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

Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi ‍[J. ‍ACM, 65(6):37:1–55, 2018]) 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 ‍[Theory ‍Comput., 17: 1-88, 2021]), 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.