lec5 - Introduction Power from an unexpected source we know...

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1 CS151 Complexity Theory Lecture 5 April 12, 2011 April 12, 2011 2 Introduction Power from an unexpected source? we know P EXP , which implies no poly- time algorithm for Succinct CVAL poly-size Boolean circuits for Succinct CVAL ?? Does NP have linear-size, log-depth Boolean circuits ?? April 12, 2011 3 Outline Boolean circuits and formulas uniformity and advice the NC hierarchy and parallel computation the quest for circuit lower bounds a lower bound for formulas April 12, 2011 4 Boolean circuits C computes function f:{0,1} n {0,1} in natural way identify C with function f it computes circuit C directed acyclic graph nodes: AND ( ); OR ( ); NOT ( ); variables x i x 1 x 2 x 3 x n April 12, 2011 5 Boolean circuits size = # gates depth = longest path from input to output formula (or expression) : graph is a tree every function f:{0,1} n {0,1} computable by a circuit of size at most O(n2 n ) AND of n literals for each x such that f(x) = 1 OR of up to 2 n such terms April 12, 2011 6 Circuit families circuit works for specific input length we‟re used to f:∑ * ! {0,1} circuit family : a circuit for each input length C 1 , C 2 , C 3 , … = “{C n }” • “{C n } computes f” iff for all x C |x| (x) = f(x) • “{C n } decides L”, where L is the language associated with f
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