lect06 - 6.841 Advanced Complexity Theory Feb 23, 2009...

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Unformatted text preview: 6.841 Advanced Complexity Theory Feb 23, 2009 Lecture 6 Lecturer: Madhu Sudan Scribe: Michael Forbes The goal of this lecture is to give alternate proof of PARITY / AC , following the outline of Razborov and Smolensky. 0.1 Probability Review Probability focuses on the probability of events and random variables. There are several facts that will be used throughout the course. Consult the appropriate textbook for proof. As this is a computer science course, we will restrict ourselves to the simpler case of discrete events, and discrete random variables. The results do generalize but we need not go there. Lemma 1 (Linearity of Expectation) For random variables X and Y , and real numbers a , b R , E [ aX + bY ] = a E [ X ] + b E [ Y ] . It is important to recall that this fact holds regardless of whether X and Y are independent or not. Lemma 2 For independent random variables X and Y , E [ XY ] = E [ X ] E [ Y ] . Lemma 3 (Union Bound) For events E 1 and E 2 , Pr[ E 1 E 2 ] Pr[ E 1 ] + Pr[ E 2 ] . Recall that E 1 and E 2 are called independent if Pr[ E 1 E 2 ] Pr[ E 1 ]Pr[ E 2 ]. Two discrete random variables are said to be independent if their joint probability distribution decomposes into a product of marginal distributions. That is, the probabilities of the variables taking certain values is what you would expect. Theorem 4 (Markovs Inequality) If X is a non-negative random variable then Pr[ X k E [ X ]] 1 k . Theorem 5 (Chebychevs Inequality) For random variable X with variance 2 , Pr[ | X- E [ X ] | k ] 1 k 2 . Theorem 6 (Chernoff Bound) For X 1 ,...,X n independent, identically distributed (possibly continuous) random variables with expectation , such that X i [0 , 1] , Pr n i =1 X i n- exp(- 2 n ) 0.2 Algebra Finite fields are finite sets equipped with the usual notions of addition and multiplication. That is, a finite field F q of size q is such that addition forms an abelian group, with an identity 0, and F q \ { } forms an abelian group with multiplication with identity element 1. These two operations are related through the distributive axiom. Abstract algebra gives us that there is a unique (up to isomorphism) finite field of size q if q is a prime power, and no finite field for other values of q . For q prime, these finite fields are the familiar structures of the integers modulo q . For most computer science purposes, the prime fields are enough. A basic fact about finite fields is their relation with polynomials. If we recall Fermats Little theorem, then we know that x p = x when working over F p (for prime p ). This leads us to think that no polynomial over F p need have degree in a single variable above p ....
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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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lect06 - 6.841 Advanced Complexity Theory Feb 23, 2009...

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