 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: https://drops.dagstuhl.de/opus/volltexte/2022/16875/
 Go to the corresponding LIPIcs Volume Portal

### On Uniformization in the Full Binary Tree

 pdf-format:

### 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},
DROPS-Home | Fulltext Search | Imprint | Privacy 