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Regular Choice Functions and Uniformisations For countable Domains

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Vincent Michielini and Michał Skrzypczak. Regular Choice Functions and Uniformisations For countable Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 69:1-69:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.69

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.

Subject Classification

ACM Subject Classification
• Theory of computation → Regular languages
• Theory of computation → Automata over infinite objects
Keywords
• Uniformisation
• Countable words

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