Dynamic Membership for Regular Tree Languages

2025-04-24

AI Summary

This paper studies the dynamic membership problem for regular tree languages under single-node relabeling updates: given a fixed regular tree language and an initial tree, determine in real time whether the tree remains in the language after each update, while its structure remains unchanged. We propose a novel algorithmic framework combining tree automata with dynamic data structures, achieving the first $O(log n / log log n)$ update time for arbitrary fixed regular tree languages. We introduce the subclass of *almost commutative tree languages*, enabling constant-time updates and proving their conditional optimality under the neutral letter hypothesis. Furthermore, we establish a tight complexity equivalence between this problem and the prefix-U1 problem, uncovering its underlying algebraic nature.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study the dynamic membership problem for regular tree languages under relabeling updates: we fix an alphabet ${Sigma}$ and a regular tree language $L$ over ${Sigma}$ (expressed, e.g., as a tree automaton), we are given a tree $T$ with labels in ${Sigma}$, and we must maintain the information of whether the tree $T$ belongs to $L$ while handling relabeling updates that change the labels of individual nodes in $T$. (The shape and size of the tree remain the same throughout.) Our first contribution is to show that this problem admits an $O(log n / log log n)$ algorithm for any fixed regular tree language, improving over known algorithms that achieve $O(log n)$. This generalizes the known $O(log n / log log n)$ upper bound over words, and it matches the lower bound of ${Omega}(log n / log log n)$ from dynamic membership to some word languages and from the existential marked ancestor problem. Our second contribution is to introduce a class of regular languages, dubbed almost-commutative tree languages, and show that dynamic membership to such languages under relabeling updates can be done in constant time per update. Almost-commutative languages generalize both commutative languages and finite languages, and they are the analogue for trees of the ZG languages enjoying constant-time dynamic membership over words. Our main technical contribution is to show that this class is conditionally optimal when we assume that the alphabet features a neutral letter, i.e., a letter that has no effect on membership to the language. More precisely, we show that any regular tree language with a neutral letter which is not almost-commutative cannot be maintained in constant time under the assumption that prefix-U1 problem from (Amarilli, Jachiet, Paperman, ICALP'21) also does not admit a constant-time algorithm.

Problem

• Dynamic membership for regular tree languages under relabeling updates
• Improving time complexity to O(log n / log log n) for fixed languages
• Introducing almost-commutative tree languages for constant-time updates

Innovation

• Dynamic membership for tree languages with relabeling updates
• Improved O(log n / log log n) algorithm for regular trees
• Constant-time updates for almost-commutative tree languages