Uniformization on thin trees

David Hutchison, Takeo Kanade, Josef Kittler, Jon M. Kleinberg, Friedemann Mattern, John C. Mitchell, Moni Naor, C. Pandu Rangan, Bernhard Steffen, Demetri Terzopoulos, Doug Tygar, Gerhard Weikum, Michał Skrzypczak

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

As the axiom of choice implies, for every relation R ⊆ X × Y there exists a graph of a total function f: πX (R) → Y that is contained in R (such a graph is called a uniformization of R). A natural question asks in which cases such a function f is definable. A particular instance of this problem is, when R is an mso-definable set of pairs of trees and we ask about mso-definable f. This question is known as Rabin’s uniformization question. The negative answer to this question was given by Gurevich and Shelah [GS83] (see [CL07] for a simplified proof). They proved that there is no mso formula ψ(x, X) that chooses from every non-empty subset X of the complete binary tree a unique element x of X. This result is known as undefinability of a choice function on the complete binary tree. On the other hand, the formula saying that x is the ≤-minimal element of X is a choice formula on ω-words. In [Sie75, LS98, Rab07] it is proved that any mso-definable relation on ω-words admits an mso-definable uniformization.

Original languageEnglish
Title of host publicationDescriptive Set Theoretic Methods in Automata Theory
Subtitle of host publicationDecidability and Topological Complexity
PublisherSpringer Verlag
Chapter8
Pages137-156
Number of pages20
DOIs
StatePublished - 6 Aug 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9802 LNCS

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

Fingerprint

Dive into the research topics of 'Uniformization on thin trees'. Together they form a unique fingerprint.

Cite this