New Bounds for Energy Complexity of Boolean Functions

TL;DR

This paper establishes new bounds and relationships for the energy complexity of Boolean functions, connecting it to decision tree depth, sensitivity, and Karchmer-Wigderson complexity, advancing understanding of circuit efficiency.

Contribution

It introduces novel bounds linking energy complexity to decision tree depth, sensitivity, and monotone Karchmer-Wigderson measures, and defines positive sensitivity as a new parameter.

Findings

Energy complexity is at most cubic in decision tree depth.

Positive sensitivity provides a lower bound on energy complexity.

For monotone functions, energy complexity relates to Karchmer-Wigderson complexity.

Abstract

$\newcommand{\EC}{\mathsf{EC}}\newcommand{\KW}{\mathsf{KW}}\newcommand{\DT}{\mathsf{DT}}\newcommand{\psens}{\mathsf{psens}} \newcommand{\calB}{{\cal B}} $ For a Boolean function $f:\{0,1\}^n \to \{0,1\}$ computed by a circuit $C$ over a finite basis $\mathcal{B}$, the energy complexity of $C$ (denoted by $\EC_{\calB}(C)$) is the maximum over all inputs $\{0,1\}^n$ the numbers of gates of the circuit $C$ (excluding the inputs) that output a one. Energy Complexity of a Boolean function over a finite basis $\calB$ denoted by $\EC_\calB(f):= \min_C \EC_{\calB}(C)$ where $C$ is a circuit over $\calB$ computing $f$. We study the case when $\calB = \{\land_2, \lor_2, \lnot\}$, 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 $3n(1+\epsilon(n))$ for a small $ \epsilon(n)$(which we observe is improvable to $3n-1$). We show several new results and connections between energy complexity and other well-studied parameters of Boolean functions. * For all Boolean functions $f$, $\EC(f) \le O(\DT(f)^3)$ where $\DT(f)$ is the optimal decision tree depth of $f$. * We define a parameter \textit{positive sensitivity} (denoted by $\psens$), a quantity that is smaller than sensitivity and defined in a similar way, and show that for any Boolean circuit $C$ computing a Boolean function $f$, $ \EC(C) \ge \psens(f)/3$. * For a monotone function $f$, we show that $\EC(f) = \Omega(\KW^+(f))$ where $\KW^+(f)$ is the cost of monotone Karchmer-Wigderson game of $f$. * Restricting the above notion of energy complexity to Boolean formulas, we show $\EC(F) = \Omega\left (\sqrt{L(F)}-depth(F)\right )$ where $L(F)$ is the size and $depth(F)$ is the depth of a formula $F$.