On the Relationship Between Energy Complexity and Other Boolean Function Measures

Abstract

In this work we investigate energy complexity, a Boolean function measure related to circuit complexity. 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, Dinesh et al. [3] gave ${{\,\mathrm{EC}\,}}(f)$ an upper bound in terms of the decision tree complexity, ${{\,\mathrm{EC}\,}}(f)=O(\text {D}(f)^3)$. They also showed that ${{\,\mathrm{EC}\,}}(f)\le 3n-1$, where n is the input size. For the lower bound, they show that ${{\,\mathrm{EC}\,}}(f)\ge \frac{1}{3}{{\,\mathrm{psens}\,}}(f)$, where ${{\,\mathrm{psens}\,}}(f)$ is the positive sensitivity. They asked whether ${{\,\mathrm{EC}\,}}(f)$ can be lower bounded by a polynomial of $\text {D}(f)$. We improve both the upper and lower bounds in this paper. For upper bounds, We show that ${{\,\mathrm{EC}\,}}(f)\le \min \{\frac{1}{2}\text {D}(f)^2+O(\text {D}(f)),n+2\text {D}(f)-2\}$. For the lower bound, we answer Dinesh et al.’s question by proving that ${{\,\mathrm{EC}\,}}(f)=\varOmega (\sqrt{\text {D}(f)})$. For non-degenerated functions, we also give another lower bound ${{\,\mathrm{EC}\,}}(f)=\varOmega (\log {n})$ where n is the input size. These two lower bounds are incomparable to each other. Besides, we examine the energy complexity of $\mathtt {OR}$ 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 [3] asking for non-trivial lower bound for energy complexity of $\mathtt {OR}$ functions.

This work was supported in part by the National Natural Science Foundation of China Grants No. 61433014, 61832003, 61761136014, 61872334, 61502449, 61602440, 61801459, the 973 Program of China Grant No. 2016YFB1000201, K.C. Wong Education Foundation.