VDM meets LCF: Domain-Theoretic and Topological Aspects of VDM

We discuss the domain-theoretic and topological content of the operator calculus used in the Irish School of the Vienna Development Method (VDM♣) of formal systems development. Thus, we examine the Scott continuity, or otherwise, of the basic operators used in this calculus when viewed as operators on the domain (X → Y ) of partial functions mapping X into Y . It turns out that the override, one of the more important of the basic operators, is not Scott continuous, and in order to overcome this problem we introduce another topology, which we call here the strong Cantor topology, by means of the topological tool of convergence classes. Indeed, the strong Cantor topology is the smallest topology which refines the Scott and Lawson topologies and is such that, with respect to it, all the basic operators we consider are continuous. Furthermore, we examine the role of the strong Cantor topology in relation to indexed monoids, both with and without units, and display them as topological monoids in the strong Cantor topology. The totality of our results gives considerable support to the view that the strong Cantor topology is the topology of formal methods.