Improved Low-Degree Testing and its Applications

Abstract

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! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu % HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr % 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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu % 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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu % 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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu % HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr % fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9 % 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.