Decidability of Closeness and Distance between Finite State Transducers

Core Concepts

The distance between two transductions defined by finite state transducers can be computed by deciding their closeness and k-closeness problems for various edit distance metrics.

Abstract

The paper introduces a framework to meaningfully compare two transductions (word-to-word functions) defined by finite state transducers beyond just equivalence.
The key insights are:

The distance between two transductions is defined as the supremum of the distances of their respective outputs over all inputs. This allows comparing transducers that are not equivalent.

Two transducers are close (resp. k-close) if their distance is finite (resp. at most k) with respect to a given metric.

For common integer-valued edit distances like Hamming, transposition, conjugacy, and Levenshtein family, the closeness and k-closeness problems are decidable for functional transducers. This implies the distance between such transducers is computable.

The distance between transducers is equivalent to computing the diameter of a rational relation and both are a specific instance of the index problem of rational relations.

The decision procedures involve designing weighted automata that count the number of edit operations required to transform one output to the other. Additional techniques are used for specific edit distances like checking conjugacy of loops.

The results establish the computational complexity of comparing transductions defined by finite state machines, which is useful in applications like encoders, decoders, spell checkers, etc.

Stats

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Key Insights Distilled From

Edit Distance of Finite State Transducers

by C. Aiswarya,... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16518.pdf

Deeper Inquiries

What are some potential applications of the framework for comparing transductions defined by finite state transducers

The framework for comparing transductions defined by finite state transducers has various potential applications in different fields. One application could be in natural language processing, specifically in machine translation systems. By comparing the transductions of different language pairs, the framework can help improve the accuracy and efficiency of translation algorithms. Additionally, in speech recognition systems, comparing transductions can aid in refining the phonetic models used for converting speech to text. Moreover, in bioinformatics, the framework can be utilized to analyze genetic sequences and identify similarities or differences in biological data. Overall, the framework can be valuable in various domains where pattern recognition and transformation of data are essential.

How can the techniques developed in this work be extended to handle more expressive models of transducers beyond finite state

The techniques developed in this work can be extended to handle more expressive models of transducers beyond finite state by considering more complex operations and transformations. One approach could be to incorporate probabilistic transducers, which assign probabilities to different output sequences, allowing for a more nuanced comparison of transductions. Additionally, extending the framework to handle weighted transducers, where each transition has an associated weight, can enable the evaluation of transductions based on different criteria such as cost or importance. Furthermore, integrating deep learning models with transducers can enhance the framework's capabilities by leveraging the power of neural networks for more sophisticated pattern recognition and transformation tasks.

Are there other notions of distance or similarity between transductions that could be meaningful and efficiently computable

There are several other notions of distance or similarity between transductions that could be meaningful and efficiently computable. One such notion is the alignment-based distance, where the goal is to find the optimal alignment between the outputs of two transducers. This alignment can reveal structural similarities and differences between the transductions, providing valuable insights for comparison. Another approach could be to explore the concept of edit distance between transducers based on specific edit operations tailored to the nature of the transduction task. Additionally, considering the concept of divergence between transductions, inspired by information theory, can offer a measure of dissimilarity that takes into account the complexity and information content of the outputs. These alternative notions of distance or similarity can complement the existing framework and provide a more comprehensive understanding of the relationships between transductions.