An Order on Sets of Tilings Corresponding to an Order on Languages

TL;DR

This paper explores a generalized concept of tilings through sets of patterns, establishing a correspondence between an order on subshifts and an order on languages based on computability, linking dynamical systems and formal language theory.

Contribution

It introduces a new framework connecting orders on subshifts with orders on languages of forbidden patterns, expanding the understanding of tilings in a dynamical context.

Findings

Established a correspondence between subshift orders and language orders

Linked dynamical transformations to computability properties of languages

Extended tiling theory to more general pattern sets

Abstract

Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view, leading to the notion of subshift. In this article we establish a correspondence between an order on subshifts based on dynamical transformations on them and an order on languages of forbidden patterns based on computability properties.