Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Mathematical Foundations of Computer Science

MFCS 1981: Mathematical Foundations of Computer Science 1981 pp 46–60Cite as

A survey of some recent results on computational complexity in weak theories of arithmetic

  • Invited Lectures
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 118))

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. Adleman, L. and Manders, K., "Computational complexity of decision procedures for polynomials," Proceedings of the 17th Annual IEEE Symp on Switching and Automata Theory, pp. 169–177 (1976).

    Google Scholar 

  2. Baker, T. P., "On ‘provable’ analogs of P and NP," Math Systems Theory Vol. 12 pp. 213–218 (1979).

    Google Scholar 

  3. Berman, L. and Hartmanis, J., "On the isomorphism and density of NP and other complete sets," SIAM J Comput Vol. 6 pp. 305–322 (Dec 1977).

    Google Scholar 

  4. Blum, M., "A machine independent theory of the complexity of the recursive functions," J Association Comput Mach Vol. 14(2) pp. 322–336 (1967).

    Google Scholar 

  5. Brainerd, W. and Landweber, L., Theory of Computation, John Wiley & Sons, New York (1974).

    Google Scholar 

  6. Csirmaz, L., "Structure of program runs of non-standard time," Acta Cybernetica Vol. 4(4) pp. 325–331 (1980).

    Google Scholar 

  7. Csirmaz, L., "Programs and program verification in a general setting," Theor Comput Sci, (to appear).

    Google Scholar 

  8. DeMillo, R. A. and Lipton, R. J., "Some connections between mathematical logic and complexity theory," Proceedings of the 11th Symposium on the Theory of Computation, pp. 153–159 (1979).

    Google Scholar 

  9. DeMillo, R. A. and Lipton, R. J., "The consistency of ‘P = NP’ and related problems with fragments of number theory," Proceedings of the 12th Symposium on the Theory of Computing, pp. 45-57 (May 1980).

    Google Scholar 

  10. Dimitracopolous, D. and Paris, J., "Truth definitions for delta-zero formulas," Preprint, (1980).

    Google Scholar 

  11. Fischer, P. C., "Theory of provably recursive functions," Trans Amer Math Soc Vol. 117(5) pp. 494–520 (May 1965).

    Google Scholar 

  12. Fortune, S., Topics in Computational Complexity, Cornell University (1979). (PhD Dissertation)

    Google Scholar 

  13. Fortune, S., Leivant, D., and O'Donnell, M., "The expressiveness of simple and second order type structures," IBM Research Report #RC 8542, (1980).

    Google Scholar 

  14. Goldrei, D. C., Macintyre, A., and Simmons, H., "The forcing companions of number theories," Israel J Math Vol. 14 pp. 317–337 (1973).

    Google Scholar 

  15. Gordon, D., "Complexity classes of provably recursive functions," J Comput and System Sci Vol. 18(3) pp. 294–303 (June 1979).

    Google Scholar 

  16. Grant, P. W., "Some more independence results in complexity theory," Theor Comput Sci Vol. 12 pp. 119–126 (1980).

    Google Scholar 

  17. Hajek, P., "Arithmetical complexity of some problems in computer science," pp. 282–287 in Mathematical Foundations of Computer Science, ed. J. Gruska,Springer-Verlag (1977).

    Google Scholar 

  18. Hajek, P., "Arithmetical hierarchy and complexity of computation," Theor Comput Sci Vol. 8 pp. 227–237 (1979).

    Google Scholar 

  19. Harrow, K., "The bounded arithmetic hierarchy," Inf and Control Vol. 36 pp. 102–117 (1978).

    Google Scholar 

  20. Hartmanis, J., "Relation between diagonalization, proof systems and complexity gaps," Proceedings of the 9th Symposium on the Theory of Computing, pp. 223–227 (1977).

    Google Scholar 

  21. Hartmanis, J. and Hopcroft, J. E., "Independence results in computer science," SIGACT News Vol. 8(4) pp. 13–24 (1976).

    Google Scholar 

  22. Hirschfeld, J. and Wheeler, W. H., Forcing, arithmetic, and division rings, Springer-Verlag Lecture Notes in Mathematics #454, Berlin & New York (1975).

    Google Scholar 

  23. Homer, S. and Reif, J. H., "The complexity of provable properties of first order theories," Preprint, (May 1980).

    Google Scholar 

  24. Joseph, D., On the power of formal systems for analyzing linear and polynomial time program behavior, Purdue University (1981). (Ph.D. Dissertation)

    Google Scholar 

  25. Joseph, D. and Young, P., "Fast programs for initial segments and polynomial time computation in weak models of arithmetic (preliminary abstract)," Proceedings of the 13th Symposium on the Theory of Computing, pp. 55–61 (May 1981).

    Google Scholar 

  26. Joseph, D. and Young, P., "Independence results in computer science?," J Comput Systems Sci, (to appear). (preliminary version in Proc. 12th SIGACT Symp Theor Comput, 1980)

    Google Scholar 

  27. Karp, R., "Reducibility among combinatorial problems," in Complexity of Computer Computations, ed. Miller and Thatcher,Plenum Press (1972).

    Google Scholar 

  28. Kent, C. and Hodgson, B., "An arithmetical characterization of NP," Lakehead Univ Technical Report, #6-80, (1980). (Dept of Mathematics, Thunder Bay, Ontario, Can.)

    Google Scholar 

  29. Kirby, L., Private communication

    Google Scholar 

  30. Kirby, L. and Paris, J., "Initial segments of models of Peano's axioms," in Proceedings Conference Set Theory & Hierarchy Theory V; Bieutowice, Poland, Springer-Verlag Lecture Notes in Mathematics #619, Berlin & New York (1976).

    Google Scholar 

  31. Leivant, D., "The complexity of parameter passing in polymorphic procedures," Proceedings of the 13th Symposium on the Theory of Computing, pp. 38–45 (May 1981).

    Google Scholar 

  32. Leivant, D., "Unprovability of theorems of complexity theory in weak number theories," Theor Comput Sci, (to appear).

    Google Scholar 

  33. Lipton, R. J., "Model theoretic aspects of complexity theory," Proceedings of the 19th Symposium of the Foundations of Computer Science, pp. 193–200 (1978).

    Google Scholar 

  34. Macintyre, A. and Simmons, H., "Algebraic properties of number theories," Israel J Math Vol. 22 pp. 7–27 (1975).

    Google Scholar 

  35. Mahaney, S., "Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis," Proc. of the 21st Symposium on the Foundations of Computer Science, pp. 54–60 (1980).

    Google Scholar 

  36. Manders, K., "Computational complexity of decision problems in elementary number theory," in Proceedings of the Set Theory and Hierarchy Theory Conference, Springer-Verlag, Karpacz (1979). (Lecture Notes in Mathematics Series.)

    Google Scholar 

  37. Matijesevic, Y., "Enumerable sets are diophantine," Dokl Akad Nauk SSSR Vol. 191 pp. 279–282 (1970).

    Google Scholar 

  38. McAloon, K.(ed.), Modeles de l'Arithmetique, Societe Mathematique de France, Paris (1980).

    Google Scholar 

  39. O'Donnell, M., "A programming language theorem which is independent of Peano Arithmetic," Proceedings of the 11th Symposium on the Theory of Computation, pp. 176–188 (1979).

    Google Scholar 

  40. Paris, J. and Wilkie, A., "Models of arithmetic and rudimentary sets," Preprint, (1980).

    Google Scholar 

  41. Paris, J. B., "Some independence results for Peano Arithmetic," J Symbolic Logic Vol. 43 pp. 725–730 (Dec. 1978).

    Google Scholar 

  42. Paris, J. B. and Harrington, L., "A mathematical incompleteness in Peano arithmetic," in Handbook of Mathematical Logic, ed. J. Barwise,North-Holland, New York (1978).

    Google Scholar 

  43. Phillips, N. C. K., "The P = NP question and recent independence results," Quaestiones Informaticae Vol. 1 pp. 26–29 (June 1979).

    Google Scholar 

  44. Rabin, M., "Degree of difficulty of computing a function and a partial ordering of recursive sets," Technical Report Hebrew Univ, (1960).

    Google Scholar 

  45. Rogers, H., The Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).

    Google Scholar 

  46. Simmons, H., "Existentially closed models of basic number theory," pp. 325–369 in Logic Colloquium 76, ed. R. Gandy and J. M. R. Hyland,North-Holland (1977).

    Google Scholar 

  47. Smullyan, R., "Theory of formal systems," in Annals of Mathematics no. 47, Princeton Univ Press (1961).

    Google Scholar 

  48. Stockmeyer, L., "The polynomial-time hierarchy," Theo Comput Sci Vol. 3 pp. 1–22 (1977).

    Google Scholar 

  49. Wilkie, A., "to appear," in Proceedings of the Set Theory and Hierarchy Theory Conference, Springer-Verlag, Karpacz (1979). (Lecture Notes in Mathematics Series.)

    Google Scholar 

  50. Wilkie, A., "Applications of complexity theory to sigma-zero definability problems in arithmetic," Preprint, (1980).

    Google Scholar 

  51. Wrathall, C., "Rudimentary predicates and relative computability," Siam J Comput Vol. 7 pp. 194–209 (1978).

    Google Scholar 

  52. Young, P., "Optimization among provably equivalent programs," J Assoc Comput Mach Vol. 24(4) pp. 693–700 (1977). (Prelim abstract in IEEE SWAT Proc 1973).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Cornell University, 14853, Ithaca, New York, USA

    Deborah Joseph

  2. Computer Sciences and Mathematics Departments, Purdue University, 47907, Lafayette, Indiana, USA

    Paul Young

Authors
  1. Deborah Joseph
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Young
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jozef Gruska Michal Chytil

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Joseph, D., Young, P. (1981). A survey of some recent results on computational complexity in weak theories of arithmetic. In: Gruska, J., Chytil, M. (eds) Mathematical Foundations of Computer Science 1981. MFCS 1981. Lecture Notes in Computer Science, vol 118. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-10856-4_73

Download citation

  • DOI: https://doi.org/10.1007/3-540-10856-4_73

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10856-6

  • Online ISBN: 978-3-540-38769-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation