lecture13

# lecture13 - 1 COMPLEXITY THEORY CSci 5403 LECTURE XIII WHY...

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Unformatted text preview: 1 COMPLEXITY THEORY CSci 5403 LECTURE XIII: WHY CONSTANTS STILL DON’T MATTER RP = { A | exists polytime TM M: x 2 A ) Pr r [M(x,r) accepts] ¸ ½ x ∉ A ) Pr r [M(x,r) accepts] = 0. } coRP = { A | exists polytime TM M: x 2 A ) Pr r [M(x,r) accepts] = 1 x ∉ A ) Pr r [M(x,r) accepts] < ½ . } BPP = { A | exists polytime TM M: x 2 A ) Pr r [M(x,r) accepts] ¸ 2/3 x ∉ A ) Pr r [M(x,r) accepts] · 1/3. } Definition. A 2 ZPP if there exists PTM M so that 1. x 2 A ) M accepts x 2. x ∉ A ) M rejects x 3. For all x, the expected running time of M(x) is poly(|x|). Theorem. ZPP = RP Å coRP. Proof. A 2 RP \ coRP means 9 M, M’ : M(x) accepts ) x 2 A, Pr[M(x) accepts] ¸ ½ . M’(x) rejects ) x ∉ A, Pr[M’(x) rejects] ¸ ½ . while(true): run M(x). If M(x)=True: return True. run M’(x). If M’(x)=False: return False. Definition. A 2 ZPP if there exists PTM M so that 1. x 2 A ) M accepts x 2. x ∉ A ) M rejects x 3. For all x, the expected running time of M(x) is poly(|x|)....
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lecture13 - 1 COMPLEXITY THEORY CSci 5403 LECTURE XIII WHY...

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