# hw7 - CSci 5403 Spring 2010 Homework 7 due 1 Fool me once...

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Unformatted text preview: CSci 5403, Spring 2010 Homework 7 due: April 27, 2010 1. Fool me once. . . Deﬁne the function fMAJ : {0, 1}n × {0, 1}n → {0, 1} to return 1 iﬀ at least n + 1 bits of its input are 1. Give a fooling set of size n for fMAJ . 2. Rank my matrix Let fIP (x, y ) = i xi yi mod 2. Show that rank (MIP ) = 2n . 3. Parity Circuits Prove that any depth-two circuit that computes parity of t bits must either be a DNF with 2t−1 terms or a CNF with 2t−1 clauses. 4. Restricted sequences Complete the proof of H˚ astad’s switching lemma given in class: rs prove that |Seq(r, s)| ≤ ln 2 , where Seq(r, s) is the set of sequences of nonzero r-bit strings with total hamming weight s. 5. The Monotone Blues Prove that a boolean function f : {0, 1}n → {0, 1} is monotone if and only if it is computable by a monotone circuit. 1 ...
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