Computing in the Fractal Cloud: Modular Generic Solvers for SAT and Q-SAT Variants

Duchier, Denys; Durand-Lose, Jérôme; Senot, Maxime

doi:10.1007/978-3-642-29952-0_42

Cited by 10 publications

(13 citation statements)

References 24 publications

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“…They are a powerful tool to accelerate a computation and provide hypercomputation [Durand-Lose, 2009a]. Recently, it has been proved that, starting from a rational machine and configuration, the locations of isolated accumulations have to be d-c.e (difference of computably enumerable) real numbers and that any such number can be reached [Durand-Lose, 2012].…”

Section: Introductionmentioning

confidence: 99%

Abstract geometrical computation 8: Small machines, accumulations & rationality

Becker

Chapelle

Durand-Lose

et al. 2018

Journal of Computer and System Sciences

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Section: Introductionmentioning

confidence: 99%

Abstract geometrical computation 8: Small machines, accumulations & rationality

Becker

Chapelle

Durand-Lose

et al. 2018

Journal of Computer and System Sciences

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“…Accumulations provide a powerful tool to accelerate a computation, to do hyper-computation and analog computation [9]. Fractals can be generated and their construction modied so as to achieve massive parallelism and the capability to solve eciently NSPACE-complete problems (Q-SAT in [17]).…”

Section: Introductionmentioning

confidence: 99%

Irrationality Is Needed to Compute with Signal Machines with Only Three Speeds

Durand-Lose

2013

Lecture Notes in Computer Science

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Space-time diagrams of signal machines on nite congurations are composed of interconnected line segments in the Euclidean plane. As the system runs, a network emerges. If segments extend only in one or two directions, the dynamics is nite and simplistic. With four directions, it is known that fractal generation, accumulation and any Turing computation are possible. This communication deals with the three directions/speeds case. If there is no irrational ratio (between initial distances between signals or between speeds) then the network follows a mesh preventing accumulation and forcing a cyclic behavior. With an irrational ratio (here, the Golden ratio) between initial distances, it becomes possible to provoke an accumulation that generates innitely many interacting signals in a bounded portion of the Euclidean plane. This behavior is then controlled and used in order to simulate a Turing machine and generate a 25-state 3-speed Turinguniversal signal machine.

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“…This has been used to do fractal parallelism (Duchier et al, 2010(Duchier et al, , 2012. More-over, Zeno effects can be implemented to generate unbounded acceleration; in particular to allow infinitely many discrete transitions during a finite duration.…”

Section: Introductionmentioning

confidence: 99%

Abstract geometrical computation 7: geometrical accumulations and computably enumerable real numbers

Durand-Lose

2012

Nat Comput

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Abstract geometrical computation 7: Geometrical accumulations and computably enumerable real numbersJérôme Durand-Lose To cite this version:Jérôme Durand-Lose. Abstract geometrical computation 7: Geometrical accumulations and computably enumerable real numbers. Natural Computing, Springer Verlag, 2012, 11 (4) Abstract 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. Constructions exist to unboundedly accelerate a computation and provide, in a finite duration, exact analog values as limits/accumulations.Starting with rational numbers for coordinates and speeds, the time of any isolated accumulation is a c.e. (computably enumerable) real number. There is a signal machine and an initial configuration that accumulates at any c.e. time. Similarly, the spatial positions of isolated 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 depending only on the initial configuration.These existence results rely on a two-level construction: an inner structure simulates a Turing machine that output orders to the outer structure which handles the accumulation. * http://www.univ-orleans.fr/lifo/Members/Jerome.Durand-Lose, Jerome.DurandLose@univ-orleans.fr † This work was partially funded by the ANR project AGAPE, ANR-09-BLAN-0159-03.1

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Computing in the Fractal Cloud: Modular Generic Solvers for SAT and Q-SAT Variants

Cited by 10 publications

References 24 publications

Abstract geometrical computation 8: Small machines, accumulations & rationality

Abstract geometrical computation 8: Small machines, accumulations & rationality

Irrationality Is Needed to Compute with Signal Machines with Only Three Speeds

Abstract geometrical computation 7: geometrical accumulations and computably enumerable real numbers

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