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Ce mémoire se place dans l'étude des modèles du calcul continus. Nous y montrons que la géométrie plane permet de calculer. Nous définissons un calcul géométrique et utilisons la continuité de l'espace et du temps pour stocker de l'information au point de provoquer des accumulations.Dans le monde des automates cellulaires, on parle souvent de particules ou de signaux (qui forment des lignes discrètes sur les diagrammes espace-temps) tant, pour analyser une dynamique que, pour concevoir des automates cellulaires particuliers. Le point de départ de nos travaux est d'envisager des versions continues de ces signaux. Nous définissons un modèle de calcul continu, les machines à signaux, qui engendre des figures géométriques suivant des règles strictes. Ce modèle peut se comprendre comme une extension continue des automates cellulaires. Le mémoire commence par une présentation des automates cellulaires et des particules. Nous faisons ensuite une classification des différents modèles de calcul existants et mettons en valeur leurs aspects discrets et continus. À notre connaissance, notre modèle est le seul à temps et espace continus mais à valeurs et mises à jour discrètes.
Dans la première partie du mémoire, nous présentons ce modèle, les machines à signaux, et montrons comment y mener tout calcul au sens de Turing (par la simulation de tout automate à deux compteurs). Nous montrons comment modifier une machine de manière à réaliser des transformations géométriques (translations, homothéties) sur les diagrammes engendrés. Nous construisons également les itérations automatiques de ces constructions de manière à contracter le calcul à une bande (espace borné) puis, à un triangle (temps également borné).
Dans la seconde partie du mémoire, nous cherchons à caractériser les points d'accumulation. Nous reformulons de manière topologique les diagrammes espace-temps: pour chaque position, la valeur doit correspondre au voisinage sur un ouvert suffisamment petit. Muni de cet outil, nous regardons les plus simples accumulations possibles (les singularités isolées) et proposons un critère pour y prolonger le calcul; mais le déterminisme peut être perdu dans le cône d'influence. Enfin, en construisant pour tout automate à deux compteurs une machine à signaux et une configuration initiale simulant l'automate pour toutes les valeurs possibles, nous montrons que le problème de la prévision de l'apparition d'une accumulation est Σ20-complet.
Le mémoire se conclut par la présentation de nombreuses perspectives de recherches.
Cette thèse s'intéresse dans un premier temps aux automates cellulaires réversibles, et dans un second temps aux tas de sable linéaires.Nous construisons diverses simulations reliant les automates cellulaires aux automates à partitions, en particulier celle des automates cellulaires réversibles par les automates à partitions réversibles, ce qui était une conjecture depuis 1990. Par des constructions successives, nous montrons que le “Billiard ball model” de Toffoli et Margolus est capable de simuler tous les automates à partitions réversibles de dimension 2. En rassemblant ces résultats, nous montrons qu'il existe des automates cellulaires réversibles capables de simuler tous les automates cellulaires réversibles de même dimension.
Dans un espace linéaire, “Tas de sable” et “Chip firing game” sont équivalents. Nous portons notre attention sur le cas où les grains tombent un à un. Des motifs délimités par des signaux apparaissent au sein des configurations engendrées. Nous étudions la dynamique du système et démontrons un équivalent asymptotique. Nous étendons nos méthodes et nos résultats à d'autres types de configurations initiales. Dans chaque cas étudié, le temps parallèle est inférieur au temps séquentiel dans un rapport de l'ordre du nombre de piles mises en oeuvre.
Using rules to automatically extend a drawing on an Euclidean space might lead to accumulating drawings into a single point. Such points are characterized in the context of Abstract geometrical computation.Colored line segments (traces of signals) are drawn according to rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. Time and space are continuous and accumulations can happen. They can be devised to unboundedly accelerate a computation and provide, in a finite duration, exact analog values as limits.
Starting with rational numbers for coordinates and speeds, the time of any isolated accumulation is a c.e.-R (computably enumerable) real number. There is a signal machine and an initial configuration that accumulates at any c.e.-R time. Similarly, the spatial positions of isolated accumulations are exactly the d.-c.e.-R (difference of computably enumerable) numbers. Moreover, there is a signal machine that can accumulate at any c.e.-R time or d.-c.e.-R position depending only on the initial configuration.
This relies on a two-level construction: an inner structure simulates a Turing machine that output orders to the outer structure which handles the accumulation.
Extended Signal machines are proven capable to compute any computable function in the understanding of recursive/computable analysis (CA), represented here with type-2 Turing machines (T2-TM) and signed binary. This relies on a mixed representation of any real number as an integer (in signed binary) plus an exact value in (-1,1). This permits to have only finitely many signals present simultaneously. Extracting a (signed) bit, improving the precision by one bit and iterating a T2-TM only involve standard signal machines. For exact CA-computations, T2-TM have to deal with an infinite entry and to run through infinitely many iterations to produce an infinite output. This infinite duration can be provided by an infinite acceleration construction. Extracting/encoding an infinite sequence of bits is achieved as the limit of the approximation process with a careful handling of accumulations.
This article provides several very small signal machines able to perform any computation -in the classical understanding- generated from Turing machines, cellular automata and cyclic tag systems. A halting universal signal machine with 13 meta-signals and 21 collision rules is presented (resp. 15 and 24 for a robust version). If infinitely many signals are allowed to be present in the initial configuration, 5 meta-signals and 7 collision rules are enough to achieve non-halting weak universality (resp. 6 and 9 for a robust version).
In the context of Abstract geometrical computation, it has been proved that black hole model (and SAD computers) can be implemented. To be more physic-like, it would be interesting that the construction is reversible and preserves some energy. There is already a (energy) conservative and reversible two-counter automaton simulation.In the present paper, based on reversible and conservative stacks, reversible Turing machines are simulated. Then a shrinking construction that preserves these properties is presented. All together, a black hole model implementation that is reversible and conservative (both the shrinking structure and the universal Turing machine) is provided.
Abstract geometrical computation involves drawing colored line segments (traces of signals) according to rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. Time and space are continuous and accumulations can be devised to unlimitedly accelerate a computation and provide, in a finite duration, exact analog values as limits.In the present paper, we show that starting with rational numbers for coordinates and speeds, the time of any accumulation is a c.e. (computably enumerable) real number and moreover, there is a signal machine and an initial configuration that accumulates at any c.e. time. Similarly, we show that the spatial positions of accumulations are exactly the d-c.e. (difference of computably enumerable) numbers. Moreover, there is a signal machine that can accumulate at any c.e. time or d-c.e. position.
Abstract geometrical computation can solve NP-complete problems efficiently: any boolean constraint satisfaction problem, instance of SAT, can be solved in bounded space and time with simple geometrical constructions involving only drawing parallel lines on a Euclidean space-time plane. Complexity as the maximal length of a sequence of consecutive segments is quadratic. The geometrical algorithm achieves massive parallelism: an exponential number of cases are explored simultaneously. The construction relies on a fractal pattern and requires the same amount of space and time independently of the SAT formula.
Abstract geometrical computation naturally arises as a continuous counterpart of cellular automata. It relies on signals (dimensionless points) traveling at constant speed in a continuous space in continuous time. When signals collide, they are replaced by new signals according to some collision rules. This simple dynamics relies on real numbers with exact precision and is already known to be able to carry out any (discrete) Turing-computation. The Blum, Shub and Small (BSS) model is famous for computing over (considered here as a unlimited register machine) by performing algebraic computations. We prove that signal machines (set of signals and corresponding rules) and the infinite-dimension linear (multiplications are only by constants) BSS machines can simulate one another.
In Abstract geometrical computation for black hole computation (MCU '04, LNCS 3354), the author provides a setting based on rational numbers, abstract geometrical computation, with super-Turing capability: any recursively enumerable set can be decided in finite time. To achieve this, a Zeno-like construction is used to provide an accumulation similar in effect to the black holes of the black hole model.We prove here that forecasting an accumulation is Σ02-complete (in the arithmetical hierarchy) even if only energy conserving signal machines are addressed (as in the cited paper). The Σ02-hardness is achieved by reducing the problem of deciding whether a recursive function (represented by a 2-counter automaton) is strictly partial. The Σ02-membership is proved with a logical characterization.
Abstract geometrical computation can solve PSPACE-complete problems efficiently: any quantified boolean formula, instance of Q-SAT - the problem of satisfiability of quantified boolean formula - can be decided in bounded space and time with simple geometrical constructions involving only drawing parallel lines on an Euclidean space-time. Complexity as the maximal length of a sequence of consecutive segments is quadratic. We use the continuity of the real line to cover all the possible boolean valuations by a recursive tree structure relying on a fractal pattern: an exponential number of cases are explored simultaneously by a massive parallelism.
In the cellular automata (CA) literature, discrete lines in discrete space-time diagrams are often idealized as Euclidean lines in order to design CA or analyze their dynamic behavior. In this paper, we present a parallel model of computation corresponding to this idealization: dimensionless particles move uniformely at fixed velocities along the real line and are transformed when they collide. Like CA, this model is parallel, uniform in space-time and uses local updating. The main difference is the use of the continuity of space and time, which we proceed to illustrate with a construction to solve Q-SAT, the satisfiability problem for quantified boolean formulae, in bounded space and time, and quadratic collision depth.
In the context of Abstract geometrical computation, it has been proved that black hole model (and SAD computers) can be implemented. To be more physic-like, it would be interesting that the construction is reversible and preserves some energy. There is already a (energy) conservative and reversible two-counter automaton simulation.In the present paper, based on reversible and conservative stacks, reversible Turing machines are simulated. Then a shrinking construction that preserves these properties is presented. All together, a black hole model implementation that is reversible and conservative (both the shrinking structure and the universal Turing machine) is provided.
This paper has a flow. Please look at the corrected version: [?].In Abstract geometrical computation, Turing computability is provided by simples machines involving drawing colored line segments, called signals, accordin g to simple rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. These signal machines also provide a very powerful model of analog computation following both the approaches of computable analysis and of Blum, Shub and S male. The key is that accumulations can be devised to accelerate the computation and provide an exact analog values as limits in finite time.
In the present paper, we show that starting with rational numbers for coordinates and speeds, the collections of positions of accumulations in both space and time are exactly the computable real numbers (as defined by computable analysis). Moreover, there is a signal machine that can provide an accumulation at any computable place and date.
Abstract geometrical computation (AGC) naturally arises as a continuous counterpart of cellular automata. It relies on signals (dimensionless points) traveling and colliding. It can carry out any Turing computation, but since it works with continuous time and space, some analog computing capability exists. In Abstract Geometrical Computation and the Linear BSS Model (CiE 2007, LNCS 4497, p. 238-247), it is shown that AGC without any accumulation has the same computing capability as the linear BSS model.An accumulation brings infinitely many time steps in a finite duration. This has been used to implement the black-hole model of computation (Fundamenta Informaticae 74(4), p. 491-510). It also makes it possible to multiply two variables, thus simulating the full BSS. Nevertheless a BSS uncomputable function, the square root, can also be implemented, thus proving that the computing capability of AGC with isolated accumulations is strictly beyond the one of BSS.
Signal machines can be seen as a way to automatically extends a drawing consisting of line segments in Euclidean spaces. Dealing with a continuous setting, accumulation points might occur. We characterize exactly the possible localizations of accumulation points as enumerable computable real and difference of such. These reals are natural extension of computable reals of computable analysis.