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Regular tree grammars and regular path expressions constitute core constructswidely used in programming languages and type systems. Nevertheless, there hasbeen little research so far on reasoning frameworks for path expressions wherenode cardinality constraints occur along a path in a tree. We present a logiccapable of expressing deep counting along paths which may include arbitraryrecursive forward and backward navigation. The counting extensions can be seenas a generalization of graded modalities that count immediate successor nodes.While the combination of graded modalities, nominals, and inverse modalitiesyields undecidable logics over graphs, we show that these features can becombined in a tree logic decidable in exponential time.

A Tree Logic with Graded Paths and Nominals