New Bounds for Energy Complexity of Boolean Functions

Abstract

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 EC 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 B denoted by EC B ( f ) def *

On the Relationship Between Energy Complexity and Other Boolean Function Measures

“…In this section, we will show upper bounds of energy complexity with respect to decision tree complexity, which improves the result in [3].…”

“…Recently, Dinesh, Otiv, Sarma [3] discovered a new upper bound which relates energy complexity to decision tree complexity, a well-studied Boolean function complexity measure. In fact, they proved that for any Boolean function f : {0, 1} n → {0, 1}, psens(f ) 3 ≤ EC(f ) ≤ min O(D(f ) 3 ), 3n − 1 holds, where the function psens(f ) is defined as the positive sensitivity of f , i.e., the maximum of the number of sensitive bits i ∈ {1, 2, . .…”

“…ORn circuit Remark 2. [3] showed that EC(AND n ) = Θ(n), so it is rather intriguing that the energy complexity of AND n and OR n are different while they are basically same under other Boolean function measures such as decision tree complexity, sensitivity, block sensitivity, etc.…”

“…The energy complexity of a Boolean function f , denoted by EC(f ), is the minimum of EC(C) over all circuits C computing f . Recently, K. Dinesh et al [3] gave EC(f ) an upper bound in terms of the decision tree complexity, EC(f ) = O(D(f ) 3 ). They also showed that EC(f ) ≤ 3n−1, where n is the input size.…”

Asymptotically Best Synthesis Methods for Reflexive-Recursive Circuits

“…In addition to complexity defined as the number or total "weight" of circuit elements, many authors (see, e.g., [9][10][11][12]) have considered the power consumption of the circuits. These authors deal, for instance, with the static activity of circuits, which is defined as the maximum number of ones on the outputs of the functional elements, or dynamic activity, which characterizes the number of value changes on the outputs of the functional elements following a change of the input vector.…”