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Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle

Authors Aaron Potechin , Aaron Zhang

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Author Details

Aaron Potechin
  • University of Chicago, IL, USA
Aaron Zhang
  • The Voleon Group, Berkeley, CA, USA

Funding

  • Potechin, Aaron: NSF grant CCF-2008920
  • Zhang, Aaron: NDSEG Fellowship 9422254702

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Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.117

Abstract

We show that the minimum total coefficient size of a Nullstellensatz proof of the pigeonhole principle on n+1 pigeons and n holes is 2^{Θ(n)}. We also investigate the ordering principle and construct an explicit Nullstellensatz proof for the ordering principle on n elements with total coefficient size 2ⁿ - n.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof complexity
  • Nullstellensatz
  • pigeonhole principle
  • coefficient size

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References

  1. Yaroslav Alekseev. A Lower Bound for Polynomial Calculus with Extension Rule. In Proceedings of the 36th Computational Complexity Conference, CCC '21, Dagstuhl, DEU, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.21.
  2. Yaroslav Alekseev, Dima Grigoriev, Edward A. Hirsch, and Iddo Tzameret. Semi-algebraic proofs, IPS lower bounds, and the tau-conjecture: can a natural number be negative? In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 54-67, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384245.
  3. Albert Atserias and Tuomas Hakoniemi. Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs. In Proceedings of the 34th Computational Complexity Conference, CCC '19, Dagstuhl, DEU, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.24.
  4. Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh K Kothari, Ankur Moitra, and Aaron Potechin. A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem. SIAM Journal on Computing, 48(2):687-735, 2019. Google Scholar
  5. P. Beame, R. Impagliazzo, J. Krajicek, T. Pitassi, and P. Pudlak. Lower Bounds on Hilbert’s Nullstellensatz and Propositional Proofs. In Proceedings 35th Annual Symposium on Foundations of Computer Science, pages 794-806, 1994. URL: https://doi.org/10.1109/SFCS.1994.365714.
  6. Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The Relative Complexity of NP Search Problems. J. Comput. Syst. Sci., 57(1):3-19, August 1998. URL: https://doi.org/10.1006/jcss.1998.1575.
  7. Eli Ben-Sasson and Avi Wigderson. Short Proofs are Narrow—Resolution made Simple. J. ACM, 48(2):149-169, March 2001. URL: https://doi.org/10.1145/375827.375835.
  8. Olaf Beyersdorff, Nicola Galesi, and Massimo Lauria. A lower bound for the pigeonhole principle in tree-like resolution by asymmetric Prover–Delayer games. Information Processing Letters, 110(23):1074-1077, 2010. URL: https://doi.org/10.1016/j.ipl.2010.09.007.
  9. Ilario Bonacina and Maria Luisa Bonet. On the Strength of Sherali-Adams and Nullstellensatz as Propositional Proof Systems. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '22, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3531130.3533344.
  10. María Luisa Bonet, Jordi Levy, and Felip Manyà. Resolution for Max-SAT. Artificial Intelligence, 171(8):606-618, 2007. URL: https://doi.org/10.1016/j.artint.2007.03.001.
  11. W. Dale Brownawell. Bounds for the Degrees in the Nullstellensatz. Annals of Mathematics, 126(3):577-591, 1987. URL: http://www.jstor.org/stable/1971361.
  12. S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlák, A. A. Razborov, and J. Sgall. Proof Complexity in Algebraic Systems and Bounded Depth Frege Systems with Modular Counting. Comput. Complex., 6(3):256-298, December 1997. URL: https://doi.org/10.1007/BF01294258.
  13. Sam Buss and Toniann Pitassi. Resolution and the weak pigeonhole principle. In Mogens Nielsen and Wolfgang Thomas, editors, Computer Science Logic, pages 149-156, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg. Google Scholar
  14. Samuel R. Buss. Lower Bounds on Nullstellensatz Proofs via Designs. In Paul Beame and Samuel R. Buss, editors, Proof Complexity and Feasible Arithmetics, Proceedings of a DIMACS Workshop, New Brunswick, New Jersey, USA, April 21-24, 1996, volume 39 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 59-71. DIMACS/AMS, 1996. URL: https://doi.org/10.1090/dimacs/039/04.
  15. S.R. Buss and T. Pitassi. Good Degree Bounds on Nullstellensatz Refutations of the Induction Principle. In Proceedings of Computational Complexity (Formerly Structure in Complexity Theory), pages 233-242, 1996. URL: https://doi.org/10.1109/CCC.1996.507685.
  16. Matthew Clegg, Jeffery Edmonds, and Russell Impagliazzo. Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 174-183, New York, NY, USA, 1996. Association for Computing Machinery. URL: https://doi.org/10.1145/237814.237860.
  17. S. Dantchev and S. Riis. Tree resolution proofs of the weak pigeon-hole principle. In Proceedings 16th Annual IEEE Conference on Computational Complexity, pages 69-75, 2001. URL: https://doi.org/10.1109/CCC.2001.933873.
  18. Stefan Dantchev, Barnaby Martin, and Mark Rhodes. Tight rank lower bounds for the Sherali–Adams proof system. Theoretical Computer Science, 410(21):2054-2063, 2009. URL: https://doi.org/10.1016/j.tcs.2009.01.002.
  19. Susanna F. de Rezende, Jakob Nordström, Or Meir, and Robert Robere. Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling. In Amir Shpilka, editor, 34th Computational Complexity Conference (CCC 2019), volume 137 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:16, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.18.
  20. M. Goos, A. Hollender, S. Jain, G. Maystre, W. Pires, R. Robere, and R. Tao. Separations in Proof Complexity and TFNP. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1150-1161, Los Alamitos, CA, USA, November 2022. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS54457.2022.00111.
  21. Joshua A. Grochow and Toniann Pitassi. Circuit Complexity, Proof Complexity, and Polynomial Identity Testing: The Ideal Proof System. J. ACM, 65(6), November 2018. URL: https://doi.org/10.1145/3230742.
  22. Armin Haken. The intractability of resolution. Theoretical Computer Science, 39:297-308, 1985. Third Conference on Foundations of Software Technology and Theoretical Computer Science. URL: https://doi.org/10.1016/0304-3975(85)90144-6.
  23. Tuomas Hakoniemi. Monomial size vs. Bit-Complexity in Sums-of-Squares and Polynomial Calculus. In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '21, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1109/LICS52264.2021.9470545.
  24. Grete Hermann. Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Mathematische Annalen, 95:736-788, 1926. Google Scholar
  25. Russell Impagliazzo, Pavel Pudlák, and Jiří Sgall. Lower Bounds for the Polynomial Calculus and the Gröbner Basis Algorithm. Comput. Complex., 8(2):127-144, November 1999. URL: https://doi.org/10.1007/s000370050024.
  26. Kazuo Iwama and S. Miyazaki. Tree-Like Resolution is Superpolynomially Slower than DAG-Like Resolution for the Pigeonhole Principle. In International Symposium on Algorithms and Computation, 1999. Google Scholar
  27. János Kollár. Sharp Effective Nullstellensatz. Journal of the American Mathematical Society, pages 963-975, 1988. Google Scholar
  28. Toniann Pitassi and Robert Robere. Lifting Nullstellensatz to Monotone Span Programs over Any Field. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1207-1219, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3188745.3188914.
  29. Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle and the Ordering Principle, 2022. URL: https://arxiv.org/abs/2205.03577.
  30. Alexander A Razborov. Lower bounds for the polynomial calculus. computational complexity, 7(4):291-324, 1998. Google Scholar
  31. S. F. De Rezende, A. Potechin, and K. Risse. Clique is Hard on Average for Unary Sherali-Adams. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 12-25, Los Alamitos, CA, USA, November 2023. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS57990.2023.00008.
  32. Gunnar Stålmarck. Short resolution proofs for a sequence of tricky formulas. Acta Informatica, 33(3):277-280, 1996. Google Scholar
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