We resolve three long-standing open problems, namely the (algorithmic) decidability of network coding, the decidability of conditional information inequalities, and the decidability of conditional independence implication among random variables, by showing that these problems are undecidable. The proof utilizes a construction inspired by Herrmann's arguments on embedded multivalued database dependencies, a network studied by Dougherty, Freiling and Zeger, together with a novel construction to represent group automorphisms on top of the network.

翻译：我们解决了三个长期存在的公开问题,即网络编码(算法)的可变性、有条件信息不平等的可变性、有条件独立的影响在随机变量中的可变性,通过表明这些问题是不可变的。 证据利用了Hermann关于嵌入的多价值数据库依赖性(由Dougherty、Freiling和Zeger研究的网络)的推理,以及代表网络顶端群体自定义性的新构思。