数理解析研究所講究録 第 1554 巻 2007 年 131-138 131 Negation-Limited Complexity of Parity and Inverters 森住大樹 1 垂井淳 2 岩間一雄 1 1 京都大学大学院情報学研究科, {morizumi, iwama}@kuis.kyoto-u.ac.jp 2 電気通信大学情報通信工学科,
[email protected] Abstract 1 Introduction and Summary In negation-limited complexity, one considers circuits Although exponential lower bounds are known for the with a limited number of NOT gates, being mo- monotone circuit size [4], [6], at present we cannot tivated by the gap in our understanding of mono- prove a superlinear lower bound for the size of circuits tone versus general circuit complexity, and hop- computing an explicit Boolean function: the largest ing to better understand the power of NOT gates. known lower bound is $5n-o(n)[\eta, |10],$ $[8]$ . It is We give improved lower bounds for the size (the natural to ask: What happens if we allow a limited number of $AND/OR/NOT$) of negation-limited cir- number of NOT gatae? The hope is that by the study cuits computing Parity and for the size of negation- of negation-limited complexity of Boolean functions limited inverters. An inverter is a circuit with inputs under various scenarios [3], [17], [15]. [2], $|1$ ], $[13]$ , we $x_{1},$ $\ldots,x_{\mathfrak{n}}$ $\neg x_{1},$ $po\mathfrak{n}^{r}er$ and outputs $\ldots,$ $\neg x_{n}$ . We show that understand the of NOT gates better. (a) For $n=2^{f}-1$ , circuits computing Parity with $r-1$ We consider circuits consisting of $AND/OR/NOT$ NOT gates have size at least $6n-\log_{2}(n+1)-O(1)$, gates.