Approximate Problems for Finite Transducers

2025-04-24

AI Summary

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.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Finite (word) state transducers extend finite state automata by defining a binary relation over finite words, called rational relation. If the rational relation is the graph of a function, this function is said to be rational. The class of sequential functions is a strict subclass of rational functions, defined as the functions recognised by input-deterministic finite state transducers. The class membership problems between those classes are known to be decidable. We consider approximate versions of these problems and show they are decidable as well. This includes the approximate functionality problem, which asks whether given a rational relation (by a transducer), is it close to a rational function, and the approximate determinisation problem, which asks whether a given rational function is close to a sequential function. We prove decidability results for several classical distances, including Hamming and Levenshtein edit distance. Finally, we investigate the approximate uniformisation problem, which asks, given a rational relation $R$, whether there exists a sequential function that is close to some function uniformising $R$. As for its exact version, we prove that this problem is undecidable.

Problem

• Decide if a rational relation is close to a rational function
• Determine if a rational function is near a sequential function
• Investigate undecidability of approximate uniformisation problem

Innovation

• Decidable approximate functionality for rational relations
• Decidable approximate determinisation for rational functions
• Undecidable approximate uniformisation for rational relations