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lect11 - 6.841 Advanced Complexity Theory Lecture 11...

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6.841 Advanced Complexity Theory Mar 11, 2009 Lecture 11 Lecturer: Madhu Sudan Scribe: Colin Jia Zheng 1 Recap We defined RP as the class of languages accepted by PPT machine with one-sided error bounded below 1 / 3, BPP with two-sided error with gap 1 / 3. RP was shown to be robust in the following sense. Define RP e such that L RP e if for some poly-time TM M and random bits y , x L Pr[ M ( x, y ) rejects] e ( | x | ) x / L Pr[ M ( x, y ) accepts] = 0 Then RP 1 1 / poly( n ) = RP = RP 1 / 2 poly( n ) (the two poly’s may be different polynomials), yet RP 1 1 / 2 n = NP . We will see that BPP is robust in the similar sense. Define BPP c,s such that L BPP c,s if for some poly-time TM M and random bits y , x L Pr[ M ( x, y ) accepts] c ( | x | ) x / L Pr[ M ( x, y ) accepts] s ( | x | ) Let us assume that, as often necessary, that s is “nice”, ie fully time constructible. (Quick note: If c s then BPP c,s would contain every language. While it is not required that c ( n ) 0 . 5 and s ( n ) 0 . 5, one can shift the probability by proper amount so that c, s do straddle 0.5.) 2 Amplification for BPP Using Chernoff bound we will see that BPP f ( n )+1 / poly( n ) ,f ( n ) 1 / poly( n ) = BPP = BPP 1 2 - poly( n ) , 2 - poly( n ) . Theorem 1 (Chernoff bound) Let X 1 , . . . , X k [0 , 1] be independent random variables and X = i X i /t .
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