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Unformatted text preview: 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 ) ≥ . 5 and s ( n ) ≤ . 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 ) ....
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.
- Spring '09