Regularity, closure operations, and quantitative complexity of positive separation width

Determine which additional regularity assumptions permit canonical witness choices for positive separation width, which operations preserve finite positive separation width, and what additional information relates witness size to observation or mistake complexity in language generation in the limit.

Background

The paper characterizes generation in the limit through finite witness assignments whose active target families have infinite common intersection, and introduces positive separation width to measure the required witness sizes. Its finite-level and width-omega constructions establish the possible hierarchy of witness complexity, but the definition is global and does not yield a general intrinsic rank or quantitative relationship with the time or number of observations needed to encounter a witness.

The authors explicitly identify unresolved structural and quantitative issues: determining regularity conditions that make witness choices canonical, understanding preservation of finite width under operations on language families, and relating witness size to observation or mistake complexity. These questions concern extensions beyond the existence characterization proved in the paper.

References

It remains useful to ask which additional regularity assumptions permit canonical witness choices, which operations preserve finite width, and what extra information would relate witness size to observation or mistake complexity.

— Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarch  (2609.10525 - Li et al., 9 Sep 2026) in Section 6, paragraph “Combinatorial meaning and remaining questions”

Efficient witness discovery and active-intersection selection remain separate algorithmic questions.

— Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarch  (2609.10525 - Li et al., 9 Sep 2026) in Section 6, paragraph “Combinatorial meaning and remaining questions”