Functional Lower Bounds in Algebraic Proofs: Symmetry, Lifting, and Barriers

TL;DR

This paper advances algebraic proof complexity by developing symmetry-based methods to generate new hard instances for IPS and similar systems, while also revealing the method's limitations in proving bounds for Boolean formulas.

Contribution

It introduces a symmetry-based approach to produce new hard instances for algebraic proof systems and demonstrates the method's limitations in establishing bounds for Boolean formulas.

Findings

New hard instances over finite fields and for non-Subset Sum problems.

Stronger constant-depth lower bounds achieved.

Limitations of the functional lower bound method for Boolean formulas.

Abstract

Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi [GP18]) offer a general model for deriving polynomials in an ideal and refuting unsatisfiable propositional formulas, subsuming most standard propositional proof systems. A major approach for lower bounding the size of IPS refutations is the Functional Lower Bound Method (Forbes, Shpilka, Tzameret and Wigderson [FSTW21]), which reduces the hardness of refuting a polynomial equation f(x) = 0 with no Boolean solutions to the hardness of computing the function 1/f(x) over the Boolean cube with an algebraic circuit. Using symmetry, we provide a general way to obtain many new hard instances against fragments of IPS via the functional lower bound method. This includes hardness over finite fields and hard instances different from Subset Sum variants, both of which were unknown before, and stronger constant-depth lower bounds. Conversely, we expose the limitation of this method by showing it cannot lead to proof complexity lower bounds for any hard Boolean instance (e.g., CNFs) for any sufficiently strong proof systems.