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 DONT MATTER RP = { A | exists polytime TM M: x 2 A ) Pr r [M(x,r) accepts] 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] < . } 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 coRP. Proof. A 2 RP \ coRP means 9 M, M : M(x) accepts ) x 2 A, Pr[M(x) accepts] . M(x) rejects ) x A, Pr[M(x) rejects] . 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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