This paper proposes first-order modal $\xi$-calculus as well as genealogical Kripke models. Inspired by modal $\mu$-calculus, first-order modal $\xi$-calculus takes a quite similar form and extends its inductive expressivity onto a different dimension. We elaborate on several vivid examples that demonstrate this logic's profound utility, especially for depicting genealogy of concurrent computer processes. Bisimulation notion for the logic has also been thoroughly examined.

翻译：本文提出了第一级模式$\x1$计算模型以及基因Kripke模型。 受第一级模式$\mu1$-计算模型,第一级模式$\x1$-计算模型的启发,其形式非常相似,将其感应表达性扩大到不同的层面。 我们详细阐述了几个生动的例子,这些例子显示了这一逻辑的深刻作用,特别是用来描述并行计算机过程的基因学。 逻辑的分离模拟概念也得到了彻底研究。