Do Accelerating Turing Machines Compute the Uncomputable?
- 382 Downloads
Accelerating Turing machines have attracted much attention in the last decade or so. They have been described as “the work-horse of hypercomputation” (Potgieter and Rosinger 2010: 853). But do they really compute beyond the “Turing limit”—e.g., compute the halting function? We argue that the answer depends on what you mean by an accelerating Turing machine, on what you mean by computation, and even on what you mean by a Turing machine. We show first that in the current literature the term “accelerating Turing machine” is used to refer to two very different species of accelerating machine, which we call end-stage-in and end-stage-out machines, respectively. We argue that end-stage-in accelerating machines are not Turing machines at all. We then present two differing conceptions of computation, the internal and the external, and introduce the notion of an epistemic embedding of a computation. We argue that no accelerating Turing machine computes the halting function in the internal sense. Finally, we distinguish between two very different conceptions of the Turing machine, the purist conception and the realist conception; and we argue that Turing himself was no subscriber to the purist conception. We conclude that under the realist conception, but not under the purist conception, an accelerating Turing machine is able to compute the halting function in the external sense. We adopt a relatively informal approach throughout, since we take the key issues to be philosophical rather than mathematical.
KeywordsAccelerating Turing machine Supertask Halting problem ATM paradox Hypercomputation External and internal computation Epistemic embedding Ontology of computing Turing-machine purism Turing-machine realism Thompson lamp paradox
Copeland’s research was supported in part by the Royal Society of New Zealand Marsden Fund, grant UOC905, and Shagrir’s research was supported by the Israel Science Foundation, grant 725/08.
- Barker-Plummer, D. (2004). Turing machines. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. http://www.plato.stanford.edu/archives/spr2005/entries/turing-machine.
- Boolos, G. S., & Jeffrey, R. C. (1980). Computability and logic (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
- Copeland, B. J. (1998a). Even Turing machines can compute uncomputable functions. In C. S. Calude, J. Casti, & M. J. Dinneen (Eds.), Unconventional models of computation (pp. 150–164). Singapore: Springer.Google Scholar
- Copeland, B. J. (2002b). Hypercomputation. In B. J. Copeland (Ed.) (2002–2003), 461–502.Google Scholar
- Copeland, B. J. (Ed.) (2002–2003). Hypercomputation. Special issue of Minds and Machines, 12(4), 13(1).Google Scholar
- Copeland, B. J. (2005). Comments from the chair: Hypercomputation and the Church-Turing thesis. Paper delivered at the American Philosophical Society Eastern Division Meeting, New York City.Google Scholar
- Copeland, B. J. (2010). Colossus: Breaking the German “Tunny” code at Bletchley Park. An illustrated history. The Rutherford Journal: The New Zealand Journal for the History and Philosophy of Science and Technology, 3, http://www.rutherfordjournal.org.
- 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
- Fearnley, L. G. (2009). On accelerated Turing machines. Honours thesis in Computer Science, University of Auckland.Google Scholar
- Hamkins, J. D. (2002). Infinite time Turing machines. In B. J. Copeland (Ed.) (2002–2003), 521–539.Google Scholar
- Hogarth, M. L. (1994). Non-Turing computers and non-Turing computability. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 1, 126–138.Google Scholar
- Pitowsky, I. (1990). The physical Church thesis and physical computational complexity. Iyyun, 39, 81–99.Google Scholar
- Potgieter, P. H., & Rosinger, E. E. (2010). Output concepts for accelerated Turing machines. Natural Computing, 9, 853–864.Google Scholar
- Russell, B. A. W. (1915). Our knowledge of the external world as a field for scientific method in philosophy. Chicago: Open Court.Google Scholar
- Schaller, M., & Svozil, K. (2009). Zeno squeezing of cellular automata. arXiv:0908.0835.Google Scholar
- Shagrir, O. (2011). Supertasks do not increase computational power. Natural Computing (forthcoming).Google Scholar
- Shagrir, O., & Pitowsky, I. (2003). Physical hypercomputation and the Church-Turing thesis. In B. J. Copeland (Ed.) (2002–2003), 87–101.Google Scholar
- Steinhart, E. (2003). The physics of information. In L. Floridi (Ed.), The Blackwell guide to the philosophy of computing and information (pp. 178–185). Oxford: Blackwell.Google Scholar
- Svozil, K. (1998). The Church-Turing thesis as a guiding principle for physics. In C. S. Calude, J. Casti, & M. J. Dinneen (Eds.), Unconventional models of computation (pp. 371–385). London: Springer.Google Scholar
- Thomson, J. F. (1970). Comments on professor Benacerraf’s paper. In W. C. Salmon (Ed.), Zeno’s paradoxes (pp. 130–138). Indianapolis: Bobbs-Merrill.Google Scholar
- Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series 2, 42, 230–265. (In The essential Turing (Copeland 2004a); page references are to the latter.)Google Scholar
- Turing, A. M. (1948). Intelligent machinery. National Physical Laboratory report. In The essential Turing (Copeland 2004a). A digital facsimile of the original document may be viewed in the Turing Archive for the History of Computing. http://www.AlanTuring.net/intelligent_machinery.
- Turing, A. M. (1950). Computing machinery and intelligence. Mind, 59, 433–60. (In The essential Turing (Copeland 2004a); page references are to the latter.)Google Scholar