Skip to main content
SpringerLink
Log in
Book cover

Category Theory and Computer Science pp 182–197Cite as

Graded multicategories of polynomial-time realizers

  • Conference paper
  • First Online:

  • 158 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 389))

Abstract

We present a logical calculus which imposes a grading on a sequent-style calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomial-time realizability. An extension of the grading is also considered, giving a notion of “dependant grades”, which is also sound. Furthermore, we define a notion of closed graded multicategory, and show how the structure of polynomial-time realizers has that structure.

This work was done following a visit to Monash University and is based on the joint work done there by J. N. Crossley, G.L. Mathai, and the author, who wishes to express his thanks to those two for their many kindnesses during his visit.

Research partially supported by a grant from Le Fonds F.C.A.R., Québec.

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. S. Buss, Bounded Arithmetic, Doctoral dissertation, Princeton University, 1985.

    Google Scholar 

  2. S. Buss, “The polynomial time hierarchy and intuitionistic bounded arithmetic”, Proceedings of the First Symposium on Structures and Complexity, 1986, (IEEE Publications).

    Google Scholar 

  3. S. Cook and A. Urquhart, “Functional interpretations of feasibly constructive arithmetic”, Technical Report 210/88, Department of Computer Science, University of Toronto, 1988.

    Google Scholar 

  4. J.N. Crossley, “Proofs, programs and run-times”, Preprint, Monash University, 1989.

    Google Scholar 

  5. J.N. Crossley, G.L. Mathai, and R.A.G. Seely, “A logical calculus for polynomial time realizability”, Preprint, Monash University, 1989, (final version in preperation).

    Google Scholar 

  6. G. Gentzen, “Investigations into logical deductions”, in M.E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, 1969.

    Google Scholar 

  7. J.-Y. Girard, “Linear logic”, J. Theoretical Computer Science 50 (1987), 1–102.

    Google Scholar 

  8. J. Lambek, “Deductive systems and categories I”, J. Math. Systems Theory 2 (1968), 278–318.

    Google Scholar 

  9. J. Lambek, “Deductive systems and categories II”, Springer LNM 86 (1969), 76–122.

    Google Scholar 

  10. J. Lambek, “Multicategories revisited”, Proceedings of the A.M.S. Summer Conference on Categories in Logic and Computer Science, Boulder 1987.

    Google Scholar 

  11. A. Nerode, J.B. Remmel, and A. Scedrov, “Polynomially graded logic”, Proceedings of the Fourth Symposium on Logic in Computer Science, Asilomar 1989 (IEEE Publications).

    Google Scholar 

  12. R.A.G. Seely, “Hyperdoctrines, natural deduction, and the Beck condition”, Zeitsch. f. math. Logik und Grundlagen d. Math. 29 (1983), 505–542.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, John Abbott College, Ste. Anne de Bellevue, Canada

    R. A. G. Seely

  2. McGill University, Montréal, Québec

    R. A. G. Seely

Authors
  1. R. A. G. Seely
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

David H. Pitt David E. Rydeheard Peter Dybjer Andrew M. Pitts Axel Poigné

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Seely, R.A.G. (1989). Graded multicategories of polynomial-time realizers. In: Pitt, D.H., Rydeheard, D.E., Dybjer, P., Pitts, A.M., Poigné, A. (eds) Category Theory and Computer Science. Lecture Notes in Computer Science, vol 389. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0018352

Download citation

  • DOI: https://doi.org/10.1007/BFb0018352

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51662-0

  • Online ISBN: 978-3-540-46740-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation