Direct Prime Subshifts and Canonical Covers

TL;DR

The paper introduces a new criterion based on Fischer graphs to determine when certain non-sofic subshifts are direct prime, and applies it to various classes including n-Dyck, beta-shifts, and S-gap shifts.

Contribution

It provides a novel conjugacy invariant criterion for direct primeness of half-synchronized subshifts and demonstrates its application to multiple classes of non-sofic shifts.

Findings

All n-Dyck shifts are direct prime.

New proofs for non-sofic beta-shifts and S-gap shifts being direct prime.

Constructed non-sofic synchronized direct prime subshifts with reversible cellular automata.

Abstract

We present a new sufficient criterion to prove that a non-sofic half-synchronized subshift is direct prime. The criterion is based on conjugacy invariant properties of Fischer graphs of half-synchronized shifts. We use this criterion to show as a new result that all n-Dyck shifts are direct prime, and we also give new proofs of direct primeness of non-sofic beta-shifts and non-sofic S-gap shifts. We also construct a class of non-sofic synchronized direct prime subshifts which additionally admit reversible cellular automata with all directions sensitive.