Subshifts as models for MSO logic

Abstract

International audienceWe 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

Subshifts, MSO Logic, and Collapsing Hierarchies

“…We continue the line of research of [5,6], and define subshifts by monadic secondorder (MSO) formulas. We now introduce the terminology used in these articles, and then expand upon it.…”

“…In this section, we recall the definition of a hierarchy of subshift classes defined in [5,6], and then generalize it. We also state some general lemmas.…”

“…In [6], the u-MSO hierarchy was denoted by the letter C, but we use the longer name for clarity. In the rest of this article, C denotes an arbitrary class of subshifts, unless otherwise noted.…”

“…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.…”

“…In this article, we follow the approach of [5,6] and define two-dimensional subshifts by monadic second-order (MSO) logical formulas. We show that certain hierarchies obtained by counting quantifier alternations are finite, solving an open problem posed in [6]. Classes of finite structures defined by MSO formulas have been studied extensively.…”