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Elementary realizability

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Abstract

A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π10-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π10 formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π10-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount of computational resources required can be bounded in advance.

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REFERENCES

  1. Boolos, G. S. and Jeffrey, R. C.: Computability and Logic, 2nd edition, Cambridge University Press, 1980.

  2. Brainerd, W. S. and Landweber, L. H.: Theory of Computation, J. Wiley & Sons, New York, 1974.

    Google Scholar 

  3. Buss, S. R. and Scott, P. J. (eds): Feasible Mathematics, Birkhäuser, Boston–Basel–Berlin, 1990.

    Google Scholar 

  4. Cleave, J. P.: A Hierarchy of Primitive Recursive Functions, Zeitschrift für mathematische Logik und Grundlagen der Mathematik Bd. 9 (1963), 331–345.

    Google Scholar 

  5. Clote, P. and Takeuti, G.: Exponential Time and Bounded Arithmetic (Extended Abstract), in: A. L. Selman (ed), Structure in Complexity Theory: Proceedings of the Conference held at the University of California, Berkeley, California, June 1986, Lecture Notes in Computer Science vol. 223, Springer-Verlag, Berlin–Heidelberg–New York, 1986.

    Google Scholar 

  6. Cobham, A.: The Intrinsic Computational Difficulty of Functions, Y. Bar-Hillel (ed), Proceedings of the 1964 Congress on Logic, Methodology and Philosophy of Science, North-Holland, Amsterdam, 1965.

    Google Scholar 

  7. Cook, S. A. and Urquhart, A.: Functional interpretations of feasibly constructive arithmetic (Extended Abstract), in: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, New York: Association for Computing Machinery, 1989, 107–112.

    Google Scholar 

  8. Cutland, N. J.: Computability: An Introduction to Recursive Function Theory, Cambridge University Press, 1980.

  9. Damnjanovic, Z.: Elementary Functions and Loop Programs, in: Notre Dame Journal of Formal Logic vol. 35, No. 3 (1994).

  10. Damnjanovic, Z.: Strictly primitive recursive realizability, I, in: Journal of Symbolic Logic vol. 59 (1994), 1210–1227.

    Google Scholar 

  11. Damnjanovic, Z.: Minimal realizability of intuitionistic arithmetic and elementary analysis, in Journal of Symbolic Logic vol. 60 (1995), 1208–1241.

    Google Scholar 

  12. Harrow, K.: The Bounded Arithmetic Hierarchy, Information and Control vol. 36 (1978), 102–117.

    Google Scholar 

  13. Kleene, S. C.: Introduction to Metamathematics, Van Nostrand, New York, 1952.

    Google Scholar 

  14. Machtey, M. and Young, P.: An Introduction to the General Theory of Algorithms, North-Holland, New York–Oxford, 1979.

    Google Scholar 

  15. Parikh, R.: Existence and Feasibility in Arithmetic, Journal of Symbolic Logic vol. 39 (1974), 494–508.

    Google Scholar 

  16. Parsons, C.: Hierarchies of Primitive Recursive Functions, Zeitschrift für mathematische Logik und Grundlagen der Mathematik Bd. 14 (1968), 357–376.

    Google Scholar 

  17. Ritchie, R. W.: Classes of Predictably Computable Functions, Transactions of the American Mathematical Society vol. 89 (1963), 139–173.

    Google Scholar 

  18. Rose, H. E.: Subrecursion: Functions and Hierarchies, Clarendon Press, Oxford, 1984.

    Google Scholar 

  19. Rödding, D.: Über die Eliminierbarkeit von Definitionsschemata in der Theorie der rekursiven Funktionen, Zeitschrift für mathematische Logik und Grundlagen der Mathematik Bd. 10 (1964), 315–330.

    Google Scholar 

  20. Rödding, D.: Klassen rekursiver Funktionen, in: M. H. Löb (ed), Proceedings of the Summer School in Logic, Leeds 1967, Springer-Verlag, Berlin–Heidelberg–New York, 1968.s

    Google Scholar 

  21. Smullyan, R.: Theory of Formal Systems, Princeton University Press, 1961.

  22. Tait, W. W.: Finitism, Journal of Philosophy vol. 78 (1981), 524–546.

    Google Scholar 

  23. Troelstra, A. S.: Constructivism in Mathematics: An Introduction Vol. 1, North-Holland, Amsterdam, 1988.

    Google Scholar 

  24. Weinstein, S.: The intended interpretation of intuitionistic logic, in: Journal of Philosophical Logic vol. xii (1983), 261–270.

    Google Scholar 

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Damnjanovic, Z. Elementary realizability. Journal of Philosophical Logic 26, 311–339 (1997). https://doi.org/10.1023/A:1017994504149

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