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- W4385291850 abstract "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 $TILE equiv_m ILL equiv_m WTILE$, and thereby both $TILE$ and $WTILE$ are $Sigma^1_1$-complete. We also show that the opposite problems, $neg TILE$ and $SNT$ (short for `Strongly Not Tile') are such that $neg TILE equiv_m WELL equiv_m SNT$ and so both $neg TILE$ and $SNT$ are both $Pi^1_1$-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 $(Sigma^1_1 wedge Pi^1_1)$. 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, $C_{omega^omega}$. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to $C_{omega^omega}$. 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." @default.
- W4385291850 created "2023-07-27" @default.
- W4385291850 creator A5040472640 @default.
- W4385291850 date "2023-07-20" @default.
- W4385291850 modified "2023-09-25" @default.
- W4385291850 title "Computability and Tiling Problems" @default.
- W4385291850 doi "https://doi.org/10.48550/arxiv.2307.13075" @default.
- W4385291850 hasPublicationYear "2023" @default.
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