arXiv subjects

Benjamin Hellouin de Menibus

Publications and source records attributed to Benjamin Hellouin de Menibus.

At least 19 recordsLinked to original sources

arXiv2026

On the cohomology of homshifts

We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour $\mathbb{Z}^d$ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt (Pacific J. Math. 170 (1995), no.1, 237-269) we give a necessary and sufficient condition for homshifts to be cohomological trivial. This condition is expressed in terms of the topology of a natural two-dimensional CW complex arising from the shift space which can be analyzed in many natural cases. However, we prove that in general, cohomological triviality is algorithmically undecidable for homshifts.

math.DS↗

arXiv2026

String attractors and bi-infinite words

String attractors are a combinatorial tool coming from the field of data compression. It is a set of positions within a word which captures an occurrence of every factor. While one-sided infinite words admitting a finite string attractor are eventually periodic, the situation is different for two-sided infinite words. In this article, we characterise the bi-infinite words admitting a finite string attractor as the characteristic Sturmian words and their morphic images. For words that do not admit finite string attractors, we study the structure and properties of their infinite string attractors.

math.CO↗

arXiv2026

Undecidability of the block gluing classes of homshifts

A homshift is a $d$-dimensional shift of finite type which arises as the space of graph homomorphisms from the grid graph $\mathbb Z^d$ to a finite connected undirected graph $G$. While shifts of finite type are known to be mired by the swamp of undecidability, homshifts seem to be better behaved and there was hope that all the properties of homshifts are decidable. In this paper we build on the work by Gangloff, Hellouin de Menibus and Oprocha (arxiv:2211.04075) to show that finer mixing properties are undecidable for reasons completely different than the ones used to prove undecidability for general multidimensional shifts of finite type. Inspired by the work of Gao, Jackson, Krohne and Seward (arxiv:1803.03872) and elementary algebraic topology, we interpret the square cover introduced by Gangloff, Hellouin de Menibus and Oprocha topologically. Using this interpretation, we prove that it is undecidable whether a homshift is $Θ(n)$-block gluing or not, by relating this problem to the one of finiteness for finitely presented groups.

math.DS↗

arXiv2026

Parametrized complexity of relations between multidimensional subshifts

We study the parametrized complexity of fundamental relations between multidimensional subshifts, such as equality, conjugacy, inclusion, and embedding, for subshifts of finite type (SFTs) and effective subshifts. We build on previous work of E. Jeandel and P. Vanier on the complexity of these relations as two-input problems, by fixing one subshift as parameter and taking the other subshift as input. We study the impact of various dynamical properties related to periodicity, minimality, finite type, etc. on the computational properties of the parameter subshift, which reveals interesting differences and asymmetries. Among other notable results, we find choices of parameter that reach the maximum difficulty for each problem; we find nontrivial decidable problems for multidimensional SFT, where most properties are undecidable; and we find connections with recent work relating having computable language and being minimal for some property, showing in particular that this property may not always be chosen conjugacy-invariant.

math.DS↗

arXiv2025

Decision problems on geometric tilings

We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes, extending previous results on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set. Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that this problem is undecidable even in a simple setting (square shapes with small modifications).

cs.DM↗

arXiv2025

Short-range and long-range order: a transition in block-gluing behavior in Hom shifts

Hom shifts form a class of multidimensional shifts of finite type (SFT) and consist of colorings of the grid Z2 where adjacent colours must be neighbors in a fixed finite undirected simple graph G. This class includes several important statistical physics models such as the hard square model. The gluing gap measures how far any two square patterns of size n can be glued, which can be seen as a measure of the range of order, and affects the possibility to compute the entropy (or free energy per site) of a shift. This motivates a study of the possible behaviors of the gluing gap. The class of Hom shifts has the interest that mixing type properties can be formulated in terms of algebraic graph theory, which has received a lot of attention recently. Improving some former work of N. Chandgotia and B. Marcus, we prove that the gluing gap either depends linearly on n or is dominated by log(n). We also find a Hom shift with gap Θ(log(n)), infirming a conjecture formulated by R. Pavlov and M. Schraudner. The physical interest of these results is to better understand the transition from short-range to long-range order (respectively sublogarithmic and linear gluing gap), which is reflected in whether some parameter, the square cover, is finite or infinite.

math.DS↗

arXiv2025

Minimality and computability of languages of G-shifts

Motivated by the notion of strong computable type for sets in computable analysis, we define the notion of strong computable type for $G$-shifts, where $G$ is a finitely generated group with decidable word problem. A $G$-shift has strong computable type if one can compute its language from the complement of its language. We obtain a characterization of $G$-shifts with strong computable type in terms of a notion of minimality with respect to properties with a bounded computational complexity. We provide a self-contained direct proof, and also explain how this characterization can be obtained from an existing similar characterization for sets by Amir and Hoyrup, and discuss its connexions with results by Jeandel on closure spaces. We apply this characterization to several classes of shifts that are minimal with respect to specific properties. This provides a unifying approach that not only generalizes many existing results but also has the potential to yield new findings effortlessly. In contrast to the case of sets, we prove that strong computable type for G-shifts is preserved under products. We conclude by discussing some generalizations and future directions.

cs.FL↗

arXiv2025

Symbol Frequencies in Surjective Cellular Automata

We study the behavior of probability measures under iteration of a surjective cellular automaton. We solve the following question in the negative: if the initial measure is ergodic and has full support, do all weak-* limit points of the sequence of measures have full support as well? The initial measure of our solution is not a product measure, and in this case the question remains open. To this end, we present a tool for studying the frequencies of symbols in preimages of surjective cellular automata, and prove some basic results about it. However, we show that by itself it is not enough to solve the stricter question in the positive.

math.DS↗

arXiv2025

Two-player Domino games

We introduce a 2-player game played on an infinite grid, initially empty, where each player in turn chooses a vertex and colours it. The first player aims to create some pattern from a target set, while the second player aims to prevent it. We study the problem of deciding which player wins, and prove that it is undecidable. We also consider a variant where the turn order is not alternating but given by a balanced word, and we characterise the decidable and undecidable cases.

cs.CC↗

arXiv2025

Subshifts defined by nondeterministic and alternating plane-walking automata

Plane-walking automata were introduced by Salo & Törma to recognise languages of two-dimensional infinite words (subshifts), the counterpart of $4$-way finite automata for two-dimensional finite words. We extend the model to allow for nondeterminism and alternation of quantifiers. We prove that the recognised subshifts form a strict subclass of sofic subshifts, and that the classes corresponding to existential and universal nondeterminism are incomparable and both larger that the deterministic class. We define a hierarchy of subshifts recognised by plane-walking automata with alternating quantifiers, which we conjecture to be strict.

cs.FL↗

arXiv2023

Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automaton

The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic "rock-paper-scissors" prey-predator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; asymptotically, any finite region is filled with a uniform state that is, after some time, driven out by its predator, each state taking its turn in dominating the region (heteroclinic cycles). We consider the situation where each site in the initial configuration is chosen independently at random with a different probability for each state. We prove that the asymptotic probability that a state dominates a finite region corresponds to the initial probability of its prey. The proof methods are based on discrete probability tools, mainly particle systems and random walks.

math.PR↗

arXiv2023

Sturmian and infinitely desubstitutable words accepted by an ω-automaton

Given an $ω$-automaton and a set of substitutions, we look at which accepted words can also be defined through these substitutions, and in particular if there is at least one. We introduce a method using desubstitution of $ω$-automata to describe the structure of preimages of accepted words under arbitrary sequences of homomorphisms: this takes the form of a meta-$ω$-automaton. We decide the existence of an accepted purely substitutive word, as well as the existence of an accepted fixed point. In the case of multiple substitutions (non-erasing homomorphisms), we decide the existence of an accepted infinitely desubstitutable word, with possibly some constraints on the sequence of substitutions e.g. Sturmian words or Arnoux-Rauzy words). As an application, we decide when a set of finite words codes e.g. a Sturmian word. As another application, we also show that if an $ω$-automaton accepts a Sturmian word, it accepts the image of the full shift under some Sturmian morphism.

cs.FL↗

arXiv2023

The Domino problem is undecidable on every rhombus subshift

We extend the classical Domino problem to any tiling of rhombus-shaped tiles. For any subshift X of edge-to-edge rhombus tilings, such as the Penrose subshift, we prove that the associated X-Domino problem is $Π^0_1$ -hard and therefore undecidable. It is $Π^0_1$ -complete when the subshift X is given by a computable sequence of forbidden patterns.

cs.DM↗

arXiv2022

The aperiodic Domino problem in higher dimension

The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift. arXiv:1805.08829 proved that this problem is co-recursively enumerable ($Π_0^1$-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic ($Σ_1^1$-complete), in higher dimension: $d \geq 4$ in the finite type case, $d \geq 3$ for sofic and effective subshifts. The reduction uses a subshift embedding universal computation and two additional dimensions to control periodicity. This complexity jump is surprising for two reasons: first, it separates 2- and 3-dimensional subshifts, whereas most subshift properties are the same in dimension 2 and higher; second, it is unexpectedly large.

cs.DM↗

arXiv2019

Necessary conditions for tiling finitely generated amenable groups

We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group. Piantadosi gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group. We then consider two other conditions: the first, also given by Piantadosi, is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al., is a necessary condition to decide if a set of Wang tiles gives a tiling of $\mathbb Z^2$. We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel.

math.DS↗

arXiv2018

Self-organisation in cellular automata with coalescent particles: qualitative and quantitative approaches

This article introduces new tools to study self-organisation in a family of simple cellular automata which contain some particle-like objects with good collision properties (coalescence) in their time evolution. We draw an initial configuration at random according to some initial $σ$-ergodic measure $μ$, and use the limit measure to descrbe the asymptotic behaviour of the automata. We first take a qualitative approach, i.e. we obtain information on the limit measure(s). We prove that only particles moving in one particular direction can persist asymptotically. This provides some previously unknown information on the limit measures of various deterministic and probabilistic cellular automata: 3 and 4-cyclic cellular automata (introduced in [Fis90b]), one-sided captive cellular automata (introduced in [The04]), N. Fat{è}s' candidate to solve the density classification problem [Fat13], self stabilization process toward a discrete line [RR15]... In a second time we restrict our study to to a subclass, the gliders cellular automata. For this class we show quantitative results, consisting in the asymptotic law of some parameters: the entry times (generalising [KFD11]), the density of particles and the rate of convergence to the limit measure.

math.DS↗

arXiv2018

Maximal Matching and Path Matching Counting in Polynomial Time for Graphs of Bounded Clique Width

In this paper, we provide polynomial-time algorithms for different extensions of the matching counting problem, namely maximal matchings, path matchings (linear forest) and paths, on graph classes of bounded clique-width. For maximal matchings, we introduce matching-cover pairs to efficiently handle maximality in the local structure, and develop a polynomial time algorithm. For path matchings, we develop a way to classify the path matchings in a polynomial number of equivalent classes. Using these, we develop dynamic programing algorithms that run in polynomial time of the graph size, but in exponential time of the clique-width. In particular, we show that for a graph $G$ of $n$ vertices and clique-width $k$, these problems can be solved in $O(n^{f(k)})$ time where $f$ is exponential in $k$ or in $O(n^g(l))$ time where $g$ is linear or quadratic in $l$ if an $l$-expression for $G$ is given as input.

cs.DM↗

arXiv2018

Aperiodic points in $\mathbb Z^2$-subshifts

We consider the structure of aperiodic points in $\mathbb Z^2$-subshifts, and in particular the positions at which they fail to be periodic. We prove that if a $\mathbb Z^2$-subshift contains points whose smallest period is arbitrarily large, then it contains an aperiodic point. This lets us characterise the computational difficulty of deciding if an $\mathbb Z^2$-subshift of finite type contains an aperiodic point. Another consequence is that $\mathbb Z^2$-subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some $\mathbb Z$-subshift. Finally, we use this result to characterize sets of possible slopes of periodicity for $\mathbb Z^3$-subshifts of finite type.

cs.DM↗