  We study the following generalization of the well-known model of broadcasting
on trees. Consider an infinite directed acyclic graph (DAG) with a unique
source node $X$. Let the collection of nodes at distance $k$ from $X$ be called
the $k$th layer. At time zero, the source node is given a bit. At time $k\geq
1$, each node in the $(k-1)$th layer inspects its inputs and sends a bit to its
descendants in the $k$th layer. Each bit is flipped with a probability of error
$\delta \in \left(0,\frac{1}{2}\right)$ in the process of transmission. The
goal is to be able to recover the original bit with probability of error better
than $\frac{1}{2}$ from the values of all nodes at an arbitrarily deep layer
$k$.
  Besides its natural broadcast interpretation, the DAG broadcast is a natural
model of noisy computation. Some special cases of the model represent
information flow in biological networks, and other cases represent noisy finite
automata models.
  We show that there exist DAGs with bounded degree and layers of size
$\omega(\log(k))$ that permit recovery provided $\delta$ is sufficiently small
and find the critical $\delta$ for the DAGs constructed. Our result
demonstrates a doubly-exponential advantage for storing a bit in bounded degree
DAGs compared to trees. On the negative side, we show that if the DAG is a
two-dimensional regular grid, then recovery is impossible for any $\delta \in
\left(0,\frac{1}{2}\right)$ provided all nodes use either AND or XOR for their
processing functions.
