# Fixed Parameter Undecidability for Wang Tilesets

@inproceedings{Jeandel2012FixedPU, title={Fixed Parameter Undecidability for Wang Tilesets}, author={Emmanuel Jeandel and Nicolas Rolin}, booktitle={AUTOMATA \& JAC}, year={2012} }

Deciding if a given set of Wang tiles admits a tiling of the plane is decidable if the number of Wang tiles (or the number of colors) is bounded, for a trivial reason, as there are only finitely many such tilesets. We prove however that the tiling problem remains undecidable if the difference between the number of tiles and the number of colors is bounded by 43. One of the main new tool is the concept of Wang bars, which are equivalently inflated Wang tiles or thin polyominoes.

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## References

SHOWING 1-7 OF 7 REFERENCES

### Tiling the Plane with a Fixed Number of Polyominoes

- MathematicsLATA
- 2009

It is proved that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes, and in the case of tiling by translations only, the problem is Undecidable for sets of 11 polyominees.

### The undecidability of the domino problem

- Medicine
- 1966

A way to break the boredom in reading is choosing undecidability of the domino problem as the reading material.

### Proving theorems by pattern recognition — II

- Mathematics
- 1961

Theoretical questions concerning the possibilities of proving theorems by machines are considered here from the viewpoint that emphasizes the underlying logic. A proof procedure for the predicate…

### Proving theorems by pattern recognition I

- Computer ScienceCommun. ACM
- 1960

It now appears that as more progress is made, strategies in the search for proofs, or what are often called heuristic methods, will also gradually become part of the subject matter of inferential analysis.