Classify intersections of incomparable deterministic Parikh automata models

Classify the intersections of the language classes recognized by the pairwise incomparable deterministic variants studied in the paper (including deterministic limit PA, deterministic reachability-regular PA, deterministic weak reset PA, deterministic strong reset PA, deterministic safety PA, deterministic co-Büchi PA, and deterministic Büchi PA) to obtain a fine-grained map of Parikh-recognizable ω-languages.

Background

The paper establishes numerous incomparability results among deterministic Parikh automata models on infinite words and provides an inclusion diagram highlighting the relationships. While closure properties, expressiveness, and (un)decidability results are analyzed, the precise characterization of intersections between these incomparable classes is left unresolved.

The authors explicitly point out that this classification task remains to be done, framing it as a direction to provide a fine-grained "map of the universe" for Parikh-recognizable ω-languages.