Establish a Converse Compilation from CMTT to Environment Classifiers

Establish a compilation from contextual modal types to environment classifiers that supports contextual code scopes, including contexts that do not arise from nested code binders.

Background

The paper proves that contextual modal types are at least as expressive as environment classifiers by compiling environment-classifier scopes into contextual typing contexts. The converse direction is not established.

The difficulty identified by the authors is that environment classifiers can name only scopes generated by nested binders in code, whereas contextual modal types may describe arbitrary typing contexts. Consequently, a converse compilation would need to account for contextual scopes that have no corresponding nested nominal scope.

References

We conjecture that the converse direction is difficult, because a classifier can only name a scope that arises from nested binders in code, whereas a contextual modal type may mention any context.

— Compiling Environment Classifiers to Contextual Modal Types  (2610.25741 - Chen et al., 19 Oct 2026) in Section 1, Introduction, subsection “Metatheoretical results” (paragraph beginning “We prove in Agda…”; repeated in Section 7, Conclusion)