Abstract
Turing was awarded the Order of the British Empire in June of 1946. Most people thought that the award was a well deserved mark of recognition honoring the mathematician who had given a successful definition of mechanical procedure, had introduced the universal machine capable of simulating all mechanical procedures, and had settled in the negative Hilbert’s Entscheidungsproblem , the problem whether mathematics is fully mechanizable. Few knew that Turing had received the OBE mainly because of his outstanding services as a cryptographer at Bletchley Park during the Second World War.
This essay is about Turing’s mathematical achievements. We will, as informally as possible, analyze his 1936 paper “On computable numbers with an application to the Entscheidungsproblem” and highlight the main ideas contained in this classical paper. Naturally, we will also discuss the broader context of Turing’s definitions and theorems, their fundamental interaction with Gödel’s completeness and incompleteness theorems, and their basic role in the proof of the Cook-Levin NP-completeness theorem and in the formulation of the P/NP problem.
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Notes
- 1.
Note that in Turing’s remark below this contemporary understanding of reduction is “reversed”: Turing’s “reducing the unsolvability of A to the unsolvability of” amounts to “reducing the problem B to the problem A”.
- 2.
Machines are here taken to be real physical mechanisms that consist of finitely many parts and operate in discrete steps; Gandy called them discrete mechanical devices and contrasted them to analogue machines. Parallel machines locally operate on parts of a given configuration; the results of the local operations are then joined into the next configuration. The paper (Mundici 1983) discusses Turing machines as physical objects under the effect of special relativity. Further papers by Mundici, quoted in this paper, discuss also quantum mechanical limitations. None of these papers, however, aims at motivating the boundedness and locality conditions for (Turing or parallel) machines.
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Acknowledgments
The authors are grateful to Rossella Lupacchini and Guglielmo Tamburrini for their valuable comments and suggestions towards clarity and understandability. This paper has appeared in earlier form in Italian, as (2014) “Turing, il Matematico”, in Contributi del Centro Linceo Interdisciplinare “Beniamino Segre”, Accademia Nazionale dei Lincei, Rome, Vol. 129, pp. 85–120. Convegno Per il Centenario di Alan Turing Fondatore Dell’Informatica, 22 November 2012, Scienze e Lettere Editore Commerciale, Rome.
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Mundici, D., Sieg, W. (2017). Turing, the Mathematician. In: Floyd, J., Bokulich, A. (eds) Philosophical Explorations of the Legacy of Alan Turing. Boston Studies in the Philosophy and History of Science, vol 324. Springer, Cham. https://doi.org/10.1007/978-3-319-53280-6_2
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