Fixed-point tile sets and their applications

Abstract

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many fields, ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleeneʼs fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumannʼs self-reproducing automata; similar ideas were also used by P. Gács in the context of error-correcting computations. This construction is rather flexible, so it can be used in many ways. We show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (in which any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, to characterize effectively closed one-dimensional subshifts in terms of two-dimensional subshifts of finite type (an improvement of a result by M. Hochman), to construct a tile set that has only complex tilings, and to construct a “robust” aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter, we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: There exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed. Some of these results were included in the DLT extended abstract (Durand et al., 2008 [9]) and in the ICALP extended abstract (Durand et al., 2009 [10]).