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.
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by C. Aiswarya,... at arxiv.org 04-26-2024
https://arxiv.org/pdf/2404.16518.pdfDeeper Inquiries