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                <editor>Davide Sangiorgi</editor>
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              <p>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 &amp; Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.</p>
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