Sum of Squares Bounds for the Ordering Principle

Author Aaron Potechin

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Aaron Potechin
  • University of Chicago, IL, USA


The author would like to thank Marc Vinyals for proposing this problem. The author would also like to thank Prahladh Harsha, Toniann Pitassi, and anonymous reviewers for helpful comments on this paper.

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Aaron Potechin. Sum of Squares Bounds for the Ordering Principle. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 38:1-38:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on n elements (which has N = Θ(n²) variables). We prove that degree O(√nlog(n)) SOS can prove the ordering principle. We then show that this upper bound is essentially tight by proving that for any ε > 0, SOS requires degree Ω(n^(1/2 - ε)) to prove the ordering principle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • sum of squares hierarchy
  • proof complexity
  • ordering principle


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  1. Albert Atserias and Tuomas Hakoniemi. Size-degree trade-offs for sums-of-squares and positivstellensatz proofs. In Computational Complexity Conference, 2019. Google Scholar
  2. Boaz Barak, Fernando G.S.L. Brandao, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, STOC ’12, page 307–326, New York, NY, USA, 2012. Association for Computing Machinery. URL:
  3. Boaz Barak and David Steurer. Sum-of-squares proofs and the quest toward optimal algorithms. Electronic Colloquium on Computational Complexity (ECCC), 21:59, 2014. Google Scholar
  4. Boaz Barak and David Steurer. Proofs, beliefs, and algorithms through the lens of sum-of-squares, 2016. URL:
  5. Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149–169, March 2001. URL:
  6. Maria Luisa Bonet and Nicola Galesi. Optimality of size-width tradeoffs for resolution. Comput. Complex., 10(4):261–276, May 2002. URL:
  7. Nicola Galesi and Massimo Lauria. Optimality of size-degree tradeoffs for polynomial calculus. ACM Trans. Comput. Logic, 12(1), November 2010. URL:
  8. Russell Impagliazzo, Pavel Pudlák, and Jirí Sgall. Lower bounds for the polynomial calculus and the gröbner basis algorithm. computational complexity, 8:127-144, 1999. Google Scholar
  9. Pravesh Kothari and David Steurer. Sum-of-squares: proofs, beliefs, and algorithms, 2016. URL:
  10. Balakrishnan Krishnamurthy. Short proofs for tricky formulas. Acta Inf., 22(3):253–275, August 1985. Google Scholar
  11. Massimo Lauria. Dd3013 sum of squares and integer programming relaxations, 2014. URL:
  12. Aaron Potechin. Cmsc 39600 1 (autumn 2018) topics in theoretical computer science: The sum of squares hierarchy, 2018. URL:
  13. Aaron Potechin. Sum of squares lower bounds from symmetry and a good story. In ITCS, 2019. Google Scholar
  14. Annie Raymond, James Saunderson, Mohit Singh, and Rekha R. Thomas. Symmetric sums of squares over k-subset hypercubes. Math. Program., 167(2):315–354, February 2018. URL:
  15. Alexander Razborov. Guest column: Proof complexity and beyond. SIGACT News, 47(2):66–86, June 2016. URL:
  16. Gilbert Stengle. A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207:87-97, 1974. Google Scholar
  17. Gunnar Stålmarck. Short resolution proofs for a sequence of tricky formulas. Acta Inf., 33(3):277–280, May 1996. URL:
  18. Wikipedia. Chebyshev polynomials, Accessed May 30, 2020. URL:
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