Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Logical Foundations of Computer Science

LFCS 1994: Logical Foundations of Computer Science pp 227–239Cite as

Predicative recurrence in finite types

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 813))

Abstract

We consider the functionals defined using an extension to higher types of predicative recurrence, introduced in [Lei90b, BC92, Lei93b]. Three styles of predicative recurrence over a free algebra A are examined: equational recurrence, applicative programs with recurrence operators, and purely applicative higher-type programs. We show that, for every free algebra A and each one of these styles, the functions defined by predicative recurrence in finite types are precisely the functions over A that are computable in a number of steps elementary in the size of the input. This should be contrasted with unrestricted higher type recurrence over ℕ, which yields all provably recursive functions of first order arithmetic [Göd58].

The same equivalences holds for natural notions of computing relative to a class of functions (as oracles). This shows that every class of functions closed under composition with elementary functions is closed under any higher order functional defined by predicative recurrence. Among such functionals are natural generalizations of recurrence, such as recurrence with parameter substitution. This generalizes a result of [Sim88].

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Corrado Böhm and Allessandro Berarducci. Automatic synthesis of typed λ-programs on term algebras. Theoretical Computer Science, 39:135–154, 1985.

    Google Scholar 

  2. Stephen Bellantoni and Stephen Cook. A new recursion-theoretic characterization of the poly-time functions, 1992.

    Google Scholar 

  3. Stephen Bellantoni. Predicative recursion and the polytime hierarchy. In Peter Clote and Jeffery Remmel, editors, Feasible Mathematics II, Perspectives in Computer Science. Birkhäuser, 1994.

    Google Scholar 

  4. Stephen Bloch. Functional characterizations of uniform log-depth and polylog-depth circuit families. In Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pages 193–206. IEEE Computer Society Press, 1992.

    Google Scholar 

  5. A. Cobham. The intrinsic computational difficulty of functions. In Y. Bar-Hillel, editor, Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, pages 24–30. North-Holland, Amsterdam, 1962.

    Google Scholar 

  6. Kurt Gödel. Über eine bisher noch nicht benüte Erweiterung des finiten Standpunktes. Dialectica, 12:280–287, 1958. Republished with English translation and explanatory notes by A. S. Troelstra in Kurt Gödel: Collected Works, Vol. II. S. Feferman, ed. Oxford University Press, 1990.

    Google Scholar 

  7. W.G. Handley. Bellantoni and Cook's characterization of polynomial time functions. Manuscript, 1993.

    Google Scholar 

  8. J.van Heijenoort. From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA, 1967.

    Google Scholar 

  9. David Hilbert. Über das unendliche. Mathematische Annalen, 95:161–190, 1925. English translation in [Hei67], pages 367–392.

    Google Scholar 

  10. David Isles. What evidence is there that 2∧65536 is a natural number? Notre Dame Journal of Formal Logic, 1993.

    Google Scholar 

  11. L. Kalmar. Egyszerü példa eldönthetetlen aritmetikai problémára. Mate. és Fizikai Lapok, 50:1–23, 1943.

    Google Scholar 

  12. Daniel Leivant. Computationally based set existence principles. In P. Lorents, G. Mints, and E. Tyugu, editors, Proceedings of the COLOG'88 Conference, pages 43–57, Tallinn, 1988. Institute of Cybernetics of the Academy of Sciences of Estonia.

    Google Scholar 

  13. Daniel Leivant. Computationally based set existence principles. In W. Sieg, editor, Logic and Computation, Volume 106 of Contemporary Mathematics, pages 197–211. American Mathematical Society, Providence, RI, 1990.

    Google Scholar 

  14. Daniel Leivant. Subrecursion and lambda representation over free algebras. In Samuel Buss and Philip Scott, editors, Feasible Mathematics, Perspectives in Computer Science, pages 281–291. Birkhauser-Boston, New York, 1990.

    Google Scholar 

  15. Daniel Leivant. A foundational delineation of computational feasibility. In Proceedings of the Sixth IEEE Conference on Logic in Computer Science, Washington, D.C., 1991. IEEE Computer Society Press.

    Google Scholar 

  16. Daniel Leivant. Semantic characterization of number theories. In Y. Moschovakis, editor, Logic from Computer Science, pages 295–318. Springer-Verlag, New York, 1991.

    Google Scholar 

  17. Daniel Leivant. Functions over free algebras definable in the simply typed lambda calculus. Theoretical Computer Science, 121:309–321, 1993.

    Google Scholar 

  18. Daniel Leivant. Stratified functional programs and computational complexity. In Conference Record of the Twentieth Annual ACM Symposium on Principles of Programming Languages, New York, 1993. ACM.

    Google Scholar 

  19. Daniel Leivant. A foundational delineation of poly-time. Information and Computation, 1994. (Special issue of selected papers from LICS'91, edited by G. Kahn).

    Google Scholar 

  20. Daniel Leivant. Peano theories and the formalization of feasible mathematics, 1994. To appear.

    Google Scholar 

  21. Daniel Leivant. Predicative recurrence and computational complexity I: Word recurrence and poly-time. 1994. To appear.

    Google Scholar 

  22. Daniel Leivant and Jean-Yves Marion. Lambda-calculus characterizations of poly-time. Fundamenta Informaticae, 1993.

    Google Scholar 

  23. Daniel Leivant and Jean-Yves Marion. Predicative recurrence over free algebras and computability in polynomial space. 1994. to appear.

    Google Scholar 

  24. Edward Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.

    Google Scholar 

  25. Rósza Péter. Rekursive Funktionen. Akadémiai Kiadó, Budapest, 1966. English translation: Recursive Functions, Academic Press, New York, 1967.

    Google Scholar 

  26. H.E. Rose. Subrecursion. Clarendon Press (Oxford University Press), Oxford, 1984.

    Google Scholar 

  27. Harold Simmons. The realm of primitive recursion. Archive for Mathematical Logic, 27:177-, 1988.

    Google Scholar 

  28. Richard Statman. The typed λ-calculus is not elementary recursive. Theoretical Computer Science, 9:73–81, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Indiana University, 47405, Bloomington, IN

    Daniel Leivant

Authors
  1. Daniel Leivant
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Anil Nerode Yu. V. Matiyasevich

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Leivant, D. (1994). Predicative recurrence in finite types. In: Nerode, A., Matiyasevich, Y.V. (eds) Logical Foundations of Computer Science. LFCS 1994. Lecture Notes in Computer Science, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58140-5_23

Download citation

  • DOI: https://doi.org/10.1007/3-540-58140-5_23

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58140-6

  • Online ISBN: 978-3-540-48442-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation