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13 Lemma (Kannan 1981) For every integer k ≥ 1, there is a boolean function of n variables such that f can be written as a DNF with n2k+1 monomials, but C(f ) > nk . Proof. We view a circuit computing a boolean function f as accepting the set of vectors f −1 (1) ⊆ {0, 1}n, and rejecting the remaining vectors. Fix a subset T ⊆ {0, 1}n of size |T | = nt2 = n2k+1 . 12, we know that 2k at most 2n < 2|T | distinct subsets of T can be accepted by circuits of size at most nk . Thus, some subset S ⊆ T cannot be accepted by a circuit of size nk .

For example, in combinatorics it is known that a random graph on n vertices is a Ramsey-graph, that is, has no cliques or independent sets on more than t = 2 log n vertices. But where such “mystical” graphs are? The best known explicit construction of non-bipartite t-Ramsey √ graphs due to Frankl and Wilson only achieves a much larger value t about exp( log n log log n). 5 So where are the complex functions? 37 the bipartite case, t-Ramsey graphs with t = n1/2 can be obtained from Hadamard matrices: Lindsey’s Lemma (see Appendix A) implies that such a matrix can have a monochromatic a × b submatrix only if ab ≤ n.

6) The proof actually gives that the o(1) factor is equal to O(1/ log n); such a lower bound was also proved by Lutz (1992). 4 Almost all functions are complex 27 Redkin (2004) considered the behavior of the Shannon function when restricted to boolean functions accepting a small number of input vectors. Let C(n, K) denote the smallest number t such that every boolean function f of n variables such that |f −1 (1)| = K can be computed by a circuit over {∧, ∨, ¬} of size at most t. Redkin (2004) proved that, if 2 ≤ K ≤ log2 n − c log2 log2 n holds for some constant c > 1, then C(n, K) ∼ 2n .

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Boolean function complexity by Jukna S.

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