1 research outputs found
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.Comment: 25 pages, 3 figure