an:05285664
Zbl 1141.03021
Durand, Bruno; Levin, Leonid A.; Shen, Alexander
Complex tilings
EN
J. Symb. Log. 73, No. 2, 593-613 (2008).
00219999
2008
j
03D80 68Q30 03D28 52C20
tilings; Kolmogorov complexity; recursion theory; Turing degrees of unsolvability
Summary: We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with \(\mathcal O(n)\) Kolmogorov complexity of its (\(n\times n\))-squares. We construct tile sets for which this bound is tight: all (\(n\times n\))-squares in all tilings have complexity \(\Omega (n)\). This adds a quantitative angle to classical results on non-recursivity of tilings -- that we also develop in terms of Turing degrees of unsolvability.