Counting in Trees along Multidirectional Regular Paths

We propose a tree logic capable of expressing simple cardinality constraints on the number of nodes selected by an arbitrarily deep regular path with backward navigation. Specifically, a sublogic of the alternation-free μ−calculus with converse for finite trees is extended with a counting operator in order to reason on the cardinality of node sets. Also, we developed a bottom-up tableau-based satisfiability-checking algorithm, which resulted to have the same complexity than the logic without the counting operator: a simple exponential in the size of a formula. This result can be seen as an extension of the so-called graded-modalities introduced in [18], which allows counting constraints only on immediate successors, with conditions on the number of nodes accessible by an arbitrary recursive and multidirectional path. This work generalizes the optimal complexity bound: 2 where n is the length of the formula, shown in [11], for satisfiability of the logic extended with such counting constraints. Finally, we identify a decidable XPath fragment featuring cardinality constraints on paths with upward/downward recursive navigation, in the presence of XML types.