New Bounds for Energy Complexity of Boolean Functions

$\newcommand{\EC}{\mathsf{EC}}\newcommand{\KW}{\mathsf{KW}}\newcommand{\DT}{\mathsf{DT}}\newcommand{\psens}{\mathsf{psens}} \newcommand{\calB}{{\cal B}}$ 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 $F$.Comment: 25 pages, 4 figures. Improved presentation of Theorem 1.5. Added an improvement due to Sun et.al. and a comparison to their result (in Section 6