# Jacques Mazoyer's research while affiliated with Aix-Marseille Université and other places

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## Publications (41)

This chapter is dedicated to classic tools and methods involved in cellular transformations and constructions of signals and of functions by means of signals, which will be used in subsequent chapters. The term "signal" is widely used in the field of cellular automata (CA). But, as it arises from different levels of understanding, a general definit...

This chapter shows how simple, common algorithms (multiplication and prime number sieve) lead to very natural cellular automata implementations. All these implementations are built with some natural basic tools: signals and grids. Attention is first focussed on the concept of signals and how simple and rich they are to realize computations. Looking...

This paper is the first part of a series of two papers dealing with bulking: a quasi-order on cellular automata comparing space–time diagrams up to some rescaling. Bulking is a generalization of grouping taking into account universality phenomena, giving rise to a maximal equivalence class. In the present paper, we discuss the proper components of...

This paper is the second part of a series of two papers dealing with bulking: a way to define quasi-order on cellular automata by comparing space-time diagrams up to rescaling. In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the wh...

Lorsqu'on observe des orbites de certains automates cellulaires, on
peut penser qu'elles apparaissent comme des mélanges d'orbites
d'autres automates (composants). Dans cet article, nous tentons de
comprendre ce phénomène en construisant un hybride de deux automates
au moyen d'un troisième. Deux types d'automates cellulaires sont
introduits :...

We show simple methods in order to construct cellular automata with a defined behavior. To achieve this goal, we explain how to move local information through the networks and to set up their meeting in order to get the wished global behavior. Some well-known examples are given: Fischer’s prime construction, Firing Squad Synchronization Problem and...

We consider the Turing Machine as a dynamical system and we study a particular partition projection of it. In this way, we define a language (a subshift) associated to each machine. The classical definition of Turing Machines over a one-dimensional tape is generalized to allow for a tape in the form of a Cayley Graph. We study the complexity of the...

A grouped instance of a cellular automaton (CA) is another one obtained by grouping several states into blocks and by letting
interact neighbor blocks. Based on this operation a preorder ≤ on the set of one dimensional CA is introduced. It is shown
that (CA,≤) admits a global minimum and that on the bottom of (CA,≤) very natural equivalence classes...

In a previous work we began to study the question of “how to compare” cellular automata (CA). In that context it was introduced
a preorder (CA,≤) admitting a global minimum and it was shown that all the CA satisfying very simple dynamical properties
as nilpotency or periodicity are located “on the bottom of (CA,≤)”. Here we prove that also the (alg...

Cellular automata are a formal model of locally interacting systems. They are syntactically simple but can present extremely complex behaviors, which make them suitable to study complex systems in general. Many classifications have been proposed in literature [1], often relying on the observation of dynamics. In a first part, we present more recent...

Fine studies on how powerful are two-dimensional computing devices are restricted by the difficulty to set up a relevant notion of language of figures. In the case of two-dimensional cellular automata, several solutions have been proposed, but we adopt here another point of view: to the automaton is added a one-to-one “locally connected” mapping, τ...

In this paper, we consider the pebble automata introduced by Blum and Hewitt, but now moving through the unbounded plane Z^2. We are interested in their ability to recognize families of dotted figures. Contrary to the bounded case studied by Blum and Hewitt, the hierarchy collapses: there are families recognized with 0, 1, 2 and 3 pebbles, but each...

Peu de résultats sont connus sur la puissance de calcul des automates cellulaires 2D. Dans l'approche “reconnaissance de langages”, la difficulté provient de la multiplicité des plongements possibles d'un mot sur le plan discret. Une question très simple comme la reconnaisance des langages rationnels en temps réel devient vraiment délicate. Nous mo...

We are interested in the power of cellular automata as parallel devices, and in the complexity classes they determine. With the intention of starting research on parallel recognition of languages on cellular automata of dimension higher than or equal 2, we present the state of art for dimension 1, in parallel and sequential mode, for bounded and un...

We present two cellular algorithms, in O(n) and respectively in O(w(2)), for the leader election problem on finite connected rings F and respectively finite connected subsets of Z(d), of eccentricity w, for any fixed d. The problem consists of finding an algorithm such that when setting the elements of F to a special state, and all the others to a...

The chapter studies some families of signals generated by cellular automata and shows their application in cellular algorithms. As it does not seem to be possible to give a general definition of a signal, we choose, in this chapter, to think about signals as trajectories of information quanta. Various cellular automata that support propagation of t...

We first present a protocol to elect a unique leader of a given cycle of the planar square lattice on vertices of which synchronous communicating finite automata are placed, which know the directions of its successor and its predecessor. Thereafter, we extend it to finite parts of the planar grid, giving in both cases the time complexity (twice the...

We consider the infinite versions of the usual computational complexity questions LogSpace ≟ P, NLogSpace ≟ P by studying the comparison of their descriptive logics on infinite partially ordered structures rather than restricting
ourselves to finite structures. We show that the infinite versions of those famous class separation questions are consis...

We consider the infinite versions of the usual computational complexity questions LOGSPACE = P, NLOGSPACE = P by studying the comparison of their descriptive logics on infinite partially ordered structures rather than restricting ourselves to finite structures. We show that the infinite versions of those famous class separation questions are consis...

A great amount of work has been deveted to the understanding of the long-time behavior of cellular automata (CA). As for any other kind of dynamical system, the long-time behavior of a CA is described by its attractors. In this context, it has been proved that it is undecidable whether every circular configuration of a given CA evolves to some fixe...

In this paper, we are interested in signals, from the data can be transmitted in a cellular automaton. We study generation of some signals. In this aim, we investigate a notion of constructibility of increasing functions related to the production of words on the initial cell (in the sense of Fisher for the prime numbers). We establish some closure...

We study how the underlying graph of dependencies of one dimensional cellular automaton may be used in order to move and compose areas of computations. This allows us to define complex cellular automata, relaxing in some way the inherent synchronism of such networks.

We first present a protocol to elect a unique leader of a givencycle of the planar square lattice on vertices of which synchronouscommunicating finite automata are placed, whichknow the directions of its successor and its predecessor.Thereafter, we extend it to finite parts of the planar grid,giving in both cases the time complexity (twice the leng...

Language recognition is a powerful tool to evaluate the computational power of devices. In case of cellular automata, a specific problematics appears in the one dimensional case where the space is bounded. A state of the art is presented. In the two dimensional case, few results are known. The main difficulty is to inject an order on a plane of cel...

A great amount of work has been devoted to the understanding of the longtime behavior of cellular automata (CA). As for any other kind of dynamical system, the long-time behavior of a CA is described by its attractors. In this context, it has been proved that it is undecidable to know whether every circular configuration of a given CA evolves to so...

Let X be a one-dimensional cellular automaton. A “power of X” is another cellular automaton obtained by grouping several states of X into blocks and by considering as local transitions the “natural” interactions between neighbor blocks. Based on this operation a preorder ⩽ on the set of one-dimensional cellular automata is introduced. We denote by...

Cellular automata may be viewed as a modelization of synchronous parallel computation. Even in the one-dimensional case, they are known as capable of universal computations. The usual proof uses a simulation of a universal Turing machine. In this paper, we present how a one-dimensional cellular automata can simulate any recursive function in such a...

The author considers the one-dimensional firing squad synchronization problem (FSSP). He raises the following question: What is the minimal number of states and information flow needed to solve the FSSP in minimal time? After proving that the sets of solutions and of minimal time solutions to the FSSP are not recursively enumerable, he constructs p...

Ibarra (1985) showed that, given a cellular automaton of range 1 recognizing some language in time n+1+R(n), we can obtain another CA of range 1 recognizing exactly the same language but in time n+1+R(n)/k (k⩾2 arbitrary). Their proof proceeds indirectly (through the simulation of CAs by a special kind of sequential machines, the STMs) and we think...

From Balzer's work (1967), we know that the firing squad synchronization problem has a minimal-time solution with eight states. We show that such a solution exists with only six states. Our method is somewhat different from all previous ones: the initial line is iteratively divided in two inequal parts so that each new right part can be treated as...

## Citations

... The grouping quasi-order was introduced by Mazoyer and Rapaport [12] as a successful tool to classify CA according to algebraic properties [11]. However, grouping fails to capture several geometrical properties of CA that one would like to see classified by such a geometric classification. ...

Reference: Bulking I: an Abstract Theory of Bulking

... Dans la littérature, les programmes d'automates cellulaires ne sont que rarement donnés explicitement. On y préfère souvent, pour un problème donné, présenter une stratégie sous forme de signaux et collisions et donner le programme de l'automate uniquement de manière informelle [7,10,11,14,33,34,51]. Cette présentation sous forme de signaux, même si elle ne donne pas explicitement le programme de l'AC, spécifie des schémas d'induction sur lesquels peut se baser un tel programme. ...

... Dans la littérature, les programmes d'automates cellulaires ne sont que rarement donnés explicitement. On y préfère souvent, pour un problème donné, présenter une stratégie sous forme de signaux et collisions et donner le programme de l'automate uniquement de manière informelle [7,10,11,14,33,34,51]. Cette présentation sous forme de signaux, même si elle ne donne pas explicitement le programme de l'AC, spécifie des schémas d'induction sur lesquels peut se baser un tel programme. ...

... Mazoyer [5]). As in Mazoyer [6] , we use Minsky's dichotomy process set up by the following three signals: 1. Signal S is launched by cell 1 (the general) at time 0 and goes rightwards at maximum speed. Thus, S reaches cell i at time D 1,i . ...

... In other words, the simulation of all these 1 + +log n computations is in the same order of n 2 sites than the real time computation on x 1 · · · x n . Here we do not recall the tricky construction of the CA A which allows to compress and set up the 1 + +log n evolutions of the CA A. See [10] or the self contained CA solution in [5] for a complete proof of the lemma. Lemma 1 provides the technical tool to prove Theorem 1. Actually, we may remark that L 2 belongs to rtIA and get the following corollary: ...

Reference: Closure properties of cellular automata

... Signals store and transmit information to start a process, to synchronise, etc. The use of signals in the context of CA is widespread in the literature: collision computing [Adamatzky, 2002], gliders Jin and Chen [2016], solitons [Jakubowski et al., 1996, 2000, 2017, Siwak, 2001, particles [Boccara et al., 1991, Mitchell, 1996, Hordijk et al., 1998], Turing-computation [Lindgren andNordahl, 1990, Cook, 2004], synchronisation [Varshavsky et al., 1970, Yunès, 2007, geometrical constructions [Cook, 2004], signals Terrier, 1999, Delorme andMazoyer, 2002], etc. ...

... In this article, we consider the application of CCA to shape generation. The use of CA in shape generation has been considered before [3,4,8,12], but due to the single cell model, the shape formation is not easy to achieve. Delormea et al. [4], for example, used signals in grouped cellular automata based on the mathematical definition of the shapes, whilst Chavoya and Duthen [3] evolved a shape from a single cell in a 2D grid, by using genetic algorithms to update lookup tables for the CA. ...

... In 1966 Waksman [30] gave a 16-state minimal-time solution, and Balzer [1] independently produced an 8-state solution using the same ideas. In 1987 Mazoyer produced a 6-state solution to a restricted version of the problem in which the initiator is always located at the left endpoint of the array [10]. In 1968 Moore and Langdon introduced the generalized problem in which the initiator can be located anywhere in the array and gave a 17-state minimal-time solution [12]. ...

... The grouping quasi-order was introduced by Mazoyer and Rapaport [12] as a successful tool to classify CA according to algebraic properties [11]. However, grouping fails to capture several geometrical properties of CA that one would like to see classified by such a geometric classification. ...

Reference: Bulking I: an Abstract Theory of Bulking

... Historically , the notion of universality used for CA was more or less an adaptation of the classical Turing-universality. Later, a stronger notion called intrinsic universality was proposed: A CA is intrinsically universal if it is able to simulate any other CA [3] [9] [12] through a uniform and regular encoding based on rescaling. This definition of intrinsic universality may seem very restrictive. ...