The chase procedure for existential rules is an indispensable tool for several database applications, where its termination guarantees the decidability of these tasks. Most previous studies have focused on the skolem chase variant and its termination analysis. It is known that the restricted chase variant is a more powerful tool in termination analysis provided a database is given. But all-instance termination presents a challenge since the critical database and similar techniques do not work. In this paper, we develop a novel technique to characterize the activeness of all possible cycles of a certain length for the restricted chase, which leads to the formulation of a framework of parameterized classes of the finite restricted chase, called $k$-$\mathsf{safe}(\Phi)$ rule sets. This approach applies to any class of finite skolem chase identified with a condition of acyclicity. More generally, we show that the approach can be applied to the hierarchy of bounded rule sets previously only defined for the skolem chase. Experiments on a collection of ontologies from the web show the applicability of the proposed methods on real-world ontologies.

中文翻译：

---

存在规则的受限追逐终止：分层方法和实验

存在规则的追踪过程是几个数据库应用程序不可或缺的工具，它的终止保证了这些任务的可判定性。大多数先前的研究都集中在 skolem 追逐变体及其终止分析上。众所周知，限制追逐变体是终止分析中更强大的工具，只要给定数据库。但是由于关键数据库和类似技术不起作用，所以全实例终止提出了挑战。在本文中，我们开发了一种新技术来表征受限追逐的一定长度的所有可能循环的活跃性，这导致了有限受限追逐的参数化类框架的制定，称为$k$-$\mathsf{安全}(\Phi)$规则集。这种方法适用于任何类别的具有非循环性条件的有限 skolem 追逐。更一般地说，我们表明该方法可以应用于有界规则集以前只为 skolem 追逐定义。对来自网络的一组本体进行的实验表明了所提出的方法在现实世界本体上的适用性。