An exercise in proving undecidability Balder ten Cate Bertinoro 15/12/2006

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(  )

Complexity For conjunctive queries , the problem is in LOGSPACE (by “unfolding”) For FO queries , it’s 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/...)

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.

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 > 1) ?...

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.

First attempt Source schema: –A unary relation TT listing tile types –Binary relations COMP H and COMP V specifying horizontal and vertical compatibility The GAV mapping:  x (TTx  TT’x)  x (R H x  R H ’ x)  x (R V x  R V ’ x) Before we continue: What is wrong with this attempt?

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”

The correct reduction: Source schema: –A unary relation TT listing tile types –Binary relations INCOMP H and INCOMP V 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).

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  R’xyz) –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.