Subshifts as Models for MSO Logic

TL;DR

This paper explores the connection between Monadic Second Order logic and subshifts in the discrete plane, providing characterizations of logical classes via subshift properties and highlighting differences from tiling picture cases.

Contribution

It offers a novel characterization of MSO logic fragments in terms of subshifts and tilings, and compares these with previous tiling picture results.

Findings

Characterization of existential MSO as projections of tilings

Universal sentences characterized by pattern counting subshifts

Distinct separation results from tiling picture studies

Abstract

We study the Monadic Second Order (MSO) Hierarchy over colourings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings, and of universal sentences in terms of combinations of "pattern counting" subshifts. Conversely, we characterise logic fragments corresponding to various classes of subshifts (subshifts of finite type, sofic subshifts, all subshifts). Finally, we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by Giammarresi et al.