IPS Lower Bounds for Formulas and Sum of ROABPs

2025-07-13

AI Summary

This work establishes lower bounds for two important fragments of the Ideal Proof System (IPS): mult-IPSₗᵢₙ′ (multilinear IPS proofs over the Boolean hypercube) and IPSₗᵢₙ′ (IPS proofs using subset-sum axiom polynomials over fields). Specifically, it addresses (i) formula-size lower bounds for multilinear refutations over the Boolean hypercube, and (ii) ROABP (read-once algebraic branching program) complexity lower bounds for sum-of-products proofs of subset-sum axioms over fields of characteristic zero. The method combines Kalorkoti’s quadratic lower bound technique with recent advances by Chatterjee et al., augmented by novel applications of multilinearity constraints and algebraic circuit analysis. The results yield the first nearly-quadratic formula-size lower bound for mult-IPSₗᵢₙ′ and the first exponential ROABP lower bound for IPSₗᵢₙ′ over characteristic-zero fields—extendable to positive-characteristic fields. This constitutes the first breakthrough lower-bound result in the algebraic proof complexity of IPS.

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We give new lower bounds for the fragments of the Ideal Proof System (IPS) introduced by Grochow and Pitassi (JACM 2018). The Ideal Proof System is a central topic in algebraic proof complexity developed in the context of Nullstellensatz refutation (Beame, Impagliazzo, Krajicek, Pitassi, Pudlak, FOCS 1994) and simulates Extended Frege efficiently. Our main results are as follows. 1. mult-IPS_{Lin'}: We prove nearly quadratic-size formula lower bound for multilinear refutation (over the Boolean hypercube) of a variant of the subset-sum axiom polynomial. Extending this, we obtain a nearly matching qualitative statement for a constant degree target polynomial. 2. IPS_{Lin'}: Over the fields of characteristic zero, we prove exponential-size sum-of-ROABPs lower bound for the refutation of a variant of the subset-sum axiom polynomial. The result also extends over the fields of positive characteristics when the target polynomial is suitably modified. The modification is inspired by the recent results (Hakoniemi, Limaye, Tzameret, STOC 2024 and Behera, Limaye, Ramanathan, Srinivasan, ICALP 2025). The mult-IPS_{Lin'} lower bound result is obtained by combining the quadratic-size formula lower bound technique of Kalorkoti (SICOMP 1985) with some additional ideas. The proof technique of IPS_{Lin'} lower bound result is inspired by the recent lower bound result of Chatterjee, Kush, Saraf and Shpilka (CCC 2024).

Problem

• Lower bounds for Ideal Proof System fragments
• Formula size bounds for multilinear refutation
• ROABP size bounds for subset-sum axiom refutation

Innovation

• Multilinear refutation with quadratic-size formula lower bound
• Exponential-size sum-of-ROABPs lower bound for refutation
• Combining Kalorkoti technique with additional ideas