Polynomial Identity Testing and the Ideal Proof System: PIT Is in NP If and Only If IPS Can Be p-Simulated by a Cook-Reckhow Proof System

The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018) is an algebraic proof system that uses algebraic circuits to refute the solvability of unsatisfiable systems of polynomial equations. One potential drawback of IPS is that verifying an IPS proof is only known to be doable using Polynomial Identity Testing (PIT), which is solvable by a randomized algorithm, but whose derandomization, even into NSUBEXP, is equivalent to strong lower bounds. However, the circuits that are used in IPS proofs are not arbitrary, and it is conceivable that one could get around general PIT by leveraging some structure in these circuits. This proposal may be even more tempting when IPS is used as a proof system for Boolean Unsatisfiability, where the equations themselves have additional structure. Our main result is that, on the contrary, one cannot get around PIT as above: we show that IPS, even as a proof system for Boolean Unsatisfiability, can be p-simulated by a deterministically verifiable (Cook-Reckhow) proof system if and only if PIT is in NP (with some technical caveats about the relationship between the field used for IPS and that used for PIT). We use our main result to propose a potentially new approach to derandomizing PIT into NP.