lect11

Lect11 - 6.841 Advanced Complexity Theory Lecture 11 Lecturer Madhu Sudan Scribe Colin Jia Zheng 1 Recap We defined RP as the class of languages

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ) ....
View Full Document

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.

Page1 / 2

Lect11 - 6.841 Advanced Complexity Theory Lecture 11 Lecturer Madhu Sudan Scribe Colin Jia Zheng 1 Recap We defined RP as the class of languages

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online