  One of the most fundamental problems in tiling theory is the domino problem:
given a set of tiles and tiling rules, decide if there exists a way to tile the
plane using copies of tiles and following their rules. The problem is known to
be undecidable in general and even for sets of Wang tiles, which are unit
square tiles wearing colours on their edges which can be assembled provided
they share the same colour on their common edge, as proven by Berger in the
1960s. In this paper, we focus on Wang tilesets. We prove that the domino
problem is decidable for robust tilesets, i.e. tilesets that either cannot tile
the plane or can but, if so, satisfy some particular invariant provably. We
establish that several famous tilesets considered in the literature are robust.
We give arguments that this is true for all tilesets unless they are produced
from non-robust Turing machines: a Turing machine is said to be non-robust if
it does not halt and furthermore does so non-provably.
  As a side effect of our work, we provide a sound and relatively complete
method for proving that a tileset can tile the plane.
  Our analysis also provides explanations for the observed similarities between
proofs in the literature for various tilesets, as well as of phenomena that
have been observed experimentally in the systematic study of tilesets using
computer methods.
