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Unformatted text preview: Notes on Complexity Theory Last updated: October, 2011 Lecture 11 Jonathan Katz 1 NonUniform Complexity 1.1 Circuit Lower Bounds for a Language in Σ 2 ∩ Π 2 We have seen that there exist “very hard” languages (i.e., languages that require circuits of size (1 ε )2 n /n ). If we can show that there exists a language in NP that is even “moderately hard” (i.e., requires circuits of superpolynomial size) then we will have proved P 6 = NP . (In some sense, it would be even nicer to show some concrete language in NP that requires circuits of superpolynomial size. But mere existence of such a language is enough.) Here we show that for every c there is a language in Σ 2 ∩ Π 2 that is not in size ( n c ). Note that this does not prove Σ 2 ∩ Π 2 6⊆ P / poly since, for every c , the language we obtain is different. (Indeed, using the time hierarchy theorem, we have that for every c there is a language in P that is not in time ( n c ).) What is particularly interesting here is that (1) we prove a nonuniform lower bound and (2) the proof is, in some sense, rather simple. Theorem 1 For every c , there is a language in Σ 4 ∩ Π 4 that is not in size ( n c ) . Proof Fix some c . For each n , let C n be the lexicographically first circuit on n inputs such that (the function computed by) C n cannot be computed by any circuit of size at most n c . By the nonuniform hierarchy theorem (see [1]), there exists such a C n of size at most n c +1 (for n large enough). Let L be the language decided by { C n } , and note that we trivially have L 6∈ size ( n c ). We claim that L ∈ Σ 4 ∩ Π 4 . Indeed, x ∈ L iff (let  x  = n ): 1. There exists a circuit C of size at most n c +1 such that 2. For all circuits C (on n inputs) of size at most n c , and for all circuits B (on n inputs) lexicographically preceding C , 3. There exists an input x ∈ { , 1 } n such that C ( x ) 6 = C ( x ), and there exists a circuit B of size at most n c such that 4. For all w ∈ { , 1 } n it holds that B ( w ) = B ( w ) and 5. C ( x ) = 1....
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This note was uploaded on 01/13/2012 for the course CMSC 652 taught by Professor Staff during the Fall '08 term at Maryland.
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