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lect05 - 6.841 Advanced Complexity Theory Feb 18, 2009...

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Unformatted text preview: 6.841 Advanced Complexity Theory Feb 18, 2009 Lecture 5 Lecturer: Madhu Sudan Scribe: Yang Cai 1 Overview • PARITY / ∈ AC . • Random Restriction • Switching Lemma DNF → CNF 2 Introduction We first introduce two sets of classes. • AC k : Class of functions computable by polynomial size and O ((log n ) k ) depth circuits over {∞ - AND, ∞ - OR,NOT } gates. • NC k : Class of functions computable by polynomial size and O ((log n ) k ) depth circuits over { 2- AND, 2- OR,NOT } gates. We know that for any k , AC k ⊆ NC k +1 ⊆ AC k +1 , so S k AC k = S k NC k . Through this lecture, we assume all circuits in AC k are organized to have alternating levels of AND and OR gates. Because all NOT gates can be moved to the first level, and since the AND and OR gates have infinite fan-in, we can combine any consecutive AND or OR levels. So we can consider the depth as the number of AND and OR levels. The goal of this lecture is to prove the following theorem PARITY / ∈ AC . By PARITY we mean PARITY ( x 1 ,x 2 ,...,x n ) = X i x i ( mod 2) . Theorem 1 [Furst, Saxe, Sipser; Ajtai; Yao; H¨astad; Razborov; Razborov-Smolensky] PARITY / ∈ AC . 3 Random Restriction If x 1 ,x 2 ,...,x n are variables, a random restriction on them is to randomly set values for most of the variables....
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.

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lect05 - 6.841 Advanced Complexity Theory Feb 18, 2009...

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