Hamming distance between finite transducers

2026-04-28

AI Summary

This study addresses the bounded discrepancy problem for two nondeterministic finite transducers, asking whether the Hamming distance between their output strings is at most a given threshold \(k\) for all inputs. Employing techniques from formal language theory, automata theory, and computational complexity, and via logspace many-one reductions, the work establishes the precise complexity classification of this problem for the first time: it is NL-complete when \(k\) is fixed, co-NP-complete when \(k\) is given in binary, and DP-complete when the Hamming distance is required to be exactly \(k\). Furthermore, the paper proves that the maximum Hamming distance between two such transducers admits a tight quadratic upper bound, which is asymptotically optimal with respect to the size of the transducers.

Abstract

We study bounded deviation of non-deterministic finite transducers under the Hamming distance: the bounded comparison problem asks, given two transducers and $k \in \mathbb{N}$, whether for every input the two transducers produce words at Hamming distance at most $k$. This problem is known to be decidable in polynomial time when $k$ is fixed, and in co-NP otherwise. We show that the problem is NL-complete when $k$ is fixed, co-NP-complete when $k$ is given in binary, and it is DP-complete to decide if the distance is exactly $k$. We also prove that if the two transducers have bounded comparison, then the maximal distance is at most quadratic in the size of both transducers, and that this bound is asymptotically tight. We prove the results on deviations problem, which asks similar questions on the distance of the pairs of input and output of a single transducer, and show that these two families of problems are logspace many-one equivalent.

Problem

• Hamming distance
• finite transducers
• bounded comparison
• deviations problem
• computational complexity

Innovation

• finite transducers
• Hamming distance
• computational complexity
• bounded comparison
• NL-completeness