete answer to a query.
• Two approaches while traversing the tree
• optimistic
• pessimistic

26
Conclusion

• Stability of queries with binding patterns
• Various classes of queries
• CQs (two algorithms)
• Unions of CQs (two algorithms)
• CQs with arithmetic comparisons (one algorithm)
• Datalog (undecidable)
• Dynamic computability of a CQs complete answer
• Another contribution decidability result of
  testing relative query containment with binding
  restrictions

27
Related work

• Answering queries using views with binding
  patterns RSU95
• Query optimization YLUGM99,FLMS99
• Computing maximal answer to queries DL97,LC00
• Our work considers whether the complete answer
  to a query is computable.