Tiling Problems A.1 Introduction

In this appendix, we prove the undecidability of the following problems: the constrained domino problem (proved undecidable by Wang): give a tile set and a tile as input, ask whether it is possible to form a tiling of the plane containing the given tile; the unconstrained domino problem (Berger's Theorem): the input is a tile set and the question is whether one can tile the plane with it; the periodic domino problem (Berger Gurevich Koryakov): the input is also a tile set, but the question is whether it can be used to form a periodic tiling of the plane. In order to prove these results, we present some recursive transformations of Turing machines into tile sets. These constructions are not independent, thus the reader may not understand the last one if he could not understand the rst ones. The last construction also provides a direct proof of the recursive inseparability result of Berger, Gurevich and Koryakov (Theorem ?? in this book and reference 2]). Its intuitive meaning is that it is not possible to separate with any computing device tile sets that cannot tile the plane and tile sets that can form a periodic tiling of the plane. We do not present here the original proofs of these theorems. They were based on Berger's construction (see 1]): we present a simpler proof inspired by Robinson's ideas in 3]. Both proofs are based on the construction of an 3 This work was done during a training period of Cyril Allauzen under the direction of Bruno Durand in summer 1995.