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

TL;DR

The paper proves that Polynomial Identity Testing (PIT) is in NP if and only if the Ideal Proof System (IPS) can be efficiently simulated by a Cook-Reckhow proof system, linking proof complexity to derandomization of PIT.

Contribution

It establishes a tight equivalence between PIT being in NP and the p-simulation of IPS by Cook-Reckhow proof systems, providing a new perspective on derandomization.

Findings

PIT in NP implies IPS can be p-simulated by Cook-Reckhow systems.

If IPS can be p-simulated by Cook-Reckhow, then PIT is in NP.

The result offers a new approach to derandomizing PIT into NP.

Abstract

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. We use our main result to propose a potentially new approach to derandomizing PIT into NP.