Hard CNF Instances for Ideal Proof Systems

TL;DR

This paper proves lower bounds for refutations of CNF formulas within the Ideal Proof System over read-once oblivious algebraic branching programs, advancing understanding of proof complexity for Boolean formulas.

Contribution

It establishes the first lower bounds for CNF formulas in IPS over roABPs using a rank-based interpolation method, extending previous algebraic results.

Findings

Proves lower bounds for CNF refutations in IPS over roABPs.

Introduces a rank-based feasible interpolation technique for this setting.

Extends Nullstellensatz refutations from degree to size measures.

Abstract

Since the introduction of the Ideal Proof System (IPS) by Grochow and Pitassi (J. ACM 2018), a substantial body of work has established size lower bounds for IPS and its fragments. In particular, Forbes, Shpilka, Tzameret, and Wigderson (Theory Comput. 2021) developed the main lower-bound frameworks for restricted IPS fragments, namely functional lower bounds and the hard multiples method, while Alekseev, Grigoriev, Hirsch, and Tzameret (SIAM J. Comput. 2024) gave a general template for conditional lower bounds for full IPS. Yet all these lower bounds apply only to purely algebraic formulas over a field, that is, non-Boolean formulas not directly expressible in propositional logic. Proving lower bounds for CNF formulas has therefore remained a central open problem in this line of work. The current work resolves this question for IPS over read-once oblivious algebraic branching programs (roABPs) by proving lower bounds for refutations of CNF formulas in this system. Our approach is a rank-based feasible interpolation argument, following the method of Pudl\'ak and Sgall (Proof Complexity and Feasible Arithmetic 1996) for monotone span programs, in which decomposing a given roABP refutation along a variable partition yields a low-dimensional space of polynomials from which we construct a span-program interpolant. We extend their result from Nullstellensatz refutations measured by degree to Nullstellensatz refutations measured by roABP size (i.e., roABP-IPS$_\text{LIN}$).