By Stasys Jukna

Boolean circuit complexity is the combinatorics of laptop technology and comprises many interesting difficulties which are effortless to kingdom and clarify, even for the layman. This e-book is a accomplished description of easy reduce sure arguments, overlaying a number of the gem stones of this “complexity Waterloo” which were chanced on over the last numerous a long time, correct as much as effects from the final yr or . Many open difficulties, marked as study difficulties, are pointed out alongside the way in which. the issues are in general of combinatorial style yet their recommendations may have nice outcomes in circuit complexity and computing device technological know-how. The booklet might be of curiosity to graduate scholars and researchers within the fields of computing device technological know-how and discrete mathematics.

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4 Almost all functions are complex 1+ 31 2 log n − O(1) 2n O(1) 2n ≤ S(n) ≤ 1 + √ . n n n n Lipatova (1989) announced an improvement of the upper bound to S(n) ≤ 1 + O 5 10 log n n 2n . 10), the behavior of Shannon’s function is somewhat diﬀerent: for some boolean functions, their formulas are at least n/ log n times larger than circuits and switching networks. When counting formulas, we have to count full binary tree, that is, binary trees where every vertex has either two children or no children.

An important result of Reingold (2004) translates to BP(fn ) ≤ S(fn )c . This is a “P = NP” type result for branching programs. 4 Almost all functions are complex 25 30 35 July 14, 2011 We still cannot prove super-linear lower bounds for circuits with AND, OR and NOT gates. This is in sharp contrast with the fact, proved more than 60 years ago by Riordan and Shannon (1942) that most boolean functions require formulas of leafsize about 2n / log n. Then Shannon (1949) showed a lower bound 2n /n for circuits.

The operator L(x) is just a set of m ≤ n parity functions, and hence, can be computed by a trivial circuit of size O(n2 ), which is o(N/n) because log N = Ω(n), by our assumption. The function h can be computed by a small circuit just because it accepts at most N/n3 vectors x ∈ D. Indeed, h(x) = 0 for all x ∈ D0 because then L(x) ∈ L(D0 ). Hence, h can accept a vector x ∈ D only if x ∈ D1 and g(L(x)) = 0, that is, if x ∈ D1 and L(x) = L(y) for some y ∈ D0 . Since the operator L is almost injective, and since 2m ≥ N n3 , there are at most 2−m N2 ≤ N/n3 pairs (y, x) ∈ D0 × D1 such that L(x) = L(y).