lec5 - CS151 Complexity Theory Lecture 5 Introduction Power...

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CS151 Complexity Theory Lecture 5 April 12, 2011
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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 ??
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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
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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
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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
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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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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 polynomial-size circuits that decides L
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April 12, 2011 8 Connection to TMs other direction? A poly-size 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…
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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
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April 12, 2011 10 Uniformity Theorem : P = languages decidable by logspace uniform, polynomial-size circuit families {C n }. Proof: already saw ( ) – ( ) on input x, generate C |x| , evaluate it and accept iff output = 1
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April 12, 2011 11 TMs that take advice family {C n } without uniformity constraint is called “ non-uniform” regard “non-uniformity” as a limited resource just like time, space, as follows: add read-only “advice” tape to TM M M “decides L with advice A(n)” iff M(x, A(|x|)) accepts x L note: A(n) depends only on |x|
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April 12, 2011 12
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