New Bounds for Energy Complexity of Boolean Functions
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For a Boolean function f:{0,1}^n →{0,1} computed by a circuit C over a finite basis B, the energy complexity of C (denoted by _(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 denoted by _(f):= _C _(C) where C is a circuit over computing f. We study the case when = {_2, _2, }, 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+ϵ(n)) for a small ϵ(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, (f) < O((f)^3) where (f) is the optimal decision tree depth of f. * We define a parameter positive sensitivity (denoted by ), 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, (C) >(f)/3. * For a monotone function f, we show that (f) = Ω(^+(f)) where ^+(f) is the cost of monotone Karchmer-Wigderson game of f. * Restricting the above notion of energy complexity to Boolean formulas, we show (F) = Ω (√(L(F))-depth(F) ) where L(F) is the size and depth(F) is the depth of a formula F.
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