2 research outputs found
On the Relationship Between Energy Complexity and Other Boolean Function Measures
In this work we investigate into energy complexity, a Boolean function
measure related to circuit complexity. Given a circuit over the
standard basis , the energy complexity of
, denoted by , is the maximum number of
its activated inner gates over all inputs. The energy complexity of a Boolean
function , denoted by , is the minimum of
over all circuits computing . This
concept has attracted lots of attention in literature. Recently, Dinesh, Otiv,
and Sarma [COCOON'18] gave an upper bound in terms of the
decision tree complexity, . They also showed
that , where is the input size. Recall that the
minimum size of circuit to compute could be as large as . We improve
their upper bounds by showing that
.
For the lower bound, Dinesh, Otiv, and Sarma defined positive sensitivity, a
complexity measure denoted by , and showed that
. They asked whether
can also be lower bounded by a polynomial of .
In this paper we affirm it by proving
. For non-degenerated functions
with input size , we give another lower bound
. All these three lower bounds are incomparable
to each other. Besides, we also examine the energy complexity of
functions and functions, which implies the tightness of our
two lower bounds respectively. In addition, the former one answers another open
question asking for a non-trivial lower bounds for the energy complexity of
functions.Comment: 15 pages, 6 figure