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Conference on Computability in Europe

CiE 2016: Pursuit of the Universal pp 78–88Cite as

Squeezing Feasibility

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

Abstract

This note explores an often overlooked question about the characterization of the notion model of computation which was originally identified by Cobham [5]. A simple formulation is as follows: what primitive operations are allowable in the definition of a model such that its time and space complexity measures provide accurate gauges of practical computational difficulty? After exploring the significance of this question in the context of subsequent work on machine models and simulations, an adaptation of Kreisel’s squeezing argument [17] for Church’s Thesis involving Gandy machines [11] is sketched which potentially bears on this question.

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Notes

  1. 1.

    Cobham characterized the class \(\mathcal {L}\) of feasibly computable functions as those computable by an informal algorithm in “a number of steps \(\ldots \) bounded by polynomials in the lengths of the numbers involved”. He proposed that \(\mathcal {L}\) coincided with the functions definable using a restricted recursion scheme known as limited recursion on notation. He then conjectured that the class of functions definable in this manner coincided with \(\mathbf {FP}\) under its contemporary definition – i.e. the class of functions computable in polynomial time by a deterministic Turing machine. This was later demonstrated by Rose [20].

  2. 2.

    “Is it harder to multiply than to add? \(\ldots \) I grant I have put [this] question loosely; nevertheless, I think the answer ought to be yes” [5], p. 24. By the mid-1960s it was already possible to marshall a good deal of evidence in favor of this hypothesis – e.g., [7].

  3. 3.

    See [27] for full definitions of \(\mathfrak {T}\) and \(\mathfrak {R}\) and references for the simulation results just summarized.

  4. 4.

    Considerable inductive support for this claim is supplied by Knuth’s [15] detailed implementations of a wide range of algorithms using a variant of \(\mathfrak {R}\) which satisfies the Invariance Thesis when its multiplication and division operations are removed.

  5. 5.

    I.e. for all \(N_0 \in \mathfrak {N}_0\) there exists \(N_1 \in \mathfrak {N}_1\) computing the same function such that \(\text {time}_{N_1}(|x|) \in O(\text {time}_{N_0}(|x|)^k)\) and \(\text {space}_{N_1}(|x|) \in O(\text {space}_{N_0}(|x|))\) and conversely.

  6. 6.

    See [27] for discussion of caveats pertaining both to how the relevant time and space measures must be formulated and the status of the requirement that the relevant overheads be achieved with respect to the same simulation.

  7. 7.

    Hartmanis and Simon showed this originally with respect to a variant of the MRAM model which allows parallel bit-wise Boolean operations in the manner described above. Bertoni et al. showed that such operations were not necessary in the presence of both addition and multiplication.

  8. 8.

    For if \(\mathfrak {M}\) satisfied the Invariance Thesis with respect to \(\mathfrak {R}\), this would also entail that there was a time and space efficient simulation of MRAM machines by multi-tape Turing machines from which \(\mathbf {P}_{\mathfrak {T}} = \mathbf {PSPACE}_{\mathfrak {T}}\) would follow.

  9. 9.

    Schönhage and Strassen [23] showed that the \(O(n^2)\) time complexity of the “naive” carry multiplication algorithm can be improved to \(O(n \log (n) \log (\log n))\), while also conjecturing a lower bound of \(\varOmega (n \log (n))\). The Schönhage-Strassen bound has recently been improved by Fürer [10] who presented an \(n \log n 2^{O(\log ^*(n))}\) algorithm.

  10. 10.

    Although I will put aside such issues here, rejoinders to these objections can be found in textbooks such as [1]. See also [9].

  11. 11.

    For instance, while \(\mathbf {FP}\) coincides with the provably recursive functions of Buss’s system \(\mathsf {S}^1_2\), similar characterizations of the Linear Time Hierarchy or Polynomial Hierarchy arise as the provably recursive functions of similar theories [3]. And while \(\mathbf {P}_{\mathfrak {T}}\) is characterizable as the set of languages describable in first-order logic with a least fixed point operator in the presence of a linear order, similar (and in many instances simpler) characterizations of narrower classes (like \(\mathsf {AC}^0\)) or broader classes (like \(\mathbf {NP}_{\mathfrak {T}}\)) are also available [14]. See [4] for function algebra descriptions of additional complexity classes such as \(\mathbf {PSPACE}_{\mathfrak {T}}\).

  12. 12.

    The graph-theoretic models of Kolomogorv and Uspensky [16] and Schönhage [22] are inspired by similar considerations. However Gandy’s model differs not only defining states in defining states more abstractly – i.e. as hereditarily finite sets (which may contain unboundedly many labels which are treated as urelements) of bounded rank – but also in that it allows for parallelism in the sense that machines may locally transform discontiguous parts of states in a single step.

  13. 13.

    In fact it seems likely that \(\mathfrak {G}\) is a not a reasonable model. For it is easy to see that Gandy machines can simulate Schönage’s storage modification machines with constant factor time overhead. As the latter model admits a linear time (i.e. O(n)) multiplication algorithm, Gandy machines do as well. And as Cook [6] (p. 403) observes, “we are forced to conclude that either multiplication is easier than we thought or that Schönhage’s machines cheat”. See Schöngage [22] (pp. 506–507) and van Emde Boas [26] (p. 110) for similar observations.

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Authors and Affiliations

  1. Department of Philosophy, University of Warwick, Coventry, CV4 7AL, UK

    Walter Dean

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  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

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Dean, W. (2016). Squeezing Feasibility. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_8

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