Presburger Constraints In Trees

The fully enriched mu-calculus is an expressive propositional modal logic with least and greatest fixed-points, nominals, inverse programs and graded modalities. Several fragments of this logic are known to be decidable in EXPTIME. However, the full logic is undecidable. Nevertheless, it has been recently shown that the fully enriched mu-calculus is decidable in EXPTIME when its models are finite trees. In the present work, we study the fully-enriched mu-calculus for trees extended with Presburger constraints. These constraints generalize graded modalities by restricting the number of children nodes with respect to Presburger arithmetic expressions. We show that this extension is decidable in EXPTIME. In addition, we also identify decidable extensions of regular tree languages (XML schemas) with interleaving and counting operators. This is achieved by a linear characterization in terms of the logic. Regular path queries (XPath) with Presburger constraints on children paths are also characterized. These results imply new optimal reasoning (emptiness, containment, equivalence) bounds on counting extensions of XPath queries and XML schemas.