We revisit the membership problem for subclasses of rational relations over finite and infinite words: Given a relation R in a class C_2, does R belong to a smaller class C_1? 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) omega-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.

翻译：本文重新研究了有限词和无限词上有理关系子类的成员关系问题：给定属于C_2类的关系R，问R是否属于更小的C_1类？我们考虑的有理关系子类包括确定性有理关系、同步关系（也称为自动机关系或正则关系）以及可识别关系。对于几乎所有版本的成员关系问题，确定其精确复杂度甚至可判定性在近二十年里一直是一个未解决的开放问题。本文给出了改进的复杂度结果和新的可判定性结果。（i）若无限词上的同步关系由确定性（非确定性）ω-自动机给出，则检验该关系是否可识别是NL-完全（PSPACE-完全）问题。这完全解决了该可识别性问题的复杂度，与有限词上相同问题的复杂度相匹配。（ii）检验确定性有理二元关系是否可识别可在多项式时间内判定，这改进了先前已知的双指数时间上界。对于更高元数的关系，我们给出了一种随机指数时间算法。（iii）我们首次给出了判定确定性有理关系是否同步的算法。对于二元关系，该算法甚至可在多项式时间内运行。