Switching activity of Boolean circuits and synthesis of Boolean circuits with asymptotically optimal complexity and linear switching activity

On the Relationship Between Energy Complexity and Other Boolean Function Measures

“…(From now on we use EC(f ) to represent EC B (f ) for the standard basis.) Towards this, Kasim-zade [7] showed that EC(f ) = O(n 2 ) for any n variable Boolean function f by constructing an explicit circuit, which was further improved by Lozhkin and Shupletsov [8] to 4n and then (3 + ǫ(n))n where lim n→∞ ǫ(n) = 0. Yet these results only connect energy complexity with the number of variables in Boolean function while disregarding other important Boolean function measures.…”

New Bounds for Energy Complexity of Boolean Functions

“…As mentioned in the introduction, Lozhkin and Shupletsov [11] showed that EC( f ) ≤ 3n(1 + (n)) by constructing a circuit of size 2 n n (1 + (n)) where (n) → 0 as n → ∞. Their idea is to construct a Boolean circuit of low energy that outputs all product terms on n variables where each of them appear exactly once in a negated or unnegated form.…”

“…We remark that the size of the circuit constructed above is exponentially in DT( f ). However, in terms of the energy of the circuit, this improves the bounds of [11] since it now depends only on DT( f ). There are several Boolean functions, for which the decision trees are very shallow -a demonstrative example is the address function 4 where the decision tree is of depth O(log n).…”

“…By an explicit circuit construction, Kasim-zade [13] showed that EC B (n) ≤ O(n 2 ). Recently, Lozhkin and Shupletsov [11] states (without proof) that the circuit construction by Kasim-zade [13] over the complete Boolean basis is of energy 4n, thus deriving that EC B (n) ≤ 4n. Lozhkin and Shupletsov improves it to 3n(1 + (n)) by constructing a circuit of size 2 n n (1 + (n)) for an (n) tending to 0 for large n. We observe that this bounds can be further improved to be at most 3n − 1 while the size is 2 O(n) by carefully following the construction in [11] (Proposition 2.1).…”

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.…”