Subshifts as Models for MSO Logic

Emmanuel Jeandel (1) , Guillaume Theyssier (2)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille - UMR 6166
2 LAMA - Laboratoire de Mathématiques

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.

Keywords: Symbolic Dynamics; Model Theory; Tilings

Domains: Discrete Mathematics [cs.DM]; Logic in Computer Science [cs.LO]

Emmanuel Jeandel, Guillaume Theyssier. Subshifts as Models for MSO Logic. 2009. ⟨hal-00439407v1⟩