Abstract geometrical computation 3: black holes for classical and analog computing

Abstract

geometrical computation 3: Black holes for classical and analog computingJérôme Durand-Lose To cite this version:Jérôme Durand-Lose. The date of receipt and acceptance will be inserted by the editorAbstract The so-called Black Hole model of computation involves a non Euclidean space-time where one device is infinitely "accelerated" on one world-line but can send some limited information to an observer working at "normal pace". The key stone is that after a finite duration, the observer has received the information or knows that no information was ever sent by the device which had an infinite time to complete its computation. This allows to decide semi-decidable problems and clearly falls out of classical computability.A setting in a continuous Euclidean space-time that mimics this is presented. Not only is Zeno effect possible but it is used to unleash the black hole power. Both discrete (classical) computation and analog computation (in the understanding of Blum, Shub and Smale) are considered. Moreover, using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies.Key words Abstract geometrical computation -Analog computationArithmetic hierarchy -Black hole model -Malament-Hogarth space-time -Hyper-computing -Zeno phenomenon Disclaimer The author is a computer scientist and not a physicist. So that it is refereed to other contributions in this issue and to the cited bibliography for enlightenment on black holes. The present article aims at providing a context of computation exhibiting the same computing capabilities as black holes and at illustrating the point of view of a computer scientist.

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

“…It does in the understandings of both Turing computability (Durand-Lose, 2005), the original Blum, Shub and Smale model (Blum et al, 1989;Durand-Lose, 2007, 2008a and Computable analysis (Weihrauch, 2000;Durand-Lose, 2009b, 2011a. The so-called Blackhole model of computation can be embedded too (Etesi and Németi, 2002;Hogarth, 2004;Lloyd and Ng, 2004;Andréka et al, 2009;Durand-Lose, 2006a, 2009a.…”

“…More-over, Zeno effects can be implemented to generate unbounded acceleration; in particular to allow infinitely many discrete transitions during a finite duration. This has been used to decide the halting problem and to implement the Black-hole model (Durand-Lose, 2009a). It has also been used to carry out exact analog computations (Durand-Lose, 2008a, 2009b.…”

Abstract Geometrical Computation and Computable Analysis

“…Unless a singularity is generated, an accumulation results in one signal (always the same). More complex rules can be devised to handle singularities and accumulations -but are not used here -so as to emulate the (nested) black hole model of computation [Hogarth, 1994, Etesi and Németi, 2002, Durand-Lose, 2006, 2009 (with the use of folding structures) as well as to achieve exact multiplication of real numbers to simulate the BSS model [DurandLose, 2008].…”

“…The presentation of the implementation of Turing machines (TM) is only illustrated; it is quite straightforward and already done in [Durand-Lose, 2009]. Type-2 TM need infinitely many iterations to complete a computation.…”

“…Type-2 TM need infinitely many iterations to complete a computation. To get the results in finite duration, TM and input are embedded into a folding structure as in [Durand-Lose, 2006, 2009. The output is an infinite convoy of signals, bringing forth singularities.…”