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Unformatted text preview: 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 • polysize Boolean circuits for Succinct CVAL ?? Does NP have linearsize, logdepth 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 2 April 12, 2011 7 Connection to TMs • given TM M running in time t(n) decides language L • can build circuit family {C n } that decides L – size of C n = O(t(n) 2 ) – Proof: CVAL construction • Conclude: L P implies family of polynomialsize circuits that decides L April 12, 2011 8 Connection to TMs • other direction? • A polysize circuit family: – C n = (x 1 x 1 ) if M n halts – C n = (x 1 x 1 ) if M n loops • decides (unary version of) HALT! • oops… April 12, 2011 9 Uniformity • Strange aspect of circuit family: – can “encode” (potentially uncomputable) information in family specification • solution: uniformity – require specification is simple to compute Definition : circuit family {C n } is logspace uniform iff TM M outputs C n on input 1 n and runs in O(log n) space April 12, 2011 10 Uniformity Theorem : P = languages decidable by logspace uniform, polynomialsize circuit families {C n }. • Proof: – already saw ( ) – ( ) on input x, generate C x , evaluate it and accept iff output = 1 April 12, 2011 11 TMs that take advice • family {C n } without uniformity constraint is called “ nonuniform” • regard “nonuniformity” as a limited resource just like time, space, as follows: – add readonly “advice” tape to TM M – M “decides L with advice A(n)” iff M(x, A(x)) accepts x...
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 Fall '09

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