Short PCPs with Polylog Query Complexity

We give constructions of probabilistically checkable proofs (PCPs) of length $n \cdot polylog n$ proving satisfiability of circuits of size $n$ that can be verified by querying $polylog n$ bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping $n$ information bits to $n\cdot polylog n$ bit long codewords that are testable with $polylog n$ queries. Our constructions rely on new techniques revolving around properties of codes based on relatively high-degree polynomials in one variable, i.e., Reed-Solomon codes. In contrast, previous constructions of short PCPs, beginning with [L. Babai, L. Fortnow, L. Levin, and M. Szegedy, Checking computations in polylogarithmic time, in Proceedings of the 23rd ACM Symposium on Theory of Computing, ACM, New York, 1991, pp. 21-31] and until the recent [E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan, Robust PCPs of proximity, shorter PCPs, and applications to coding, in Proceedings of the 36th ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 13-15], relied extensively on properties of low-degree polynomials in many variables. We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a Reed-Solomon codeword, i.e., a univariate polynomial of specified degree. This reduction also gives an alternative to using the “sumcheck protocol” [C. Lund, L. Fortnow, H. Karloff, and N. Nisan, J. ACM, 39 (1992), pp. 859-868]. We then give a new PCP for the special task of proving that a function is close to being a Reed-Solomon codeword. The resulting PCPs are not only shorter than previous ones but also arguably simpler. In fact, our constructions are also more natural in that they yield locally testable codes first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite direction of getting locally testable codes from PCPs.