License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2022.77
URN: urn:nbn:de:0030-drops-168757
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16875/
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Rabinovich, Alexander

On Uniformization in the Full Binary Tree

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Abstract

A function f uniformizes a relation R(X,Y) if R(X,f(X)) holds for every X in the domain of R. The uniformization problem for a logic L asks whether for every L-definable relation there is an L-definable function that uniformizes it. Gurevich and Shelah proved that no Monadic Second-Order (MSO) definable function uniformizes relation "Y is a one element subset of X" in the full binary tree. In other words, there is no MSO definable choice function in the full binary tree.
The cross-section of a relation R(X,Y) at D is the set of all E such that R(D,E) holds. Hence, a function that uniformizes R chooses one element from every non-empty cross-section. The relation "Y is a one element subset of X" has finite and countable cross-sections.
We prove that in the full binary tree the following theorems hold:
▶ Theorem (Finite cross-sections) If every cross-section of an MSO definable relation is finite, then it has an MSO definable uniformizer.
▶ Theorem (Uncountable cross-section) There is an MSO definable relation R such that every MSO definable relation included in R and with the same domain as R has an uncountable cross-section.

BibTeX - Entry

@InProceedings{rabinovich:LIPIcs.MFCS.2022.77,
  author =	{Rabinovich, Alexander},
  title =	{{On Uniformization in the Full Binary Tree}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16875},
  URN =		{urn:nbn:de:0030-drops-168757},
  doi =		{10.4230/LIPIcs.MFCS.2022.77},
  annote =	{Keywords: Monadic Second-Order Logic, Uniformization}
}

Keywords: Monadic Second-Order Logic, Uniformization
Collection: 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
Issue Date: 2022
Date of publication: 22.08.2022

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