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
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Reliable cellular automata with self-organization
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Aperiodic Tilings: Breaking Translational Symmetry
The Computer Journal
Fixed Point and Aperiodic Tilings
DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
Forbidden substrings, kolmogorov complexity and almost periodic sequences
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Fixed-point tile sets and their applications
Journal of Computer and System Sciences
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Tile sets and tilings of the plane appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). The idea is to enforce some global properties (of the entire tiling) by means of local rules (for neighbor tiles). A fundamental question: Can local rules enforce a complex (highly irregular) structure of a tiling? The minimal (and weak) notion of irregularity is aperiodicity . R. Berger constructed a tile set such that every tiling is aperiodic. Though Berger's tilings are not periodic, they are very regular in an intuitive sense. In [3] a stronger result was proven: There exists a tile set such that all n ×n squares in all tilings have Kolmogorov complexity *** (n ), i.e., contain *** (n ) bits of information. Such a tiling cannot be periodic or even computable. In the present paper we apply the fixed-point argument from [5] to give a new construction of a tile set that enforces high Kolmogorov complexity tilings (thus providing an alternative proof of the results of [3]). The new construction is quite flexible, and we use it to prove a much stronger result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.