An exercise in proving undecidability Balder ten Cate

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 COMPH and COMPV specifying horizontal and vertical compatibility • The GAV mapping: x (TTx TT’x) x (RHx RH’ x) x (RVx RV’ 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 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).

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.