By Stasys Jukna

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

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Of boolean functions of n variables such that C(fn ) ≤ t(n). 17 (Circuit Hierarchy Theorem) If n ≤ t(n) ≤ 2n−2 /n then Size(t) 10 15 Size(4t) . Proof: Fix the maximal m ∈ {1, . . , n} such that t(n) ≤ 2m /m ≤ 2 · t(n). This is possible: if m is the largest number with 2m /m ≤ 2 · t(n), then 2m+1 /(m + 1) > 2 · t(n), which implies t(n) ≤ 2m /m. Consider the set Bn,m of all boolean functions of n variables that depend only on m bits of their inputs. By the Shannon–Lupanov lower bound, there exists fn ∈ Bn,m such that C(fn ) > 2m /m ≥ t(n).

Xn+1 ) be an arbitrary boolean function in Q depending on n + 1 variables. 1) yields |Q(n + 1)| ≤ |Q(n)|2 . 4 Almost all functions are complex 33 n sequence |Q(n)|1/2 is non-increasing. If Q = ∅, then n n n n 1 = 11/2 ≤ |Q(n)|1/2 ≤ (22 )1/2 = 2 . Thus Lim(Q) exists and is a number in the interval [1, 2]. 22, every invariant class Q of boolean functions deﬁnes the unique real number 0 ≤ σ ≤ 1 such that Lim(Q) = 2σ . This number is an important parameter of the invariant class characterizing its cardinality.

V v∈A∪B v∈A∪B v 5 the negated variable ¬xP . Thus Lbip (G) is exactly the minimum leafsize of a monotone DeMorgan formula of these meta-variables which accepts all edges and rejects all nonedges of G. Also, the depth of a decision tree for the graph Gf , querying the meta-variables, is exactly the communication complexity of the boolean function f (x, y), a measure which we will introduce in Chapter 3. 2 Star complexity of graphs 10 15 20 25 30 July 14, 2011 Now we consider the complexity of graphs when only special bicliques – stars – are used as generators.