  We analyze Cumulative Knowledge Processes, introduced by Ben-Eliezer,
Mikulincer, Mossel, and Sudan (ITCS 2023), in the setting of "directed acyclic
graphs", i.e., when new units of knowledge may be derived by combining multiple
previous units of knowledge. The main considerations in this model are the role
of errors (when new units may be erroneous) and local checking (where a few
antecedent units of knowledge are checked when a new unit of knowledge is
discovered). The aforementioned work defined this model but only analyzed an
idealized and simplified "tree-like" setting, i.e., a setting where new units
of knowledge only depended directly on one previously generated unit of
knowledge.
  The main goal of our work is to understand when the general process is safe,
i.e., when the effect of errors remains under control. We provide some
necessary and some sufficient conditions for safety. As in the earlier work, we
demonstrate that the frequency of checking as well as the depth of the checks
play a crucial role in determining safety. A key new parameter in the current
work is the $\textit{combination factor}$ which is the distribution of the
number of units $M$ of old knowledge that a new unit of knowledge depends on.
Our results indicate that a large combination factor can compensate for a small
depth of checking. The dependency of the safety on the combination factor is
far from trivial. Indeed some of our main results are stated in terms of
$\mathbb{E}\{1/M\}$ while others depend on $\mathbb{E}\{M\}$.

Artificial Intelligence;Data Structures and Algorithms;Social and Information Networks;Probability