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Abstract Geometrical Computation 1: Embedding Black Hole Computations with Rational Numbers
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A geometric model of computation, conservative abstract geometrical computation, that, although being based on rational numbers, has the same property: it can simulate any Turing machine and can decide any R.E. problem through the creation of an accumulation.Abstract:
The Black hole model of computation provides super-Turing computing power since it offers the possibility to decide in finite (observer's) time any recursively enumerable (R.E.) problem. In this paper, we provide a geometric model of computation, conservative abstract geometrical computation, that, although being based on rational numbers (and not real numbers), has the same property: it can simulate any Turing machine and can decide any R.E. problem through the creation of an accumulation. Finitely many signals can leave any accumulation, and it can be known whether anything leaves. This corresponds to a black hole effect.read more
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Book ChapterDOI
A Survey on Analog Models of Computation.
Olivier Bournez,Amaury Pouly +1 more
TL;DR: A survey on analog models of computations, which considers both approaches, often intertwined, with a point of view mostly oriented by computation theory.
Journal ArticleDOI
Abstract geometrical computation 3: black holes for classical and analog computing
TL;DR: Using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies and not only is Zeno effect possible but it is used to unleash the black hole power.
Book ChapterDOI
Abstract Geometrical Computation and the Linear Blum, Shub and Smale Model
TL;DR: It is proved that signal machines and the infinite-dimension linear BSS machines can simulate one another and be able to carry out any Turing-computation.
The signal point of view: from cellular automata to signal machines
TL;DR: This work shows how the use of discrete signals in the literature on cellular automata have been considered on their own, and presents their continuous counterpart: abstract geometrical computation and signal machines.
Abstract geometrical computation with accumulations: Beyond the Blum, Shub and Smale model
TL;DR: It is shown that AGC without any accumulation has the same computing capability as the linear BSS model, and a BSS uncomputable function, the square root, can be implemented, thus proving that the computing capability of AGC with isolated accumulations is strictly beyond the one of BSS.
References
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