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 $\mathbb{E}\{M\}$.

翻译：我们分析了由Ben-Eliezer、Mikulincer、Mossel和Sudan（ITCS 2023）引入的累积知识过程，该过程设定在“有向无环图”环境中，即新知识单元可通过组合多个先前知识单元推导得出。该模型的主要考量因素包括错误的作用（新单元可能包含错误）以及局部检查（当新知识单元被发现时，对其少数前驱知识单元进行核查）。前述工作定义了该模型，但仅分析了理想化且简化的“树状”场景，即新知识单元仅直接依赖于一个先前生成的知识单元。本研究的主要目标是理解一般过程何时是安全的，即错误的效应何时仍然可控。我们给出了安全性的若干必要条件和充分条件。与早期工作一致，我们证明了检查频率和检查深度在决定安全性中起关键作用。本研究的一个关键新参数是$\textit{组合因子}$，它表示一个新知识单元所依赖的旧知识单元数量$M$的分布。我们的结果表明，较大的组合因子可以补偿较小的检查深度。安全性对组合因子的依赖性远非平凡：事实上，我们的一些主要结果用$\mathbb{E}\{1/M\}$表述，而另一些则依赖于$\mathbb{E}\{M\}$。