MIT6_045JS11_lec17

MIT6_045JS11_lec17 - 6.045: Automata, Computability, and...

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Unformatted text preview: 6.045: Automata, Computability, and Complexity (GITCS) Class 17 Nancy Lynch Today Probabilistic Turing Machines and Probabilistic Time Complexity Classes Now add a new capability to standard TMs: random choice of moves. Gives rise to new complexity classes: BPP and RP Topics: Probabilistic polynomial-time TMs, BPP and RP Amplification lemmas Example 1: Primality testing Example 2: Branching-program equivalence Relationships between classes Reading: Sipser Section 10.2 Probabilistic Polynomial-Time Turing Machines, BPP and RP Probabilistic Polynomial-Time TM New kind of NTM, in which each nondeterministic step is a coin flip: has exactly 2 next moves, to each of which we assign probability . Example: 1/4 1/4 1/8 1/8 1/8 1/16 1/16 Computation on input w To each maximal branch, we assign a probability: Has accept and reject states, as for NTMs. Now we can talk about probability of acceptance or rejection, on input w. number of coin flips on the branch Probabilistic Poly-Time TMs 1/4 1/4 1/8 1/8 1/8 1/16 1/16 Computation on input w Probability of acceptance = b an accepting branch Pr(b) Probability of rejection = b a rejecting branch Pr(b) Example: Add accept/reject information Probability of acceptance = 1/16 + 1/8 + 1/4 + 1/8 + 1/4 = 13/16 Probability of rejection = 1/16 + 1/8 = 3/16 We consider TMs that halt (either accept or reject) on every branch--- deciders . So the two probabilities total 1. Acc Acc Acc Acc Rej Acc Rej Probabilistic Poly-Time TMs Time complexity: Worst case over all branches, as usual. Q : What good are probabilistic TMs? Random choices can help solve some problems efficiently. Good for getting estimates---arbitrarily accurate, based on the number of choices. f Example: Monte Carlo estimation of areas E.g, integral of a function f. Repeatedly choose a random point (x,y) in the rectangle. Compare y with f(x). Fraction of trials in which y f(x) can be used to estimate the integral of f. Probabilistic Poly-Time TMs Random choices can help solve some problems efficiently. Well see 2 languages that have efficient probabilistic estimation algorithms. Q : What does it mean to estimate a language? Each w is either in the language or not; what does it mean to approximate a binary decision? Possible answer: For most inputs w, we always get the right answer, on all branches of the probabilistic computation tree. O r : For most w, we get the right answer with high probability. Better answer: For every input w, we get the right answer with high probability. Probabilistic Poly-Time TMs Better answer: For every input w, we get the right answer with high probability....
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This note was uploaded on 12/26/2011 for the course ENGINEERIN 18.400J taught by Professor Prof.scottaaronson during the Spring '11 term at MIT.

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MIT6_045JS11_lec17 - 6.045: Automata, Computability, and...

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