6 Software composition

There is a complication that arises from the fact that a component itself may recursively refer to (extracted) subcomponents. Because a subcomponent does not make part of the component, I have introduced what I call an application type, which may be considered to be a default more abstract substitute for the subcomponent. Hence, a component not only consists of what I call a specification type, but also of an application type for each extracted subcomponent. This is a sharp characterization of a component and enables me to describe two different tasks in component-based programming: composition and replacement (or re-implementation).

• A composition is a mapping from application types to specification types and arrows indicate the mapping. If the mapping is in conformance with the lattice order, then the composition is valid.
• In order to replace a component, its specification type has to be a refinement of each application type. A constructive approach is the calculation of the GLAT (greatest lower application type), which is the meet of the application types associated by the mapping. From the calculated GLAT, an implementation may be derived by refinement.