2

In this link, where aperiodic tilings are discussed, the author mentions the following statement

Interestingly, five years before Berger's proof, Hao Wang proved that there would be an algorithm for deciding whether a given set of tiles can tile the plane if every set of tiles that tile the plane also tile the plane periodically

Or in other words the quote claims Wang proved that $$\nexists \text{ aperiodic tiling}\implies \text{the domino problem is decidable}$$ where the domino problem is asking the question whether a given set of Wang tiles can tile the plane.

I think I have seen this being posed as a conjecture but I am not sure so my question is whether Wang has proven this or only conjectured this. I also want to know if this goes both ways (is the $\implies$ a $\iff$?)

AccidentalTaylorExpansion
  • 1,736
  • 1
    In a german website , Joerg Resag mentioned this problem and said that the periodic version is decidable (not formulated as a conjecture!) The general version is know to be undecidable (it can be reduced to the halting problem). Hence you actually can replace "$\implies$" by "$\iff$". – Peter Jan 09 '23 at 14:18
  • @Peter Thanks a lot! – AccidentalTaylorExpansion Jan 09 '23 at 15:52

0 Answers0