The Weak Rank Principle: Lower Bounds and Applications

2026-08-09

AI Summary

This work addresses the longstanding challenge of deriving strong lower bounds for the weak pigeonhole principle in proof complexity by introducing its algebraic generalization—the Weak Rank principle (WRank)—and constructing multiple encodings, including perfect matching and bamboo-tree CNF formulations. By innovatively designing a scalable lower-bound generator tailored for Polynomial Calculus Resolution over $F_2$ (PCR$_{F_2}$) and a novel pseudo-expectation method adapted to the Sherali–Adams system, the paper establishes the first exponential proof-length lower bounds for WRank in PCR$_{F_2}$, resolving a major open problem. It further demonstrates that circuit lower bound formulas admit no short proofs in this system. These results establish WRank as both necessary and sufficient for proving lower bounds against NC$^2$ and AC$^{0}[p]$, thereby cementing its central role in structured proof complexity analysis.

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.

Problem

• proof complexity
• circuit lower bounds
• weak rank principle
• algebraic proof systems
• generators

Innovation

• Weak Rank Principle
• Proof Complexity Generators
• Polynomial Calculus Resolution
• Sherali–Adams
• Circuit Lower Bounds