布尔函数反演复杂性的几种推广
Some Extensions of the Inversion Complexity of Boolean Functions
摘要：
在计算布尔函数的布尔电路中，“非”门的最小数量称为该函数的求逆复杂度。1957年，A. A.马尔可夫确定了每一个布尔函数的反演复杂度，并证明了$\lceil\log_{2}(d(f)+1)\rceil$“非”门是计算任何布尔函数$f$的必要和充分条件（其中$d(f)$是在所有变量值的元组链上从1到0的值变化的最大数量）。本文研究了任意基上的布尔电路，该基由所有单调函数（权为零）和有限个非空非单调函数集（权为单位）组成。证明了布尔函数$f$的非单调门数的最小充分实现等于$\lceil\log_{2}(d(f)+1)\rceil - O(1)$。类似地推广了A. A.得到了布尔函数系的逆复杂性的马氏性。
英文摘要：
The minimum number of NOT gates in a Boolean circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that $\lceil\log_{2}(d(f)+1)\rceil$ NOT gates are necessary and sufficient to compute any Boolean function $f$ (where $d(f)$ is maximum number of value changes from 1 to 0 over all increasing chains of tuples of variables values). In this paper we consider Boolean circuits over an arbitrary basis that consists of all monotone functions (with zero weight) and finite nonempty set of non-monotone functions (with unit weight). It is shown that the minimal sufficient for a realization of the Boolean function $f$ number of non-monotone gates is equal to $\lceil\log_{2}(d(f)+1)\rceil - O(1)$. Similar extends of another classical result of A. A. Markov for the inversion complexity of system of Boolean functions has been obtained.
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