On the Relationship between Energy Complexity and other Boolean Function Measures

TL;DR

This paper explores the energy complexity of Boolean functions, establishing new upper and lower bounds related to decision tree complexity and sensitivity, and examines specific functions to demonstrate the bounds' tightness.

Contribution

It improves existing upper bounds on energy complexity, proves new lower bounds in terms of decision tree complexity and sensitivity, and analyzes specific functions for tightness.

Findings

Improved upper bounds: EC(f) ≤ min{(1/2)D(f)^2 + O(D(f)), n + 2D(f) - 2}

Lower bounds: EC(f) = Ω(√D(f)) and EC(f) = Ω(log n) for non-degenerate functions

Energy complexity of OR and ADDRESS functions analyzed, confirming bounds and answering open questions.

Abstract

In this work we investigate into energy complexity, a Boolean function measure related to circuit complexity. Given a circuit $\mathcal{C}$ over the standard basis $\{\vee_2,\wedge_2,\neg\}$, 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 $f$. This concept has attracted lots of attention in literature. Recently, Dinesh, Otiv, and Sarma [COCOON'18] gave $\mathrm{EC}(f)$ an upper bound in terms of the decision tree complexity, $\mathrm{EC}(f)=O(\mathrm{D}(f)^3)$. They also showed that $\mathrm{EC}(f)\leq 3n-1$, where $n$ is the input size. Recall that the minimum size of circuit to compute $f$ could be as large as $2^n/n$. We improve their upper bounds by showing that $\mathrm{EC}(f)\leq\min\{\frac12\mathrm{D}(f)^2+O(\mathrm{D}(f)),n+2\mathrm{D}(f)-2\}$. For the lower bound, Dinesh, Otiv, and Sarma defined positive sensitivity, a complexity measure denoted by $\mathrm{psens}(f)$, and showed that $\mathrm{EC}(f)\ge\frac{1}{3}\mathrm{psens}(f)$. They asked whether $\mathrm{EC}(f)$ can also be lower bounded by a polynomial of $\mathrm{D}(f)$. In this paper we affirm it by proving $\mathrm{EC}(f)=\Omega(\sqrt{\mathrm{D}(f)})$. For non-degenerated functions with input size $n$, we give another lower bound $\mathrm{EC}(f)=\Omega(\log{n})$. All these three lower bounds are incomparable to each other. Besides, we also 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 asking for a non-trivial lower bounds for the energy complexity of $\mathtt{OR}$ functions.