Observational Completeness on Abstract Interpretation

In the theory of abstract interpretation, we introduce the observational completeness, which extends the common notion of completeness. A domain is complete when abstract computations are as precise as concrete computations. A domain is observationally complete for an observable π when abstract computations are as precise as concrete computations, if we only look at properties in π. We prove that continuity of state-transition functions ensures the existence of the least observationally complete domain. When state-transition functions are additive, the least observationally complete domain boils down to the complete shell.