Hard CNF Instances for Ideal Proof Systems

2026-05-06

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 Pudlák 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$_\text{LIN}$).

Problem

• Ideal Proof System
• CNF formulas
• proof complexity
• lower bounds
• roABP

Innovation

• Ideal Proof System
• CNF formulas
• read-once oblivious ABPs
• feasible interpolation
• proof complexity