Existential rules are a positive fragment of first-order logic that generalizes function-free Horn rules by allowing existentially quantified variables in rule heads. This family of languages has recently attracted significant interest in the context of ontology-mediated query answering. Forward chaining, also known as the chase, is a fundamental tool for computing universal models of knowledge bases, which consist of existential rules and facts. Several chase variants have been defined, which differ on the way they handle redundancies. A set of existential rules is bounded if it ensures the existence of a bound on the depth of the chase, independently from any set of facts. Deciding if a set of rules is bounded is an undecidable problem for all chase variants. Nevertheless, when computing universal models, knowing that a set of rules is bounded for some chase variant does not help much in practice if the bound remains unknown or even very large. Hence, we investigate the decidability of the k-boundedness problem, which asks whether the depth of the chase for a given set of rules is bounded by an integer k. We identify a general property which, when satisfied by a chase variant, leads to the decidability of k-boundedness. We then show that the main chase variants satisfy this property, namely the oblivious, semi-oblivious (aka Skolem), and restricted chase, as well as their breadth-first versions.

中文翻译：

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表征追逐变体的有界性

存在规则是一阶逻辑的一个积极片段，它通过允许规则头中存在存在量化的变量来推广无函数 Horn 规则。这一系列语言最近在本体介导的查询回答的背景下引起了极大的兴趣。前向链接，也称为追逐，是计算知识库通用模型的基本工具，知识库由存在规则和事实组成。已经定义了几个追逐变体，它们在处理冗余的方式上有所不同。一组存在规则是有界的，如果它确保在追逐的深度上存在一个界限，独立于任何一组事实。确定一组规则是否有界对于所有追逐变体来说都是一个无法确定的问题。然而，在计算通用模型时，如果边界仍然未知或什至非常大，那么知道对于某些追逐变体的一组规则在实践中没有多大帮助。因此，我们研究了可判定性ķ-有界问题，它询问给定规则集的追逐深度是否以整数 k 为界。我们确定了一个一般属性，当它被追逐变体满足时，导致可判定性ķ-有界性。然后，我们展示了主要的追逐变体满足这个属性，即遗忘、半遗忘（又名 Skolem）和受限追逐，以及它们的广度优先版本。