Consistency of multidimensional combinatorial substitutions

TL;DR

This paper investigates the properties of multidimensional combinatorial substitutions, proving the undecidability of their consistency and overlap issues, and offering algorithms for specific cases.

Contribution

It establishes the undecidability of consistency and overlap problems for 2D substitutions and provides practical algorithms for certain scenarios.

Findings

Proves undecidability of consistency in 2D substitutions.

Proves undecidability of overlapping patterns.

Provides algorithms to decide these properties in specific cases.

Abstract

Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in $\mathbb Z^d$. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of the way they are glued together in the image by a substitution. Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the patterns in an image by the substitution might overlap. We prove that it is undecidable whether a two-dimensional substitution is consistent or overlapping, and we provide practical algorithms to decide these properties in some particular cases.