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Languages of higher-dimensional automata

Published online by Cambridge University Press:  18 October 2021

Uli Fahrenberg*
École Polytechnique, Palaiseau, France
Christian Johansen
Norwegian University of Science and Technology, Norway
Georg Struth
University of Sheffield, UK
Krzysztof Ziemiański
University of Warsaw, Poland
*Corresponding author. Email:


We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step, we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.

© The Author(s), 2021. Published by Cambridge University Press

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