Some applications of topology to program semantics

The relationship between programs and the set of partial correctness assertions that they satisfy, constitutes a Galois connection. The topology resulting from this Galois connection is closely related to the Lindenbaum baum topology for the language in which these partial correctness assertions are stated. This relationship provides us with a tool for understanding the incompleteness of Hoare Logics and for answering certain natural questions about the connection between the relational semantics and the partial correctness assertion semantics for programs, especially in connection with the question of modularity of programs. Two questions to which we shall find topological answers in this paper are “When is a language expressive for a program?”, and “When can we have rules of inference which are adequate to infer the properties of the complex program ±#β from those of its components ±,β?” We also obtain a natural answer to the question “What can the set{(A, B)|{A}±{B} is true) look like for arbitraryα?”.