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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 polynomial-time 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 non-deterministic 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 (read-only) random tape . For the latter approach, one can consider either one-way or two-way 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 time-complexity classes; in the following, ppt stands for “prob- abilistic, polynomial time” (where this is measured as worst-case 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 non-deterministic 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
- Analysis of algorithms, Computational complexity theory, Randomized algorithm, Probabilistic Turing machine, ppt machine