I'm interested in tilings of the plane by squares, with labels on the edges. It's well known that (1) the question "can one tile the plane with the following finite set of tiles?" is undecidable, and (2) there are finite sets of tiles that tile the plane, but only aperiodically. Also, (1) implies (2).


Is the question "can one tile the plane *periodically* with the following finite set of tiles?" decidable? 



Decidability of periodic tilings of the plane
Deciding whether a set of tiles admits a periodic tiling or no tiling at all is undecidable as well.


This has been shown in [Y.S. Gurevich, I.I. Koryakov, Remarks on Berger's paper on the domino problem, J Sib Math J 13, 319–321 (1972)](https://scholar.google.com/scholar?cluster=2757770576292515424). The results can also be found in the book The Classical Decision Problem by Egon Börger, Erich Grädel, Yuri Gurevich, where Theorem 3.1.7 states



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> The set of domino systems that admit, respectively, no tiling and a periodic tiling of $\mathbb Z\times \mathbb Z$ or $\mathbb N\times \mathbb N$ are recursively inseparable.
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