AI 中文总结
本文研究弱秩原理（WRank），证明其在PCR_{F₂}和Sherali–Adams（SA）中存在指数规模下界，解决了PCR_{F₂}相关开放问题，还明确其对NC²电路下界必要、对AC⁰[p]下界充分。
AI 中文摘要
给定两个维度为m×n和n×m的符号矩阵X和Y，*弱秩原理*（WRank）指出，当m>n且矩阵A的秩超过n时，方程XY=A不可满足。我们将该原理作为弱鸽巢原理（WPHP）的代数推广进行研究。作为WPHP的强化形式，它在WPHP无已知证明复杂度下界的场景中可提供证明复杂度下界，同时仍支持类似应用。*PCR_{F₂}的生成器*：我们证明了WRank的代数、完美匹配和竹树编码在PCR_{F₂}中存在指数规模下界，其中竹树编码与电路下界公式的应用最为相关，相关研究见于Alekhnovich、Ben-Sasson、Razborov和Wigderson（《SIAM计算期刊》，2004）以及Razborov（《数学年刊》，2015）的工作。利用标准迭代技术，我们将拉伸放大至指数级，解决了PCR_{F₂}中具有良好拉伸的证明复杂度生成器构造这一开放问题。*Sherali–Adams的生成器*：我们开发了一种新的规模下界技术，表明编码为竹树CNF的WRank可作为SA的证明复杂度生成器，我们的方法引入了专门针对秩原理设计（且与WPHP不兼容）的伪期望。*电路下界公式*：我们证明PCR_{F₂}不存在针对布尔电路或弱代数电路模型的下界陈述的短证明，解决了Razborov（《数学年刊》，2015）提出的关于此类下界在PCR_{F₂}中可证明性的开放问题。*弱秩原理的强度*：最后，我们证明WRank对于证明NC²电路下界是*必要*的，且对于奇素数p，在对应AC⁰[p]的理论内，它对于推导AC⁰[p]下界是*充分*的。
英文摘要
Given two symbolic matrices $X$ and $Y$ of dimensions $m\times n$ and $n\times m$, the *weak rank principle* (WRank) states the equation $XY = A$ is unsatisfiable when $m>n$ and rank of $A$ exceeds $n$. We study this principle as an algebraic generalisation of the weak pigeonhole principle (WPHP). As a strengthening of WPHP, it admits proof complexity lower bounds in settings where none are known for WPHP, while still supporting analogous applications. *Generators for PCR$_{F_2}$*: We prove exponential size lower bounds for algebraic, perfect matching, and bamboo-tree encodings of WRank in PCR$_{F_2}$. The latter encoding is the most relevant for applications to circuit lower-bound formulas, as considered by Alekhnovich, Ben-Sasson, Razborov, and Wigderson (SIAM J. Comput., 2004) and Razborov (Ann. Math., 2015). Using a standard iteration technique we amplify the stretch to exponential. This resolves the open problem concerning the construction of proof complexity generators with good stretch for PCR$_{F_2}$. *Generators for Sherali--Adams:* We develop a new size lower-bound technique showing that WRank, encoded as a bamboo-tree CNF, serves as a proof complexity generator for SA. Our method introduces a pseudoexpectation tailored specifically to the rank principle (and incompatible with WPHP). *Circuit lower bound formulas:* We show that PCR$_{F_2}$ does not admit short proofs of lower-bound statements against Boolean circuits, nor against weak models of algebraic circuits. This settles the open problem raised by Razborov (Ann. Math., 2015) concerning the provability of such lower bounds in PCR$_{F_2}$. *Strength of the weak rank principle:* Finally, we show that WRank is *necessary* for proving NC$^2$ circuit lower bounds and, for odd primes $p$, *sufficient* within the theory corresponding to AC$^{0}[p]$ for deriving AC$^{0}[p]$ lower bounds.
Journal refProceedings of the 58th Annual ACM Symposium on Theory of Computing (ACM. STOC) 2026 (pp. 138-149)
