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 -