  For zero-error function computation over directed acyclic networks, existing
upper and lower bounds on the computation capacity are known to be loose. In
this work we consider the problem of computing the arithmetic sum over a
specific directed acyclic network that is not a tree. We assume the sources to
be i.i.d. Bernoulli with parameter $1/2$. Even in this simple setting, we
demonstrate that upper bounding the computation rate is quite nontrivial. In
particular, it requires us to consider variable length network codes and relate
the upper bound to equivalently lower bounding the entropy of descriptions
observed by the terminal conditioned on the function value. This lower bound is
obtained by further lower bounding the entropy of a so-called \textit{clumpy
distribution}. We also demonstrate an achievable scheme that uses variable
length network codes and in-network compression.
