Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

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

  • Published:
Natural Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. Throughout the article location is to be understood as a position in space and time and position as only spatial.

  2. This is why we do not talk about any analytic hierarchy.

  3. For example think about the so-called blue-shift effect and how it might be countered (Németi and Dávid 2006, Sub. 5.3.1).

References

  • Blum L, Shub M, Smale S (1989) On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull Am Math Soc 21(1): 1–46

    Article  MATH  MathSciNet  Google Scholar 

  • Blum L, Cucker F, Shub M, Smale S (1998) Complexity and real computation. Springer, New York

    Google Scholar 

  • Bournez O (1999a) Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theor Comp Sci 210(1): 21–71

    Article  MATH  MathSciNet  Google Scholar 

  • Bournez O (1999b) Some bounds on the computational power of piecewise constant derivative systems. Theory Comput Syst 32(1):35–67

    Article  MATH  MathSciNet  Google Scholar 

  • Bournez O, Emmanuel H (2007) On the computational capabilities of several models. In: Durand-Lose J, Margenstern M (eds) Machines, Computations, and Universality, MCU ’07, volume 4664 of LNCS. Springer, pp 12–23

  • Chadzelek T, Hotz G (1999) Analytic machines. Theor Comp Sci 219(1–2):151–167

    Article  MATH  MathSciNet  Google Scholar 

  • Copeland JB (2002) Hypercomputation. Minds Mach 12(4):461–502

    Article  MATH  Google Scholar 

  • Durand-Lose J (2006a) Abstract geometrical computation 1: embedding black hole computations with rational numbers. Fundam Inform 74(4):491–510

    MATH  MathSciNet  Google Scholar 

  • Durand-Lose J (2006b) Forcasting black holes in abstract geometrical computation is highly unpredictable. In: Cai J-Y, Cooper SB, Li A (eds) Theory and applications of models of computations (TAMC ’06), number 3959 in LNCS. Springer, pp 644–653

  • Durand-Lose J (2007) Abstract geometrical computation and the linear Blum, Shub and Smale model. In: Cooper SB, Löwe B, Sorbi A (eds) Computation and logic in the real world, 3rd Conference on Computability in Europe (CiE ’07), number 4497 in LNCS. Springer, pp 238–247

  • Durand-Lose J (2008a) Abstract geometrical computation with accumulations: beyond the Blum, Shub and Smale model. In: Beckmann A, Dimitracopoulos C, Löwe B (eds) Logic and theory of algorithms, CiE 2008 (abstracts and extended abstracts of unpublished papers). University of Athens, pp 107–116

  • Durand-Lose J (2008b) The signal point of view: from cellular automata to signal machines. In: Durand B (ed) Journées automates cellulaires (JAC ’08), pp 238–249. URL http://www.lif.univ-mrs.fr/jac/

  • Etesi G, Németi I (2002) Non-Turing computations via Malament-Hogarth space-times. Int J Theor Phys 41(2):341–370 gr-qc/0104023

    Google Scholar 

  • Hamkins JD (2002) Infinite time Turing machines: supertask computation. Minds Mach, 12(4):521–539 arXiv:math.LO/0212047

    Google Scholar 

  • Hamkins JD (2007) A survey of infinite time Turing machines. In: Durand-Lose J, Margenstern M (eds) Machines, Computations and Universality (MCU ’07), number 4664 in LNCS. Springer, pp 62–71

  • Hamkins JD, Lewis A (2000) Infinite time Turing machines. J Symb Log 65(2):567–604. arXiv:math.LO/9808093

    Google Scholar 

  • Hogarth ML (1994) Non-Turing computers and non-Turing computability. In: Biennial Meeting of the Philosophy of Science Association, pp 126–138

  • Hogarth ML (2004) Deciding arithmetic using SAD computers. Br J Philos Sci 55:681–691

    Article  MATH  MathSciNet  Google Scholar 

  • Huckenbeck U (1989) Euclidian geometry in terms of automata theory. Theor Comp Sci 68(1):71–87

    Article  MATH  MathSciNet  Google Scholar 

  • Huckenbeck U (1991) A result about the power of geometric oracle machines. Theor Comp Sci 88(2):231–251

    Article  MATH  MathSciNet  Google Scholar 

  • Jacopini G, Sontacchi G (1990) Reversible parallel computation: an evolving space-model. Theor Comp Sci 73(1):1–46

    Article  MATH  MathSciNet  Google Scholar 

  • Kawamura A (2005) Type-2 computability and Moore’s recursive functions. Electr Notes Theor Comput Sci 120:83–95

    Article  MathSciNet  Google Scholar 

  • Mycka J (2003a) Infinite limits and \({\mathbb{R}}\) -recursive functions. Acta Cybern 16(1):83–91

    MATH  MathSciNet  Google Scholar 

  • Mycka J (2003b) μ-recursion and infinite limits. Theor Comp Sci 302(1–3):123–133

    Article  MATH  MathSciNet  Google Scholar 

  • Mycka J (2006) Analog computation beyond the Turing limit. Appl Math Comput 178(1):103–117

    Article  MATH  MathSciNet  Google Scholar 

  • Mycka J, Costa JF (2007) A new conceptual framework for analog computation. Theor Comp Sci 374(1–3):277–290

    Article  MATH  MathSciNet  Google Scholar 

  • Naughton TJ, Woods D (2001) On the computational power of a continuous-space optical model of computation. In: Margenstern M (ed) Machines, Computations, and Universality (MCU ’01), volume 2055 of LNCS, pp 288–299

  • Németi I, Andréka H (2006) Can general relativistic computers break the Turing barrier? In: Beckmann A, Berger U, Löwe B, Tucker JV (eds) Logical approaches to computational barriers, 2nd Conference on Computability in Europe, CiE ’06, volume 3988 of LNCS. Springer, pp 398–412

  • Németi I, Dávid G (2006) Relativistic computers and the Turing barrier. Appl Math Comput 178(1):118–142

    Article  MATH  MathSciNet  Google Scholar 

  • Weihrauch K (2000) Introduction to computable analysis. Texts in Theoretical computer science. Springer, Berlin

    Google Scholar 

  • Welch PD (2008) The extentent of computation in malament-hogarth spacetime. Br J Philos Sci 59(4):659–674. doi:10.1093/bjps/axn031

    Article  MATH  Google Scholar 

  • Ziegler M (2007) (Short) Survey of real hypercomputation. In: Barry Cooper S, Löwe B, Sorbi A (eds) Computation and logic in the real world, 3rd Conference on Computability in Europe, CiE ’07. volume 4497 of LNCS. Springer, pp 809–824

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jérôme Durand-Lose.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Durand-Lose, J. Abstract geometrical computation 3: black holes for classical and analog computing. Nat Comput 8, 455–472 (2009). https://doi.org/10.1007/s11047-009-9117-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11047-009-9117-0

Keywords

Navigation