Hypercomputational Models

  • Mike Stannett
Chapter

Summary

Hypercomputers are physical or conceptual machines capable of performing non-recursive tasks; their behavior lies beyond the so-called “Turing limit.” Recent decades have seen many hypercomputational models in the literature, but in many cases we know neither how these models are related to one another, nor the precise reasons why they are so much more powerful than Turing machines. In this chapter we start by considering Turing’s machine-based model of computation, and identify various structural constraints. By loosening each of these constraints in turn, we identify various classes of hypercomputational device, thereby generating a basic taxonomy for hypercomputation itself.

Keywords

Manifold Coherence Posit Lution Prep 

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References

  1. 1.
    F. G. Abramson. Effective computation over the real numbers. In Twelfth Annual Symposium on Switching and Automata Theory. Institute of Electrical and Electronics Engineers, Northridge, CA, 1971.Google Scholar
  2. 2.
    G.S. Boolos and R. C Jeffrey. Computability and Logic. Cambridge University Press, Cambridge, 1974.MATHGoogle Scholar
  3. 3.
    C. S. Calude. Incompleteness, complexity, randomness and beyond. Minds and Machines, 12:503–517, 2002.MATHCrossRefGoogle Scholar
  4. 4.
    B. J. Copeland. Super Turing-machines. Complexity, 4(l):30–32, 1998.MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. J. Copeland. Accelerating Turing machines. Minds and Machines, 12:281–301, 2002.MATHCrossRefGoogle Scholar
  6. 6.
    B. J. Copeland. Hypercomputation. Minds and Machines, 12:461–502, 2002.MATHCrossRefGoogle Scholar
  7. 7.
    B. J. Copeland and D. Proudfoot. Alan Turing’s forgotten ideas in computer science. Scientifíc American, 280(4) :77–81, April 1999.Google Scholar
  8. 8.
    D. Deutsch. Quantum theory, the Church-Turing principle of the Universal Quantum Computer. Proceedings of the Royal Society of London, A400:97–117, 1985.MathSciNetGoogle Scholar
  9. 9.
    J. Doyle. What is Church’s thesis? An outline. Technical report, Laboratory for Computer Science, MIT, 1982.Google Scholar
  10. 10.
    J. Doyle. What is Church’s thesis? An outline. Minds and Machines, 12:519–520, 2002.MATHCrossRefGoogle Scholar
  11. 11.
    J. Earman. Bangs, Crunches, Whimpers, and ShrieksSingularities and Acausalities in Relativistic Spacetimes. Oxford University Press, Oxford, UK, 1995.Google Scholar
  12. 12.
    J. Earman and J. D. Norton. Forever is a day: Supertasks in Pitowsky and Malament-Hogarth spacetimes. Philosophy of Science, 60:22–42, 1993.MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Earman and J. D. Norton. Infinite pains: The trouble with supertasks. In A. Morton and S. P. Stich, editors, Benacerraf and his Critics. Blackwell, Oxford, UK, 1996.Google Scholar
  14. 14.
    A. P. French and E. F. Taylor. An Introduction to Quantum Physics. Chapman and Hall, London, UK, 1990.Google Scholar
  15. 15.
    R. Geroch and J. B. Hartle. Computability and physical theories. Foundations of Physics, 16:533–550, 1986.MathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Gödel. Über formal unendscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik, 38:173–198, 1931.Google Scholar
  17. 17.
    E. M. Gold. Limiting recursion. Journal of Symbolic Logic, 30:28–48, 1965.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    M. L. Hogarth. Does general relativity allow an observer to view an eternity in a finite time? Foundations of Physics Letters, 5:173–181, 1992.MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. L. Hogarth. Non-Turing computers and non-Turing computability. PAS, 1:126–138, 1994. Available online: http://hypercomputation.net/resources.html.Google Scholar
  20. 20.
    M. L. Hogarth. Predictability, Computability and Spacetime. PhD thesis, Cambridge University, UK, 1996. Available online:Google Scholar
  21. 21.
    M. L. Hogarth. Deciding arithmetic in Malament-Hogarth spacetimes. Available online: http://hypercomputation.net/resources.html, 2002.Google Scholar
  22. 22.
    G. Kampis. Self-Modifying Systems in Biology and Cognitive Science: A New-Framework for Dynamics, Information and Complexity. Pergamon, Oxford, UK, 1991.Google Scholar
  23. 23.
    G. Kampis. Computability, self-reference, and self-amendment. Communications and Cognition-Artificial Intelligence, 12:91–110, 1995.Google Scholar
  24. 24.
    R. M. Karp and R. J. Lipton. Turing machines that take advice. In E. Engeler et al., editor, Logic and Algorithmic. L’Enseignement Mathématique, Genève, Switzerland, 1982.Google Scholar
  25. 25.
    A. Komar. Undecidability of macroscopically distinguishable states in quantum field theory. Physical Review, 133B:542–544, 1964.MathSciNetCrossRefGoogle Scholar
  26. 26.
    G. Kreisel. Mathematical logic. In T. L. Saaty, editor, Lectures on Modern Mathematics, volume 3. John Wiley, New York, 1965.Google Scholar
  27. 27.
    G. Kreisel. Mathematical logic: What has it done for the philosophy of mathematics? In R. Schoenman, editor, Bertrand Russell: Philosopher of the Century. George Allen and Unwin, London, UK, 1967.Google Scholar
  28. 28.
    G. Kreisel. Hilbert’s programme and the search for automatic proof procedures. In M. Laudet et al., editor, Symposium on Automatic Demonstration, volume 125 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1970.Google Scholar
  29. 29.
    G. Kreisel. Some reasons for generalising recursion theory. In R. O. Gandy and C. M. E. Yates, editors, Logic Colloquium ’69. North-Holland, Amsterdam, 1971.Google Scholar
  30. 30.
    G. Kreisel. Which number theoretic problems can be solved in recursive progressions on math-paths through 0? Journal of Symbolic Logic, 37:311–334, 1972.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    G. Kreisel. A notion of mechanistic theory. Synthese, 29:11–26, 1974.MATHCrossRefGoogle Scholar
  32. 32.
    G. Kreisel. Review of Pour-El and Richards. Journal of Symbolic Logic, 47:900–902, 1982.CrossRefGoogle Scholar
  33. 33.
    G. Kreisel. Church’s thesis and the ideal of formal rigour. Notre Dame Journal of Formal Logic, 28:499–519, 1987.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    S. M. Krylov. Formal technology and universal systems (part 1 and 2). Cybernetics, 4 and 5:85–89 and 28–31, 1986.Google Scholar
  35. 35.
    P. Kugel. Thinking may be more than computing. Cognition, 22:137–198, 1986.CrossRefGoogle Scholar
  36. 36.
    B. J. MacLennan. Technology-independent design of neurocomputers: The universal field computer. In M. Caudill and C. Butler, editors, Proceedings of the IEEE First International Conference on Neural Networks, San Diego, CA, volume 3, pages 39–49. IEEE Press, 1987.Google Scholar
  37. 37.
    B. J. MacLennan. Logic for the new AI. In J. H. Fetzer, editor, Aspects of Artificial Intelligence, pages 163–192. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.CrossRefGoogle Scholar
  38. 38.
    B. J. MacLennan. Field computation: A theoretical framework for massively parallel analog computation, parts I-IV. Technical Report CS-90–100, Dept of Computer Science, University of Tennessee, 1990. Available online: http://www.cs.utk.edu/~mclennan.Google Scholar
  39. 39.
    B. J. MacLennan. Characteristics of connectionist knowledge representation. Information Sciences, 70:119–143, 1993.CrossRefGoogle Scholar
  40. 40.
    B. J. MacLennan. Field computation in the brain. In K. H. Pribram, editor, Rethinking Neural Networks: Quantum Fields and Biological Data, pages 199–232. Lawrence Erlbaum, Hillsdale, NJ, 1993.Google Scholar
  41. 41.
    B. J. MacLennan. Grounding analog computers. Think, 2:48–51, 1993.Google Scholar
  42. 42.
    B. J MacLennan. Continuous computation and the emergence of the discrete. In K. H. Pribram, editor, Origins: Brain &: Self-Organisation, pages 121–151. Lawrence Erlbaum, Hillsdale, NJ, 1994.Google Scholar
  43. 43.
    B. J. MacLennan. Continuous symbol systems: The logic of connectionism. In M. Aparicio D. S. Levine, editor, Neural Networks for Knowledge Representation and Inference, pages 121–151. Lawrence Erlbaum, Hillsdale, NJ, 1994.Google Scholar
  44. 44.
    B. J. MacLennan. Continuous formal systems: A unifying model in language and cognition. In Proceedings of the IEEE Workshop on Architectures for Semiotic Modeling and Situation Analysis in Large Complex Systems, Monterey, CA, pages 161–172. IEEE Press, 1995.Google Scholar
  45. 45.
    B. J. MacLennan. Field computation in motor control. In: P. G. Morasso and V. Sanguineti, editors, Self-Organization, Computational Maps and Motor Control, pages 37–73. Elsevier, Amsterdam, The Netherlands, 1997.CrossRefGoogle Scholar
  46. 46.
    B. J. MacLennan. Field computation in natural and artificial intelligence. Information Sciences, 119:73–89, 1999.MathSciNetCrossRefGoogle Scholar
  47. 47.
    B. J. MacLennan. Can differential equations compute? Technical Report UT-CS-01–459, Dept of Computer Science, University of Tennessee, Knoxville, USA, 2001. Available online: http://www.cs.utk.edu/~mclennan.Google Scholar
  48. 48.
    B. J. MacLennan. Transcending Turing computability. Minds and Machines, 13(l):3–22, 2003.MATHCrossRefGoogle Scholar
  49. 49.
    J. Myhill. A recursive function, defined on a compact interval and having a continuous derivative that is not recursive. Michigan Math. J., 18:97–98, 1971.MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    M. H. A. Newman. Alan Mathison Turing 1912–1954. Biographical Memoirs of the Fellows of the Royal Society, 1:253–263, November 1955.CrossRefGoogle Scholar
  51. 51.
    R. Penrose. The Emperor’s New Mind. Oxford University Press, Oxford, UK, 1989.Google Scholar
  52. 52.
    R. Penrose. Précis of the emperor’s new mind: Concerning computers, minds, and the laws of physics. Behavioural and Brain Sciences, 13:643–655 and 692–705, 1990.CrossRefGoogle Scholar
  53. 53.
    R. Penrose. Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press, Oxford, UK, 1994.Google Scholar
  54. 54.
    I. Pitowsky. The physical Church thesis and physical computational complexity. Iyuun, 39:81–99, 1990.Google Scholar
  55. 55.
    M. B. Pour-El. Abstract computability versus analog-comput ability (a survey). In Cambridge Summer School in Mathematical Logic, volume 337 of Springer Lecture Notes in Mathematics, pages 345–360. Springer-Verlag, Berlin, 1971.Google Scholar
  56. 56.
    M. B. Pour-El. Abstract computability and its relation to the general purpose analog computer. Trans. American Math. Soc, 199:1–28, 1974.MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    M. B. Pour-El and J. I. Richards. A computable ordinary differential equation which possesses no computable solution. Annals of Mathematical Logic, 17:61–90, 1979.MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    M. B. Pour-El and J. I. Richards. The wave equation with computable initial data such that its unique solution is not computable. Advances in Mathematics, 39:215–239, 1981.MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    M. B. Pour-El and J. I. Richards. Computability in Analysis and Physics. Springer-Verlag, Berlin, 1989.MATHGoogle Scholar
  60. 60.
    H. Putnam. Trial and error predicates and the solution of a problem of Mostowski. Journal of Symbolic Logic, 30:49–57, 1965.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    H. Putnam. Renewing Philosophy. Harvard University Press, Cambridge, MA, 1992.Google Scholar
  62. 62.
    L. A. Rubel. The brain as an analog computer. Journal of Theoretical Neurobiology, 4:73–81, 1985.Google Scholar
  63. 63.
    L. A. Rubel. Some mathematical limitations of the general-purpose analog computer. Advances in Applied Mathematics, 9:22–34, 1988.MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    L. A. Rubel. Digital simulation of analog computation and Church’s thesis. Journal of Symbolic Logic, 54:1011–1017, 1989.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    K. Sabbagh. Dr Riemann’s Zeros. Atlantic Books, London, UK, 2002.Google Scholar
  66. 66.
    B. Scarpellini. Zwei Unentscheitbare Probleme der Analysis. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 9:265–289, 1963.MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    B. Scarpellini. Comments on two undecidable problems of analysis. Minds and Machines, 13(l):79–85, 2003.MATHCrossRefGoogle Scholar
  68. 68.
    B. Scarpellini. Two undecidable problems of analysis. Minds and Machines, 13(l):49–77, 2003.MATHCrossRefGoogle Scholar
  69. 69.
    O. Shagrir and I. Pitowsky. Physical hypercomputation and the Church-Turing thesis. Minds and Machines, 13(1):87–101, 2003.MATHCrossRefGoogle Scholar
  70. 70.
    H. T. Siegelmann. Computation beyond the Turing limit. Science, 268(5210):545–548, April 1995.CrossRefGoogle Scholar
  71. 71.
    H. T. Siegelmann. Analog computational power. Science, 271(19) :373, January 1996.CrossRefGoogle Scholar
  72. 72.
    H. T. Siegelmann. The simple dynamics of super Turing theories. Theoretical Computer Science, 168:461–472, 1996.MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    H. T. Siegelmann. Neural Networks and Analog Computation: Beyond the Turing Limit. Progress in Theoretical Computer Science. Birkhauser Verlag, November 1998.Google Scholar
  74. 74.
    H. T. Siegelmann. Stochastic analog networks and computational complexity. Journal of Complexity, 15:451–475, 1999.MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    H. T. Siegelmann. Neural and super-Turing computing. Minds and Machines, 13(1):103–114, 2003.MATHCrossRefGoogle Scholar
  76. 76.
    H. T. Siegelmann and E. D. Sontag. On the computational power of neural nets. In Proceedings of the 5 th Annual ACM Workshop on Computational Learning Theory, Pittsburgh, pages 440–449, 1992.Google Scholar
  77. 77.
    H. T. Siegelmann and E. D. Sontag. Analog computation via neural networks. Theoretical Computer Science, 131:331–360, 1994.MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    M. Stannett. X-machines and the halting problem: Building a super-Turing machine. Formal Aspects of Computing, 2(4):331–341, 1990.MATHCrossRefGoogle Scholar
  79. 79.
    M. Stannett. An introduction to post-Newtonian and non-Turing computation. Technical Report CS-91–02, Department of Computer Science, Sheffield University, UK, 1991. Available online: http://www.hypercomputation.net/resources.html.Google Scholar
  80. 80.
    M. Stannett. Computation over arbitrary models of time (a unified model of discrete, analog, quantum and hybrid computation). Technical Report CS-01–08, Dept of Computer Science, University of Sheffield, UK, 2001. Available online: http://www.hypercomputation.net/resources.html.Google Scholar
  81. 81.
    M. Stannett. Hypercomputation is physically irrefutable. Technical Report CS-01–04, Dept of Computer Science, University of Sheffield, UK, 2001. Available online: http://www.hypercomputation.net/resources.html.Google Scholar
  82. 82.
    M. Stannett. Computation and hypercomputation. Minds and Machines, 13:115–153, 2003.MATHCrossRefGoogle Scholar
  83. 83.
    I. Stewart. Deciding the undecidable. Nature, 352:664–665, 1991.CrossRefGoogle Scholar
  84. 84.
    I. Stewart. The dynamics of impossible devices. Nonlinear Science Today, 1:8–9, 1991.MathSciNetGoogle Scholar
  85. 85.
    A. M. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, series 2, 42:230–265, 1936–37.Google Scholar
  86. 86.
    A. M. Turing. Systems of Logic Based on Ordinals. PhD thesis, Princeton University, USA, 1938.Google Scholar
  87. 87.
    A. M. Turing. Systems of logic based on ordinals. In Proceedings of the London Mathematical Society, series 2, 45:161–228, 1939.Google Scholar
  88. 88.
    A. M. Turing. A method for the calculation of the zeta-function. Proceedings of the London Mathematical Society, 2(48): 180, 1943.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Mike Stannett
    • 1
  1. 1.Department of Computer ScienceUniversity of SheffieldUK

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