An exercise in proving undecidability

1
An exercise in proving undecidability

• Balder ten Cate
• Bertinoro 15/12/2006

2
Query answering under GAV mappings

• Input a GAV mapping m S?T
• a source instance I
• a target query ?
• Output the certain answers
• ?(I,J) m J(?)

3
Complexity

• For conjunctive queries ?, the problem is in
  LOGSPACE (by unfolding)
• For FO queries ?, its undecidable.
• This talk There a fixed FO query ? for which
  computing the certain answers is undecidable.

(Corrolary CERT(m, ?) is not definable in
FO/datalog/...)
4
More precisely

• Fact There is a GAV mapping m S?T and a Boolean
  FO query ? over T such that the following is
  undecidable
• Given a source instance I, is Yes a certain
  answer to ? ?
• Proof by reduction from an undecidable tiling
  problem.

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Periodic tiling

• An undecidable problem
• Given a finite set of tile types
• Can we tile any n ? n square with these tiles so
  that (a) neighboring tiles match, (b) the first
  and last column coincide, and (c) the first and
  last row coincide (n gt 1) ?

...
6
Reduction to GAV answering

• Basic idea
• The source instance I specifies the set of tile
  types
• The GAV mapping m (which is fixed) simply copies
  all the information
• The FO query ? (which is fixed) describes a
  periodic tiling with the given tile types.
• Yes is a certain answer to ?? on source
  instance I iff the set of tile types specified by
  I admits no periodic tiling.

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First attempt

• Source schema
• A unary relation TT listing tile types
• Binary relations COMPH and COMPV specifying
  horizontal and vertical compatibility
• The GAV mapping
• ?x (TTx ? TTx)
• ?x (RHx ? RH x)
• ?x (RVx ? RV x)
• Before we continue
• What is wrong with this attempt?

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Bug fix

• We need to make sure that ...
• no compatibilities are added in the target
• Solution represent incompatibilities
• no new tile types are added in the target
• Solution use extra relations so that tampering
  can be detected

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The correct reduction

• Source schema
• A unary relation TT listing tile types
• Binary relations INCOMPH and INCOMPV specifying
  horizontal and vertical incompatibility
• Two binary relations coding a linear ordering of
  the tile types and a corresponding successor
  relation.
• The GAV mapping copies everything (as before)
• The target query ? describes a periodic tiling
  using the given tile types (homework exercise,
  for the solution see Börger- Grädel-Gurevich) .

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Added in print

• Prof. Kolaitis found a simpler and more elegant
  proof by reduction of the undecidable embedding
  problem for finite semi-groups given a partial
  binary function, can it be extended to a
  semi-group (over a possible larger but finite
  carrier set)?
• Source schema a single ternary relation R
• Target schema a single ternary relation R
• GAV mapping ?xyz (Rxyz ? Rxyz)
• The target query ? expresses that R is an
  associative total function (this can be expressed
  in FO logic, even using only ??-formulas).
• Yes is a certain answer to ?? on source
  instance I iff the I(R) cannot be extended to a
  finite semi-group.