Skip to main content

On the structure of valiant's complexity classes

Complexity II

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1373)

Abstract

In [25,27] Valiant developed an algebraic analogue of the theory of NP-completeness for computations with polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay [1], Ladner [18], and Schöning [23,24].

We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor VNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for VP in VNP.

Over finite fields, we give a specific example of a family of polynomials which is neither VNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.

We define relativized complexity classes VPh and VNPh and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VPh = VNPh.

Keywords

  • Turing Machine
  • Complexity Class
  • Hamilton Cycle
  • Minimal Pair
  • Boolean Circuit

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/BFb0028561
  • Chapter length: 11 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-69705-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   149.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Baker, J. Gill, and R. Solovay. Relativizations of the P =?NP question. SIAM J. Comp., 4:431–442, 1975.

    MATH  CrossRef  MathSciNet  Google Scholar 

  2. S. Ben-David, K. Meer, and C. Michaux. A note on non-complete problems in NP. Preprint, 1996.

    Google Scholar 

  3. C.H. Bennett and J. Gill. Relative to a random oracle A, PA ≠ NPA ≠ co-NPA with probability 1. SIAM J. Comp., 10:96–113, 1981.

    MATH  CrossRef  MathSciNet  Google Scholar 

  4. L. Blum, F. Cucker, M. Shub, and S. Smale. Algebraic Settings for the Problem “PNP?”. In The mathematics of numerical analysis, number 32 in Lectures in Applied Mathematics, pages 125–144. Amer. Math. Soc., 1996.

    Google Scholar 

  5. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers. Bull. Amer. Math. Soc., 21:1–46, 1989.

    MATH  CrossRef  MathSciNet  Google Scholar 

  6. P. Bürgisser. Cook's versus Valiant's hypothesis. Preprint, University of Zurich, 1997.

    Google Scholar 

  7. P. Bürgisser. Some complete families of polynomials. Manuscript, 1997.

    Google Scholar 

  8. P. Bürgisser, M. Clausen, and M.A. Shokrollahi. Algebraic Complexity Theory. Number 315 in Grundlehren der mathematischen Wissenschaften. Springer Verlag, 1996.

    Google Scholar 

  9. J. Cai and L.A. Hemachandra. On the power of parity polynomial time. In Proc. STACS'89, number 349 in LNCS, pages 229–239. Springer Verlag, 1989.

    Google Scholar 

  10. O. Chapuis and P. Koiran. Saturation and Stability in the Theory of Computation over the Reals. Preprint, 1997.

    Google Scholar 

  11. S.A. Cook. The complexity of theorem proving procedures. In Proc. 3rd ACM STOC, pages 151–158, 1971.

    Google Scholar 

  12. T. Emerson. Relativizations of the P=?NP question over the reals (and other ordered rings). Theoret. Comp. Sci., 133:15–22, 1994.

    MATH  CrossRef  MathSciNet  Google Scholar 

  13. W. Feller. An introduction to probability theory and its applications, volume 2. John Wiley & Sons, 1971.

    Google Scholar 

  14. J. von zur Gathen. Feasible arithmetic computations: Valiant's hypothesis. J. Symb. Comp., 4:137–172, 1987.

    MATH  CrossRef  Google Scholar 

  15. J. Heintz and J. Morgenstern. On the intrinsic complexity of elimination theory. Journal of Complexity, 9:471–498, 1993.

    MATH  CrossRef  MathSciNet  Google Scholar 

  16. R.M. Karp and R.J. Lipton. Turing machines that take advice. In Logic and Algorithmic: An international Symposium held in honor of Ernst Specker, pages 255–273. Monogr. No. 30 de l'Enseign. Math., 1982.

    Google Scholar 

  17. P. Koiran. Hilbert's Nullstellensatz is in the polynomial hierarchy. J. Compl., 12:273–286, 1996.

    MATH  CrossRef  MathSciNet  Google Scholar 

  18. R.E. Ladner. On the structure of polynomial time reducibility. J. ACM, 22:155–171, 1975.

    MATH  CrossRef  MathSciNet  Google Scholar 

  19. Landweber, Lipton, and Robertson. On the structure of sets in NP and other complexity classes. Theoret. Comp. Sci., 15:181–200, 1981.

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. G. Malajovich and K. Meer. On the structure of NP. SIAM J. Comp. to appear.

    Google Scholar 

  21. C.H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.

    Google Scholar 

  22. C.H. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proc. 6th GI conference in Theoretical Computer Science, number 145 in LNCS, pages 269–276. Springer Verlag, 1983.

    Google Scholar 

  23. U. Schöning. A uniform approach to obtain diagonal sets in complexity classes. Theoret. Comp. Sci., 18:95–103, 1982.

    MATH  CrossRef  Google Scholar 

  24. U. Schöning. Minimal pairs for P. Theoret. Comp. Sci., 31:41–48, 1984.

    MATH  CrossRef  Google Scholar 

  25. L.G. Valiant. Completeness classes in algebra. In Proc. 11th ACM STOC, pages 249–261, 1979.

    Google Scholar 

  26. L.G. Valiant. The complexity of computing the permanent. Theoret. Comp. Sci., 8:189–201, 1979.

    MATH  CrossRef  MathSciNet  Google Scholar 

  27. L.G. Valiant. Reducibility by algebraic projections. In Logic and Algorithmic: an International Symposium held in honor of Ernst Specker, volume 30, pages 365–380. Monographies de l'Enseignement Mathématique, 1982.

    Google Scholar 

  28. L.G. Valiant and V.V. Vazirani. NP is as easy as detecting unique solutions. Theoret. Comp. Sci., 47:85–93, 1986.

    MATH  CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1998 Springer-Verlag

About this paper

Cite this paper

Bürgisser, P. (1998). On the structure of valiant's complexity classes. In: Morvan, M., Meinel, C., Krob, D. (eds) STACS 98. STACS 1998. Lecture Notes in Computer Science, vol 1373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028561

Download citation

  • DOI: https://doi.org/10.1007/BFb0028561

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64230-5

  • Online ISBN: 978-3-540-69705-3

  • eBook Packages: Springer Book Archive