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Do Accelerating Turing Machines Compute the Uncomputable?

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Abstract

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.

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Notes

  1. The idea of relativistic machines originated in Pitowsky (1990). For a discussion of the relations between ATMs and relativistic machines see Shagrir (2011); for a discussion of their physical possibility see Earman and Norton (1993, 1996), Copeland and Shagrir (2007), Andréka et al. (2009).

  2. The variant term “accelerated Turing machine” (see e.g. Calude and Staiger (2010), Fearnley (2009), Potgieter and Rosinger (2010)) is from Copeland (1998a).

  3. The “halting problem” was introduced and named by Davis (1958: 70) (not by Turing himself, contrary to popular belief: see Copeland (2004a: 40)).

  4. Another way to put the difference is in terms of whether the specification of the machine refers to a last computation step (end-stage-in machine) or not (end-stage-out machine); see also note 7. In our terminology, a step of the computation consists in the execution of either an atomic operation or a subroutine, and is analogous to a step in the proof-theoretic sense, i.e. the application of a primitive or derived rule; a stage of the computing is analogous to a stage in the construction of a proof (see e.g. Kripke 1959). A state of the accelerating machine is either an m-configuration (in Turing’s sense—see the quotations below) or a limit-state (see below).

  5. See Boolos and Jeffrey (1980: 23), and Lewis and Papadimitriou (1981: 172). See also Turing (1936: 59), quoted in the next section; and note 9.

  6. See Copeland (2002a) and Shagrir (2004) for a discussion of the Thomson lamp in connection with accelerating Turing machines.

  7. Following Benacerraf (1962: 772; and see also Fraser and Akl 2008) we can say that an end-stage-in machine performs a super-duper task, whereas an end-stage-out machine performs only a supertask. If performing a super-task includes a sequence of infinitely many atomic operations (of order type ω), then performing a super-duper task also includes another, last, operation at an extra limit stage.

  8. The term “hypercomputer” was introduced by Copeland in Scientific American in 1999 (Copeland and Proudfoot 1999); see further Copeland (1997, 1998c, 2000, 2002b, 2002–2003, 2004c), Copeland and Sylvan (1999).

  9. Note that the configuration as defined by Turing in this quotation does not include every symbol contained on the tape at that stage (only the currently scanned symbol). Sometimes the term “total configuration” is used to distinguish the sense of “configuration” in which the total tape content (at that stage) is included.

  10. Shagrir (2011) argues that similar considerations apply to relativistic machines (e.g., Pitowsky 1990; Hogarth 1992, 1994, 2004; Shagrir and Pitowsky 2003) and to shrinking machines (e.g., Davies 2001; Beggs and Tucker 2006; Schaller and Svozil 2009). These machines also perform supertasks and exhibit hypercomputational power; yet, like ℋ+, they do not have the computational structure of a Turing machine. Their hypercomputational power is the result (at least partly) of a computational structure that differs in crucial respects from the computational structure of a Turing machine.

  11. Copeland (1998a, b); Svozil (1998).

  12. See Shagrir (2004, 2011). One could also challenge the physical feasibility of accelerating Turing machines (for discussion see Steinhart 2003, and Fearnley 2009), but the claim argued here is that ℋ does not compute the halting function even if all problems of physical implementation are resolved.

  13. See Lewis and Papadimitriou (1981:170). A “halted configuration” is one whose state component is a halt state (p. 172). There may be other means (other than being in a “halt state”) to indicate that the machine has halted, such as being in a halted configuration; see Boolos and Jeffrey (1980: 22–23), and Turing (1936).

  14. We use the halting function as an example of an uncomputable function; however, our arguments, here and in what follows, are general in nature and apply to uncomputable functions both below and beyond the halting function.

  15. See Copeland (1998a, 2002a).

  16. The distinction and terminology were introduced in Copeland (1998a).

  17. It is important that the time-interval be a proper (i.e. delimited) interval and not simply the whole of (non-transfinite) time, since otherwise the reference to an external clock (or some other time-keeping arrangement) is rendered otiose. For example, if the time “interval” under consideration is the whole of time, then ℋ’s non-accelerating counterpart computes the halting function in the external sense: the value is 1 if and only if the initial “0” is replaced by “1” at some time; and the value is 0 otherwise.

  18. As emphasized above, ℋ’s specification does not include a configuration of the machine at the end of the second moment.

  19. Copeland (2005).

  20. The situation described is similar to that with Hogarth’s (1992) example of a Turing machine travelling through anti-de Sitter spacetime, which Earman and Norton (1993) showed to be subject to an observer-destroying blue shift.

  21. Copeland (1998a).

  22. See, for example, the definition of a Turing machine in Lewis and Papadimitriou (1981: 170ff.; see also our next section); and the entry “Turing machines” in the Stanford Encyclopedia of Philosophy—e.g. “Turing machines are not physical objects but mathematical ones” (Barker-Plummer 2004).

  23. Quine (1960), Chap. 6.

  24. See Lewis and Papadimitriou (1981: 170–171) for the details.

  25. The subroutines are described on pp. 63–66 of Turing (1936).

  26. Draft précis of “On Computable Numbers” (undated, 2 pp.; in the Turing Papers, Modern Archive Centre, King’s College Library, Cambridge, catalogue reference K 4). In French; translation by Copeland.

  27. For biographical information on Newman see Copeland (2004b, 2010).

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Acknowledgments

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.

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  1. University of Canterbury, Christchurch, New Zealand

    B. Jack Copeland

  2. Hebrew University of Jerusalem, Jerusalem, Israel

    Oron Shagrir

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Correspondence to B. Jack Copeland or Oron Shagrir.

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Jack Copeland, B., Shagrir, O. Do Accelerating Turing Machines Compute the Uncomputable?. Minds & Machines 21, 221–239 (2011). https://doi.org/10.1007/s11023-011-9238-y

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