Revisiting Membership Problems in Subclasses of Rational Relations

TL;DR

This paper investigates the membership problem for subclasses of rational relations, providing new complexity and decidability results, including polynomial-time algorithms and settling open problems over infinite words.

Contribution

It offers the first polynomial-time algorithm for recognizing deterministic rational relations and settles the complexity of recognizability over infinite words.

Findings

Recognizability of synchronous relations over infinite words is NL-complete for deterministic automata.

Deciding recognizability of deterministic rational binary relations is polynomial-time.

First polynomial-time algorithm for checking if a deterministic rational relation is synchronous.

Abstract

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.