Skip to main content
SpringerLink
Log in

On parallel hierarchies and R ik

  • Conference paper
  • First Online:
Logic and Computational Complexity (LCC 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 960))

Included in the following conference series:

  • 152 Accesses

Abstract

This paper defines natural hierarchies of function and relation classes, constructed from parallel complexity classes in a manner analogous to the polynomial-time hierarchy. A number of structural results about these classes are proven: relationships between them and the levels of PH, a Buss-style witnessing theorem relating the levels of these hierarchies to definability in the bounded arithmetic theories R ik (generalizing [1] and improving on [9]), a conservation result between S ik andR i+kk , and results analogous to those of [18, 8, 16] relating conservationbetween theories of bounded arithmetic to the collapse of complexityclasses

This work was sponsored in part by Judy Goldsmith's NSERC operating grant OGP0121527 while the author was at the University of Manitoba; further work was done at the University of Kentucky.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. William Allen. Arithmetizing uniform NC. Annals of Pure and Applied Logic, 53(1):1–50, 1991. See also Divide and Conquer as a Foundation of Arithmetic, Ph.D. thesis, University of Hawaii at Manoa, 1988.

    Article  Google Scholar 

  2. Stephen Bellantoni and Stephen Cook. A new recursion-theoretic characterization of the polytime functions. computational complexity, 2:97–110, Dec 1992.

    Article  Google Scholar 

  3. Stephen Bloch. Divide and Conquer in Parallel Complexity and Proof Theory. PhD thesis, University of California, San Diego, 1992.

    Google Scholar 

  4. Stephen Bloch. Function-algebraic characterizations of log and polylog parallel time, computational complexity, 4(2):175–205, 1994. See also Proceedings of the Seventh Annual Structure in Complexity Theory Conference, 193–206, 1992.

    Google Scholar 

  5. Stephen Bloch. Parameter-free induction in bounded arithmetic. In preparation for submission, 1995.

    Google Scholar 

  6. Samuel R. Buss. Bounded Arithmetic. Number 3 in Studies in Proof Theory. Bibliopolis (Naples), 1986.

    Google Scholar 

  7. Samuel R. Buss. Axiomatizations and conservation results for theories of bounded arithmetic. In Proceedings of a Workshop in Logic and Computation, AMS Contemporary Mathematics, May 1987.

    Google Scholar 

  8. Samuel R. Buss. Relating the bounded arithmetic and polynomial time hierarchies. Manuscript, 1994.

    Google Scholar 

  9. Samuel R. Buss, Jan Krajíček, and Gaisi Takeuti. Provably total functions in bounded arithmetic theories R i3 , U i2 and V i2 . In P. Clote and J. Krajíček, editors, Arithmetic, Proof Theory and Computational Complexity, pages 116–161. Oxford University Press, 1993.

    Google Scholar 

  10. A. Chandra, D. Kozen, and L. Stockmeyer. Alternation. Journal of the ACM, 28(1):114–133, January 1981.

    Article  Google Scholar 

  11. Peter Clote. A first order theory for the parallel complexity class NC. Technical Report BCCS-8901, Boston College, 1989.

    Google Scholar 

  12. Peter Clote and Gaisi Takeuti. Bounded arithmetic for NC, ALogTIME, L and NL. Annals of Pure and Applied Logic, 56:73–117, 1992.

    Article  Google Scholar 

  13. A. Cobham. The intrinsic computational difficulty of functions. In Y. Bar-Hillel, editor, Logic, Methodology and Philosophy of Science II, pages 24–30. North-Holland, 1965.

    Google Scholar 

  14. Ronald Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In Richard M. Karp, editor, Complexity of Computations, volume 7 of SIAM-AMS Proceedings, pages 43–73. 1974.

    Google Scholar 

  15. Jay Hook. A many-sorted approach to predicative mathematics. PhD thesis, Princeton University, 1983.

    Google Scholar 

  16. Jan Krajíček. Fragments of bounded arithmetic and bounded query classes. Transactions of the AMS, 338(2):587–598, August 1993.

    Google Scholar 

  17. Jan Krajíček, Pavel Pudlák, and Jiři Sgall. Interactive computations of optimal solutions. In Mathematical Foundations of Computer Science, volume 452 of Lecture Notes in Computer Science, pages 48–60. Springer, 1990.

    Google Scholar 

  18. J. Krajíček, P. Pudlák, and G. Takeuti. Bounded arithmetic and the polynomial hierarchy. Annals of Pure and Applied Logic, 52:143–153, 1991.

    Google Scholar 

  19. 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. Birkhäuser, 1990.

    Google Scholar 

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

    Google Scholar 

  21. Rohit J. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494–508, 1971.

    Google Scholar 

  22. Alexander A. Razborov. Bounded arithmetic and lower bounds in Boolean complexity. Manuscript, 1994.

    Google Scholar 

  23. W. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences, 22:365–383, 1981.

    Article  Google Scholar 

  24. C.P. Schnorr. Satisfiability is quasilinear complete in NQL. Journal of the ACM, 25:136–145, 1978.

    Google Scholar 

  25. Christopher B. Wilson. Relativized circuit complexity. Journal of Computer and System Sciences, 31:169–181, 1985.

    Google Scholar 

  26. Christopher B. Wilson. Relativized NC. Math. Systems Theory, 20:13–29, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Math/Computer Science Dept, Adelphi University, USA

    Stephen Bloch

Authors
  1. Stephen Bloch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Daniel Leivant

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bloch, S. (1995). On parallel hierarchies and R ik . In: Leivant, D. (eds) Logic and Computational Complexity. LCC 1994. Lecture Notes in Computer Science, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60178-3_79

Download citation

  • DOI: https://doi.org/10.1007/3-540-60178-3_79

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60178-4

  • Online ISBN: 978-3-540-44720-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation