Prahladh Harsha

Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding

We continue the study of the trade-off between the length of probabilistically checkable proofs (PCPs) and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size

Some 3CNF Properties Are Hard to Test

n

The Communication Complexity of Correlation

): 1. We present PCPs of length

Bounding the Sensitivity of Polynomial Threshold Functions

\exp(o(\log\log n)^2)\cdot n

Some 3CNF properties are hard to test

that can be verified by making

Sound 3-Query PCPPs Are Long

o(\log\log n)

Small PCPs with Low Query Complexity

Boolean queries. 2. For every \epsilon>0, we present PCPs of length

Composition of Low-Error 2-Query PCPs Using Decodable PCPs

\exp(\log^\epsilon n)\cdot n

Communication Versus Computation

that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi polylogarithmic in the first case (of query complexity

On Polynomial Approximations to AC 0

o(\log\log n)