Multidimensional Shifts And Finite Matrices

TL;DR

This paper introduces an algorithm that characterizes multidimensional shifts of finite type using sequences of finite matrices, extending the approach from 2D to arbitrary dimensions.

Contribution

It provides a novel matrix-based characterization and algorithm for generating and analyzing multidimensional shifts of finite type, generalizing previous 2D results to higher dimensions.

Findings

Algorithm effectively generates all elements of 2D shift space.

Extension of matrix characterization to d-dimensional shifts.

Provides theoretical proof of the characterization's correctness.

Abstract

Let $X$ be a $2$-dimensional subshift of finite type generated by a finite set of forbidden blocks (of finite size). We give an algorithm for generating the elements of the shift space using sequence of finite matrices (of increasing size). We prove that the sequence generated yields precisely the elements of the shift space $X$ and hence characterizes the elements of the shift space $X$. We extend our investigations to a general $d$-dimensional shift of finite type. In the process, we prove that that elements of $d$-dimensional shift of finite type can be characterized by a sequence of finite matrices (of increasing size).