Higher-dimensional automata (HDA) are a formalism to model the behaviour of concurrent systems. They are similar to ordinary automata but allow transitions in higher dimensions, effectively enabling multiple actions to happen simultaneously. For ordinary automata, there is a correspondence between regular languages and finite automata. However, regular languages are inherently sequential and one may ask how such a correspondence carries over to HDA, in which several actions can happen at the same time. It has been shown by Fahrenberg et al. that finite HDA correspond with interfaced interval pomset languages generated by sequential and parallel composition and non-empty iteration. In this paper, we seek to extend the correspondence to process replication, also known as parallel Kleene closure. This correspondence cannot be with finite HDA and we instead focus here on locally compact and finitely branching HDA. In the course of this, we extend the notion of interval ipomset languages to arbitrary HDA, show that the category of HDA is locally finitely presentable with compact objects being finite HDA, and we prove language preservation results of colimits. We then define parallel composition as a tensor product of HDA and show that the repeated parallel composition can be expressed as locally compact and as finitely branching HDA, but also that the latter requires infinitely many initial states.

翻译：高维自动机（HDA）是一种用于建模并发系统行为的 formalism。它们与普通自动机类似，但允许在更高维度上进行转换，从而能够同时实现多个动作。对于普通自动机，正则语言与有限自动机之间存在对应关系。然而，正则语言本质上是顺序的，因此需要研究这种对应关系如何推广到可同时发生多个动作的高维自动机。Fahrenberg 等人已证明，有限高维自动机对应于由顺序和并行组合以及非空迭代生成的接口区间偏序集语言。本文旨在将这种对应关系扩展到过程复制（也称为并行 Kleene 闭包）。这种对应关系无法通过有限高维自动机实现，因此我们转而关注局部紧致且有限分支的高维自动机。在此过程中，我们将区间 ipomset 语言的概念推广到任意高维自动机，证明高维自动机范畴是局部有限可表示的且其紧致对象为有限高维自动机，并验证了余极限的语言保持性质。随后，我们将并行组合定义为高维自动机的张量积，证明重复并行组合既可表示为局部紧致的高维自动机，也可表示为有限分支的高维自动机，但后者需要无穷多个初始状态。