Interpretations in Trees with Countably Many Branches

Abstract

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.Index Terms-Composition method, finite-set interpretations, infinite scattered trees, monadic second order logic.

Regular Languages of Thin Trees

“…Some authors use different names to denote tree-like structures with countably many branches. We recall here two of them: a forest that is a tree is thin if and only if it is a scattered tree in the meaning of [19]. A forest is thin if and only if it is, up to isomorphism, a tame tree in the meaning of [14].…”

“…, u n in t, exactly one of them that has maximal CB-rank: if u i ∈σ then for every j = i we have rank CB (t u i ) ≥ rank CB (t u j ). Clearlyσ defined this way satisfies (19).…”