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Strictifying and taming directed paths in Higher Dimensional Automata

Published online by Cambridge University Press:  15 September 2021

Martin Raussen Open the ORCID record for Martin Raussen [Opens in a new window]
Martin Raussen*
Affiliation:
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, Aalborg Øst DK-9220, Denmark Email: [email protected]
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Abstract

Directed paths have been used by several authors to describe concurrent executions of a program. Spaces of directed paths in an appropriate state space contain executions with all possible legal schedulings. It is interesting to investigate whether one obtains different topological properties of such a space of executions if one restricts attention to schedulings with “nice” properties, e.g. involving synchronisations. This note shows that this is not the case, i.e. that one may operate with nice schedulings without inflicting any harm. Several of the results in this note had previously been obtained by Ziemiański in Ziemiański (2017. Applicable Algebra in Engineering, Communication and Computing28 497–525; 2020a. Journal of Applied and Computational Topology4 (1) 45–78). We attempt to make them accessible for a wider audience by giving an easier proof for these findings by an application of quite elementary results from algebraic topology; notably the nerve lemma.

Keywords

Higher Dimensional Automatad-pathstricttameserialparallelhomotopy equivalencenerve lemma
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 2 , February 2021 , pp. 193 - 213
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

The author thanks Uli Fahrenberg (École Polytechnique, Paris) and Krzysztof Ziemiański (Warsaw) for helpful conversations; Ziemiański particularly for pointing out several uncorrect statements in previous versions. Thanks are also due to the anonymous referees for several hints leading to improvements of the presentation.

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