Complex Tilings

TL;DR

This paper explores the minimal Kolmogorov complexity of plane tilings with specific tile sets, establishing tight bounds and connecting tiling properties with recursion theory and Turing degrees.

Contribution

It introduces tile sets where all n-by-n squares have complexity at least n, providing a quantitative perspective on classical non-recursivity results in tilings.

Findings

Existence of tile sets with minimal complexity bounds

Connection between tiling complexity and Turing degrees

Quantitative analysis of non-recursive tilings

Abstract

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 O(n) Kolmogorov complexity of its n-by-n squares. We construct tile sets for which this bound is tight: all n-by-n squares in all tilings have complexity at least 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. Keywords: Tilings, Kolmogorov complexity, recursion theory