Program-Structural Characterization of Exact Achievability

Characterize directly in terms of a pGCL program and its multiobjective postexpectations when every point in the Scott-closed multiobjective preexpectation is exactly achievable by a determinization, extending the corresponding single-objective existence criterion.

Background

The multiobjective preexpectation transformer computes the Scott closure of the points achievable by determinizations. Consequently, it includes limit points that may only be approximated and not attained exactly. The paper gives Scott closedness of the achievable set as a sufficient condition for exact achievability, while noting that this condition is not necessary: in the diminishing-increment example, the supremum is unattainable but every strictly smaller value is attainable.

For single-objective expectations, the paper provides program-level sufficient conditions based on demonically almost-sure termination or demonically certain termination. The unresolved question is whether an analogous criterion can be formulated directly from the structure of the program and its multiobjective postexpectations, rather than by assuming Scott closedness of the achievable set.