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Computability and Tiling Problems Computability and Tiling Problems Mark Richard Carney University of Leeds School of Mathematics Submitted in accordance with the requirements for the degree of Doctor of Philosophy October 2019 Intellectual Property Statement The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copy- right material and that no quotation from the thesis may be pub- lished without proper acknowledgement. The right of Mark Richard Carney to be identified as Author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. c October 2019 The University of Leeds and Mark Richard Carney. i 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 T ILE, or whether it has infinite connected but not necessarily total tilings, W T ILE (short for ‘weakly tile’). We show that both T ILE ≡m ILL ≡m W T ILE, and thereby both T ILE 1 and W T ILE are Σ1-complete. We also show that the opposite problems, :T ILE and SNT (short for ‘Strongly Not Tile’) are such that :T ILE ≡m W ELL ≡m 1 SNT and so both :T ILE and SNT are both Π1-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 P T ile of periodic tilings, and AT ile of aperiodic tilings. We then show that both of these sets are 1 1 complete for the class of problems of the form (Σ1 ^ Π1). 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!! . We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to C!! . 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. iii Dedicated to Prof. S. Barry Cooper v Acknowledgements I wish to thank my supervisors Dr. Paul Shafer and Prof. Michael Rathjen, both of whom have guided and inspired me along this journey. In particular, Dr. Shafer’s regular engagement and inspiring passion for computability, logic, and mathematics, matched with his rigorous approach and choice quality coffee, has been one of the most profound privileges to work with. The generosity of time, expertise, and guidance from both of my supervisors has made this Ph.D. possible. I am grateful to my colleagues and friends in the Logic Group at the University of Leeds for their support in answering questions and commenting on ideas, in particular: Giovanni Solda, Emanuele Frittaion, Alberto Marcone, Marta Fiori Carones, John Truss, Stan Wainer, Andrew Brooke-Taylor, Charles Harris, Anton Freund, Martin Krombholz, Bjarki Geir Benediktsson, John Howe, Jakob Vidmar, Richard Matthews, Rosario Mennuni, James Gay, James Riley, Anja Komatar, Cong Chen, Richard Whyman, Sarah Sigley, Cesare Gallozzi, and Petra Staynova. Of note is Emanuele Frittaion, whose observation regarding Weihrauch degrees after a seminar I gave spawned the work in Chapter 5, and Andrew Brooke-Taylor for the references on Vopenka’sˇ Principle in Chapter 4. My thanks to the University of Leeds and the School of Mathematics for their hosting me, and to the EPSRC for their financial assistance. It has been gratefully received. My thanks to my parents Pauline and Richard, my brother Edward, and my family - in particular my mum, who has lingered through several conversations I am certain she was not expecting to understand all of. My love and thanks to you all for supporting me in many ways. I acknowledge the plentiful support of my friends and colleagues in the tech- nology scene, including the participants at Leeds Hackspace, DC151, and DC11331, and the DEFCON and BSides family. I honour the marvellous work of the baristas of Leeds. Lastly, my thanks to my first supervisor on this Ph.D., Prof. S. Barry Cooper, to whom this thesis is dedicated. Without you, Barry, I would not have been so encouraged and prepared to undertake this task in the beginning. Cheers, Barry, and thank you, my friends. Contents Abstract i Dedication iii Acknowledgements v Contents vii List of figures ix List of Figures ix List of tables x List of Tables xi Introduction 1 Background to the Thesis 1 The Current Literature on Tiling Problems and Logic 3 Outline of the Thesis and Main Results 5 Glossary of Sets and Constructions 11 Chapter 1. Computability, Trees, and Preliminary Concepts 15 1.1. Preliminaries 15 1.2. Computability 16 1.3. Computable Trees 27 1 1.4. Kleene’s O and Π1-Completeness 29 1.5. Trees, Ordinals, and the Arithmetical and Analytic Hierarchies 40 Chapter 2. Tilings - Concepts and Results 43 2.1. Tilings of the Plane 43 2.2. The Domino Problem 48 2.3. Undecidability of the Domino Problem 53 2.4. Implications of TM Tilings 59 vii viii CONTENTS 1 Chapter 3. Σ1-Complete Tilings 61 3.1. Computable Trees and Computable Tilings 61 1 3.2. Π1 Properties of Tilings 62 3.3. Domino Problems for Infinite Computable Sets of Prototiles 66 1 1 3.4. Π1 and Σ1 Domino Problems 67 Chapter 4. Aperiodicity, Tilings, and Logical Complexity 83 1 1 4.1. Aperiodic Tilings and Σ1/Π1 Sets 83 4.2. Periodicity and Aperiodicity of ILL 90 4.3. Completeness of P T ile and AT ile 101 4.4. Aperiodicity and Periodicity for Finite Prototile Sets 105 Chapter 5. Weihrauch Reducibility and Tiling Problems 109 5.1. Weihrauch Reducibility 109 5.2. Weihrauch Reducibility and Choice Principles 113 5.3. Weihrauch Reducibility and Tiling Problems 117 5.4. Weihrauch Reductions for Weak Planar Tilings 122 5.5. General Weihrauch Reducibility for Wang Domino Problems 130 Chapter 6. Small ECA Tilings 135 6.1. Elementary Cellular Automata 135 6.2. Some Results about ECAs 136 6.3. Elementary Cellular Automata and Tilings 143 6.4. A 15 Prototile ECA Tiling 146 Chapter 7. Conclusion 153 7.1. Conclusions from Results 153 7.2. Open Problems and Further Work 154 Bibliography 157 Index 161 List of Figures 1 Edge Conditions in the von Neumann Neighbourhood surrounding a Wang tile. 50 1 Overall shape of our tiling construction in the proof of 3.4.1. 71 2 Tile Path Construction 73 3 Weakly Tiling Path Construction 79 1 A Penrose Tiling - generated online at https://misc.0o0o.org/ penrose/ 87 2 A set of 13 aperiodic Wang prototiles due to Culik [17]. 88 3 A set of 11 aperiodic Wang prototiles due to Jeandel and Rao [39]. 89 4 P T ile for e 2 ILL Construction 93 5 Overall shape of our tiling construction in the proof of 3.4.1. 94 1 The schematic diagram for Cook’s encoding of Cyclic Tag Systems in Rule 110, taken from [28] 142 f f 2 A sample tiling of S30. NB: Indicators O and 1 are omitted for clarity. 146 3 A 15 prototile set of tiles that encodes the behaviour of the Rule 30 ECA in the lower half-plane. 150 4 Example few rows of a hexagon and lozenge tiling of Rule 30. 151 ix List of Tables 1 Rule 30 Automaton Rules 136 2 This table shows the development of a cyclic tag system for initial d of 11 and Pi’s in sequence as given in the text. The development of the contents of d is given at each line. 141 xi Introduction In this thesis we will explore the connections between tiling problems and logic, specifically in relation to, and through the lens of, computability theory. Background to the Thesis Broadly speaking, the tiling problems we study fall into two categories, for given prototile set S: (1) Domino Problems - the question of whether S tiles the plane. (2) Tiling Properties - do all/any S-tilings have some specific property, e.g. are they all periodic or aperiodic? We will construct well defined versions of both of these problems, and study their relationships to various areas of computability theory. This thesis builds on results that the author first presented in their MSc disser- tation [12] as part of their MSc Mathematics at the University of Leeds. In that work, we presented some ways to code various results in computability, as well as elementary cellular automata, into sets of Wang prototiles. In building on these results, we explore with much more depth the ways in which the classes of tiling problems listed above relate to various aspects of com- putability. We ask questions along the following lines: • What are the computable parts of a given tiling problem? • How do tiling problems fit into existing computability hierarchies? We also present improved versions of the Elementary Cellular Automata tilings using an original tile schema that we have constructed for this purpose.
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