Goodman-Strauss theorem revisited

TL;DR

This work addresses the problem of enforcing substitution tilings by finite local rules (soficity) and proposes a streamlined set of sufficient conditions $R0$–$R8$ that guarantee soficity for both hierarchical tilings and the underlying substitution tilings. The authors develop a generalization of Mozes' theorem by analyzing a power $\sigma=\tau^m$, constructing macrotiles on a fixed $k\times k$ grid, and encoding composition history via a decorated tile framework, including a key Dichotomy Lemma and Composition Lemma. They prove that decorated hierarchies faithfully capture $\tau$-hierarchical tilings and that, under their assumptions, the $\tau$-substitution tilings are also sofic. The paper also explains how two simple tricks—increasing the substitution power and joining tiles into larger macrotiles—extend the result to a broad class of substitutions, including those with square prototiles, and discusses the scope and limitations of applicability. Overall, the results provide a rigorous path to finite local rules enforcing a wide range of substitution and hierarchical tilings, with practical guidance for constructing markups in challenging cases.

Abstract

The Goodman-Strauss theorem states that for ``almost every" substitution, the family of substitution tilings is sofic, that is, it can be defined by local rules for some decoration of tiles. The conditions on the substitution that guarantee the soficity are quite complicated. In this paper we propose a version of Goodman-Strauss theorem with simpler conditions on the substitution. Although the conditions are quite restrictive, we show that, in combination with two simple tricks (taking a sufficiently large power of the substitution and combining small tiles into larger ones), our version of Goodman-Strauss theorem can also prove the soficity of the family of substitution tilings for ``almost every'' substitution. We also prove a similar theorem for the family of hierarchical tilings associated with the given substitution. A tiling is called hierarchical if it has a composition under the substitution, such that this composition also has a composition, and so on, infinitely many times. Every substitution tiling is hierarchical, but the converse is not always true. Fernique and Ollinger formulated some conditions on the substitution that guarantee that the family of hierarchical tilings is sofic. However, their technique does not prove this statement under such general conditions as in their paper. In the present paper, we show that under the same assumptions, as for our version of the Goodman-Strauss theorem, their technique works.

Figures (27)

• Figure 1: First substitution
• Figure 2: Second substitution
• Figure 3: Third substitution. The first prototile can be rotated by 90, 180, and 270 degrees. Therefore, the total number of prototiles is 6.
• Figure 4: The figure shows a substitution and supertiles of order 1, 2 and 3.
• Figure 5: An example of substitution for which there is an acyclic central type (type 1) but the Cyclicity Requirement is satisfied. Central types are in the middle. On the right there is the graph of the function $c_\sigma(u)$.

Theorems & Definitions (42)

• Theorem 1
• Definition 1
• Lemma 1
• proof
• Theorem 2
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• proof