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Unformatted text preview: 6.045J/18.400J: Automata, Computability and Complexity Nancy Lynch Homework 12.2 (Fake) Due: Never Elena Grigorescu Readings: Sipser, Section 10.2. Problem 1 : Sipser problem 10.11. Let M be a probabilistic polynomial time TM and let C be a language where, for some fixed 0 < ǫ 1 < ǫ 2 < 1, 1. w negationslash∈ C implies Pr[M acceps w] ≤ ǫ 1 2. w ∈ C implies Pr[M accepts w] ≥ ǫ 2 . Show that C ∈ BPP . (Hint: Use Lemma 10.5) Solution 1 : Since the error probability of M is ≤ ǫ 2 we can use the amplification algorithm given in Lemma 10.5 with ǫ = ǫ 2 to obtain a BPP algorithm with error probability 2- poly ( n ) < 1 / 3. Problem 2 : Define the language class PP as follows: A language L ∈ PP if and only if there exists a probabilistic polynomial time Turing machine such that: · If w ∈ L , then Pr[ M accepts w ] ≥ 1 2 . · If w negationslash∈ L , then Pr[ M accepts w ] < 1 2 ....
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This note was uploaded on 08/12/2009 for the course CS 430 taught by Professor Nancylynch during the Spring '07 term at New Mexico Junior College.
- Spring '07