eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-18
69:1
69:13
10.4230/LIPIcs.MFCS.2020.69
article
Regular Choice Functions and Uniformisations For countable Domains
Michielini, Vincent
1
Skrzypczak, Michał
1
https://orcid.org/0000-0002-9647-4993
University of Warsaw, Poland
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol170-mfcs2020/LIPIcs.MFCS.2020.69/LIPIcs.MFCS.2020.69.pdf
Uniformisation
Monadic Second-order logic
Countable words