Antoine Amarilli, Corentin Barloy, Louis Jachiet, Charles Paperman
备注：
44 pages including 16 pages of main text and 2 pages of bibliography. This is the full version with proofs of the MFCS'25 article: the main text of the article is identical to the MFCS'25 version up to minor changes
We study the dynamic membership problem for regular tree languages under relabeling updates: we fix an alphabet $Σ$ and a regular tree language $L$ over $Σ$ (expressed, e.g., as a tree automaton), we are given a tree $T$ with labels in $Σ$, 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 $Ω(\log n / \log \log n)$ 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.
HOM问题，它询问在给定的树同态下正则树语言的图像是否再次正则，已知是可判定的[Godoy & Giménez：The HOM problem is decidable. JACM 60（4），2013]。然而，这个问题仍然是开放的常规加权树语言。证明了在无权设置中使用的主要概念，具有等式和不等式约束的树自动机，可以直接推广到加权设置，并且可以表示任何正则加权树语言在任何非删除和非擦除树同态下的图像。本文还研究了由带约束的加权树自动机生成的加权树语言的若干闭包性质和判定问题。
