Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Science Logic

CSL 2002: Computer Science Logic pp 367–381Cite as

Implicit Computational Complexity for Higher Type Functionals

Extended Abstract

  • Conference paper
  • First Online:

  • 533 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2471))

Abstract

In previous works we argued that second order logic with comprehension restricted to positive formulas can be viewed as the core of Feasible Mathematics. Indeed, the equational programs over strings that are provable in this logic compute precisely the poly-time computable functions.

Here we investigate the provable functionals of this logic, and show that they are precisely Cook and Urquhart’s basic feasible functionals, BFF. This further confirms the stability of BFF as a notion of computational feasibility in higher type.

Using a formula-as-type morphism, we also show that BFF consists precisely of the functionals that are lambda representable in F 2 restricted to positive type arguments (and trivially augmented with basic constructors and destructors).

Research partially supported by NSF grant CCR-0105651.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Samuel Buss. Bounded Arithmetic. Bibliopolis, Naples, 1986.

    Google Scholar 

  2. Samuel Buss. The polynomial hierarchy and intuitionistic bounded arithmetic. In Structure in Complexity, LNCS 233, pages 77–103, Berlin, 1986. Springer-Verlag.

    Google Scholar 

  3. Peter Clote. A note on the relation between polynomial time functionals and constable’s class κ. In Hans Kleine-Büuning, editor, Computer Science Logic, LNCS 1092, pages 145–160, Berlin, 1996. Springer-Verlag.

    Google Scholar 

  4. 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 

  5. Robert Constable. Type 2 computational complexity. In Fifth Annual ACM Symposium on Theory of Computing, pages 108–121, New York, 1973. ACM.

    Google Scholar 

  6. Stephen Cook. Computability and complexity of higher type functions. In Y. Moschovakis, editor, Logic from Computer Science, pages 51–72. Springer-Verlag, New York, 1991.

    Google Scholar 

  7. Stephen A. Cook and Alasdair Urquhart. Functional interpretations of feasible constructive arithemtic (extended abstract). In Proceedings of the 21st ACM Symposium on Theory of Computing, pages 107–112, 1989.

    Google Scholar 

  8. Stephen A. Cook and Alasdair Urquhart. Functional interpretations of feasible constructive arithemtic. Annals of Pure and Applied Logic, 63:103–200, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  9. Steven Fortune. Topics in computational complexity. Phd dissertation, Cornell University, 1979.

    Google Scholar 

  10. Steven Fortune, Daniel Leivant, and Michael O’Donnell. The expressiveness of simple and second-order type structures. Journal of the ACM, 30(1):151–185, January 1983.

    Google Scholar 

  11. Jean-Yves Girard. Une extension de l’interprétation de Gödel à l’anal yse, et son application a l’élimination des coupures dans l’analyse et la théorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 63–92, Amsterdam, 1971. North-Holland.

    Google Scholar 

  12. Kurt Gödel. Uber eine bisher noch nicht benutzte erweiterung des finiten standpunktes. Dialectica, 12:280–287, 1958.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Grzegorczyk. Some classes of recursive functions. In Rozprawy Mate. IV. Warsaw, 1953.

    Google Scholar 

  14. R. Irwin, B. M. Kapron, and J. Royer. On characterizations of the basic feasible functionals part i. Journal of Functional Programming, 11:117–153, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. M. Kapron and S. A. Cook. A new characerization of type-2 feasibility. SIAM Journal of Computing, 25:117–132, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  16. Bruce Kapron and Stephen Cook. Characterizations of the basic feasible functionals of finite type. In S. Buss and P. Scott, editors, Feasible Mathematics, pages 71–95. Birkhauser-Boston, 1990.

    Google Scholar 

  17. Daniel Leivant. Contracting proofs to programs. In P. Odifreddi, editor, Logic and Computer Science, pages 279–327. Academic Press, London, 1990.

    Google Scholar 

  18. Daniel Leivant. A foundational delineation of poly-time. Information and Computation, 110:391–420, 1994. (Special issue of selected papers from LICS’91, edited by G. Kahn). Preminary report: A foundational delineation of computational feasibility, in Proceedings of the Sixth IEEE Conference on Logic in Computer Science, IEEE Computer Society Press, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  19. Daniel Leivant. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Peter Clote and Jeffrey Remmel, editors, Feasible Mathematics II, Perspectives in Computer Science, pages 320–343. Birkhauser-Boston, New York, 1994.

    Google Scholar 

  20. Daniel Leivant. Termination proofs and complexity certification. In N. Kobayashi and B. Pierce, editors, Theoretical aspects of computer software, volume 2215 of LNCS, pages 183–200, Berlin, 2001. Springer-Verlag.

    Chapter  Google Scholar 

  21. Daniel Leivant. Calibrating computational feasibility by abstraction rank. In Gordon Plotkin, editor, Seventeenth IEEE Annual Symposium on Logic in Computer Science. IEEE Computer society Press, 2002.

    Google Scholar 

  22. Daniel Leivant. Intrinsic reasoning about functional programs I: first order theories. Annals of Pure and Applied Logic, 114:117–153, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  23. Daniel Leivant and Jean-Yves Marion. Lambda calculus characterizations of poly-time. Fundamenta Informaticae, 19:167–184, 1993.

    MATH  MathSciNet  Google Scholar 

  24. Kurt Mehlhorn. Polynomial and abstract subrecursive classes. JCSS, 12:147–178, 1976.

    MATH  MathSciNet  Google Scholar 

  25. Michael O’Donnell. A programming language theorem which is independent of Peano Arithmetic. In Eleventh Annual ACM Symposium on Theory of Computing. ACM, 1979.

    Google Scholar 

  26. John Reynolds. Towards a theory of type structures. In J. Loeckx, editor, Conference on Porgrammin, pages 408–425, Berlin, 1974.

    Google Scholar 

  27. Anil Seth. Some desirable conditions for feasible functionals of type 2. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 320–331, Washington, DC, 1993. IEEE Computer Society Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Daniel Leivant

Authors
  1. Daniel Leivant
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratory for Foundations of Computer Science Division of Informatics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Julian Bradfield

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Leivant, D. (2002). Implicit Computational Complexity for Higher Type Functionals. In: Bradfield, J. (eds) Computer Science Logic. CSL 2002. Lecture Notes in Computer Science, vol 2471. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45793-3_25

Download citation

  • DOI: https://doi.org/10.1007/3-540-45793-3_25

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44240-0

  • Online ISBN: 978-3-540-45793-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation