Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
An aperiodic set of 13 Wang tiles
Discrete Mathematics
A small aperiodic set of Wang tiles
Discrete Mathematics
Finite fields
Reliable cellular automata with self-organization
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Theory of Self-Reproducing Automata
Theory of Self-Reproducing Automata
Aperiodic Tilings: Breaking Translational Symmetry
The Computer Journal
Two-by-Two Substitution Systems and the Undecidability of the Domino Problem
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
Fixed Point and Aperiodic Tilings
DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
High Complexity Tilings with Sparse Errors
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Aperiodicity Measure for Infinite Sequences
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Forbidden substrings, kolmogorov complexity and almost periodic sequences
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
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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 Kleene@?s fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann@?s self-reproducing automata; similar ideas were also used by P. Gacs 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]).