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Sub-constant error probabilistically checkable proof of almost-linear size. (English) Zbl 1213.68317

Summary: We show a construction of a \(PCP\) with both sub-constant error and almost-linear size. Specifically, for some constant \(0 < \alpha < 1\), we construct a \(PCP\) verifier for checking satisfiability of Boolean formulas that on input of size \(n\) uses \(\log n+O((\log n)^{1-\alpha})\) random bits to make 7 queries to a proof of size \(n\cdot 2^{O((\log n)^{1-\alpha})}\), where each query is answered by \(O((\log n)^{1-\alpha})\) bit long string, and the verifier has perfect completeness and error \(2^{-\Omega((\log n)^{\alpha})}\).
The construction is by a new randomness-efficient version of the aggregation through curves technique. Its main ingredients are a recent low degree test with both sub-constant error and almost-linear size and a new method for constructing a short list of balanced curves.

MSC:

68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q60 Specification and verification (program logics, model checking, etc.)
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