TY - GEN

T1 - On Uniformization in the Full Binary Tree

AU - Rabinovich, Alexander

N1 - Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

PY - 2022/8/1

Y1 - 2022/8/1

N2 - 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:.

AB - 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:.

KW - Monadic Second-Order Logic

KW - Uniformization

UR - http://www.scopus.com/inward/record.url?scp=85137549570&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2022.77

DO - 10.4230/LIPIcs.MFCS.2022.77

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AN - SCOPUS:85137549570

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022

A2 - Szeider, Stefan

A2 - Ganian, Robert

A2 - Silva, Alexandra

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022

Y2 - 22 August 2022 through 26 August 2022

ER -