Finitely Presentable Higher-Dimensional Automata and the Irrationality of Process Replication

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.