Minds and Machines

, Volume 21, Issue 1, pp 83–96 | Cite as

On the Possibilities of Hypercomputing Supertasks

  • Vincent C. Müller


This paper investigates the view that digital hypercomputing is a good reason for rejection or re-interpretation of the Church-Turing thesis. After suggestion that such re-interpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with “Zeno-machines”, i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zeno-machines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though non-effective methods or special rules for semi-decidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the Church-Turing thesis in its traditional interpretation.


Computing Computability Hypercomputing Effective computing Supertask Church-Turing thesis Copeland Benacerraf Thomson Zeno Zeno-machine Accelerated Turing machine 


  1. Barrow, J. D. (2005). The infinite book: A short guide to the boundless, timeless and endless. New York: Pantheon Books.Google Scholar
  2. Benacerraf, P. (1962). Tasks, supertasks, and the modern Eleatics. Journal of Philosophy, 765–784.Google Scholar
  3. Boolos, G., Burghess, J. P., & Jeffrey, R. C. (2007). Computability and logic. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  4. Church, A. (1936). An unsolvable problem in elementary number theory. The American Journal of Mathematics, 58, 345–363.MathSciNetCrossRefGoogle Scholar
  5. Churchland, P. M. (2005). Functionalism at forty: A critical retrospective. Journal of Philosophy, 33–50.Google Scholar
  6. Copeland, J. B. (1993). Artificial intelligence: A philosophical introduction. Oxford: Blackwell.Google Scholar
  7. Copeland, J. B. (1997). The broad conception of computation. American Behavioral Scientist, 690–716.Google Scholar
  8. Copeland, J. B. (1998). Turing’s O-machines, Penrose, Searle and the brain. Analysis, 128–138.Google Scholar
  9. Copeland, J. B., & Proudfoot D. (1999). Review of ‘The Legacy of Alan Turing’. In Peter Millican, & Andy Clark (Ed.), Mind, pp. 187–195.Google Scholar
  10. Copeland, J. B. (2000a). Narrow versus wide mechanism, including a re-examination of Turing’s views on the mind-machine issue. Journal of Philosophy, 97, 5–32.MathSciNetCrossRefGoogle Scholar
  11. Copeland, J. B., & Proudfoot D. (2000). What Turing did after he invented the universal Turing machine. Journal of Logic, Language and Information, 491–509.Google Scholar
  12. Copeland, J. B. (2002a). Accelerating Turing machines. Minds and Machines, 281–301.Google Scholar
  13. Copeland, J. B. (2002b). Hypercomputation. Minds and Machines, 461–502.Google Scholar
  14. Copeland, J. B. (2003). Computation. In L. Floridi (Ed.), The Blackwell guide to the philosophy of computing and information (pp. 3–17). Oxford: Blackwell.Google Scholar
  15. Copeland, J. B. (2004). Hypercomputation: Philosophical issues. Theoretical Computer Science, 317, 251–267.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Cotogno, P. (2003). Hypercomputation and the physical Church-Turing thesis. British Journal for the Philosophy of Science, 181–223.Google Scholar
  17. Cotogno, P. (2009). A brief critique of pure hypercomputation. Minds and Machines, 19, 391–405.CrossRefGoogle Scholar
  18. Davies, M. (2000). The universal computer: The road from Leibniz to Turing. New York: W. W. Norton.Google Scholar
  19. Davies, E. B. (2001). Building infinite machines. British Journal for the Philosophy of Science, 671–682.Google Scholar
  20. Deutsch, D. (2004). It from qubit. In J. D. Barrow, P. C. W. Davies, & C. L. Harper (Eds.), Science and ultimate reality: Quantum theory, cosmology, and complexity (Festschrift for John A. Wheeler) (pp. 90–102). Cambridge: Cambridge University Press.Google Scholar
  21. Earman, J., & Norton, J. D. (1996). Infinite pains: The trouble with supertasks. In A. Morton & S. P. Stich (Eds.), Benacerraf and his critics (pp. 231–261). Oxford: Blackwell.Google Scholar
  22. Floridi, L. (1999). Philosophy and computing: An introduction. London: Routledge.zbMATHGoogle Scholar
  23. Fodor, J. A. (2000). The mind doesn’t work that way: The scope and limits of computational psychology. Cambridge, Mass: MIT Press.Google Scholar
  24. Gandy, R. (1980). ‘Church’s thesis and principles of mechanics’. In J. Barwise, H. J. Keisler, & K. Kunen, (Eds.), The Kleene symposium (pp. 123–148). Amsterdam: North-Holland.Google Scholar
  25. Hamkins, J. D., & Lewis, A. (2000). Infinite time Turing machines. The Journal of Symbolic Logic, 65, 567–604.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Harel, D. (2000). Computers ltd.: What they really can’t do. Oxford: Oxford University Press.zbMATHGoogle Scholar
  27. Hodges, A. (2006). Did Church and Turing have a thesis about machines? In A. Olszewski, J. Woleński, & R. Janusz (Eds.), Church’s Thesis after 70 years (pp. 242–252). Frankfurt: Ontos.Google Scholar
  28. Kieu, T. D. (2002). Quantum hypercomputability. Minds and Machines, 541–561.Google Scholar
  29. Kieu, T. D. (2004). Hypercomputation with quantum adiabatic processes. Theoretical Computer Science, 317, 93–104.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Müller, V. C. (2008). Representation in digital systems. In A. Briggle, K. Waelbers, & P. Brey (Eds.), Current issues in computing and philosophy (pp. 116–121). Amsterdam: IOS Press.Google Scholar
  31. Ord, T., & Kieu, T. D. (2005). The diagonal method and hypercomputation. British Journal for the Philosophy of Science, 147–156.Google Scholar
  32. Penrose, R. (1989). The emperor’s new mind: Concerning computers, minds and the laws of physics. London: Vintage.Google Scholar
  33. Piccinini, G. (2004). Functionalism, computationalism, and mental contents. Studies in the History and Philosophy of Science, 35, 811–833.MathSciNetCrossRefGoogle Scholar
  34. Piccinini, G. (2007). Computationalism, the Church-Turing thesis, and the Church-Turing fallacy. Synthese, 154, 97–120.MathSciNetzbMATHCrossRefGoogle Scholar
  35. Pinker, S. (2005). So how does the mind work? Mind and Language, 20, 1–24.CrossRefGoogle Scholar
  36. Potgieter, P. H. (2006). Zeno machines and hypercomputation. Theoretical Computer Science, 358, 23–33.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Scheutz, M. (Ed.). (2002). Computationalism: New directions. Cambridge: Cambridge University Press.Google Scholar
  38. Shagrir, O. (2004). Super-tasks, accelerating Turing machines and uncomputability. Theoretical Computer Science, 317, 105–114.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Shagrir, O., & Pitowsky, I. (2003). Physical hypercomputation and the Church-Turing thesis. Minds and Machines, 13, 87–101.zbMATHCrossRefGoogle Scholar
  40. Siegelmann, H. T. (1995). Computation beyond the Turing limit. Science, 545–548.Google Scholar
  41. Siegelmann, H. T. (1997). Neural networks and analog computation: Beyond the Turing limit. Basel: Birkhäuser.Google Scholar
  42. Siegelmann, H. T., & Sontag, E. D. (1994). Analog computation via neural nets. Theoretical Computer Science, 331–360.Google Scholar
  43. Thomson, J. F. (1954). Tasks and super-tasks. Analysis, 1–13.Google Scholar
  44. Turing, A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42, 230–256.zbMATHCrossRefGoogle Scholar
  45. Turing, A. (1992). Collected works: Mechanical intelligence. A. Sevenster, (Ed.), Amsterdam: North-Holland.Google Scholar
  46. Welch, P. D. (2004). On the possibility, or otherwise, of hypercomputation. British Journal for the Philosophy of Science, 739–746.Google Scholar
  47. Weyl, H. (1927). Philosophie der Mathematik und Naturwissenschaft. Munich: Oldenbourg.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.ACT Department of Humanities and Social SciencesAnatolia CollegePylaiaGreece

Personalised recommendations