Geometrical accumulations and computably enumerable real numbers (extended abstract)

Abstract : 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.
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Jérôme Durand-Lose. Geometrical accumulations and computably enumerable real numbers (extended abstract). UC '11, Jun 2011, Turku, Finland. pp.101-112, ⟨10.1007/978-3-642-21341-0⟩. ⟨hal-00601746⟩

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