lecture25 - Bounds on circuit computation so far CSci 5403...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XXV: CIRCUIT LOWER BOUNDS “Bounds” on circuit computation so far: Size Hierarchy Theorem: For every ƒ(n) = o(2 n /n) there exists a function with complexity ω (ƒ(n)) Shannon bounds: “most” functions have complexity Θ (2 n /n) Relationships to uniform classes ( k, PH languages in SIZE( Θ (n k )), X P/poly Amazing(X), BPP P/poly, NC 1 L NL NC 2 …) This week: concrete bounds, specific languages. Theorem. Define (x 1 ,…,x n ) = i x i mod 2. Then any constant-depth circuit family for (with unbounded fan-in/out ( , ,¬ gates) has exponential size. Corollary. AC 0 . Proposition. NC 1 . Corollary. AC 0 NC 1 . Exponential lower bounds! For a simple function! Idea. A constant-depth ( , ,¬) poly(n)-size circuit can be set to a constant by fixing “few” input bits: But fixing even n-1 bits of the input to does not make the function constant!
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2 Definition. A restriction ρ {0,1,*} n of function ƒ : {0,1} n {0,1} is the function given by ρ (x) i = ρ i , if ρ i {0,1} x i , if ρ i = * We let ƒ| ρ denote the map x ƒ( ρ (x)) Proposition. | ρ {0,1} iff ρ {0,1} n .
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