Advances in Unconventional Computing pp 73-115

Part of the Emergence, Complexity and Computation book series (ECC, volume 22) | Cite as

An Analogue-Digital Model of Computation: Turing Machines with Physical Oracles

  • Tânia Ambaram
  • Edwin Beggs
  • José Félix Costa
  • Diogo Poças
  • John V. Tucker
Chapter

Abstract

We introduce an abstract analogue-digital model of computation that couples Turing machines to oracles that are physical processes. Since any oracle has the potential to boost the computational power of a Turing machine, the effect on the power of the Turing machine of adding a physical process raises interesting questions. Do physical processes add significantly to the power of Turing machines; can they break the Turing Barrier? Does the power of the Turing machine vary with different physical processes? Specifically, here, we take a physical oracle to be a physical experiment, controlled by the Turing machine, that measures some physical quantity. There are three protocols of communication between the Turing machine and the oracle that simulate the types of error propagation common to analogue-digital devices, namely: infinite precision, unbounded precision, and fixed precision. These three types of precision introduce three variants of the physical oracle model. On fixing one archetypal experiment, we show how to classify the computational power of the three models by establishing the lower and upper bounds. Using new techniques and ideas about timing, we give a complete classification.

References

  1. 1.
    Siegelmann, H.T., Sontag, E.D.: Analog computation via neural networks. Theor. Comput. Sci. 131(2), 331–360 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Woods, D., Naughton, T.J.: An optical model of computation. Theor. Comput. Sci. 334(1–3), 227–258 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bournez, O., Cosnard, M.: On the computational power of dynamical systems and hybrid systems. Theor. Comput. Sci. 168(2), 417–459 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Carnap, R.: Philosophical Foundations of Physics. Basic Books, New York (1966)Google Scholar
  5. 5.
    Beggs, E., Costa, J., Tucker, J.V.: Computational models of measurement and Hempel’s axiomatization. In: Carsetti, A. (ed.), Causality, Meaningful Complexity and Knowledge Construction, vol. 46. Theory and Decision Library A, pp. 155–184. Springer, Berlin (2010)Google Scholar
  6. 6.
    Geroch, R., Hartle, J.B.: Computability and physical theories. Found. Phys. 16(6), 533–550 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Beggs, E., Costa, J., Tucker, J.V.: Three forms of physical measurement and their computability. Rev. Symb. Log. 7(4), 618–646 (2014)Google Scholar
  8. 8.
    Hempel, C.G.: Fundamentals of concept formation in empirical science. Int. Encycl. Unified Sci. II, 7 (1952)Google Scholar
  9. 9.
    Krantz, D.H., Suppes, P., Luce, R.D., Tversky, A.: Foundations of Measurement. Dover, New York (2009)Google Scholar
  10. 10.
    Beggs, E., Costa, J.F., Poças, D., Tucker, J.V.: Oracles that measure thresholds: the Turing machine and the broken balance. J. Log. Comput. 23(6), 1155–1181 (2013)Google Scholar
  11. 11.
    Beggs, E., Costa, J.F., Poças, D., Tucker, J.V.: Computations with oracles that measure vanishing quantities. Math. Struct. Comput. Sci., p. 49 (in print)Google Scholar
  12. 12.
    Beggs, E.J., Costa, J.F., Tucker, J.V.: Limits to measurement in experiments governed by algorithms. Math. Struct. Comput. Sci. 20(06), 1019–1050 (2010)Google Scholar
  13. 13.
    Beggs, E., Costa, J.F., Tucker, J.V.: The impact of models of a physical oracle on computational power. Math. Struct. Comput. Sci. 22(5), 853–879 (2012)Google Scholar
  14. 14.
    Beggs, E., Costa, J.F., Loff, B., Tucker, J.V.: Computational complexity with experiments as oracles. Proceedings of the Royal Society, Series A (Mathematical,Physical and Engineering Sciences), 464(2098), 2777–2801 (2008)Google Scholar
  15. 15.
    Beggs, E., Costa, J.F., Loff, B., Tucker, J.V.: Computational complexity with experiments as oracles. II. Upper bounds. Proceedings of the Royal Society, Series A (Mathematical,Physical and Engineering Sciences) 465(2105), 1453–1465 (2009)Google Scholar
  16. 16.
    Beggs, E.J., Costa, J.F., Tucker, J.V.: Axiomatizing physical experiments as oracles to algorithms. Philosophical Transactions of the Royal Society, Series A(Mathematical, Physical and Engineering Sciences) 370(12), 3359–3384 (2012)Google Scholar
  17. 17.
    Balcázar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I, vol. 11. Theoretical Computer Science. Springer, Berlin (1990)Google Scholar
  18. 18.
    Balcázar, José L., Hermo, Montserrat: The structure of logarithmic advice complexity classes. Theor. Comput. Sci. 207(1), 217–244 (1998)Google Scholar
  19. 19.
    Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing limit. Birkhäuser, Boston (1999)Google Scholar
  20. 20.
    Beggs, E., Costa, J.F., Poças, D., Tucker, J.V.: An analogue-digital Church-Turing thesis. Int. J. Found. Comput. Sci. 25(4), 373–389 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Tânia Ambaram
    • 1
  • Edwin Beggs
    • 2
  • José Félix Costa
    • 1
  • Diogo Poças
    • 3
  • John V. Tucker
    • 2
  1. 1.Department of MathematicsInstituto Superior Técnico, Universidade de LisboaLisboaPortugal
  2. 2.College of ScienceSwansea UniversityWalesUK
  3. 3.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations