1 / 10

An exercise in proving undecidability

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.

zoey
Download Presentation

An exercise in proving undecidability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Query answering under GAV mappings Input: a GAV mappingm: ST a source instanceI a target query Output: the “certain answers” (I,J) |= mJ()

  3. 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/...)

  4. More precisely • Fact: There is a GAV mappingm: 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.

  5. 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) ? ...

  6. Reduction to GAV answering • Basic idea: • The source instanceI specifies the set of tile types • The GAV mappingm(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 Iiff the set of tile types specified by I admits no periodic tiling.

  7. First attempt • Source schema: • A unary relation TT listing tile types • Binary relations COMPHandCOMPVspecifying 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?

  8. 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”

  9. The correct reduction: • Source schema: • A unary relation TT listing tile types • Binary relations INCOMPHandINCOMPVspecifying 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) .

  10. 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 Iiff the I(R) cannot be extended to a finite semi-group.

More Related