Subshifts, Languages and Logic

Jeandel, Emmanuel; Theyssier, Guillaume

doi:10.1007/978-3-642-02737-6_23

Cited by 6 publications

(8 citation statements)

References 18 publications

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“…Aubrun and Kari have shown that the Baumslag Solitar groups have undecidable domino problem [1]. Ballier and Stein [2], building on results from several authors [19][20] [16] [15], observe that every virtually free group has decidable domino problem, and conjecture that these are the only such groups.…”

Section: Known Resultsmentioning

confidence: 99%

The large scale geometry of strongly aperiodic subshifts of finite type

Cohen

2017

Advances in Mathematics

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A subshift on a group G is a closed, G-invariant subset of A G , for some finite set A. It is said to be a subshift of finite type (SFT) if it is defined by a finite collection of "forbidden patterns", to be strongly aperiodic if all point stabilizers are trivial, and weakly aperiodic if all point stabilizers are infinite index in G. We show that groups with at least 2 ends have a strongly aperiodic SFT, and that having such an SFT is a QI invariant for finitely presented torsion free groups. We show that a finitely presented torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino problem on G asks whether the SFT specified by a given set of forbidden patterns is empty. We show that decidability of the domino problem is a QI invariant.

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Section: Known Resultsmentioning

confidence: 99%

The large scale geometry of strongly aperiodic subshifts of finite type

Cohen

2017

Advances in Mathematics

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“…In this paper, we will prove how various classes of formula correspond to well known classes of subshifts. Some of the results of this paper were already presented in [26].…”

Section: Introductionmentioning

confidence: 79%

Subshifts as models for MSO logic

Jeandel¹,

Theyssier²

2013

Information and Computation

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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

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“…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.…”

Section: Logical Formulasmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Subshifts, MSO Logic, and Collapsing Hierarchies

Törmä

2014

Advanced Information Systems Engineering

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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.

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Subshifts, Languages and Logic

Cited by 6 publications

References 18 publications

The large scale geometry of strongly aperiodic subshifts of finite type

The large scale geometry of strongly aperiodic subshifts of finite type

Subshifts as models for MSO logic

Subshifts, MSO Logic, and Collapsing Hierarchies

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