On Powerdomains and Modality

Abstract This note shows a simple connection between powerdomains and modal assertions that can be made about nondeterministic computations. We consider three kinds of powerdomains: the Plotkin powerdomain, the Smyth powerdomain, and one christened the Hoare powerdomain by Plotkin because it captures the partial correctness of a nondeterministic program. The modal operators are □ for ‘inevitably’ and ♦ for ‘possibly’. It is shown in a precise sense how the Smyth powerdomain is built up from assertions about the inevitable behaviour of a process, the Hoare powerdomain is built up from assertions about the possible behaviour of a process, while the Plotkin powerdomain is built up from both kinds of assertions taken together.