Abstract geometrical computation 7: Geometrical accumulations and computably enumerable real numbers - Archive ouverte HAL Access content directly
Journal Articles Natural Computing Year : 2012

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

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. 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.
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Dates and versions

hal-00691466 , version 1 (07-11-2012)

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Jérôme Durand-Lose. Abstract geometrical computation 7: Geometrical accumulations and computably enumerable real numbers. Natural Computing, 2012, 11 (4), pp.609-622. ⟨10.1007/s11047-012-9335-8⟩. ⟨hal-00691466⟩
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