Subshifts, MSO Logic, and Collapsing Hierarchies
  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. 
Ilkka T\"orm\"a
math.DS cs.FL cs.LO math.LO