Edit Distance of Finite State Transducers

TL;DR

The paper addresses how to compare finite-state transductions by lifting word metrics to transductions, defining d(𝒯,𝒮) as the supremum over inputs, with ∞ whenever domains differ. It shows that for several integer-valued edit distances (including the Levenshtein family, Hamming, transposition, and conjugacy), closeness and k-closeness are decidable for functional transducers, making the transducer distance computable in these cases. By connecting distance, diameter, and index via Nivat's theorem, it relates transducer distance to the diameter of rational relations and to the index in a composition closure, yielding a unified framework with decision procedures. The work provides concrete complexity characterizations—polynomial-time for closeness in Hamming/Transposition cases, and co-NP/NP-type bounds for distance computations—offering a rigorous basis for robust, scalable comparison of transductions in automata theory and related applications.

Abstract

We lift metrics over words to metrics over word-to-word transductions, by defining the distance between two transductions as the supremum of the distances of their respective outputs over all inputs. This allows to compare transducers beyond equivalence. Two transducers are close (resp. $k$-close) with respect to a metric if their distance is finite (resp. at most $k$). Over integer-valued metrics computing the distance between transducers is equivalent to deciding the closeness and $k$-closeness problems. For common integer-valued edit distances such as, Hamming, transposition, conjugacy and Levenshtein family of distances, we show that the closeness and the $k$-closeness problems are decidable for functional transducers. Hence, the distance with respect to these metrics is also computable. Finally, we relate the notion of distance between functions to the notions of diameter of a relation and index of a relation in another. We show that computing edit distance between functional transducers is equivalent to computing diameter of a rational relation and both are a specific instance of the index problem of rational relations.

Figures (3)

• Figure 1: $\mathcal{T}_1$ outputs letters at the odd positions, $\mathcal{T}_2$ outputs letters at the even positions and $\mathcal{T}_3$ outputs only $a$'s.
• Figure 2: The Boundedness preorder of edit distances
• Figure 3: $\mathcal{T}_4$ (left) outputs $0$ & $1$ for each block of $0$'s & $1$'s resp. whereas $\mathcal{T}_5$ (right) outputs $1$ & $0$ for each block of $0$'s & $1$'s resp.

Theorems & Definitions (39)

• Lemma 1
• Definition 2: Metric on transductions
• Proposition 3
• Remark 4
• Example 5
• Example 6
• Proposition 7
• Remark 8
• Proposition 9
• Proposition 10