Relating Multifunctions and Predicate Transformers through Closure Operators

We study relations between predicate transformers and mul-tifunctions in a topological setting based on closure operators. We give topological deenitions of safety and liveness predicates and using these predicates we deene predicate transformers. State transformers are mul-tifunctions with values in the collection of xed points of a closure operator. We derive several isomorphisms between predicate transformers and multifunctions. By choosing diierent closure operators we obtain multi-functions based on the usual power set construction, on the Hoare, Smyth and Plotkin power domains, and based on the compact and closed metric power constructions. Moreover, they are all related by isomorphisms to the predicate transformers.