Lecture Space-Bounded Derandomization
نویسنده
چکیده
We now prove Theorem 1. Let M be a probabilistic machine running in space S (and time 2S), using R random bits, and deciding a language L with two-sided error. (Note that S, R are functions of the input length n, and the theorem requires S = Ω(log n).) We will assume without loss of generality that M always uses exactly R random bits on all inputs. Fixing an input x and letting ` be some parameter, we will view the computation of Mx as a random walk on a multi-graph in the following way: the nodes of the graph correspond to all N def = 2O(S) possible configurations of Mx, and there is an edge from a to b labeled by the string r ∈ {0, 1}` if and only if Mx moves from configuration a to configuration b after reading r as its next ` random bits. Computation of Mx is then equivalent to a random walk of length R/` on this graph, beginning from the node corresponding to the initial configuration of Mx. if x ∈ L then the probability that this random BPL is the two-sided-error version of RL. 2SC stands for “Steve’s class”, and captures computation that simultaneously uses polynomial time and polylogarithmic space.
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