Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T18:35:08.417Z Has data issue: false hasContentIssue false
  • English
  • Français

A second order version of S 2 i and U 2 1

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti
Gaisi Takeuti*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Get access Rights & Permissions [Opens in a new window]

Extract

In [1] S. Buss introduced systems of bounded arithmetic , , , (i = 1, 2, 3, …). and are first order systems and and are second order systems. and are closely related to and respectively in the polynomial hierarchy, and and are closely related to PSPACE and EXPTIME respectively. One of the most important problems in bounded arithmetic is whether the hierarchy of bounded arithmetic collapses, i.e. whether = or = for some i, or whether = , or whether is a conservative extension of S 2 = ⋃i . These problems are relevant to the problems whether the polynomial hierarchy PH collapses or whether PSPACE = PH or whether PSPACE = EXPTIME. It was shown in [4] that = implies and consequently the collapse of the polynomial hierarchy. We believe that the separation problems of bounded arithmetic and the separation problems of computational complexities are essentially the same problem, and the solution of one of them will lead to the solution of the other.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 1038 - 1063
Copyright
Copyright © Association for Symbolic Logic 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1] Buss, S., Bounded arithmetic, Bibliopolis, Napoli, 1986.Google Scholar
[2] Buss, S., Axiomatizations and conservation results for fragments of bounded arithmetic, Logic and computation, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence, Rhode Island, 1990, pp. 57–84.CrossRefGoogle Scholar
[3] Clote, P. and Takeuti, G., Bounded arithmetic for NC, AlogTime, L and NL, Manuscripts, 1989.Google Scholar
[4] Krajíček, J., Pudlák, P., and Takeuti, G., Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic (to appear).Google Scholar
[5] Savage, J. E., The complexity of computing, Wiley, New York, 1976; reprint, Krieger, Melbourne, Florida, 1987.Google Scholar
[6] Takeuti, G., Proof theory, 2nd ed., North-Holland, Amsterdam, 1980.Google Scholar
[7] Takeuti, G., Bounded arithmetic and truth definition, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 75–104.CrossRefGoogle Scholar
[8] Takeuti, G., Sharply bounded arithmetic and the function a ∸ 1, Logic and computation, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence, Rhode Island, 1990, pp. 281–288.CrossRefGoogle Scholar
[9] Takeuti, G., and (BD), Archive for Mathematical Logic, vol. 29 (1990), pp. 149–169.CrossRefGoogle Scholar