Download e-book for kindle: Boolean Function Complexity: Advances and Frontiers by Stasys Jukna

By Stasys Jukna

Boolean circuit complexity is the combinatorics of machine technology and includes many exciting difficulties which are effortless to country and clarify, even for the layman. This ebook is a finished  description of uncomplicated reduce certain arguments, protecting some of the gemstones of this “complexity Waterloo” which were stumbled on during the last numerous many years, 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 generally of combinatorial taste yet their suggestions can have nice outcomes in circuit complexity and desktop technological know-how. The booklet may be of curiosity to graduate scholars and researchers within the fields of laptop technology and discrete mathematics.

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6) implies that C(fn ) > 2n /n. Comparing these bounds, we can conclude that the sequence F cannot be contained in any invariant class Qσ with σ < 1. ⊓ ⊔ This result serves as an indication that there (apparently) is no other way to construct a most-complex sequence of boolean function other than to do n a “brute force search” (or “perebor” in Russian): just try all 22 boolean functions. 5 So where are the complex functions? 25 30 35 July 14, 2011 Unfortunately, the results above are not quite satisfactory: we know that almost all boolean functions are complex, but no specific (or explicit) complex function is known.

5), we can write f (x) as a disjunction 2n /m f (x) = a Ka (x1 , . . , xk ) ∧ fa,i (xk+1 , . . , xn ) , i=1 where a ranges over {0, 1}k , and each fa,i belongs to Hn−k,m (i). We will use this representation to design the desired circuit for f . The circuit consists of five subcircuits (see Fig. 9). The first subcircuit F1 computes all elementary conjunctions of the first k variables. 14, this circuit has size L(F1 ) ≤ 2k + 2k2k/2 . 4 Almost all functions are complex 25 The second subcircuit F2 also computes all elementary conjunctions of the remaining n − k variables.

It makes sense, therefore, to look more carefully at the graph structure of boolean functions: that is, to move from a “bit level” to a more global one and consider a given boolean function as a matrix or as a bipartite graph. The concept of graph complexity, as we will describe it below, was introduced by Pudlák, Rödl and Savický (1988), and was later considered by Razborov (1988, 1990), Chashkin (1994), Lokam (2003), Jukna (2006, 2009, 2010b), Drucker (2011), and other authors. A circuit for a given boolean function f generates this function starting from simplest “generators” – variables and their negations.

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Boolean Function Complexity: Advances and Frontiers by Stasys Jukna

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