Approximate Problems for Finite Transducers
This paper investigates the rational relation approximation problem for finite-state transducers (FSTs) under Hamming and Levenshtein distances, focusing on three central questions: approximate functionality (whether a rational relation is close to a rational function), approximate determinizability (whether it is close to a sequential function), and approximate uniformization (whether a sequential function exists that approximates the relation uniformly). We establish, for the first time, the decidability of the first two problems under both distance metrics and provide constructive algorithmic frameworks. In contrast, we prove strict undecidability of approximate uniformization—resolving a fundamental open question in rational relation approximation theory. Our approach integrates automata-theoretic techniques, including synchronous products, automaton trimming, and reachability analysis, to precisely characterize the decidability boundaries for membership in these approximation classes.