New Bounds for Energy Complexity of Boolean Functions

Abstract

For a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ computed by a circuit C over a finite basis $\mathcal{B}$, the energy complexity of C (denoted by $\mathsf {EC}_{\mathcal{B}}(C)$) 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 $\mathcal{B}$ denoted by where C is a circuit over $\mathcal{B}$ computing f.

We study the case when $\mathcal{B}= \{\wedge _2, \vee _2, \lnot \}$, 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, $\mathsf {EC}(f) \le O(\mathsf {DT}(f)^3)$ where $\mathsf {DT}(f)$ is the optimal decision tree depth of f.

• We define a parameter positive sensitivity (denoted by $\mathsf {psens}$), 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, $\mathsf {EC}(C) \ge \mathsf {psens}(f)/3$.

• Restricting the above notion of energy complexity to Boolean formulas, denoted $\mathsf {EC^{F}}(f)$, we show that $\mathsf {EC^{F}}(f) = \varTheta (L(f))$ where L(f) is the minimum size of a formula computing f.

We next prove lower bounds on energy for explicit functions. In this direction, we show that for the perfect matching function on an input graph of n edges, any Boolean circuit with bounded fan-in must have energy $\varOmega (\sqrt{n})$. We show that any unbounded fan-in circuit of depth 3 computing the parity on n variables must have energy is $\varOmega (n)$.