Asymptotic Hausdorff and Language Similarity

TL;DR

This paper introduces the Asymptotic Hausdorff lifting, a novel method for measuring asymptotic similarity between sets, especially languages, by extending element-level metrics to set-level metrics that are insensitive to finite deviations.

Contribution

The paper presents the Asymptotic Hausdorff lifting framework, which overcomes limitations of classical Hausdorff approaches and applies it to language similarity using normalized edit distances.

Findings

The framework captures asymptotic similarity in infinite domains.

It provides a metric between languages reflecting long-term edit behavior.

Analysis of regular and bounded context-free languages under this metric.

Abstract

We introduce the \textit{Asymptotic Hausdorff} lifting, denoted $\mathbb{AH}_{d}$, a general method for lifting an element-level metric $d$ to a (pseudo-) metric on sets, that captures asymptotic similarity in infinite domains equipped with a notion of size. The construction is designed to be insensitive to finite deviations and to avoid the limitations of classical Hausdorff-based approaches, which are often overly sensitive to outliers and fail to reflect asymptotic behavior. Formal languages provide a central motivating instance of this framework, where elements are words and sets are languages. When applied to normalized edit distances, the Asymptotic Hausdorff lifting yields metric-valued distances between languages that reflect asymptotic edit behavior while preserving metric structure. We study the equivalence classes of regular languages induced by $\mathbb{AH}_{d}$ for normalized edit distances $d$, and characterize their asymptotic essence. Focusing in particular on the normalized edit distance of Marzal and Vidal, $\textsf{ned}$, we investigate the computation of $\mathbb{AH}_{\textsf{ned}}$ for regular languages and for bounded context-free languages.