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COCOON 2018: Computing and Combinatorics pp 738-750

# New Bounds for Energy Complexity of Boolean Functions

• Krishnamoorthy Dinesh
• Samir Otiv
• Jayalal Sarma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10976)

## 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 $$\varOmega (n)$$.

## Keywords

Energy complexity Boolean circuits Decision trees

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## Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Krishnamoorthy Dinesh
• 1
• Samir Otiv
• 2
• Jayalal Sarma
• 1
1. 1.Indian Institute of Technology MadrasChennaiIndia
2. 2.Maximl LabsChennaiIndia