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Theses for Computation and Recursion on Concrete and Abstract Structures

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Turing’s Revolution

Abstract

The main aim of this article is to examine proposed theses for computation and recursion on concrete and abstract structures. What is generally referred to as Church’s Thesis or the Church-Turing Thesis (abbreviated CT here) must be restricted to concrete structures whose objects are finite symbolic configurations of one sort or another. Informal and principled arguments for CT on concrete structures are reviewed. Next, it is argued that proposed generalizations of notions of computation to abstract structures must be considered instead under the general notion of algorithm. However, there is no clear general thesis in sight for that comparable to CT, though there are certain wide classes of algorithms for which plausible theses can be stated. The article concludes with a proposed thesis RT for recursion on abstract structures.

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Notes

  1. 1.

    Gandy [1] is an excellent introductory source to those developments; cf. also [2].

  2. 2.

    Soare in his articles [2, 5] has justifiably made considerable efforts to reconfigure the terminology of the subject so as to emphasize its roots in the notion of computation rather than recursion, for example to write ‘c.e.’ for ‘computably enumerable’ in place of ‘r.e.’ for ‘recursively enumerable’, but they do not seem to have overcome the weight of tradition.

  3. 3.

    Kleene’s wording derives from Turing [3, p. 249]: “No attempt has yet been made to show that the ‘computable’ numbers include all numbers which would naturally be regarded as computable.” [Italics mine]

  4. 4.

    Cf. Copeland [7] for an introductory article on the Church-Turing thesis.

  5. 5.

    For convenience I shall only refer to Sieg [18] in the following, though there is considerable overlap with Dawson [17].

  6. 6.

    Gödel was not long after to change his mind about that; cf. Gödel [70].

  7. 7.

    Church [4, pp. 100–102] had another “step-by-step” argument for the Thesis, but there is a semi-circularity involved; cf. Shagrir [23, p. 224].

  8. 8.

    Dershowitz and Gurevich [33, p. 305] state that the aim of their work is “to provide a small number of convincing postulates in favor of Church’s Thesis”; in that same article, pp. 339–342, they provide a comprehensive survey of the literature sharing that aim, going back to work of Kolmogorov in [30].

  9. 9.

    Cf. the Postscriptum to Sieg [35] for a detailed critique of this work.

  10. 10.

    For a systematic treatment of computability on concrete structures see Tucker and Zucker [36].

  11. 11.

    The literature on generalized recursion theory is very extensive and could use an up-to-date survey. Lacking that, some initial sources can be found in the bibliographies in the works of Barwise [37], Fenstad [38], Sacks [39], and Tucker and Zucker [11].

  12. 12.

    In addition, the reference Tucker and Zucker [10] is not as widely available as their year 2000 survey.

  13. 13.

    See also Blass and Gurevich [49].

  14. 14.

    Note that a partial recursive functional F need not have total values when restricted to total arguments.

  15. 15.

    There is a considerable literature on computation on the real numbers under various approaches related to the effective approximation one via Cauchy representations. A more comprehensive one is that given by Kreitz and Weihrauch [54, 55] and Weihrauch [56]; that features surjective representations from a subset of ℕℕ to ℝ. Bauer [57] introduced a still more general theory of representations via a notion of realizability, that allows one to consider classical structures and effective structures of various kinds (including those provided by domain theory) under a single framework; cf. also Bauer and Blanck [58]. The work of Pour-El surveyed in her article [59] contains interesting applications of the effective approximation approach to questions of computability in physical theory.

  16. 16.

    Cf. also Blum [60], to which Braverman and Cook [50] responds more directly. Actually, the treatment of a number of examples from numerical analysis in terms of the BSS model that takes up Part II of Blum et al. [41] via the concept of the “condition number” of a procedure in a way brings it in closer contact with the effective approximation model. As succinctly explained to me in a personal communication from Lenore Blum, “[r]oughly, ‘condition’ connects the BSS/BCSS theory with the discrete theory of computation/complexity in the following way: The ‘condition’ of a problem instance measures how outputs will vary under perturbations of the input (think of the condition as a normed derivative).” The informative article, Blum [61], traces the idea of the condition number back to a paper by Turing [62] on rounding-off errors in matrix computations from where it became a basic common concept in various guises in numerical analysis. (An expanded version of Blum [61] is forthcoming.) It may be that the puzzle of how the algebraic BSS model serves to provide a foundation for the mathematics of the continuous, at least as it appears in numerical analysis, is resolved by noting that the verification of the algorithms it employs requires in each case specific use of properties of the reals and complex numbers telling which such are “well-conditioned.”

  17. 17.

    Platek [63] also used the LFP approach to subsume recursion theory on the ordinals under the theory of recursion in the Sup functional.

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Acknowledgements

I wish to thank Lenore Blum, Andrej Bauer, John W. Dawson, Jr., Nachum Dershowitz, Yuri Gurevich, Grigori Mints, Dana Scott, Wilfried Sieg, Robert Soare, John V. Tucker, and Jeffery Zucker for their helpful comments on an early draft of this article. Also helpful were their pointers to relevant literature, though not all of that extensive material could be accounted for here.

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  1. Department of Mathematics, Stanford University, Stanford, CA, USA

    Solomon Feferman

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  1. Department of Humanities, Social and Political Sciences, ETH Zurich, Zurich, Switzerland

    Giovanni Sommaruga

  2. Institute of Computer Science and Applied Mathematics, University of Bern, Bern, Switzerland

    Thomas Strahm

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Feferman, S. (2015). Theses for Computation and Recursion on Concrete and Abstract Structures. In: Sommaruga, G., Strahm, T. (eds) Turing’s Revolution. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22156-4_4

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