Interpretations in trees with countably many branches

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Abstract

We study the expressive power of logical interpretations on the class of scattered trees, namely those with countably many infinite branches. Scattered trees can be thought of as the tree analogue of scattered linear orders. Every scattered tree has an ordinal rank that reflects the structure of its infinite branches. We prove, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank. We consider a quite general notion of interpretation: each element of the interpreted structure is represented by a set of tuples of subsets of the interpreting tree. Our trees are countable, not necessarily finitely branching, and may have finitely many unary predicates as labellings. We also show how to replace injective set-interpretations in (not necessarily scattered) trees by 'finitary' set-interpretations.

Original languageEnglish
Title of host publicationProceedings of the 2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2012
Pages551-560
Number of pages10
DOIs
StatePublished - 2012
Event2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2012 - Dubrovnik, Croatia
Duration: 25 Jun 201228 Jun 2012

Publication series

NameProceedings of the 2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2012

Conference

Conference2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2012
Country/TerritoryCroatia
CityDubrovnik
Period25/06/1228/06/12

Keywords

  • Composition method
  • finite-set interpretations
  • infinite scattered trees
  • monadic second order logic

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