We introduce the {Asymptotic Hausdorff} lifting, denoted {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 {AH}_{d} for normalized edit distances {d}, and characterize their asymptotic essence. Focusing in particular on the normalized edit distance of Marzal and Vidal, {ned}, we investigate the computation of {AH}_{ned} for regular languages and for bounded context-free languages.

The talk is based on a joint work with Gal Meirom, accepted to ICALP'26.

https://arxiv.org/abs/2605.09668