Finding a homomorphism from some hypergraph $\mathcal{Q}$ (or some relational structure) to another hypergraph $\mathcal{D}$ is a fundamental problem in computer science. We show that an answer to this problem can be maintained under single-edge changes of $\mathcal{Q}$, as long as it stays acyclic, in the DynFO framework of Patnaik and Immerman that uses updates expressed in first-order logic. If additionally also changes of $\mathcal{D}$ are allowed, we show that it is unlikely that existence of homomorphisms can be maintained in DynFO.

翻译：从某种高射量 $\ mathcal\ $( 或某种关联结构) 到另一种高射量 $\ mathcal{D} $( 或某种关联结构) 寻找同质性是计算机科学中的一个基本问题。 我们显示,只要在Patnaik 和 Immerman 的 DynFO 框架中, 使用第一阶逻辑表达的更新, 在单向变化 $\ mathcal{D} $( 或某种关联结构) 中, 这个问题的答案可以维持在单向变化 $\ mathcal{D} $( ) 下。 我们显示, 只要它保持周期性, 在 DynFO 中, 就不可能存在同质性现象 。