TR11-130 Authors: Sergei Lozhkin, Alexander Shiganov

Publication: 25th September 2011 23:13

Downloads: 3361

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In this paper we suggest a modification of classical Lupanov's method [Lupanov1958]

that allows building circuits over the basis $\{\&,\vee,\neg\}$ for Boolean functions of $n$ variables with size at most

$$

\frac{2^n}{n}\left(1+\frac{3\log n + O(1)}{n}\right),

$$

and with more uniform distribution of outgoing arcs by circuit gates.

For almost all Boolean functions of $n$ variables in the circuits for

these functions, which are built using our method, the fraction of gates

with fan-out 2 is asymptotically at least 1/32. This fact disproves upper bound [Yamamoto2011]

on the number of circuits with exact number of gates with fan-out at least 2.