Advertisement

Approximate Outputs of Accelerated Turing Machines Closest to Their Halting Point

  • Sebastien Mambou
  • Ondrej KrejcarEmail author
  • Ali Selamat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11431)

Abstract

The accelerated Turing machine (ATM) which can compute super-tasks are devices with the same computational structure as Turing machines (TM) and they are also defined as the work-horse of hypercomputation. Is the final output of the ATM can be produced at the halting state? We supported our analysis by reasoning on Thomson’s paradox and by looking closely the result of the Twin Prime conjecture. We make sure to avoid unnecessary discussion on the infinite amount of space used by the machine or considering Thomson’s lamp machine, on the difficulty of specifying a machine’s outcome. Furthermore, it’s important for us that a clear definition counterpart for ATMs of the non-halting/halting dichotomy for classical Turing must be introduced. Considering a machine which has run for a countably infinite number of steps, this paper addresses the issue of defining the output of a machine close or at the halting point.

Keywords

Thomson’s paradox Super-task Halting problem Ultrafilter accepting computations Accelerating turing machines Non-standard output concepts 

Notes

Acknowledgement

The work and the contribution were supported by the SPEV project “Smart Solutions in Ubiquitous Computing Environments”, 2019, University of Hradec Kralove, Faculty of Informatics and Management, Czech Republic.

References

  1. 1.
    Copeland, B.: Accelerating turing machines. Mind. Mach. 12(2), 281–300 (2002)zbMATHGoogle Scholar
  2. 2.
    Hamkins, J.: Infinite time turing machines. Mind. Mach. 12(4), 521–539 (2002)zbMATHGoogle Scholar
  3. 3.
    Comfort, W.: Ultrafilters: some old and some new results. Bull. Am. Math. Logic Q. 83, 417–456 (1977)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hamkins, J.D., Seabold, D.E.: Infinite time turing machines with only one tape. Math. Log. Q. 47, 271–287 (2001).  https://doi.org/10.1002/1521-3870(200105)47:2<271::AID-MALQ271>3.0.CO;2-6MathSciNetzbMATHGoogle Scholar
  5. 5.
    Welch, P.: Eventually infinite time turing machine degrees: infinite time decidable reals. J. Symbolic Logic 65(03), 1193–1203 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Burgin, M.: Algorithmic complexity as a criterion of unsolvability. Theor. Comput. Sci. 383(2), 244–259 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Baer, R., Leeuwen, J.: The halting problem for linear turing assemblers. J. Comput. Syst. Sci. 13(2), 119–135 (1976)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Copeland, B., Shagrir, O.: Do accelerating turing machines compute the uncomputable. Mind. Mach. 21(2), 221–239 (2011)Google Scholar
  9. 9.
    Potgieter, P.: Zeno machines and hypercomputation. Theor. Comput. Sci. 358(1), 23–33 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ord, T.: The many forms of hypercomputation. Appl. Math. Comput. 178(1), 143–153 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hogarth, M.: Does general relativity allow an observer to view an eternity in a finite time? Found. Phys. Lett. 5, 173–181 (1992)MathSciNetGoogle Scholar
  12. 12.
    Shagrir, O.: Super-tasks, accelerating turing machines and uncomputability. Theor. Comput. Sci. 317(1), 105–114 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Pitowsky, I.: The physical church thesis and physical computational complexity. Iyyun: Jerusalem Philos. Q. 39, 81–99 (1990)Google Scholar
  14. 14.
    Turing, A.: On Computable Numbers, with an Application to. http://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
  15. 15.
    Turing, A.: On computable numbers, with an application to the “Entscheidungsproblem”. Proc. Lond. Math. Soc. 42, 230–265 (1937)MathSciNetzbMATHGoogle Scholar
  16. 16.
    How Light Bulbs Work. In: HowStuffWorks. https://home.howstuffworks.com/light-bulb.htm
  17. 17.
  18. 18.
    Graves, A., Wayne, G., Danihelka, I.: Neural turing machines. arXiv: Neural and Evolutionary Computing (2014)Google Scholar
  19. 19.
    Mambou, S., Krejcar, O., Kuca, K., Selamat, A.: Novel cross-view human action model recognition based on the powerful view-invariant features technique. Future Internet 10(9), 89 (2018)Google Scholar
  20. 20.
    Mambou, S., Maresova, P., Krejcar, O., Selamat, A., Kuca, K.: Breast cancer detection using infrared thermal imaging and a deep learning model. Sensors 18, 2799 (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sebastien Mambou
    • 1
  • Ondrej Krejcar
    • 1
    Email author
  • Ali Selamat
    • 1
    • 2
  1. 1.Center for Basic and Applied Research, Faculty of Informatics and ManagementUniversity of Hradec KraloveHradec KraloveCzech Republic
  2. 2.Malaysia Japan International Institute of Technology (MJIIT), Universiti Teknologi MalaysiaKuala LumpurMalaysia

Personalised recommendations