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lecnotesoct82 - Recent advances in Complexity CIS 6930/CIS...

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Recent advances in Complexity CIS 6930/CIS 4930 October 10, 2002 Lecture 14 Lecturer: Dr. Meera Sitharam Scribe: Erwin Jansen 1 Introduction Today we finished the lower bound proof for monotone functions. Lemma 1. For every monotone circuit C , the number of negative test graphs for which C ˆ C does not hold is at most size ( C ) · m 2 · ( l 2 ) k - 1 · p · ( k - 1) n Proof. Look at the approximations of and gates and show that at each stage of the construction of ˆ C fails. I.e. it gives another output than the original output of or in C in at most m 2 · ( ( l 2 ) k - 1 ) p · ( k - 1) n negative test graphs. We will start by looking at gates. Consider the gate with two inputs: A = r i =1 d X i e and B = s i =1 d Y i e . From the previous lecture we know that the new approximator is obtained by performing ar most 2 m pluckings on A B . Now we are going to look at how many negative test graphs are destroyed by each plucking. More specifically we will look at the following: Let X 1 , ..., X p
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