skip to main content
research-article

On Proof Complexity of Resolution over Polynomial Calculus

Published:22 July 2022Publication History
Skip Abstract Section

Abstract

The proof system Res (PCd,R) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree d multivariate polynomials over a ring R with Boolean variables. Proving super-polynomial lower bounds for the size of Res(PC1,R)-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for Res(PC1,𝔽) when 𝔽 is a finite field, such as 𝔽2. In this article, we investigate Res(PCd,R) and tree-like Res(PCd,R) and prove size-width relations for them when R is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows:

(1)

We prove almost quadratic lower bounds for Res(PCd,𝔽)-refutations for every fixed d. The new lower bounds are for the following CNFs:

(a)

Mod q Tseitin formulas (char(𝔽)≠ q) and Flow formulas,

(b)

Random k-CNFs with linearly many clauses.

(2)

We also prove super-polynomial (more than nk for any fixed k) and also exponential (2 for an ϵ > 0) lower bounds for tree-like Res(PCd,𝔽)-refutations based on how big d is with respect to n for the following CNFs:

(a)

Mod q Tseitin formulas (char(𝔽)≠ q) and Flow formulas,

(b)

Random k-CNFs of suitable densities,

(c)

Pigeonhole principle and Counting mod q principle.

The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case d=1. The lower bounds for the tree-like systems were known for the case d=1 (except for the Counting mod q principle, in which lower bounds for the case d> 1 were known too). Our lower bounds extend those results to the case where d> 1 and also give new proofs for the case d=1.

REFERENCES

  1. [1] Ajtai Miklós. 1994. The complexity of the pigeonhole principle. Combinatorica 14, 4 (1994), 417433. Google ScholarGoogle ScholarCross RefCross Ref
  2. [2] Ajtai Miklós. 1994. The independence of the modulo p counting principles. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC’94). 402411.Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. [3] Alekhnovich Michael and Razborov Alexander A.. 2001. Lower bounds for polynomial calculus: Non-Binomial case. In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’01). 190199.Google ScholarGoogle ScholarCross RefCross Ref
  4. [4] Beame Paul, Impagliazzo Russell, Krajíček Jan, Pitassi Toniann, and Pudlák Pavel. 1994. Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS’94). 794806.Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. [5] Ben-Sasson Eli and Wigderson Avi. 1999. Short proofs are narrow—Resolution made simple. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC’99). 517526.Google ScholarGoogle Scholar
  6. [6] Buss Sam, Grigoriev Dima, Impagliazzo Russell, and Pitassi Toniann. 2001. Linear gaps between degrees for the polynomial calculus modulo distinct primes. J. Comput. System Sci. 62, 2 (2001), 267289.Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. [7] Chvátal Vašek and Szemerédi Endre. 1988. Many hard examples for resolution. J. ACM 35, 4 (Oct. 1988), 759768.Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. [8] Clegg Matthew, Edmonds Jeffery, and Impagliazzo Russell. 1996. Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC’96). 174183.Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. [9] Garlík Michal and Kołodziejczyk Leszek Aleksander. 2018. Some subsystems of constant-depth Frege with parity. ACM Trans. Comput. Logic 19, 4 (Nov. 2018).Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. [10] Gryaznov Svyatoslav. 2019. Notes on resolution over linear equations. In Proceedings of the Conference on Computer Science—Theory and Applications (CSR’19). 168179.Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. [11] Itsykson Dmitry and Sokolov Dmitry. 2020. Resolution over linear equations modulo two. Ann. Pure Appl. Logic 171, 1 (2020). Google ScholarGoogle ScholarCross RefCross Ref
  12. [12] Krajíček Jan. 2019. Proof Complexity. Encyclopedia of Mathematics and Its Applications, Vol. 170. Cambridge University Press.Google ScholarGoogle ScholarCross RefCross Ref
  13. [13] Krajícek Jan and Oliveira Igor Carboni. 2018. On monotone circuits with local oracles and clique lower bounds. Chic. J. Theor. Comput. Sci. (2018).Google ScholarGoogle ScholarCross RefCross Ref
  14. [14] Krajíček Jan, Pudlák Pavel, and Woods Alan R.. 1995. An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Struct. Algor. 7, 1 (1995), 1540. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. [15] Krajíček Jan. 1997. Lower bounds for a proof system with an exponential speed-up over constant-depth Frege systems and over polynomial calculus. In Proceedings of the Conference on Mathematical Foundations of Computer Science (MFCS’97). 8590.Google ScholarGoogle ScholarCross RefCross Ref
  16. [16] Krajíček Jan. 2018. Randomized feasible interpolation and monotone circuits with a local oracle. J. Math. Logic 18, 2 (2018).Google ScholarGoogle ScholarCross RefCross Ref
  17. [17] Lubotzky Alexander, Phillips Ralph, and Sarnak Peter. 1988. Ramanujan graphs. Combinatorica 8, 3 (1988), 261277.Google ScholarGoogle ScholarCross RefCross Ref
  18. [18] Part Fedor and Tzameret Iddo. 2020. Resolution with counting: Dag-Like lower bounds and different moduli. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS’20), Vol. 151. 137.Google ScholarGoogle Scholar
  19. [19] Pitassi Toniann, Beame Paul, and Impagliazzo Russell. 1993. Exponential lower bounds for the pigeonhole principle. Comput. Complex. 3 (1993), 97140. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. [20] Pudlák Pavel and Impagliazzo Russell. 2000. A lower bound for DLL algorithms for k-SAT (Preliminary Version). In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’00). 128136.Google ScholarGoogle Scholar
  21. [21] Raz Ran and Tzameret Iddo. 2008. Resolution over linear equations and multilinear proofs. Ann. Pure Appl. Logic 155, 3 (2008), 194224.Google ScholarGoogle ScholarCross RefCross Ref
  22. [22] Razborov Alexander A.. 1998. Lower bounds for the polynomial calculus. Comput. Complex. 7, 4 (Dec. 1998), 291324.Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. [23] Riis Sren. 1997. Count(q) does not imply Count(p). Ann. Pure Appl. Logic 90, 1 (1997), 156.Google ScholarGoogle ScholarCross RefCross Ref
  24. [24] Tzameret Iddo. 2014. Sparser random 3-SAT refutation algorithms and the interpolation problem. In Proceedings of the 41st International Colloquium on Automata, Languages and Programming (ICALP’14). 10151026.Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. On Proof Complexity of Resolution over Polynomial Calculus

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Computational Logic
        ACM Transactions on Computational Logic  Volume 23, Issue 3
        July 2022
        225 pages
        ISSN:1529-3785
        EISSN:1557-945X
        DOI:10.1145/3522734
        • Editor:
        • Anuj Dawar
        Issue’s Table of Contents

        Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 22 July 2022
        • Accepted: 1 December 2021
        • Revised: 1 October 2021
        • Received: 1 November 2020
        Published in tocl Volume 23, Issue 3

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Refereed
      • Article Metrics

        • Downloads (Last 12 months)39
        • Downloads (Last 6 weeks)5

        Other Metrics

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      Full Text

      View this article in Full Text.

      View Full Text

      HTML Format

      View this article in HTML Format .

      View HTML Format