License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.69
URN: urn:nbn:de:0030-drops-127386
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12738/
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Michielini, Vincent ; Skrzypczak, Michał

Regular Choice Functions and Uniformisations For countable Domains

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Abstract

We view languages of words over a product alphabet A x B as relations between words over A and words over B. This leads to the notion of regular relations - relations given by a regular language. We ask when it is possible to find regular uniformisations of regular relations. The answer depends on the structure or shape of the underlying model: it is true e.g. for ω-words, while false for words over ℤ or for infinite trees.
In this paper we focus on countable orders. Our main result characterises, which countable linear orders D have the property that every regular relation between words over D has a regular uniformisation. As it turns out, the only obstacle for uniformisability is the one displayed in the case of ℤ - non-trivial automorphisms of the given structure. Thus, we show that either all regular relations over D have regular uniformisations, or there is a non-trivial automorphism of D and even the simple relation of choice cannot be uniformised. Moreover, this dichotomy is effective.

BibTeX - Entry

@InProceedings{michielini_et_al:LIPIcs:2020:12738,
  author =	{Vincent Michielini and Micha{\l} Skrzypczak},
  title =	{{Regular Choice Functions and Uniformisations For countable Domains}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{69:1--69:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12738},
  URN =		{urn:nbn:de:0030-drops-127386},
  doi =		{10.4230/LIPIcs.MFCS.2020.69},
  annote =	{Keywords: Uniformisation, Monadic Second-order logic, Countable words}
}

Keywords: Uniformisation, Monadic Second-order logic, Countable words
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020

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