Data%20Exchange:%20Semantics%20and%20Query%20Answering

1
Data Exchange Semantics and Query Answering

• Ronald Fagin -- IBM Almaden Research
  Center
• Phokion G. Kolaitis -- UC Santa Cruz
• Renee J. Miller -- University of Toronto
• Lucian Popa -- IBM Almaden Research
  Center

IBM Almaden - November 12, 2002
(To appear in ICDT 2003)
2
Motivation and Overview

• Data exchange problem
• How to restructure data from a source schema to a
  target schema, according to a given specification
• Main motivation for this work
• Understanding of fundamental issues that lie
  underneath data exchange systems such as EXPRESS
  and Clio
• Main challenge
• Inherent under-specification
• Specification (as we shall see) must be simple
  and intuitive, but
• There are many ways in which the restructuring
  can be performed !
• Question did we make the right choice in the
  design of Clio ?
• Our approach (only relational case, so far)
• Define and study universal solutions
• Show this is the best way of performing data
  exchange
• Study computational aspects
• Study what happens after data exchange query
  answering

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The Data Exchange Problem
Source schema S
Target schema T
?t
?st
I
J

• Assume a data exchange setting
• source schema S,
• target schema T with a set ?t dependencies (see
  next)
• set ?st source-to-target dependencies (see next)
• The data exchange problem is the following
• Input
• source instance I
• Output
• target instance J such that ltI, Jgt ? ?st and
  J ? ?t
• (call such J a solution for I )

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Source-to-target Dependencies

• For most practical purposes, ?st contains
• source-to-target tuple-generating dependencies
  (tgds)
• ?S(x) ? ?y ?T(x, y)
• e.g.
• DeptEmp(did, mgr_name, eid)
• ? ?M. Dept (did, M, mgr_name) ? Emp (eid, did)
• (Move data from source table DeptEmp into two
  target tables, Dept and Emp. The existential
  variable M is an unspecified manager id)

Dept did mgr_id mgr_name
DeptEmp did mgr_name eid
Emp eid did
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Target Dependencies

• The second, equally important, part of the
  specification, are the target dependencies ?t
• tgds ?T(x) ? ?y ?T(x, y)
• e.g.
• Dept (did, mgr_id, mgr_name)
• ? ?D. Emp (mgr_id, D)
• (A foreign key constraint in the target)
• equality generating dependencies (egds)
• ?T(x) ? (x1x2)
• e.g.
• Emp (e, d1) ? Emp (e, d2) ? (d1 d2)
• (A target key constraint)

Dept did mgr_id mgr_name
Emp eid did
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Questions (To be Answered Next)

• When more than one solution exists, how do we
  choose a best one ?
• How do we compute a best solution ?
• Is there always a solution ? Is there always a
  best solution ?
• How does query answering on the chosen solution
  behave ?

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Universal Solutions Best Solutions
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Existence of Multiple Solutions
source
target
X0 , Y0 , Z0 represent unknown values (or
nulls)
P
A B C
T
Q
A B C
A B C
R
A B C

• There may be many solutions for the target
  instance (J, J1, J2, etc.)
• However, J seems to be more general
• there exist homomorphisms h1 J ? J1 and h2 J ?
  J2 (see definition next)
• but none from J1 or J2 to J
• intuitively, J1 and J2 have extra information

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Homomorphisms

• As we have seen, the values of a target instance
  can be either
• constants (i.e. values coming from the source
  instance), or
• nulls (unknown values)
• Definition. Assume J1 and J2 are such target
  instances. A homomorphism h J1 -gt J2 is a
  mapping from values of J1 to values of J2 such
  that
• h(c) c, for constants c
• (nulls of J1 can be mapped to
  any values of J2)
• for every tuple lta1, , angt in relation T of
  instance J1
• lt h(a1), h(an) gt must be a tuple in
  relation T of instance J2
• Example

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Universal Solution

• Definition. Assume a data exchange setting (S, T,
  ?st, ?t). Given source instance I, a universal
  solution for I is a target instance J such that
  
• (1) J is a solution for I
• (2) for every solution J for I,
• there exists homomorphism h J ? J

• For the previous example, J is a universal
  solution. J1 and J2 are not.
• Among all solutions, universal solutions are
  special
• They contain no more and no less than the amount
  of information given by the specification

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• Fact
• Uniqueness up to homomorphic equivalence
• If J1 and J2 are universal for I then there are
  homomorphisms between J1 and J2 in both
  directions
• Representation of the space of solutions
• Sol(I1) Sol(I2) iff J1 and J2 are
  homomorphically equivalent
• We adopt the universal solution as the notion of
  best solution.
• Later we will see another justification for
  universal solutions in terms of query answering.

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• When do universal solutions exist ?
• How do we compute a universal solution ?

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Computing Universal Solutions
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Chase

• We canonically generate a universal solution by
  using the chase
• Given source instance I, start with an empty
  target instance J
• Generate tuples in J by applying the
  dependencies in ?st and ?t.
• Example

Dept did mgr_id mgr_name
DeptEmp did mgr_name eid
Emp eid did
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• This process is repeatedly applied
• for all the source tuples and for the generated
  tuples,
• as long as there are dependencies that are not
  yet satisfied
• The chase may be infinite (cyclic ?t )
• or it may fail (e.g. target key constraints
  that are not possible to satisfy for the given
  source data)
• (details in the paper)
• However, if the chase successfully terminates,
  the resulting target instance is a solution.

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Canonical Generation of Universal Solutions

• Theorem. Assume a data exchange setting (S, T,
  ?st, ?t). Given source instance I
• If the chase is finite and successful then its
  result is a universal solution.
• If the chase fails then there is no solution.

• Thus, the chase is a procedure for computing
  universal solutions, provided that
• Solutions exist, and
• The chase is finite
• We call universal solutions computed by the chase
  canonical universal solutions

When can we guarantee that the chase is finite ?
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Weakly Acyclic Sets of Dependencies

• Some cyclic sets of dependencies may cause
  infinite chases
• In such case no universal solution may exist, and
  the semantics of the data exchange is undefined
• Still there are cyclic sets of dependencies that
  behave well and are quite useful
• Weakly acyclic sets of dependencies (defined in
  the paper)
• Cover many practical cases of target constraints
• Allow for restricted cyclicity
• The chase is guaranteed to be finite

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Polynomial-Time Chase
Theorem. Let ? be a weakly acyclic set of
dependencies. For every instance K, the chase of
K with ? can be computed in polynomial time.

• Corollary. Assume a data exchange setting (S, T,
  ?st, ?t) such that ?t is a weakly acyclic set of
  dependencies.
• For every source instance I, the existence of a
  solution can be checked in polynomial time
• For every source instance I, if a solution
  exists then a universal solution can be produced
  in polynomial time.

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• Next what happens after data exchange ?
• In particular, how is subsequent query answering
  affected by our choice of a solution (universal
  solution) ?

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Query Answering
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Query Evaluation on a Solution
q
Target schema T
Source schema S
?t
?st
I
J

• Assume a fixed data exchange setting with a
  source instance I. Suppose that a System 1
  chooses a solution J for data exchange.
• A query q can now be asked against the target.
• The evaluation of q, in System 1, is q(J).
• However, a System 2 materializing a different
  solution J may give a different evaluation
  q(J).
• Different choices of J (for the same I) imply
  possibly different query evaluations.
• Is there a notion of the right set of answers
  to q with respect to I ?

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Certain Answers

• We will use a notion that has been around in the
  context of data integration and incomplete
  databases, where queries are asked against a set
  of possible databases.

• Definition. Given I and q, a tuple t is a certain
  answer if
• t ? q(J), for every solution J
• Notation certain(q, I) the set of all
  certain answers

• Thus, t is certain if it is in the answer of q on
  every solution.
• The certain answers provide well-defined
  semantics to query answering because they are
  independent of the choice of a solution.

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• Can we compute the certain answers based just on
  our chosen (universal) solution ?

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Positive Queries

• Proposition. Assume a data exchange setting (S,
  T, ?st, ?t) and a source instance I.
• Let q be a positive query. If J is a universal
  solution, then certain(q, I) q(J)? .
• Let J be a solution such that for every positive
  query q we have that certain(q, I) q(J)? . Then
  J is a universal solution.
• Note In the above
• Positive query means union of SPJ queries
• q(J)? means evaluate q on J and then throw away
  tuples that contain nulls)

• Thus, the certain answers of positive queries can
  be computed by evaluating them on any universal
  solution.
• Moreover, this property characterizes universal
  solutions.

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Conjunctive Queries with Inequalities

• The situation changes when negation is involved
  (even in the very simple form of ?).
• Example

• It can be verified that
• lta0,a0gt ? q(J)?, but lta0,a0gt ? q(J2) (thus, not
  a certain answer).
• Hence certain(q, I) ? q(J)?
• The universal solution gives extra answers

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• For conjunctive queries with inequalities, we
  have seen that simple query evaluation on a
  universal solution is not enough for computing
  the certain answers.
• Question Can we find a different SQL query q
  such that when evaluated on a universal solution
  gives the set of certain answers of q ?
• There are examples for which such query q
  exists.
• However, we show next that the answer is no, in
  general.

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Complexity Two or More Inequalities
Theorem. Computing the certain answers of unions
of conjunctive queries with at most two
inequalities per term is coNP-hard, even in a
restricted data exchange setting (LAV).

• AD98 proved a similar result for the case of
  conjunctive queries with six or more
  inequalities.
• The coNP-hardness implies
• the certain answers cannot be computed by
  evaluating the query q (or any other SQL query
  q) on a polynomial-time generated universal
  solution (unless P NP).

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Complexity One Inequality
Theorem. Assume a data exchange setting (S, T,
?st, ?t) such that ?t is a weakly acyclic set of
dependencies. Let q be a union of conjunctive
queries with at most one inequality per term. Let
I be a source instance and let J be an arbitrary
universal solution for I. Then there exists a
polynomial-time algorithm with input J that
computes certain(q, I).

• Thus, computing the certain answers for such
  queries is a tractable problem.
• Moreover, this computation can take place on any
  universal solution.
• The universal solution has all the information
  needed to compute the certain answers.
• We show next that the problem of computing the
  certain answers, even for this tractable case,
  cannot be solved by means of SQL query
  evaluation.

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First-Order Inexpressibility
Theorem. There exists a data exchange setting and
a boolean conjunctive query q with one
inequality, for which there is no first-order
query q over the canonical universal solution
such that certain(q, I) q(J) .

• This is a strong inexpressibility result that
  shows that in data exchange we cannot use the
  notion of certain answers for answering queries
  with inequalities.
• (In practice, instead of going for certain
  answers, we should just use query evaluation on
  the universal solution)
• The proof uses an original combination of finite
  model theory techniques and the chase.

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Conclusions

• Universal solutions are a good candidate for
  using in data exchange
• Clio produces such universal solution (in the
  relational case)
• All universal solutions are equally good for
  answering positive queries.
• Simple query evaluation has the same semantics as
  that of the certain answers.
• For queries with inequalities, different
  universal solutions may give different query
  evaluations which may yet be different from the
  certain answers.
• There is no hope to find the certain answers by
  means of SQL query evaluation on a universal
  solution

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Future Work

• Among all universal solutions, is there a
  universal solution that approximates, in a best
  way, the certain answers ? If yes, can this be
  computed efficiently ?
• Extension to semantics and query answering for
  data exchange in the nested (XML) case.

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The End
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Relationship to Clio

• Source-to-target tgds are the same formalism that
  Clio uses internally (for the relational case)
• ?st is generated by Clio in the semantic
  translation phase VLDB02 from correspondences.
  
• User input
• Then Clio generates, based on ?st , a set of
  queries in the data translation phase VLDB02
• These queries compute a solution
• Is Clios solution a good one ? (Since other
  solutions are also possible)
• Here we try to understand, formally, the concept
  of good solutions
• ?t is more general than the target constraints
  that Clio can currently handle.