Short PCPs with Projection Queries

We construct a PCP for NTIME(2n) with constant soundness, 2npoly(n) proof length, and poly(n) queries where the verifier’s computation is simple: the queries are a projection of the input randomness, and the computation on the prover’s answers is a 3CNF. The previous upper bound for these two computations was polynomial-size circuits. Composing this verifier with a proof oracle increases the circuit-depth of the latter by 2. Our PCP is a simple variant of the PCP by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (CCC 2005). We also give a more modular exposition of the latter, separating the combinatorial from the algebraic arguments. If our PCP is taken as a black box, we obtain a more direct proof of the result by Williams, later with Santhanam (CCC 2013) that derandomizing circuits on n bits from a class C in time 2n/nω(1) yields that NEXP is not in a related circuit class C ′. Our proof yields a tighter connection: C is an And-Or of circuits from C ′. Along the way we show that the same lower bound follows if the satisfiability of the And of any 3 circuits from C ′ can be solved in time 2n/nω(1). ∗The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258. Email: [email protected] †Supported by NSF grants CCF-0845003, CCF-1319206. Email: [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 17 (2014)