Subshifts, MSO Logic, and Collapsing Hierarchies

TL;DR

This paper investigates the logical complexity of two-dimensional subshifts defined via monadic second-order logic, demonstrating that certain hierarchies collapse at a finite level, contrasting with infinite hierarchies in picture languages.

Contribution

It proves the collapse of quantifier alternation hierarchies for 2D subshifts, resolving an open problem and contrasting with known results in picture languages.

Findings

Hierarchies collapse to the third level

Solves an open problem from 2013

Contrasts with infinite hierarchies in picture languages

Abstract

We use monadic second-order logic to define two-dimensional subshifts, or sets of colorings of the infinite plane. We present a natural family of quantifier alternation hierarchies, and show that they all collapse to the third level. In particular, this solves an open problem of [Jeandel & Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.