We focus on energy complexity, a Boolean function measure related closely to Boolean circuit design. Given a circuit \(\mathcal {C}\) over the standard basis \(\{\vee _2,\wedge _2,\lnot \}\), the energy complexity of \(\mathcal {C}\), denoted by \({{\,\mathrm{EC}\,}}(\mathcal {C})\), is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function f, denoted by \({{\,\mathrm{EC}\,}}(f)\), is the minimum of \({{\,\mathrm{EC}\,}}(\mathcal {C})\) over all circuits \(\mathcal {C}\) computing f. Recently, K. Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) gave \({{\,\mathrm{EC}\,}}(f)\) an upper bound by the decision tree complexity, \({{\,\mathrm{EC}\,}}(f)=O({{\,\mathrm{D}\,}}(f)^3)\). On the input size n, They also showed that \({{\,\mathrm{EC}\,}}(f)\) is at most \(3n-1\). For the lower bound side, they showed that \({{\,\mathrm{EC}\,}}(f)\ge \frac{1}{3}{{\,\mathrm{psens}\,}}(f)\), where \({{\,\mathrm{psens}\,}}(f)\) is called positive sensitivity. A remained open problem is whether the energy complexity of a Boolean function has a polynomial relationship with its decision tree complexity.

Our results for energy complexity can be listed below.

• For the lower bound, we prove the equation that \({{\,\mathrm{EC}\,}}(f)=\varOmega (\sqrt{{{\,\mathrm{D}\,}}(f)})\), which answers the above open problem.

• For upper bounds, \({{\,\mathrm{EC}\,}}(f)\le \min \{\frac{1}{2}{{\,\mathrm{D}\,}}(f)^2+O({{\,\mathrm{D}\,}}(f)),n+2{{\,\mathrm{D}\,}}(f)-2\}\) holds.

• For non-degenerated functions, we also provide another lower bound \({{\,\mathrm{EC}\,}}(f)=\varOmega (\log {n})\) where n is the input size.

• We also discuss the energy complexity of two specific function classes, \(\mathtt {OR}\) functions and \(\mathtt {ADDRESS}\) functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question in Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) asking for non-trivial lower bound for energy complexity of \(\mathtt {OR}\) functions.

我们专注于能量复杂度，这是一个与布尔电路设计密切相关的布尔函数度量。给定的电路\（\ mathcal {C} \）在标准基\（\ {\ VEE _2，\楔_2，\ lnot \} \） ，的能量复杂\（\ mathcal {C} \） ，表示为由\（{{\\\ mathrm {EC} \，}}（\ mathcal {C}）\）表示，是所有输入上其激活的内部门的最大数量。以\（{{\，\ mathrm {EC} \，}}（f）\）表示的布尔函数f的能量复杂度是\（{{\\ mathrm {EC} \，} }（\ mathcal {C}）\）在所有电路上\（\ mathcal {C} \）计算f。最近，K.Dinesh等人。（国际计算和组合技术会议，斯普林格，柏林，738-750，2018年）为\（{{\，\ mathrm {EC} \，}}（f）\）设定了决策树复杂度\（{ {\，\ mathrm {EC} \，}}（f）= O（{{\，\ mathrm {D} \，}}（f）^ 3）\）。在输入大小n上，他们还表明\（{{\，\ mathrm {EC} \，}}（f）\）最多为\（3n-1 \）。对于下界，他们表明\（{{\，\ mathrm {EC} \，}}（f）\ ge \ frac {1} {3} {{\，\ mathrm {psens} \，}} （f）\），其中\（{{\，\ mathrm {psens} \，}}（f）\）被称为正灵敏度。一个尚待解决的问题是布尔函数的能量复杂度与其决策树的复杂度是否具有多项式关系。

我们的能源复杂性结果可以在下面列出。

• 对于下限，我们证明等式\（{{\，\ mathrm {EC} \，}}（f）= \ varOmega（\ sqrt {{{\\\ mathrm {D} \，}}（f ）}）\），它可以解决上述开放问题。

• 对于上限，\（{{\，\ mathrm {EC} \，}}（f）\ le \ min \ {\ frac {1} {2} {{\，\ mathrm {D} \，}}（ f）^ 2 + O（{{\，\ mathrm {D} \，}}（f）），n + 2 {{\，\ mathrm {D} \，}}（f）-2 \} \）持有。

• 对于非简并函数，我们还提供了另一个下界\（{{\，\ mathrm {EC} \，}}（f）= \ varOmega（\ log {n}）\），其中n是输入大小。

• 我们还讨论了两个特定函数类的能量复杂度，即\（\ mathtt {OR} \）函数和\（\ mathtt {ADDRESS} \）函数，它们分别暗示了我们两个下限的紧密性。此外，前者回答了Dinesh等人的另一个悬而未决的问题。（国际计算和组合学会议，斯普林格，柏林，738-750，2018年）要求为\（\ mathtt {OR} \）函数的能量复杂度提供一个不重要的下限。