Improved Low-Degree Testing and its Applications

NP = PCP(log n, 1) and related results crucially depend upon the close connection between the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [30]. The strongest previously known connection for this test states that a function passes the test with probability δ for some δ > 7/8 iff the function has agreement ≈ δ with a polynomial of degree d. We present a new, and surprisingly strong, analysis which shows that the preceding statement is true for arbitrarily small ≈, provided the field size is polynomially larger than d/δ. The analysis uses a version of Hilbert irreducibility, a tool of algebraic geometry.

As a consequence we obtain an alternate construction for the following proof system: A constant prover 1-round proof system for NP languages in which the verifier uses O(log n) random bits, receives answers of size O(log n) bits, and has an error probability of at most \( 2^{{ - \log ^{{1 - \in }} n}} \). Such a proof system, which implies the NP-hardness of approximating Set Cover to within Ω(log n) factors, has already been obtained by Raz and Safra [29]. Raz and Safra obtain their result by giving a strong analysis, in the sense described above, of a new low-degree test that they present.

A second consequence of our analysis is a self tester/corrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on δ fraction of inputs where \( \delta = 1/{\left| F \right|}^{ \in } \ll 0.5 \), then the tester/corrector determines δ and generates \( O{\left( {\frac{1} {\delta }} \right)} \) values for every input, such that one of them is the correct output. In fact, our results yield something stronger: Given the buggy program, we can construct \( O{\left( {\frac{1} {\delta }} \right)} \) randomized programs such that one of them is correct on every input, with high probability. Such a strong self-corrector is a useful tool in complexity theory—with some applications known.

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Correspondence to Sanjeev Arora*.

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* Supported by an NSF CAREER award, an Alfred P. Sloan Fellowship, and a Packard Fellowship.

† Part of this work was done when this author was at the IBM Thomas J. Watson Research Center.

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Arora*, S., Sudan†, M. Improved Low-Degree Testing and its Applications. Combinatorica 23, 365–426 (2003). https://doi.org/10.1007/s00493-003-0025-0

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AMS Subject Classification (2000):

  • 68Q10
  • 68Q17