Deciding membership in a fixed level of a concatenation hierarchy

Determine the decidability and design an algorithm for the membership problem at an arbitrary fixed level of a concatenation hierarchy of regular languages. Given a regular language L and a level index n (for hierarchies such as the dot-depth, Straubing–Thérien, modulo, or group hierarchies), decide whether L belongs to level n of the hierarchy.

Background

Concatenation hierarchies (e.g., dot-depth, Straubing–Thérien, modulo, group) stratify regular languages into levels built via the Boolean and polynomial closures. For the historical dot-depth hierarchy, membership is known to be decidable only at levels one and two, and this paper establishes decidability for dot-depth three under certain reductions.

Despite progress on specific hierarchies and levels, the general membership problem across levels and hierarchies has remained a central challenge in automata theory.