Uniformisation Gives the Full Strength of Regular Languages

Authors Nathan Lhote, Vincent Michielini, Michał Skrzypczak

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Nathan Lhote
  • University of Warsaw, Poland
Vincent Michielini
  • University of Warsaw, Poland
Michał Skrzypczak
  • University of Warsaw, Poland

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Nathan Lhote, Vincent Michielini, and Michał Skrzypczak. Uniformisation Gives the Full Strength of Regular Languages. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Given R a binary relation between words (which we treat as a language over a product alphabet AxB), a uniformisation of it is another relation L included in R which chooses a single word over B, for each word over A whenever there exists one. It is known that MSO, the full class of regular languages, is strong enough to define a uniformisation for each of its relations. The quest of this work is to see which other formalisms, weaker than MSO, also have this property. In this paper, we solve this problem for pseudo-varieties of semigroups: we show that no nonempty pseudo-variety weaker than MSO can provide uniformisations for its relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • pseudo-variety
  • finite word
  • semigroup
  • uniformisation
  • regular language


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