By Stasys Jukna

Boolean circuit complexity is the combinatorics of laptop technology and comprises many interesting difficulties which are effortless to country and clarify, even for the layman. This ebook is a finished description of easy reduce certain arguments, protecting the various gemstones of this “complexity Waterloo” which were stumbled on during the last a number of many years, correct as much as effects from the final yr or . Many open difficulties, marked as examine difficulties, are pointed out alongside the way in which. the issues are regularly of combinatorial taste yet their ideas can have nice results in circuit complexity and laptop technology. The ebook might be of curiosity to graduate scholars and researchers within the fields of desktop technological know-how and discrete mathematics.

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**Additional info for Boolean Function Complexity: Advances and Frontiers **

**Example text**

The last bm variables are divided into b blocks y = (y1 , . . , yb ) of length m, and the value of An is deﬁned by An (x, y) = fx (⊕m (y1 ), . . , ⊕m (yb )). 6 A 3n lower bound for circuits Existing lower bounds for general circuits were proved using the so-called “gate-elimination” argument. The proofs themselves consist of a rather involved case analysis, and we will not present them here. Instead of that we will demonstrate the main idea by proving weaker lower bounds. The gate-elimination argument does the following.

For general circuits, a lower bound 3n − o(n) is achieved by particular symmetric functions, that is, functions whose value only depends on the number of ones in the input vector. 6 A 3n lower bound for circuits 5 10 15 20 25 37 n − k variables. 1. Amano and Tarui (2008) have shown that some highly mixed boolean functions can be computed by circuits of size 5n + o(1); hence, the property of being mixed alone is not enough to improve this lower bound. 1 for the proofs). 4). The function has n = 2b + bm variables with b = log(n/2) and m = n/(2b).

Hence, Prob[a = x] = 2−n for each vector x ∈ GF(2)n . It is easy to show (see Appendix B) that Prob[ a, x = a, y ] = 1/2 holds for every two vectors x = y in GF(2)n . Now consider a random operator L(x) = Ax where A is a random m × n matrix whose rows are random vectors in GF(2)n . By the previous fact, every pair (x, y) of vectors x = y in D is not separated by L with probability 2−m . By the linearity of expectation, at most a fraction 2−m of such pairs will be not separated by L. ⊓ ⊔ Now let f be a partial boolean function of n variables deﬁned on some domain D ⊆ {0, 1}n of size N = |D|.