Hardness of conjugacy and factorization of multidimensional subshifts of finite type

TL;DR

This paper studies the computational complexity of conjugacy and factorization problems in multidimensional subshifts of finite type, establishing their exact positions in the arithmetical hierarchy.

Contribution

It proves that the factorization problem is 0-complete and the conjugacy problem is
-complete for multidimensional SFTs, clarifying their computational hardness.

Findings

Factorization problem is 0-complete.

Conjugacy problem is
-complete.

Provides a complexity classification for these problems in higher dimensions.

Abstract

We investigate here the hardness of conjugacy and factorization of subshifts of finite type (SFTs) in dimension $d>1$. In particular, we prove that the factorization problem is $\Sigma^0_3$-complete and the conjugacy problem $\Sigma^0_1$-complete in the arithmetical hierarchy.