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Higher-dimensional automata (HDA) are a formalism to model the behaviour ofconcurrent systems. They are similar to ordinary automata but allow transitionsin higher dimensions, effectively enabling multiple actions to happensimultaneously. For ordinary automata, there is a correspondence betweenregular languages and finite automata. However, regular languages areinherently sequential and one may ask how such a correspondence carries over toHDA, in which several actions can happen at the same time. It has been shown byFahrenberg et al. that finite HDA correspond with interfaced interval pomsetlanguages generated by sequential and parallel composition and non-emptyiteration. In this paper, we seek to extend the correspondence to processreplication, also known as parallel Kleene closure. This correspondence cannotbe with finite HDA and we instead focus here on locally compact and finitelybranching HDA. In the course of this, we extend the notion of interval ipomsetlanguages to arbitrary HDA, show that the category of HDA is locally finitelypresentable with compact objects being finite HDA, and we prove languagepreservation results of colimits. We then define parallel composition as atensor product of HDA and show that the repeated parallel composition can beexpressed as locally compact and as finitely branching HDA, but also that thelatter requires infinitely many initial states.

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