In this thesis we will present and discuss various results pertaining to
tiling problems and mathematical logic, specifically computability theory. We
focus on Wang prototiles, as defined in [32]. We begin by studying Domino
Problems, and do not restrict ourselves to the usual problems concerning finite
sets of prototiles. We first consider two domino problems: whether a given set
of prototiles S has total planar tilings, which we denote TILE, or whether
it has infinite connected but not necessarily total tilings, WTILE (short for
`weakly tile'). We show that both TILE≡m​ILL≡m​WTILE, and
thereby both TILE and WTILE are Σ11​-complete. We also show that
the opposite problems, ¬TILE and SNT (short for `Strongly Not Tile')
are such that ¬TILE≡m​WELL≡m​SNT and so both ¬TILE
and SNT are both Π11​-complete. Next we give some consideration to the
problem of whether a given (infinite) set of prototiles is periodic or
aperiodic. We study the sets PTile of periodic tilings, and ATile of
aperiodic tilings. We then show that both of these sets are complete for the
class of problems of the form (Σ11​∧Π11​). We also present
results for finite versions of these tiling problems. We then move on to
consider the Weihrauch reducibility for a general total tiling principle CT
as well as weaker principles of tiling, and show that there exist Weihrauch
equivalences to closed choice on Baire space, Cωω​. We also show
that all Domino Problems that tile some infinite connected region are Weihrauch
reducible to Cωω​. Finally, we give a prototile set of 15
prototiles that can encode any Elementary Cellular Automaton (ECA). We make use
of an unusual tile set, based on hexagons and lozenges that we have not see in
the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure