Read e-book online Boolean Function Complexity: Advances and Frontiers PDF

By Stasys Jukna

Boolean circuit complexity is the combinatorics of machine technology and consists of many interesting difficulties which are effortless to kingdom and clarify, even for the layman. This e-book is a finished  description of simple decrease sure arguments, overlaying the various gem stones of this “complexity Waterloo” which have been came upon over the last numerous many years, correct as much as effects from the final 12 months or . Many open difficulties, marked as study difficulties, are pointed out alongside the way in which. the issues are ordinarily of combinatorial style yet their options can have nice effects in circuit complexity and computing device technology. The publication might be of curiosity to graduate scholars and researchers within the fields of desktop technological know-how and discrete mathematics.

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Example text

In this case the complexity of a given graph turns into a monotone complexity of monotone boolean functions “representing” this graph in the following sense. Let G = (V, E) be an n-vertex graph, and let z = {zv | v ∈ V } be a set of boolean variables, one for each vertex (not for each subset P ⊆ V , as before). Say that a boolean function (or a circuit) g(z) represents the graph G if, for every input a ∈ {0, 1}n with exactly two 1s in, say, positions u = v, g(a) = 1 iff u and v are adjacent in G.

In this way we reach a sink, and the input a is accepted iff this is the 1-sink. Thus, a deterministic branching program is a nondeterministic branching program with the restriction that each non-sink node has fanout 2, and the two outgoing wires from each such node are labeled by the tests xi = 0 and xi = 1 on the same variable xi . The presence of the 0-sink is just to ensure that each input vector can reach a sink. A decision tree is a deterministic branching program whose underlying graph is a binary tree.

8 A constant factor away from P = NP? F 43 the same circuit! F AND and OR gates AND and OR gates 4m input literals 4m ORs of new variables 2n = 2 m+1 new variables Fig. 12 Having a circuit F computing a boolean function f of 2m variables, we obtain a (monotone) circuit representing the graph Gf by replacing each input literal in F by an appropriate OR of new variables. ) circuit of size nǫ = 2ǫm , which is already exponential in the number of variables of f . 6 to prove truly exponential lower bounds for unbounded-fanin depth-3 circuits with parity gates on the bottom layer.

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Boolean Function Complexity: Advances and Frontiers by Stasys Jukna

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