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Semantics-to-Syntax Analyses of Algorithms

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Turing’s Revolution

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Abstract

Alan Turing pioneered semantics-to-syntax analysis of algorithms. It is a kind of analysis where you start with a large semantically defined species of algorithms, and you finish up with a syntactic artifact, typically a computation model, that characterizes the species. The task of analyzing a large species of algorithms seems daunting if not impossible. As in quicksand, one needs a rescue point, a fulcrum. In computation analysis, a fulcrum is a particular viewpoint on computation that clarifies and simplifies things to the point that analysis become possible. We review from that point of view Turing’s analysis of human-executable computation, Kolmogorov’s analysis of sequential bit-level computation, Gandy’s analysis of a species of machine computation, and our own analysis of sequential computation.

The real question at issue is “What are the possible processes which can be carried out in computing a number?”

Turing

Give me a fulcrum, and I shall move the world.

Archimedes

AMS Classification: 68W40

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Notes

  1. 1.

    Q is our inquisitive friend Quisani, and A is the author.

  2. 2.

    Here and below a numerical function is a function f(x1, …, xj), possibly partial, of finite arity j, where the arguments xi range over natural numbers, and the values of f—when defined—are natural numbers.

  3. 3.

    “Numerical calculation in 1936 was carried out by human beings; they used mechanical aids for performing standard arithmetical operations, but these aids were not programmable” (Gandy [10, p. 12]).

  4. 4.

    This was pointed out to us by the anonymous referee.

  5. 5.

    Uspensky told us that the summary [18] of the 1953 talk was written by him after several unsuccessful attempts to make Kolmogorov to write a summary.

  6. 6.

    A set x is hereditarily finite if its transitive closure TC(x) is finite. Here TC(x) is the least set t such that x ∈ t and such that z ∈ y ∈ t implies z ∈ t.

  7. 7.

    This discussion is provoked by the anonymous referee who thought that the two sentences above are insufficient for this subsection.

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Acknowledgements

Many thanks to Andreas Blass, Bob Soare, Oron Shagrir and the anonymous referee for useful comments.

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Authors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Yuri Gurevich

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  1. Yuri Gurevich
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Correspondence to Yuri Gurevich .

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Editors and Affiliations

  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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Gurevich, Y. (2015). Semantics-to-Syntax Analyses of Algorithms. In: Sommaruga, G., Strahm, T. (eds) Turing’s Revolution. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22156-4_7

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