Theoretical Computer Science ( IF 1.1 ) Pub Date : 2020-09-11 , DOI: 10.1016/j.tcs.2020.09.003 Krishnamoorthy Dinesh , Samir Otiv , Jayalal Sarma
For a Boolean function computed by a Boolean circuit C over a finite basis , the energy complexity of C (denoted by ) is the maximum over all inputs of the number gates of the circuit C (excluding the inputs) that output a one. Energy Complexity of a Boolean function over a finite basis denoted by where C is a Boolean circuit over computing f.
We study the case when , the standard Boolean basis. It is known that any Boolean function can be computed by a circuit (with potentially large size) with an energy of at most for a small (which we observe is improvable to ). We show several new results and connections between energy complexity and other well-studied parameters of Boolean functions.
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For all Boolean functions f, where is the optimal decision tree depth of f.
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We define a parameter positive sensitivity (denoted by ), a quantity that is smaller than sensitivity [Cook et al.,SIAM Journal of Computing, 15(1):87–97, 1986] and defined in a similar way, and show that for any Boolean circuit C computing a Boolean function f, .
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For a monotone function f, we show that where is the cost of monotone Karchmer-Wigderson game of f.
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Restricting the above notion of energy complexity to Boolean formulas, we show where is the size and is the depth of a formula F.
中文翻译:
布尔函数能量复杂度的新界限
对于布尔函数 由布尔电路C在有限的基础上计算中,能量复杂的Ç(表示为通过)是所有输入的最大值 输出1的电路C的门数(不包括输入)。布尔函数在有限基础上的能量复杂度 表示为 其中C是布尔电路计算f。
我们研究的情况是 ,标准布尔基础。众所周知,任何布尔函数都可以由能量最大为零的电路(可能具有较大的尺寸)来计算 一小会儿 (我们观察到的 )。我们展示了一些新的结果以及能量复杂度和布尔函数的其他经过充分研究的参数之间的联系。
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对于所有布尔函数f, 哪里 是f的最佳决策树深度。
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我们定义一个参数正灵敏度(用),其数量小于灵敏度[Cook等人,SIAM Journal of Computing,15(1):87-97,1986],并且以类似方式定义,并且表明对于任何布尔电路C,计算布尔函数˚F,。
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对于单调函数f,我们证明 哪里 是f的单调Karchmer-Wigderson游戏的成本。
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将以上能量复杂度的概念限制为布尔公式,我们显示 哪里 是大小和 是公式F的深度。