Martin Delacourt

Directional dynamics along arbitrary curves in cellular automata

This paper studies directional dynamics on one-dimensional cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behavior of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviors 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.

The Unary Arithmetical Algorithm in Bimodular Number Systems

We analyze the performance of the unary arithmetical algorithm which computes a Moebius transformation in bimodular number systems which extend the binary signed system. We give statistical evidence that in some of these systems, the algorithm has linear average time complexity.

Finite state transducers for modular möbius number systems

Modular Mobius number systems consist of Mobius transformations with integer coefficients and unit determinant. We show that in any modular Mobius number system, the computation of a Mobius transformation with integer coefficients can be performed by a finite state transducer and has linear time complexity. As a byproduct we show that every modular Mobius number system has the expansion subshift of finite type.

Real time language recognition on 2D cellular automata: dealing with non-convex neighborhoods

In this paper we study language recognition by twodimensional cellular automata on different possible neighborhoods. Since it is known that all complete neighborhoods are linearly equivalent we focus on a natural sub-linear complexity class: the real time. We show that any complete neighborhood is sufficient to recognize in real time any language that can be recognized in real-time by a cellular automaton working on the convex hull of V.

Permutive one-way cellular automata and the finiteness problem for automaton groups

The decidability of the finiteness problem for automaton groups is a well-studied open question on Mealy automata. We connect this question of algebraic nature to the periodicity problem of one-way cellular automata, a dynamical question known to be undecidable in the general case. We provide a first undecidability result on the dynamics of one-way permutive cellular automata, arguing in favor of the undecidability of the finiteness problem for reset Mealy automata.

µ-Limit sets of cellular automata from a computational complexity perspective

We characterize the set of µ-limit sets of cellular automata.We prove that the language of these limit sets can be Σ 3 -complete.We prove a Rice theorem for µ-limit sets of cellular automata. 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 ? 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.

Construction of mu-Limit Sets of Two-dimensional Cellular Automata

We prove a characterisation of \mu-limit sets of two-dimensional cellular automata, extending existing results in the one-dimensional case. This sets describe the typical asymptotic behaviour of the cellular automaton, getting rid of exceptional cases, when starting from the uniform measure.

Characterisation of Limit Measures of Higher-Dimensional Cellular Automata

We consider the typical asymptotic behaviour of cellular automata of higher dimension (≥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 t → ∞. 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.