  A function computation problem in directed acyclic networks has been
considered in the literature, where a sink node wants to compute a target
function with the inputs generated at multiple source nodes. The network links
are error-free but capacity-limited, and the intermediate network nodes perform
network coding. The target function is required to be computed with zero error.
The computing rate of a network code is measured by the average number of times
that the target function can be computed for one use of the network, i.e., each
link in the network is used at most once. In the papers [1], [2], two cut-set
bounds were proposed on the computing rate. However, we show in this paper that
these bounds are not valid for general network function computation problems.
We analyze the arguments that lead to the invalidity of these bounds and fix
the issue with a new cut-set bound, where a new equivalence relation associated
with the inputs of the target function is used. Our bound is qualified for
general target functions and network topologies. We also show that our bound is
tight for some special cases where the computing capacity is known. Moreover,
some results in [11], [12] were proved using the invalid upper bound in [1] and
hence their correctness needs further justification. We also justify their
validity in the paper.
