## Smooth and Strong PCPs

I have been waiting for this posting for a while. It resolved a problem left open in my paper with Irit and Tom, showing that one can obtain strong PCPs that are also smooth, unlike the highly-unsmooth strong PCPs that are easily obtained in the former paper.

#### The original abstract

Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:

• A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
• A PCP is smooth if each location in a proof is queried with equal probability.
We prove that all sets in $\mathcal{NP}$ have a smooth and strong PCP of polynomial length that can be verified based on a constant number of queries. We do so by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed--Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $\mathcal{NP}$ has a smooth strong *canonical* PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $\mathcal{NP}$ witnesses to correct proofs.

This improves on the recent result of Dinur, Gur and Goldreich (ITCS, 2019) that constructs strong canonical PCPPs that are inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, proving a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019). Here *stability* means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric).

See ECCC TR19-023.

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