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Unformatted text preview: Notes on Complexity Theory Last updated: October, 2011 Lecture 12 Jonathan Katz 1 Randomized Time Complexity Is deterministic polynomialtime computation the only way to define “feasible” computation? Al lowing probabilistic algorithms, that may fail with tiny probability, seems reasonable. (In particular, consider an algorithm whose error probability is lower than the probability that there will be a hard ware error during the computation, or the probability that the computer will be hit by a meteor during the computation.) This motivates our exploration of probabilistic complexity classes. There are two different ways to define a randomized model of computation. The first is via Tur ing machines with a probabilistic transition function : as in the case of nondeterministic machines, we have a Turing machine with two transitions functions, and a random one is applied at each step. The second way to model randomized computation is by augmenting Turing machines with an additional (readonly) random tape . For the latter approach, one can consider either oneway or twoway random tapes; the difference between these models is unimportant for randomized time complexity classes, but (as we will see) becomes important for randomized space classes. Whichever approach we take, we denote by M ( x ) a random computation of M on input x , and by M ( x ; r ) the (deterministic) computation of M on input x using random choices r (where, in the first case, the i th bit of r determines which transition function is used at the i th step, and in the second case r is the value written on M ’s random tape). We now define some randomized timecomplexity classes; in the following, ppt stands for “prob abilistic, polynomial time” (where this is measured as worstcase time complexity over all inputs, and as always the running time is measured as a function of the length of the input x ). Definition 1 L ∈ RP if there exists a ppt machine M such that: x ∈ L ⇒ Pr[ M ( x ) = 1] ≥ 1 / 2 x 6∈ L ⇒ Pr[ M ( x ) = 0] = 1 . Note that if M ( x ) outputs “1” we are sure that x ∈ L ; if M ( x ) outputs “0” we cannot make any definitive claim. Viewing M as a nondeterministic machine for L , the above means that when x ∈ L at least half of the computation paths of M ( x ) accept, and when x 6∈ L then none of the computation paths of M ( x ) accept. Put differently, a random tape r for which M ( x ; r ) = 1 serves as a witness that x ∈ L . We thus have RP ⊆ NP . Symmetrically: Definition 2 L ∈ co RP if there exists a ppt machine M such that: x ∈ L ⇒ Pr[ M ( x ) = 1] = 1 x 6∈ L ⇒ Pr[ M ( x ) = 0] ≥ 1 / 2 ....
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 Fall '08
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 Analysis of algorithms, Computational complexity theory, Randomized algorithm, Probabilistic Turing machine, ppt machine

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