一句话：重新审视有理关系子类间的成员判定，改进复杂度并给出新的可判定性结果。
论文研究有限与无限字上有理关系子类的成员问题：给定类 C2 中的关系 R，判定其是否属于更小的类 C1，涵盖确定型有理、同步（自动/正则）与可识别关系。多数版本的精确复杂度或可判定性已悬置近二十年。作者给出改进：无限字上同步关系的可识别性在确定型 ω-自动机下为 NL-完全、非确定型下为 PSPACE-完全，彻底解决该问题；确定型有理二元关系的可识别性可在多项式时间判定，改进原双指数上界，高阶情形给出随机指数时间算法；首次给出判定确定型有理关系是否同步的算法，二元情形为多项式时间。
We revisit the membership problem for subclasses of rational relations over finite and infinite words: Given a relation R in a class C2, does R belong to a smaller class C1? The subclasses of rational relations that we consider are formed by the deterministic rational relations, synchronous (also called automatic or regular) relations, and recognizable relations. For almost all versions of the membership problem, determining the precise complexity or even decidability has remained an open problem for almost two decades. In this paper, we provide improved complexity and new decidability results. (i) Testing whether a synchronous relation over infinite words is recognizable is NL-complete (PSPACE-complete) if the relation is given by a deterministic (nondeterministic) ω-automaton. This fully settles the complexity of this recognizability problem, matching the complexity of the same problem over finite words. (ii) Testing whether a deterministic rational binary relation is recognizable is decidable in polynomial time, which improves a previously known double exponential time upper bound. For relations of higher arity, we present a randomized exponential time algorithm. (iii) We provide the first algorithm to decide whether a deterministic rational relation is synchronous. For binary relations the algorithm even runs in polynomial time.