On the Power of Threshold Measurements as Oracles

  • Edwin Beggs
  • José Félix Costa
  • Diogo Poças
  • John V. Tucker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7956)

Abstract

We consider the measurement of physical quantities that are thresholds. We use hybrid computing systems modelled by Turing machines having as an oracle physical equipment that measures thresholds. The Turing machines compute with the help of qualitative information provided by the oracle. The queries are governed by timing protocols and provide the equipment with numerical data with (a) infinite precision, (b) unbounded precision, or (c) finite precision. We classify the computational power in polynomial time of a canonical example of a threshold oracle using non-uniform complexity classes.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balcázar, J.L., Días, J., Gabarró, J.: Structural Complexity I, 2nd edn. Springer (1988, 1995)Google Scholar
  2. 2.
    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)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beggs, E., Costa, J.F., Loff, B., Tucker, J.V.: On the complexity of measurement in classical physics. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 20–30. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    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)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Beggs, E., Costa, J.F., Tucker, J.V.: Computational Models of Measurement and Hempel’s Axiomatization. In: Carsetti, A. (ed.) Causality, Meaningful Complexity and Knowledge Construction. Theory and Decision Library A, vol. 46, pp. 155–184. Springer (2010)Google Scholar
  6. 6.
    Beggs, E., Costa, J.F., Tucker, J.V.: Limits to measurement in experiments governed by algorithms. Mathematical Structures in Computer Science 20(06), 1019–1050 (2010) Special issue on Quantum Algorithms, Venegas-Andraca, S.E. (ed.)Google Scholar
  7. 7.
    Beggs, E., Costa, J.F., Tucker, J.V.: Axiomatising physical experiments as oracles to algorithms. Philosophical Transactions of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences) 370(12), 3359–3384 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Beggs, E., Costa, J.F., Tucker, J.V.: The impact of models of a physical oracle on computational power. Mathematical Structures in Computer Science 22(5), 853–879 (2012), Special issue on Computability of the Physical, Calude, C.S., Barry Cooper, S. (eds.)Google Scholar
  9. 9.
    Hempel, C.G.: Fundamentals of concept formation in empirical science. International Encyclopedia of Unified Science 2(7) (1952)Google Scholar
  10. 10.
    Jain, S., Osherson, D.N., Royer, J.S., Sharma, A.: Systems That Learn. An Introduction to Learning Theory, 2nd edn. The MIT Press (1999)Google Scholar
  11. 11.
    Krantz, D.H., Suppes, P., Duncan Luce, R., Tversky, A.: Foundations of Measurement. Dover (2009)Google Scholar
  12. 12.
    Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing Limit. Birkhäuser (1999)Google Scholar
  13. 13.
    von Neumann, J.: Probabilistic logics and the synthesis of reliable organisms from unreliable components. In: Automata Studies, pp. 43–98. Princeton University Press (1956)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Edwin Beggs
    • 1
  • José Félix Costa
    • 2
    • 3
  • Diogo Poças
    • 2
    • 3
  • John V. Tucker
    • 1
  1. 1.College of ScienceSwansea UniversitySwanseaUnited Kingdom
  2. 2.Department of Mathematics, Instituto Superior TécnicoUTLLisbonPortugal
  3. 3.Centro de Matemática e Aplicações FundamentaisUniversity of LisbonPortugal

Personalised recommendations