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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 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