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Unformatted text preview: Notes on Complexity Theory Last updated: October, 2011 Lecture 10 Jonathan Katz 1 NonUniform Complexity 1.1 The Power of Circuits Last time we saw that every function f : { , 1 } n → { , 1 } could be computed by a circuit of size n · 2 n , and noted that this bound could be improved to (1+ ε ) · 2 n /n for every ε > 0. We now show that this is essentially tight. Theorem 1 Let ε > and q ( n ) = (1 ε ) 2 n n . Then for n large enough there exists a function f : { , 1 } n → { , 1 } that cannot be computed by a circuit of size at most q ( n ) . Proof In fact, the fraction of functions f : { , 1 } n → { , 1 } that can be computed by circuits of size at most q ( n ) approaches 0 as n approaches infinity; this easily follows from the proof below. Let q = q ( n ). The proof is by a counting argument. We count the number of circuits of size q (note that if a function can be computed by a circuit of size at most q , then by adding useless gates it can be computed by a circuit of size exactly q ) and show that this is less than the number of nary functions. A circuit having q internal gates is defined by (1) specifying, for each internal gate, its type and its two predecessor gates, and (2) specifying the output gate. We may assume without loss of generality that each gate of the circuit computes a different function — otherwise, we can simply remove all but one copy of the gate (and rewire the circuit appropriately). Under this assumption, permuting the numbering of the internal gates does not affect the function computed by the circuit. Thus, the number of circuits with q internal gates is at most: ( 3 · ( q + n ) 2 ) q · q q ! ≤ ( 12 · ( q ) 2 ) q · q q ! . In fact, we are overcounting since some of these are not valid circuits (e.g., they are cyclic). We have: q · ( 12 · ( q + n ) 2 ) q q ! ≤ q · (36) q · q 2 q q q = q · (36 · q ) q ≤ (36 · q ) q +1 ≤ (2 n ) (1 ε )2 n /n + 1 = 2 (1 ε )2 n + n , for n sufficiently large, using Stirling’s bound q ! ≥ q q /e q ≥ q q / 3 q for the first inequality. But this is less than 2 2 n (the number of...
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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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