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Languages of higher-dimensional automata

Published online by Cambridge University Press:  18 October 2021

Uli Fahrenberg Open the ORCID record for Uli Fahrenberg [Opens in a new window] ,
Christian Johansen Open the ORCID record for Christian Johansen [Opens in a new window] ,
Georg Struth Open the ORCID record for Georg Struth [Opens in a new window]  and
Krzysztof Ziemiański Open the ORCID record for Krzysztof Ziemiański [Opens in a new window]
Uli Fahrenberg*
Affiliation:
École Polytechnique, Palaiseau, France
Christian Johansen
Affiliation:
Norwegian University of Science and Technology, Norway
Georg Struth
Affiliation:
University of Sheffield, UK
Krzysztof Ziemiański
Affiliation:
University of Warsaw, Poland
*
*Corresponding author. Email: ulrich.fahrenberg@polytechnique.edu
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Abstract

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.

Keywords

Higher-dimensional automatonconcurrency theorypomsetdirected topology

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Type
Paper
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Mathematical Structures in Computer Science , Volume 31 , Issue 5 , May 2021 , pp. 575 - 613
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© The Author(s), 2021. Published by Cambridge University Press

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