arXiv subjects

Martin Delacourt

Publications and source records attributed to Martin Delacourt.

7 recordsLinked to original sources

arXiv2025

Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences

Generalized abelian equivalence compares words by their factors up to a certain bounded length. The associated complexity function counts the equivalence classes for factors of a given size of an infinite sequence. How practical is this notion? When can these equivalence relations and complexity functions be computed efficiently? We study the fixed points of substitution of Pisot type. Each of their $k$-abelian complexities is bounded and the Parikh vectors of their length-$n$ prefixes form synchronized sequences in the associated Dumont--Thomas numeration system. Therefore, the $k$-abelian complexity of Pisot substitution fixed points is automatic in the same numeration system. Two effective generic construction approaches are investigated using the \texttt{Walnut} theorem prover and are applied to several examples. We obtain new properties of the Tribonacci sequence, such as a uniform bound for its factor balancedness together with a two-dimensional linear representation of its generalized abelian complexity functions.

cs.FL↗

arXiv2024

Linear Recurrence Sequence Automata and the Addition of Abstract Numeration Systems

Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with an additional linear recurrence sequence on each transition, are introduced to compute various $\mathbb{Z}$-rational non commutative formal series in abstract numeration systems. Under certain Pisot conditions on the recurrence sequences, the support of these series is regular. This property can be leveraged to derive various synchronized relations including a deterministic finite automaton that computes the addition relation of various Dumont-Thomas numeration systems and deterministic finite automata converting between various numeration systems. A practical implementation for Walnut is provided.

cs.FL↗

arXiv2021

Rice's theorem for generic limit sets of cellular automata

The generic limit set of a cellular automaton is a topologically dened set of congurations that intends to capture the asymptotic behaviours while avoiding atypical ones. It was dened by Milnor then studied by Djenaoui and Guillon rst, and by T{ö}rm{ä} later. They gave properties of this set related to the dynamics of the cellular automaton, and the maximal complexity of its language. In this paper, we prove that every non trivial property of these generic limit sets of cellular automata is undecidable.

cs.DM↗

arXiv2017

Characterisation of limit measures of higher-dimensional cellular automata

We consider the typical asymptotic behaviour of cellular automata of higher dimension (greater than 2). That is, we take an initial configuration at random according to a Bernoulli (i.i.d) probability measure, iterate some cellular automaton, and consider the (set of) limit probability measure(s) as time tends to infinity. In this paper, we prove that limit measures that can be reached by higher-dimensional cellular automata are completely characterised by computability conditions, as in the one-dimensional case. This implies that cellular automata have the same variety and complexity of typical asymptotic behaviours as Turing machines, and that any nontrivial property in this regard is undecidable (Rice-type theorem). These results extend to connected sets of limit measures and Cesàro mean convergence. The main tool is the implementation of arbitrary computation in the time evolution of a cellular automata in such a way that it emerges and self-organises from a random configuration.

math.DS↗

arXiv2015

$μ$-Limit Sets of Cellular Automata from a Computational Complexity Perspective

This paper concerns $μ$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $μ$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $μ$-limit sets can have a $Σ\_3^0$-hard language, second, they can contain only $α$-complex configurations, third, any non-trivial property concerning them is at least $Π\_3^0$-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.

cs.DM↗

arXiv2010

Construction of $μ$-Limit Sets

The $μ$-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to in- finity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as $μ$-limit set where $μ$ is the uniform Bernoulli measure.

cs.DM↗

arXiv2010

Directional Dynamics along Arbitrary Curves in Cellular Automata

This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers.

cs.DM↗