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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 < 1 < 2 < 1, 1. w negationslash C implies Pr[M acceps w] 1 2. w C implies Pr[M accepts w] 2 . Show that C BPP . (Hint: Use Lemma 10.5) Solution 1 : Since the error probability of M is 2 we can use the amplification algorithm given in Lemma 10.5 with = 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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- Spring '07