Regular Languages of Thin Trees

Abstract

An infinite tree is called thin if it contains only countably many infinite branches. Thin trees can be seen as intermediate structures between infinite words and infinite trees. In this work we investigate properties of regular languages of thin trees. Our main tool is an algebra suitable for thin trees. Using this framework we characterize various classes of regular languages: commutative, open in the standard topology, and definable in weak MSO logic among all trees. We also show that in various meanings thin trees are not as rich as all infinite trees. In particular we observe a collapse of the parity index to the level (1, 3) and a collapse of the topological complexity to co-analytic sets. Moreover, a gap property is shown: a regular language of thin trees is either weak MSO-definable among all trees or co-analytic-complete.

Register Games

“…It may be the case that this collapse of the index-hierarchy on these structures goes further, all the way to weak, as on words, using similar techniques to Kupferman and Vardi [KV98], or to priorities {1, 2, 3} as for trees with a countable number of infinite branches [ISB16]. We leave this as an open problem, as well as the related question of what is the simplest class of trees on which the hierarchy is strict.…”