Hamming distance between finite transducers

TL;DR

This paper analyzes the computational complexity of the bounded deviation problem for finite transducers under Hamming distance, providing complexity classifications and bounds on maximal deviation.

Contribution

It establishes complexity results for the bounded deviation problem, including NL-completeness, co-NP-completeness, and DP-completeness, and bounds on maximal deviation.

Findings

Bounded comparison implies maximal distance is at most quadratic in transducer size.

Complexity varies with the parameter k: NL-complete for fixed k, co-NP-complete for binary k.

Deviation problems for transducers and input/output pairs are logspace many-one equivalent.

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.