Abstract. We define an extension of forest algebra for ω-forests. We show how the standard algebraic notions (free object, syntactic algebra, morphisms, etc.) extend to the infinite case. To prove its usefulness, we use the framework to get an effective characterization of the ω-forest languages that are definable in the temporal logic that uses the operator EF (exists finally).

XML Schema Definitions (XSDs) can be adequately abstracted by the single-type regular tree languages. It is wellknown, that these form a strict subclass of the robust class of regular unranked tree languages. Sadly, in this respect, XSDs are not closed under the basic operations of union and set difference, complicating important tasks in schema integration and evolution. The purpose of this paper is to investigate how the union and difference of two XSDs can be approximated within the framework of single-type regular tree languages. We consider both optimal lower and upper approximations. We also address the more general question of how to approximate an arbitrary regular tree language by an XSD and consider the complexity of associated decision problems.

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.

No abstract

XML Schema Definitions (XSDs) can be adequately abstracted by the single-type regular tree languages. It is wellknown, that these form a strict subclass of the robust class of regular unranked tree languages. Sadly, in this respect, XSDs are not closed under the basic operations of union and set difference, complicating important tasks in schema integration and evolution. The purpose of this paper is to investigate how the union and difference of two XSDs can be approximated within the framework of single-type regular tree languages. We consider both optimal lower and upper approximations. We also address the more general question of how to approximate an arbitrary regular tree language by an XSD and consider the complexity of associated decision problems.