Dynamic Membership for Regular Tree Languages

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 $T$. 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 $O(\log n)$ algorithms. 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 decided in constant time per update. Almost-commutative languages generalize both commutative languages and finite languages: 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 the prefix-U1 problem from (Amarilli, Jachiet, Paperman, ICALP'21) also does not admit a constant-time algorithm.