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
% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
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% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaaIYaWa
% aWbaaSqabeaacqGHsislciGGSbGaai4BaiaacEgadaahaaadbeqaai
% aaigdacqGHsislcqGHiiIZaaWccaWGUbaaaaaa!4CB1!$$
2^{{ - \log ^{{1 - \in }} n}}
$$. 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
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% HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr
% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacqaH0oaz
% cqGH9aqpcaaIXaGaai4lamaaemaabaGaamOraaGaay5bSlaawIa7am
% aaCaaaleqabaGaeyicI4maaOGaeSOAI0JaaGimaiaac6cacaaI1aaa
% aa!50F9!$$
\delta = 1/{\left| F \right|}^{ \in } \ll 0.5
$$, then the
tester/corrector determines δ and generates
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% HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr
% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGpbWa
% aeWaaeaadaWcaaqaaiaaigdaaeaacqaH0oazaaaacaGLOaGaayzkaa
% aaaa!4880!$$
O{\left( {\frac{1}
{\delta }} \right)}
$$ 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
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% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGpbWa
% aeWaaeaadaWcaaqaaiaaigdaaeaacqaH0oazaaaacaGLOaGaayzkaa
% aaaa!4880!$$
O{\left( {\frac{1}
{\delta }} \right)}
$$ 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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