@inbook{ac87063d8d9743698bdff0c88c84e6f0,
title = "Uniformization on thin trees",
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{\textquoteright}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.",
author = "David Hutchison and Takeo Kanade and Josef Kittler and Kleinberg, {Jon M.} and Friedemann Mattern and Mitchell, {John C.} and Moni Naor and {Pandu Rangan}, C. and Bernhard Steffen and Demetri Terzopoulos and Doug Tygar and Gerhard Weikum and Micha{\l} Skrzypczak",
note = "Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2016.",
year = "2016",
month = aug,
day = "6",
doi = "https://doi.org/10.1007/978-3-662-52947-8_8",
language = "الإنجليزيّة",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "137--156",
booktitle = "Descriptive Set Theoretic Methods in Automata Theory",
address = "ألمانيا",
}