The Weak Rank Principle: Lower Bounds and Applications

Abstract

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.