Exponential Lower Bounds on the Size of ResLin Proofs of Nearly Quadratic Depth

2025-07-30

AI Summary

This work investigates depth–size trade-offs in the Res(⊕) proof system, aiming to overcome prior lower bounds restricted to nearly linear depth (O(N^{1+ε})). For Tseitin formulas over expander graphs, we introduce a novel lifting gadget that exhibits weak dependence on parity and satisfies secure affine masking—a property ensuring robustness under affine transformations. We further develop a lifting analysis framework applicable to arbitrary input distributions. Our main result establishes, for the first time, an exponential lower bound of exp(˜Ω(N^ε)) on proof size in Res(⊕) when the proof depth is O(N^{2−ε})—nearly quadratic in the number N of variables, for any ε > 0. This constitutes the deepest known superpolynomial lower bound for Res(⊕), substantially extending the frontier of understanding low-depth proof complexity in this system.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Itsykson and Sokolov identified resolution over parities, denoted by $ ext{Res}(oplus)$, as a natural and simple fragment of $ ext{AC}^0[2]$-Frege for which no super-polynomial lower bounds on size of proofs are known. Building on a recent line of work, Efremenko and Itsykson proved lower bounds of the form $ ext{exp}(N^{Ω(1)})$, on the size of $ ext{Res}(oplus)$ proofs whose depth is upper bounded by $O(Nlog N)$, where $N$ is the number of variables of the unsatisfiable CNF formula. The hard formula they used was Tseitin on an appropriately expanding graph, lifted by a $2$-stifling gadget. They posed the natural problem of proving super-polynomial lower bounds on the size of proofs that are $Ω(N^{1+ε})$ deep, for any constant $ε> 0$. We provide a significant improvement by proving a lower bound on size of the form $ ext{exp}( ildeΩ(N^ε))$, as long as the depth of the $ ext{Res}(oplus)$ proofs are $O(N^{2-ε})$, for every $ε> 0$. Our hard formula is again Tseitin on an expander graph, albeit lifted with a different type of gadget. Our gadget needs to have small correlation with all parities. An important ingredient in our work is to show that arbitrary distributions emph{lifted} with such gadgets fool emph{safe} affine spaces, an idea which originates in the earlier work of Bhattacharya, Chattopadhyay and Dvorak.

Problem

• Prove exponential lower bounds for ResLin proofs
• Address depth constraints in ResLin proof systems
• Use Tseitin formulas with expander graphs

Innovation

• Expanding graph Tseitin formula with new gadget
• Lifted gadgets with small parity correlation
• Fooling safe affine spaces via lifted distributions