Computability and Tiling Problems

TL;DR

This thesis explores the computational complexity of tiling problems, establishing their high logical complexity, and introduces a novel hexagonal prototile set capable of encoding any elementary cellular automaton.

Contribution

It provides new results on the logical and computational complexity of tiling problems, including completeness results and a novel tile set for automaton encoding.

Findings

TILE and WTILE are $ ext{Σ}^1_1$-complete.

Negations of TILE and SNT are $ ext{Π}^1_1$-complete.

A new hexagonal tile set encodes any elementary cellular automaton.

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.