Computational Complexity of $k$-Block Conjugacy

TL;DR

This paper investigates the computational complexity of $k$-block conjugacy in subshifts of finite type, providing polynomial algorithms for verification and establishing hardness results for conjugacy decision and representation reduction.

Contribution

It introduces a polynomial-time algorithm for verifying $k$-block conjugacy and proves GI-completeness and NP-hardness for related decision problems, advancing understanding of their computational complexity.

Findings

Polynomial-time algorithm for conjugacy verification

GI-completeness for deciding conjugacy

NP-hardness for reducing shift representation size

Abstract

We consider several computational problems related to conjugacy between subshifts of finite type, restricted to $k$-block codes: verifying a proposed $k$-block conjugacy, deciding if two shifts admit a $k$-block conjugacy, and reducing the representation size of a shift via a $k$-block conjugacy. We give a polynomial-time algorithm for verification, and show GI and NP-hardness for deciding conjugacy and reducing representation size, respectively. Our approach focuses on 1-block conjugacies between vertex shifts, from which we generalize to $k$-block conjugacies and to edge shifts. We conclude with several open problems.