The Myth of Hypercomputation

  • Martin Davis


Under the banner of “hypercomputation” various claims are being made for the feasibility of modes of computation that go beyond what is permitted by Turing computability. In this article it will be shown that such claims fly in the face of the inability of all currently accepted physical theories to deal with infinite-precision real numbers. When the claims are viewed critically, it is seen that they amount to little more than the obvious comment that if non-computable inputs are permitted, then non-computable outputs are attainable.


Physical Theory Turing Machine Universal Machine Computable Language Turing Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baker, Theodore P., John Gill, and Robert Solovay (1975). Relativizatons of the P =? NP Question. SIAM J. Comput. 4, 431–442.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Church, Alonzo (1937). Review of [20]. J. Symbolic Logic. 2, 42–43.CrossRefGoogle Scholar
  3. 3.
    Copeland, B. Jack (1998). Turing’s O-Machines, Penrose, Searle, and the Brain. Analysis. 58, 128–38.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Copeland, B. Jack (2000). Narrow versus Wide Mechanism: Including a Reexamination of Turing’s Views on the Mind-Machine Issue. J. of Phil. 96, 5–32.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Copeland, B. Jack and Diane Proudfoot (1999). Alan Turing’s Forgotten Ideas in Computer Science. Scientific American, New York. 253:4, 98–103.Google Scholar
  6. 6.
    Davis, Martin (1958). Computability and Unsolvability. McGraw-Hill; reprinted with an additional appendix, Dover 1983.zbMATHGoogle Scholar
  7. 7.
    Davis, Martin (1982). Why Gödel Didn’t Have Church’s Thesis. Information and Control. 54, 3–24.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Davis, Martin, ed. (1965). The Undecidable. Raven Press, New York.Google Scholar
  9. 9.
    Davis, Martin, (1987). Mathematical Logic and the Origin of Modern Computers. Studies in the History of Mathematics, pp. 137–165. Mathematical Association of America. Reprinted in The Universal Turing Machine — A Half-Century Survey, Rolf Herken, editor, pp. 149–174. Verlag Kemmerer & Unverzagt, Hamburg, Berlin 1988; Oxford University Press, Oxford, 1988.Google Scholar
  10. 10.
    Davis, Martin (2000). The Universal Computer: The Road from Leibniz to Turing. W. W. Norton, New York.Google Scholar
  11. 11.
    Davis, Martin (2001). Engines of Logic: Mathematicians and the Origin of the Computer. W. W. Norton, New York (paperback edition of [10]).Google Scholar
  12. 12.
    De Leeuw, K., E. F. Moore, C. E. Shannon, and N. Shapiro (1956). Computability by Probabilistic Machines. Automata Studies, Shannon, C. and J. McCarthy, eds., Princeton University Press, Princeton, 183–212.Google Scholar
  13. 13.
    Deutsch, David (1997). The Fabric of Reality. Allen Lane, The Penguin Press, New York.Google Scholar
  14. 14.
    Gandy, Robin (1980). Church’s Thesis and Principles for Mechanisms. In: The Kleene Symposium. Jon Barwise, ed. North-Holland, Amsterdam.Google Scholar
  15. 15.
    Hong, J. W. (1988). On Connectionist Models. Comm. Pure and Applied Math. 41, 1039–1050.zbMATHCrossRefGoogle Scholar
  16. 16.
    Park, Robert (2001). Voodoo Science. Oxford University Press, Oxford.Google Scholar
  17. 17.
    Siegelmann, Hava T. (1995). Computation Beyond the Turing Limit. Science 268, 545–548.CrossRefGoogle Scholar
  18. 18.
    Siegelmann, Hava T. (1999). Neural Networks and Analog Computation: Beyond the Turing Limit. Birkhäuser, Boston.zbMATHCrossRefGoogle Scholar
  19. 19.
    Smith, David Eugene (1929). A Source Book in Mathematics. McGraw-Hill, New York.zbMATHGoogle Scholar
  20. 20.
    Turing, A. M. (1937). On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc. 42, 230–265. Correction: Ibid. 43, 544–546. Reprinted in [8, pp. 155–222], [24, pp. 18–56].MathSciNetCrossRefGoogle Scholar
  21. 21.
    Turing, A. M. (1939). Systems of Logic Based on Ordinals. Proc. London Math. Soc. 45, 161–228. Reprinted in [8, pp. 116–154] and [24, pp. 81–148].MathSciNetCrossRefGoogle Scholar
  22. 22.
    Turing, A. M. (1947). Lecture to the London Mathematical Society on 20 February 1947. In: A. M. Turing’s ACE Report of 1946 and Other Papers. B. E. Carpenter and R.N. Doran, eds. MIT Press 106–124. Reprinted in [23, pp. 87–105].Google Scholar
  23. 23.
    Turing, A. M. (1992). Collected Works: Mechanical Intelligence. D.C. Ince, ed. North-Holland, Amsterdam.zbMATHGoogle Scholar
  24. 24.
    Turing, A. M. (2001). Collected Works: Mathematical Logic. R. O. Gandy and C. E. M. Yates, eds. North-Holland, Amsterdam.Google Scholar
  25. 25.
    Webb, Judson C. (1980). Mechanism, Mentalism, and Metamathematics. D. Reidel, Dordrecht.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Martin Davis
    • 1
    • 2
  1. 1.Courant InstituteNY UniversityUSA
  2. 2.Mathematics DepartmentUniversity of CaliforniaBerkeleyUSA

Personalised recommendations