Resynchronized Uniformization and Definability Problems for Rational Relations

TL;DR

This paper systematically studies the decidability of uniformization and definability problems for subclasses of rational relations defined by regular synchronization languages, extending known results with new decidability and undecidability findings.

Contribution

It provides a comprehensive analysis of the decidability landscape for rational relations characterized by synchronization languages, including new results that complete previous understandings.

Findings

Several new decidability results for uniformization problems.

New undecidability results for definability problems.

A unified framework rephrasing known results in this context.

Abstract

Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of uniformization and definability problems for subclasses of rational relations defined in terms of such synchronization languages. We rephrase known results in this setting and complete the picture by adding several new decidability and undecidability results.