发表机构
University of Applied Sciences and Arts of Southern Switzerland; IDSIA（南方瑞士应用科学与艺术大学; IDSIA）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多项式演算中理想成员问题的复杂度，提出基于 pp-定义的归约框架，证明在中值运算封闭的语言上可解，并完成布尔分类，推动 SoS 证明的度自动化。
AI 中文摘要
理想成员问题（IMP）询问一个多项式 f 是否属于 Q[x_1, ..., x_n] 的理想 <p_1, ..., p_m>。多项式演算（PC）通过从生成元推导出 f 来证明成员关系，度为 d 的推导最多需要 n^O(d) 步。我们将 PC-IMPd 定义为产生度有界 PC 证书的问题，并称其为可解的，当保证存在这样的证书且能在时间 n^O(d) 内找到时。在有理数域 Q 上，与有限域不同，推导可能需要指数多位。我们研究由约束满足问题产生的实例上的 PC-IMPd，并询问对于哪些约束语言 L 它是可解的。我们的主要贡献是一个针对 PC-IMPd 的归约框架，基于 pp-定义、pp-解释和 pp-编码，这镜像了 CSP 复杂度的代数方法。可解性在这些构造下保持不变，并且在代数的语言中，通过传递到子代数、有限直幂和同态像也保持不变。我们获得了在三元及更大域上的新的可处理类：在有限链上对中值运算封闭的每种语言都有可解的 PC-IMPd，通过归约到布尔多数代数，特别是 {0, 1, 2} 上对固定值多数封闭的每种语言也是如此。这也将 IMPd(L) 置于 P 中，推进了三元域上 IMPd 的分类。在此过程中，我们解决了 IMPd(L) 的布尔二分法的最后一个未决情况，并通过对 PC-IMP1 的一个实例的无条件下界完成了 PC-IMPd(L) 的布尔分类。最近的 PC 到 SoS 模拟将和-平方（Sum-of-Squares）的度自动化问题（在存在度 d SoS 证明时在时间 n^O(d) 内找到它的开放问题）归约为 PC-IMPd 的可解性。因此，每个新的可处理类都产生一族约束系统，在这些系统上 SoS 证明是度自动化的。
英文摘要
The Ideal Membership Problem (IMP) asks whether a polynomial f belongs to an ideal <p_1, ..., p_m> of Q[x_1, ..., x_n]. Polynomial Calculus (PC) certifies membership by deriving f from the generators, and a degree-d derivation needs at most n^O(d) steps. We write PC-IMPd for the problem of producing a degree-bounded PC certificate, and call it solvable when one is guaranteed to exist and can be found in time n^O(d). Over Q, unlike over finite fields, a derivation may need exponentially many bits. We study PC-IMPd on instances arising from constraint satisfaction problems, and ask for which constraint languages L it is solvable. Our main contribution is a reduction framework for PC-IMPd, based on pp-definitions, pp-interpretations, and pp-encodings, that mirrors the algebraic approach to CSP complexity. Solvability is preserved by these constructions and, in the language of algebras, by passing to subalgebras, finite direct powers, and homomorphic images. We obtain new tractable classes over ternary and larger domains: every language closed under the median operation on a finite chain has solvable PC-IMPd, by reduction to the Boolean majority algebra, and in particular so does every language over {0, 1, 2} closed under a fixed-value majority. This also places IMPd(L) in P for such languages, advancing the classification of IMPd over ternary domains. In the process, we settle the last open case of the Boolean dichotomy for IMPd(L) and complete the Boolean classification of PC-IMPd(L) with an unconditional lower bound for an instance of PC-IMP1. A recent PC-to-SoS simulation reduces degree-automatability of Sum-of-Squares (the open problem of finding a degree-d SoS proof in time n^O(d) when one exists) to solvability of PC-IMPd. Each new tractable class therefore yields a family of constraint systems on which SoS proofs are degree-automatable.
