AbstractWe exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size $${n^{\Omega{(d)}}}$$nΩ(d) for values of d = d(n) from constant all the way up to $${n^{\delta}}$$nδ for some universal constant $${\delta}$$δ. This shows that the $${{n^{{\rm O}{(d)}}}}$$nO(d) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajíček (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.

中文翻译：

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平方和证明的严格大小度界限

摘要我们在 n 个变量上展示了 4-CNF 公式族，这些变量具有程度（又名等级）d 的不可满足性的平方和 (SOS) 证明，但需要 SOS 证明的大小为 $${n^{\Omega{(d)} }}$$nΩ(d) 对于 d = d(n) 的值，从常数一直到 $${n^{\delta}}$$nδ 对于某些通用常数 $${\delta}$$δ . 这表明通过使用 Lasserre 半定规划松弛找到度 d SOS 证明获得的 $${{n^{{\rm O}{(d)}}}}$$nO(d) 运行时间是最优的指数中的常数因子。我们通过将可表示为低度 SOS 推导的 NP 归约与 Krajíček (Arch Math Log 43(4):427–441, 2004) 和 Dantchev & Riis (Proceedings of the 17th International计算机科学逻辑研讨会（CSL '03），2003），然后像 Atserias 等人一样应用限制参数。（J Symb Log 80(2):450–476, 2015；ACM Trans Comput Log 17:19:1–19:30, 2016）。这产生了一种将 SOS 度下界放大到大小下界的通用方法，并且还推广了 Atserias 等人中使用的方法。(2016) 分别从宽度、度数和秩的下限获得证明系统分辨率、多项式微积分和 Sherali-Adams 的大小下限。