Skip to main content
SpringerLink
Log in

Homogenization and the Polynomial Calculus

  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

Included in the following conference series:

Abstract

In standard implementations of the Grobner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the proof complexity of refutations of polynomials derived from Boolean formulas in both the Polynomial Calculus (PC) and Nullstellensatz systems. We show that the PC refutations of homogenized formulas give crucial information about the complexity of the original formulas. The minimum PC refutation degree of homogenized formulas is equal to the Nullstellensatz refutation degree of the original formulas, whereas the size of the homogenized PC refutation is equal to the size of the PC refutation for the originals. Using this relationship, we prove nearly linear (Ω(n/logn) vs. O(1)) separations between Nullstellensatz and PC degree, for a family of explicitly constructed contradictory 3CNF formulas. Previously, a Ω(n1/2) separation had been proved for equations that did not correspond to any CNF formulas, and a log n separation for equations derived from kCNF formulas.

Research partially supported by NSF grant CCR-9457782 and a scholarship from the Arizona Chapter of the ARCS Foundation.

Research supported by NSF grant CCR-9457782 and US-Israel BSF Grant 95-00238.

Research supported by NSF CCR-9734911, Sloan Research Fellowship BR-3311, and by a cooperative research grant INT-9600919/ME-103 from NSF and the MŠMT (Czech Republic), and USA-Israel-BSF Grant 97-00188

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.L. Bonet, J. L. Esteban, N. Galesi, and J. Johannsen. Exponential separations between restricted resolution and cutting planes proof systems. In Proceedings from 38th FOCS, pages 638–647. 1998.

    Google Scholar 

  2. [BIK+96]_Paul W. Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi, and Pavel Pudlak. Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. Proc. London Math. Soc, 73(3): 1–26, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  3. [BIK+97]_S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlak, A. A. Razborov, and J. Sgall. Proof complexity in algebraic systems and bounded depth Frege systems with modular counting. Computation Complexity, 6(3):256–298, 1997.

    Article  Google Scholar 

  4. S. R. Buss and T. Pitassi. Good degree bounds on Nullstellensatz refutations of the induction principle. In Proceedings of the Eleventh Annual Conference on Computational Complexity formerly: Structure in Complexity Theory, pages 233–242, Philadelphia, PA, May 1996. IEEE.

    Google Scholar 

  5. E. Ben Sasson and A. Wigderson. Short proofs are narrow-resolution made simple. In Proceedings of 31st ACM STOC, pages 517–526. 1999.

    Google Scholar 

  6. S. R. Buss. Lower bounds on Nullstellensatz proofs via designs. In P. W. Beame and S. R. Buss, (editors), Proof Complexity and Feasible Arithmetics, DIMACS, pages 59–71. American Math. Soc, 1997.

    Google Scholar 

  7. M. Clegg, J. Edmonds, and R. Impagliazzo. Using the Grobner basis algorithm to find proofs of unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pages 174–183, Philadelphia, PA, May 1996.

    Google Scholar 

  8. J. Cox, D. Little and D. O’Shea. Ideals, Varieties and Algoirthms. Springer-Verlag, 1992.

    Google Scholar 

  9. R. Celoni, W.J. Paul, and R.E. Tarjan. Space bounds for a game on graphs. Math. Systems Theory, 10:239–251, 1977.

    MATH  MathSciNet  Google Scholar 

  10. J. L. Esteban and Jacobo Toran. Space bounds for resolution. In Proc. 16th STACS, pages 551–561. Springer-Verlag LNCS, 1999.

    Chapter  Google Scholar 

  11. T. Pitassi. Algebraic propositional proof systems. In DIMACS Series in Discrete Mathematics, volume 31, pages 215–243. American Math. Soc, 1997.

    MathSciNet  Google Scholar 

  12. R. Raz and P. McKenzie. Separation of the monotone nc hierarchy. In Proceedings of 38th IEEE Foundations of Computer Science. 1997.

    Google Scholar 

  13. A. Urquhart. Hard examples for resolution. Journal of the ACM, 34(1):209–219, 1987.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, University of Arizona, Tucson, AZ, 85721, USA

    Josh Buresh-Oppenheim & Toniann Pitassi

  2. Computer Science and Engineering, University of California, San Diego La Jolla, CA, USA

    Matt Clegg & Russell Impagliazzo

Authors
  1. Josh Buresh-Oppenheim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Toniann Pitassi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matt Clegg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Russell Impagliazzo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Sciences, University of Pisa, Corso Italia, 40, 56125, Pisa, Italy

    Ugo Montanari

  2. Center for Computer Sciences, University of Geneva, 24, Rue Général Dufour, 1211, Geneva 4, Switzerland

    José D. P. Rolim

  3. Department of Computer Sciences, ETH Zurich, 8092, Zurich, Switzerland

    Emo Welzl

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Buresh-Oppenheim, J., Pitassi, T., Clegg, M., Impagliazzo, R. (2000). Homogenization and the Polynomial Calculus. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_78

Download citation

  • DOI: https://doi.org/10.1007/3-540-45022-X_78

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation