lecture14 - How do the probabilistic complexity classes:...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XIV: HOW DOES RANDOMNESS FIT IN? ZPP RP coRP BPP L NL NC P NP coNP PH PSPACE P/poly How do the probabilistic complexity classes: Fit in our picture of “efficient” deterministic classes: Proposition. P µ ZPP. Proposition. ZPP µ NP Å coNP Proposition. coRP µ coNP. Proposition. RP [ coRP µ BPP Conjecture. P = RP = BPP. Proposition. BPP µ PSPACE. We will see later that if there is a language L 2 DTIME(2 O(n) ) \ SIZE(2 o(n) ), then P = BPP. Theorem. BPP µ P/poly. Proof. Let L 2 BPP be decided by PPTM M. wlog, let M have error probability 2 -2n , i.e. x 2 A ) Pr r [M(x,r) accepts] ¸ 1 - 2 -2n . x A ) Pr r [M(x,r) accepts] 2 -2n . Say random tape r is “bad” for x if M(x,r) (x 2 L) Otherwise r is “good” for x. We give three “different” proofs of the following Lemma. There is a random tape r that is good for all x 2 {0,1} n .
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2 Lemma. Let M use random tapes of length p(n). 9 r 2 R = {0,1} p(n) . 8 x 2 {0,1} n . M(x,r) is correct. Proof 1. (Counting) For each x, at most 1/2 2n fraction of r’s are bad. Let B(x) = { r | r is bad for x}, then |B(x)| 2 p(n)-2n . Then | [ x B(x)| 2 n max x |B(x)| 2 p(n)-2n+n < 2 p(n) . The remaining strings in R are good for all inputs. Lemma. Let M use random tapes of length p(n). 9 r 2 R = {0,1} p(n) . 8 x 2 {0,1} n . M(x,r) is correct. Proof 2. (Probabilistic method) Pr [ (r 2 B(x 1 )) Ç (r 2 B(x 2 )) … Ç (r 2 B(x 2 )) ] i Pr[ r 2 B(x i ) ] = 2 n £ 1/2 -2n = 2 -n . So Pr[r is good for all x]
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This note was uploaded on 10/21/2011 for the course CSCI 5403 taught by Professor Sturtivant,c during the Spring '08 term at Minnesota.

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lecture14 - How do the probabilistic complexity classes:...

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