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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 Open image in new window 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

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