On the Count of Trees

TL;DR

This paper introduces a decidable logic for reasoning about tree structures with complex path expressions that include deep counting and recursive navigation, extending existing modal logics.

Contribution

It presents a novel tree logic capable of expressing deep counting along paths with recursive navigation, generalizing graded modalities while remaining decidable.

Findings

Logic can express deep counting along paths with recursion

Decidable in exponential time

Applicable to XPath query problems

Abstract

Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on frameworks for reasoning about path expressions where node cardinality constraints occur along a path in a tree. We present a logic capable of expressing deep counting along paths which may include arbitrary recursive forward and backward navigation. The counting extensions can be seen as a generalization of graded modalities that count immediate successor nodes. While the combination of graded modalities, nominals, and inverse modalities yields undecidable logics over graphs, we show that these features can be combined in a decidable tree logic whose main features can be decided in exponential time. Our logic being closed under negation, it may be used to decide typical problems on XPath queries such as satisfiability, type checking with relation to regular types, containment, or equivalence.