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A Note on Almost Perfect Probabilistically Checkable Proofs of Proximity

Probabilistically checkable proofs of proximity (PCPP) are proof systems where the verifier is given a 3SAT formula, but has only oracle access to an assignment and a proof. The verifier accepts a satisfying assignment with a valid proof, and rejects (with high enough probability) an assignment that is far from all satisfying assignments (for any given proof). In this work, we focus on the type of computation the verifier is allowed to make. Assuming P $\neq$ NP, there can be no PCPP when the verifier is only allowed to answer according to constraints from a set that forms a CSP that is solvable in P. Therefore, the notion of PCPP is relaxed to almost perfect probabilistically checkable proofs of proximity (APPCPP), where the verifier is allowed to reject a satisfying assignment with a valid proof, with arbitrary small probability. We show, unconditionally, a dichotomy of sets of allowable computations: sets that have APPCPPs (which actually follows because they have PCPPs) and sets that do not. This dichotomy turns out to be the same as that of the Dichotomy Theorem, which can be thought of as dividing sets of allowable verifier computations into sets that give rise to NP-hard CSPs, and sets that give rise to CSPs that are solvable in P.