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TR19-023 | 25th February 2019 19:48
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#### Smooth and Strong PCPs

**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).