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, closed under two variants of bisimulational equivalence, and definable in WMSO 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 parity index collapse to level (1,3) and a topological complexity collapse to co-analytic sets. Moreover, a gap property is shown: a regular language of thin trees is either WMSO-definable among all trees or co-analytic-complete.