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Computability and Recursion

  • Robert I. Soare (a1)
Abstract
Abstract

We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness”. We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas.

After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than “Recursive Function Theory.”

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[Boolos and Jeffrey, 1974] Boolos G. and Jeffrey R., Computability and logic, Cambridge University Press, Cambridge, England.
[Church, 1935] Church A., An unsolvable problem of elementary number theory, Bulletin of the American Mathematical Society, vol. 41, pp. 332333, Preliminary report (abstract).
[Church, 1936] Church A., An unsolvable problem of elementary number theory, American Journal of Mathematics, vol. 58, pp. 345363.
[Church, 1936 b] Church A., A note on the Entscheidungsproblem, Journal of Symbolic Logic, vol. 1, pp. 4041, correction 101–102.
[Church, 1937] Church A., Review of Turing 1936, Journal of Symbolic Logic, vol. 2, no. 1, pp. 4243.
[Church, 1937 b] Church A., Review of Post 1936, Journal of Symbolic Logic, vol. 2, no. 1, p. 43.
[Church, 1938] Church A., The constructive second number class, Bulletin of the American Mathematical Society, vol. 44, pp. 224232.
[Church and Kleene, 1936] Church A. and Kleene S. C., Formal definitions in the theory ofordinal numbers, Fundamenta Mathematicae, vol. 28, pp. 1121.
[Cutland, 1980] Cutiand Nigel, Computability: An introduction to recursive function theory, Cambridge University Press, Cambridge, England.
[Davis, 1958] Davis M., Computability and unsolvability, Mc-Graw-Hill, New York, reprinted in 1982 by Dover Publications.
[Davis, 1965] Davis M. (editor), The undecidable. Basicpaperson undecidablepropositions, unsolvable problems, andcomputable functions, Raven Press, Hewlett, New York.
[Davis, 1982] Davis M., Why Gödel did not have Church's thesis, Information and Control, vol. 54, pp. 324.
[Davis, 1988] Davis M., Mathematical logic and the origin of modern computers, in Herken, 1988, pp. 149174.
[Dedekind, 1872] Dedekind R., Stetigkeit und irrational Zahlen, Braunschweig, 5th ed., 1927; also in Dedekind Gesammelte mathematische Werke (Viewegand Sohn, editors), vol. III, Braunschweig, 1932, pp. 315–334; English translation by Wooster Woodruff Beman entitled Continuity and irrational numbers, Essays on the theory ofnumbers , Open Court, Chicago, 1901, pp. 1–24.
[Dedekind, 1888] Dedekind R., Was sind und was sollen die Zahlen?, Braunschweig, 6th ed., 1930; also in Dedekind Gesammelte Mathematische Werke , vol. III, pp. 335-391; English translation by Beman, The nature and meaning of numbers. loc. cit. , pp. 31–105; English translation in Dedekind, Essays on the theory ofnumbers , Open Court, Chicago, 1901, pp. 29–115.
[Epstein and Carnielli, 1989] Epstein R. L. and Carnielli W. A., Computability: computable functions, logic, and the foundations of mathematics, Brooks/Cole Advanced Books and Software, Pacific Grove, Calif.
[Feferman, 1988] Feferman S., Turing in the land of O(z), in Herken, 1988, pp. 113147.
[Feferman, 1992] Feferman S., Turing's “oracle”: From absolute to relative computability—and back, The space of mathematics (Echeverria J. et al., editors), Walter de Gruyter, Berlin, pp. 314348.
[Fitting, 1987] Fitting M., Computability theory, semantics, and logicprogramming, Oxford University Press.
[Gandy, 1980] Gandy R., Church's thesis and principles for mechanisms, The Kleene symposium, North-Holland, pp. 123148.
[Gandy, 1988] Gandy R., The confluence of ideas in 1936, in Herken, 1988, pp. 55111.
[Gödel, 193?] Gödel K., Undecidable diophantine propositions, in Gödel 1995, pp. 156–175.
[Gödel, 1931] Gödel K., Über formal unent scheidbare sätze der Principia Mathematica und verwandter systeme. I, Monatsch. Math. Phys., vol. 38, pp. 173178, English translation in Davis 1965, pp. 4–38, and in van Heijenoort, 1967, pp. 592–616.
[Gödel, 1934] Gödel K., On undecidable propositions of formal mathematical systems, notes by S. C. Kleene and J. B. Rosser on lectures at the Institute for Advanced Study, Princeton, New Jersey, 1934, reprinted in Davis 1965, pp. 39–74].
[Gödel, 1936] Gödel K., On the length of proofs, Gödel 1986, pp. 397–399; reprinted in Davis 1965, pp. 82–83, with a Remark added in proof [of the original German publication].
[Gödel, 1946] Gödel K., Remarks before the Princeton bicentennial conference of problems in mathematics, reprinted in Davis 1965, pp. 84–88.
[Gödel, 1951] Gödel K., Some basic theorems on the foundations of mathematics and their implications, in Gödel 1995, pp. 304–323. (This was the Gibbs Lecture delivered by Godel on December 26, 1951 to the American Mathematical Society).
[Gödel, 1958] Gödel K., Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes, Dialectica, vol. 12, pp. 280287, German and English translation in Godel 1986, pp. 240–251, with introductory note by A. S. Troelstra, pp. 217–241.
[Gödel, 1964] Gödel K., Postscriptum to Gödel 1931, written in 1946, printed in Davis 1965, pp. 7173.
[Gödel, 1972] Gödel K., Some remarks on the undecidability results, in Gödel 1990, pp. 305306, written in 1972.
[Gödel, 1986] Gödel K., Collected works volume I: Publications 1929–36 (Feferman S. et al., editors), Oxford University Press, Oxford.
[Gödel, 1990] Gödel K., Collected works volume II: Publications 1938–1974 (Feferman S. et al., editors), Oxford University Press, Oxford.
[Gödel, 1995] Gödel K., Collected works volume III: Unpublished essays and lectures (Feferman S. et al., editors), Oxford University Press, Oxford.
[Harrington and Soare, 1996] Harrington L. and Soare R. I., Definability, automorphisms, and dynamic properties of computably enumerable sets, this Bulletin, vol. 2, 1996, pp. 199213.
[Herken, 1988] Herken R. (editor), The universal Turing machine: A half-century survey, Oxford University Press.
[Hilbert, 1899] Hilbert D., Grundlagen der Geometrie, 7th ed., Tuebner-Verlag, Leipzig, Berlin, 1930.
[Hilbert, 1904] Hilbert D., Über die Grundlagen der Logik und der Arithmetik, in Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, Teubner, Leipzig, 1905, pp. 174185. Reprinted in van Heijenoort 1967, pp. 129–138.
[Hilbert, 1918] Hilbert D., Axiomatisches Denken, Mathematische Annalen, vol. 78, pp. 405415.
[Hilbert, 1926] Hilbert D., Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161190, English translation in van Heijenoort 1967, 367–392.
[Hilbert, 1928] Hilbert D., Abhandlungen aus dem mathematischen Seminar derHamburgischen Universität, Die Grundlagen der Mathematik, vol. 6, pp. 6585, reprinted in van Heijenoort 1967, pp. 464–479.
[Hilbert and Ackermann, 1928] Hilbert D. and Ackermann W., Grundzüge der theoretischen Logik, Springer-Verlag, Berlin, English translation of 1938 edition, Chelsea, New York, 1950.
[Hilbert and Bernays, 1934] Hilbert D. and Bernays P., Grundlagen der Mathematik I (1934), II (1939), second ed., Springer-Verlag, Berlin, I (1968), II (1970).
[Hodges, 1983] Hodges A., Alan Turing: The enigma, Burnett Books and Hutchinson, London, and Simon and Schuster, New York.
[Kleene, 1936] Kleene S. C., General recursive functions of natural numbers, Mathematische Annalen, vol. 112, pp. 727742.
[Kleene, 1936 b] Kleene S. C., λ-definability and recursiveness, Duke Mathematics Journal, vol. 2, pp. 340353.
[Kleene, 1936 c] Kleene S. C., A note on recursive functions, Bulletin of the American Mathematical Society, vol. 42, pp. 544546.
[Kleene, 1938] Kleene S. C., On notation for ordinal numbers, Journal of Symbolic Logic, vol. 3, pp. 150155.
[Kleene, 1943] Kleene S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53, pp. 4173.
[Kleene, 1944] Kleene S. C., On the forms of the predicates in the theory of constructive ordinals, American Journal of Mathematics, vol. 66, pp. 4158.
[Kleene, 1952] Kleene S. C., Introduction to metamathematics, Van Nostrand, New York, ninth reprint 1988, Walters-Noordhoff Publishing Co., Groningen and North-Holland, Amsterdam.
[Kleene, 1952 b] Kleene S. C., Recursive functions and intuitionistic mathematics, Proceedings of the international congress of mathematicians, Cambridge, Massachusetts, U.S.A., August 30-September 6, 1950, vol. 1, American Mathematical Society (Providence, R.I.), pp. 679685.
[Kleene, 1955] Kleene S. C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79, pp. 312340.
[Kleene, 1955 b] Kleene S. C., On the forms of the predicates in the theory of constructive ordinals (second paper), American Journal of Mathematics, vol. 77, pp. 405428.
[Kleene, 1955 c] Kleene S. C., Hierarchies of number-theoretical predicates, Bulletin of the American Mathematical Society, vol. 61, pp. 193213.
[Kleene, 1959] Kleene S. C., Recursive functionals and quantifiers of finite type I, Transactions of the American Mathematical Society, vol. 91, pp. 152.
[Kleene, 1962] Kleene S. C., Turing-machine computable functionals of finite types I, Logic, methodology, and philosophy of science: Proceedings of the 1960 international congress, Stanford University Press, pp. 3845.
[Kleene, 1962 b] Kleene S. C., Turing-machine computable functionals offinite types II, Proceedings of the London Mathematical Society, vol. 12, no. 3, pp. 245258.
[Kleene, 1963] Kleene S. C., Recursive functionals and quantifiers offinite type II, Transactions of the American Mathematical Society, vol. 108, pp. 106142.
[Kleene, 1981] Kleene S. C., Origins of recursive function theory, Annals of the History of Computing, vol. 3, pp. 5267.
[Kleene, 1981 b] Kleene S. C., The theory of recursive functions, approaching its centennial, Bulletin of the American Mathematical Society (n.s.), vol. 5, pp. 4361.
[Kleene, 1981 c] Kleene S. C., Algorithms in various contexts, Proc. sympos. algorithms in modern mathematics and computer science (dedicated to Al-Khowarizimi), Urgench, Khorezm Region, Uzbek, SSSR, 1979, Springer-Verlag, Berlin, Heidelberg and New York.
[Kleene, 1987] Kleene S. C., Reflections on Church's Thesis, Notre Dame Journal of Formal Logic, vol. 28, pp. 490498.
[Kleene, 1987 b] Kleene S. C., Gödel's impression on students of logic in the 1930's, Gödel remembered (Weingartner P. and Schmetterer L., editors), Bibliopolis, Naples, pp. 4964.
[Kleene, 1988] Kleene S. C., Turing's analysis of computability, and major applications of it, in Herken 1988, pp. 17–54.
[Kleene-Post, 1954] Kleene S. C. and Post E. L., The upper semi-lattice of degrees of recursive unsolvability, Annals ofMathematics, vol. 59, pp. 379407.
[Kline, 1972] Kline M., Mathematical thought from ancient to modern times, Oxford University Press, Oxford.
[Lerman, 1983] Lerman M., Degrees of unsolvability: Local and global theory, Springer-Verlag, Heidelberg, New York, Tokyo.
[Normann, 1980] Normann D., Recursion on countable functionals, Lecture notes in mathematics, no. 811, Springer-Verlag, Heidelberg, New York.
[Odifreddi, 1989] Odœreddi P., Classical recursion theory, North-Holland, Amsterdam.
[O.E.D., 1989] Simpson J. A. and Weiner E. S. C. (editors), Oxford english dictionary, second ed., Clarendon Press, Oxford.
[Peano, 1889] Peano G., Arithmetices principia, nova methodo exposita, Turin Bocca, English translation in van Heijenoort, 1967, pp. 8397.
[Peano, 1891] Peano G., Sul concetto di numéro, Rivista di Matematica, vol. 1, pp. 87–102, 256267.
[Peirce, 1960] Peirce Charles S., Book II. Speculative grammar, Volume II: Elements of logic (Hartshorne C. and Weiss P., editors), Collected Papers of Charles Sanders Peirce , The Belknap Press of Harvard University Press, Cambridge, Massachusetts and London, England.
[Penrose, 1994] Penrose R., Shadows of the mind, Oxford University Press, Oxford.
[Peter, 1934] Péter R., Über den Zussammenhang der verschiedenen Begriffe der rekursiven Funktion, Mathematische Annalen, vol. 110, pp. 612632.
[Peter, 1951] Péter R., Rekursive funktionen, Akadémaiai Kiadó (Akademische Verlag), Budapest, Recursive functions , third revised ed., Academic Press, New York, 1967.
[Platek, 1966] Platek R., Foundations of recursion theory, Ph.D. thesis , Stanford University, Stanford, CA.
[Post, 1936] Post E. L., Finite combinatory processes-formulation I, Journal of Symbolic Logic, vol. 1, pp. 103105, reprinted in Davis 1965, pp. 288–291.
[Post, 1941] Post E. L., Absolutely unsolvable problems and relatively undecidable propositions: Account of an anticipation, submitted for publication in 1941; printed in Davis 1965, pp. 340–433.
[Post, 1943] Post E. L., Formal reductions of the general combinatorial decision problem, American Journal of Mathematics, vol. 65, pp. 197215.
[Post, 1944] Post E. L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50, pp. 284316, reprinted in Davis 1965, pp. 304–337.
[Post, 1947] Post E. L., Recursive unsolvability of a problem of Thue, Journal of Symbolic Logic, vol. 12, pp. 111, reprinted in Davis 1965, pp. 292–303.
[Post, 1948] Post E. L., Degrees of recursive unsolvability: preliminary report (abstract), Bulletin of the American Mathematical Society, vol. 54, pp. 641642.
[Putnam, 1995] Putnam H., Review of Penrose 1994, Bulletin of the American Mathematical Society, vol. 32, pp. 370373.
[Sacks, 1990] Sacks G. E., Higher recursion theory, Springer-Verlag, Heidelberg, New York.
[Shoenfield, 1967] Shoenfield J. R., Mathematical logic, Addison-Wesley, Reading, Massachusetts.
[Shoenfield, 1991] Shoenfield J. R., Recursion theory, Lecture notes in logic, Springer-Verlag, Heidelberg, New York.
[Shoenfield, 1995] Shoenfield J. R., The mathematical work of S. C. Kleene, this Bulletin, vol. 1, pp. 843.
[Sieg, 1994] Sieg W., Mechanical procedures and mathematical experience, Mathematics and mind (George A., editor), Oxford University Press.
[Sieg and Byrnes, 1995] Sieg W. and Byrnes J., K-graph machines: generalizing Turing's machines and arguments, preprint.
[Skolem, 1923] Skolem T., Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbare Veränderlichen mit unendlichen Ausdehnungsbereich, Skrifter utgit av Videnskapsselskapet i Kristiania, I. Mathematisk-Naturvidenskabelig Klasse, vol. 6, English translation in van Heijenoort, 1967, pp. 302333.
[Soare, ta 1] Soare R. I., An overview of the computably enumerable sets, Handbook of computability theory, North-Holland, Amsterdam, in preparation.
[Soare, ta 2] Soare R. I., Computability and enumerability, Proceedings of the 10th international congress of logic, methodology, and philosophy of science, August 19–25, 1995, Florence, Italy, to appear.
[Soare, 1981] Soare R. I., Constructions in the recursively enumerable degrees, Recursion theory and computational complexity (Lolli G., editor), Proceedings of Centro Internazionale Matematico Estivo (C.I.M.E.), 06 14–23, 1979, in Bressanone, Italy, Liguori Editore, Naples, Italy.
[Soare, 1987] Soare R. I., Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg.
[Tamburrini, 1995] Tamburrini G., Mechanistic theories in cognitive science: The import of Turing's Thesis, Proceedings of the 10th international congress of logic, methodology, and philosophy ofscience, August 19–25, 1995, Florence, Italy, to appear.
[Turing, 1936] Turing A. M., On computable numbers, with an application to the Entscheidungsproblem, parts 3 and 4, Proceedings of the London Mathematical Society, vol. 42, pp. 230265; [Turing, 1937] A correction, Proceedings of the London Mathematical Society , vol. 43, pp. 544–546.
[Turing, 1937 b] Turing A. M., Computability and λ-definability, Journal of Symbolic Logic, vol. 2, pp. 153163.
[Turing, 1939] Turing A. M., Systems of logic basedon ordinals, part 3, Proceedings of the London Mathematical Society, vol. 45, pp. 161228, reprinted in Davis 1965, pp. 154–222.
[Turing, 1948] Turing A. M., Intelligent machinery, Machine Intelligence, vol. 5, pp. 323, written in September, 1947 and submitted to the National Physical Laboratory in 1948.
[Turing, 1949] Turing A. M., Text of a lecture by Turing on June 24, 1949, in Morris F. L. and Jones C. B., An early program proof by Alan Turing, Annals of the History of Computing, vol. 6 (1984), pp. 139143.
[Turing, 1950] Turing A. M., Computing machinery and intelligence, Mind, vol. 59, pp. 433460.
[Turing, 1950 b] Turing A. M., The word problem in semi-groups with cancellation, Annals of Mathematics, vol. 52, pp. 491505.
[Turing, 1954] Turing A. M., Solvable and unsolvable problems, Science News, vol. 31, pp. 723.
[Turing, 1986] Turing A. M., Lecture to the London Mathematical Society on 20 February 1947, A. M. Turing's ACE report of 1946 and other papers (Carpenter B. E. and Doran R.W., editors), Cambridge University Press, pp. 106124.
[van Heijenoort, 1967] Heijenoort J. van (editor), From Frege to Godel, a sourcebook in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Massachusetts.
[Webster, 1993] Gove Ph. B. (editor), Webster's third new international dictionary of the English language (unabridged), Merriam-Webster Inc., Publishers, Springfield, Massachusetts, U.S.A.
[Zabell, 1995] Zabell S. L., Alan Turing and the central limit theorem, American Mathematical Monthly, vol. 102, no. 6, pp. 483494.
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