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The Developments of the Concept of Machine Computability from 1936 to the 1960s

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Constructivity and Computability in Historical and Philosophical Perspective

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 34))

Abstract

From the 1940s to the 1960s, despite the significant work done on recursive functions (properly) and later on the lambda-calculus, the theory of calculability was developed more and more as a theory of computation by an idealized machine, or in the form of a general theory of algorithms. I will only deal here with the former aspect, a development that stems from the concepts introduced in 1936 by Turing. I will try to show how Turing’s ideas were gradually adopted, developed and modified. The Turing machine had an increasingly important role and was the object of systematic investigation. It was subsequently reworked to such an extent that a new model of machine was fashioned, the program and register machine. However, the initial model kept a significant place, and extensions of Turing’s analysis led, toward the end of the century, to profound reflections about the notion of a constructive object and the general notion of an algorithm.

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References

  • Asser, G. (1959). Turing-Maschinen und Markowsche Algorithmen. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 5, 346–365.

    Article  Google Scholar 

  • Bernays, P. (1958). Remarques sur le problème de la décision en logique élémentaire. In Le raisonnement en mathématiques et en sciences expérimentales, Colloques Internationaux du CNRS, 70 (pp. 39–44). Paris: CNRS.

    Google Scholar 

  • Boolos, G., & Jeffrey, R. (1974). Computability and logic. London: Cambridge University Press.

    Google Scholar 

  • Boone, W. J. (1958). An analysis of Turing’s ‘the word problem in semi-groups with cancellation’. Annals of Mathematics (Series 2), 67, 195–202.

    Google Scholar 

  • Boone, W. J. (1959). The word problem. Annals of Mathematics (Series 2), 70, 207–265.

    Google Scholar 

  • Büchi, R. (1962). Turing machines and the Entscheidungsproblem. Mathematische Annalen, 148, 201–213.

    Article  Google Scholar 

  • Burks, A. W. (1966). Introduction. In J. von Neumann (1966).

    Google Scholar 

  • Chomsky, N. (1963). Formal properties of grammars. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. 2, chap. 12). New York: Wiley.

    Google Scholar 

  • Church, A. (1937). Review of Turing (1936). Journal of Symbolic Logic, 2, 42–43.

    Google Scholar 

  • Davis, M. (1958). Computability and unsolvability. New York: McGraw-Hill.

    Google Scholar 

  • Davis, M. (Ed.). (1965). The undecidable. Hewlett: Raven Press.

    Google Scholar 

  • Davis, M., Putnam, H., & Robinson, J. (1961). The decision problem for exponential Diophantine equations. Annals of Matematics (Series 2), 74(3), 425–436.

    Google Scholar 

  • Ershov, A. P. (1958). Operatornye algorifmy [Algorithmes d’opérateurs]. Doklady Akademii Nauk SSSR, 122, 967–970 [Automat. Express 1 (1959), 20–23].

    Google Scholar 

  • Fischer, P. (1966). Turing machines with restricted memory access. Information and Control, 9(4), 364–379.

    Article  Google Scholar 

  • Gandy, R. (1980). Church’s thesis and principles for mechanisms. In J. Barwise, H. J. Keisler, & K. Kunen (Eds.), The Kleene symposium (pp. 123–148). Amsterdam: North-Holland.

    Chapter  Google Scholar 

  • Gandy, R. (1988). The confluence of ideas in 1936. In Herken (1988) (pp. 55–111).

    Google Scholar 

  • Gandy, R. (2001), Preface to “On computable numbers”, pp. 9–17 in Turing (2001).

    Google Scholar 

  • Gödel, K. (1934). On undecidable propositions of formal mathematical systems. In Gödel (1986) (pp. 346–371).

    Google Scholar 

  • Gödel, K. (193?). [Undecidable Diophantine propositions]. In Gödel (1995) (pp. 164–175).

    Google Scholar 

  • Gödel, K. (1986). In S. Feferman et al. (Eds.), Gödel’s collected works I. New York: Oxford University Press.

    Google Scholar 

  • Gödel, K. (1995). In S. Feferman et al. (Eds.), Gödel’s collected works III. New York: Oxford University Press.

    Google Scholar 

  • Hartmanis, J. (1981). Observations about the development of theoretical computer science. Annals of the History of Computing, 3(1), 42–51.

    Article  Google Scholar 

  • Hartmanis, J., & Stearns, R. E. (1965). On the computational complexity of algorithms. Transactions AMS, 117, 285–306.

    Article  Google Scholar 

  • Herken, R. (Ed.). (1988). The universal Turing machine. Oxford/New York: Oxford University Press (2nd ed., 1995, Wien/New York: Springer).

    Google Scholar 

  • Hermes, H. (1954). Die Universalität programmgesteuerter Rechenmaschinen. Mathematisch-Physikalische Semsterberichte (Göttingen), 4, 42–53.

    Google Scholar 

  • Hermes, H. (1961). Aufzählbarkeit, Entscheidbarkeit, Berechenbarkeit. Berlin/Göttingen/ Heidelberg: Springer.

    Book  Google Scholar 

  • Hopcroft, J., & Ullman, J. D. (1979). Introduction to automata theory, languages and computation. Reading: Addison Wesley.

    Google Scholar 

  • Kaluznin, L. A. (1959). Ob algoritmizacii matematicheskikh zadach [On algorithmization of mathematical problems]. Problémy Kybérnétiki, 2, 51–67.

    Google Scholar 

  • Kaphengst, H. (1959). Eine abstrakte programmgesteuerte Rechenmaschine. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 5, 366–379.

    Article  Google Scholar 

  • Kleene, S. C. (1952). Introduction to metamathematics. Princeton: Van Nostrand.

    Google Scholar 

  • Kleene, S. C. (1956). Representation of events in nerve nets and finite automata. Project RAND research memorandum RM-704 (1951). In C. E. Shannon & J. McCarthy (Eds.), Automata studies (pp. 3–41). Princeton: Princeton University Press.

    Google Scholar 

  • Kleene, S. C. (1967). Mathematical logic. New York: Wiley.

    Google Scholar 

  • Kleene, S. C. (1981a). Origins of recursive function theory. Annals of History of Computing, 3(1), 52–67.

    Article  Google Scholar 

  • Kleene, S. C. (1981b). The theory of recursive functions approaching its centennial. Bulletin of the American Mathematical Society (New Series), 5, 43–61.

    Article  Google Scholar 

  • Kleene, S. C. (1989). The writing of introduction to metamathematics. In Th. Drucker (Ed.), Perspectives on the history of mathematical logic. Boston: Birkhäuser.

    Google Scholar 

  • Kolmogorov, A., & Uspensky, V. (1958). On the definition of an algorithm. Uspekhi Matematicheskikh Nauk XIII, 4(82), 1–28 [AMS Translations, 21, 2 (1963), 217–245].

    Google Scholar 

  • Lambek, J. (1961). How to program an infinite abacus. Canadian Mathematical Bulletin, 4(3), 295–302.

    Article  Google Scholar 

  • Lee, C. Y. (1960). Automata and finite automata. Bell System Technical Journal, 39, 1267–1295.

    Article  Google Scholar 

  • Manna, Z. (1974). Mathematical theory of computation. New York: McGraw Hill.

    Google Scholar 

  • Matiyasevich, Yu. V. (1984). (Am. Transl.), On investigations on some algorithmic problems in algebra and number theory. Proceedings of Steklov Institute of Mathematics, 168(3) (1986), 227–252.

    Google Scholar 

  • McCulloch, W. S., & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5, 115–133. (Reprinted in McCulloch (1965), Embodiments of Mind. Cambridge: MIT)

    Google Scholar 

  • Melzak, Z. A. (1961). An informal arithmetical approach to computability and computation. Canadian Mathematical Bulletin, 4(3), 279–293.

    Article  Google Scholar 

  • Minsky, M. (1961). Recursive unsolvability of Post’s problem of ‘tag’ and other topics in the theory of Turing machines. Annals of Mathematics, 74(3), 437–454.

    Article  Google Scholar 

  • Minsky, M. (1967). Computation – Finite and infinite machines. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • von Neumann, J. (1948). The general and logical theory of automata. In Cerebral mechanisms in behavior: The Hixon symposium (pp. 1–41). New York: Wiley. (Reprinted in J. von Neumann (1963). Collected works V, pp. 288–328)

    Google Scholar 

  • von Neumann, J. (1949). Theory and organization of complicated automata. In Lectures at the University of Illinois (pp. 31–87). In von Neumann (1966).

    Google Scholar 

  • von Neumann, J. (1952–1953). The theory of automata: Construction, reproduction, homogeneity, part II. In Theory of self-reproducing automata (pp. 91–296). Urbana: University of Illinois Press.

    Google Scholar 

  • von Neumann, J. (1963). Collected works (A. H. Taub, Ed., vol. V). Oxford/London/New York/Paris: Pergamon Press.

    Google Scholar 

  • von Neumann, J. (1966). A. W. Burks (Ed. and completed), Theory of self-reproducing automata. Urbana: University of Illinois Press.

    Google Scholar 

  • Oberschelp, W. (1958). Varianten von Turingmaschinen. Archiv für Mathematische Logik und Grundlagenforschung, 4, 53–62.

    Article  Google Scholar 

  • Péter, R. (1951). Rekursive Funktionen. Budapest: Akadémiai Kiado.

    Google Scholar 

  • Péter, R. (1958). Graphschemata und rekursive Funktionen. Dialectica, 12, 373–393.

    Article  Google Scholar 

  • Péter, R. (1959), Rekursivität und Konstruktivität. In Heyting, A. (Ed.), Constructivity in mathematics, proceedings of the colloquium, Amsterdam, August 1957 (pp. 226–233). Amsterdam: North Holland.

    Google Scholar 

  • Post, E. (1936). Finitary combinatory processes. Formulation 1. Journal of Symbolic Logic, I, 103–105.

    Google Scholar 

  • Post, E. (1943). Formal reductions of the general combinatorial decision problem. American Journal of Mathematics, 65, 197–215.

    Article  Google Scholar 

  • Post, E. (1944). Recursively enumerable sets of positive integers and their decision problems. Bulletin of the AMS, 50, 284–316. (Reprinted in Davis 1965)

    Google Scholar 

  • Post, E. (1947). Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12, 1–11. (Reprinted in Davis (1965), pp. 293–303)

    Google Scholar 

  • Rabin, M. O., & Scott, D. (1959). Finite automata and their decision problems. IBM Journal of Research and Development, 3(2), 114–125. (Reprinted in E. F. Moore (Ed.), Sequential machines (pp. 63–91). Reading: Addison-Wesley)

    Google Scholar 

  • Rogers, H., Jr. (1967). Theory of recursive functions and effective computability. New York: McGraw-Hill.

    Google Scholar 

  • Scott, D. (1967). Some definitional suggestions for automata theory. Journal of Computer and System Science, 1(2), 187–212.

    Article  Google Scholar 

  • Shannon, C. E. (1938). A symbolic analysis of relay and switching circuits. AIEE Transactions, 57, 712–723.

    Google Scholar 

  • Shannon, C. E. (1956). A universal Turing machine with two internal states. In C. E. Shannon & J. McCarthy (Eds.), Automata studies (pp. 157–165). Princeton: Princeton University Press.

    Google Scholar 

  • Shannon, C. E., & McCarthy, J. (Eds.). (1956). Automata studies. Princeton: Princeton University Press.

    Google Scholar 

  • Shepherdson, J. C., & Sturgis, H. E. (1963). Computability of recursive functions. Journal of the ACM, 10 (1963), 217–255.

    Article  Google Scholar 

  • Sieg, W. (1994) Mechanical procedures and mathematical experience. In George, A. (Ed.), Mathematics and mind (pp. 71–117). Oxford: Oxford University Press.

    Google Scholar 

  • Sieg, W. (2009). Computability theory. In A. Irvine, et al. (Eds.), Handbook of the philosophy of science. Philosophy of mathematics. Amsterdam/Boston: Elsevier.

    Google Scholar 

  • Sieg, W., & Byrnes, J. (1996). K-graph machines: Generalizing Turing’s machines and arguments. In P. Hajek (Ed.), Gödel ’96 (Lectures notes in logic 6, pp. 98–119). Berlin: Springer.

    Google Scholar 

  • Sieg, W., & Byrnes, J. (1999). Gödel, Turing and K-graph machines. In A. Cantini, et al. (Eds.), Logic and foundations of mathematics (Synthese library, Vol. 280, pp. 57–66). Dordrecht: Kluwer.

    Google Scholar 

  • Soare, R. I. (1996). Computability and recursion. The Bulletin of Symbolic Logic, 2(3), 284–321.

    Article  Google Scholar 

  • Thue, A. (1914). Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln (Skrifter utgit av Videnskappselskapet i Kristiania, I. Matematisk-naturvidenskabelig Klasse 1914 n 10). Kristiania: J. Dybwad.

    Google Scholar 

  • Trakhtenbrot, B. A. (1957). Algorifmi i machinnoie rechenie zadatch. Moscou. 2d. publ., 1960. (Am. Transl., Boston: Heath, 1963. French Transl., Algorithmes et machines à calculer, Paris: Dunod, 1963)

    Google Scholar 

  • Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem. In Proceedings of the London Mathematical Society (Series 2), 42, 230–265. A correction, ibid., 43, 1937, 544–546. (Reprinted in Davis (1965), pp.115–154)

    Google Scholar 

  • Turing, A. M. (1937). Computability and λ-definability. Journal of Symbolic Logic, 2, 153–163.

    Article  Google Scholar 

  • Turing, A. M. (1939). Systems of logic based on ordinals. Proceedings of the London Mathematical Society (Series 2), 45, 161–228. (Reprinted in Davis (1965), pp. 154–222)

    Google Scholar 

  • Turing, A. M. (1950). The word problem in semi-groups with cancellation. Annals of Mathematics (Series 2), 52, 491–505.

    Google Scholar 

  • Turing, A. M. (1954). Solvable and unsolvable problems. Science News, 31, 7–23. (Reprinted in Turing (1992), pp. 99–115)

    Google Scholar 

  • Turing, A. M. (1992). J. L. Britton (Ed.), Collected works: Pure mathematics. Amsterdam: North Holland.

    Google Scholar 

  • Turing, A. M. (2001). R. Gandy & C. Yates (Eds.), Collected works: Mathematical logic. Amsterdam: Elsevier.

    Google Scholar 

  • Uspensky, V. A. (1992). Kolmogorov and mathematical logic. Journal of Symbolic Logic, 57, 383–412.

    Article  Google Scholar 

  • Wang, H. (1957a). A variant to Turing’s theory of calculating machines. Journal of the ACM, 4(1), 63–92. (Reprinted in Wang, Hao (1970), ch. VI)

    Google Scholar 

  • Wang, H. (1957b). Universal Turing machines: An exercise in coding. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3, 69–80. (Reprinted in Hao Wang (1970), ch. VII).

    Google Scholar 

  • Wang, H. (1970). Logic, computers and sets. New York: Chelsea. (Reprint of Wang, H., A survey of mathematical logic, Peking 1962/New York 1964)

    Google Scholar 

  • Wang, H. (1974). From mathematics to philosophy. London: Routledge/Kegan Paul.

    Google Scholar 

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  1. UMR 8560 IHPST – Institut d’Histoire et de Philosophie des Sciences et des Techniques, Université Paris 1 Panthéon-Sorbonne & CNRS & ENS, Paris, France

    Jean Mosconi

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Correspondence to Jean Mosconi .

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  1. Institut d’histoire et de philosophie des sciences et des techniques (IHPST; CNRS-Paris1-ENS), Paris, France

    Jacques Dubucs

  2. Institut d’histoire et de philosophie des sciences et des techniques (IHPST; CNRS-Paris1-ENS), Paris, France

    Michel Bourdeau

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Mosconi, J. (2014). The Developments of the Concept of Machine Computability from 1936 to the 1960s. In: Dubucs, J., Bourdeau, M. (eds) Constructivity and Computability in Historical and Philosophical Perspective. Logic, Epistemology, and the Unity of Science, vol 34. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9217-2_2

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