Semantics Through Pictures: towards a diagrammatic semantics for object-oriented modelling notations

An object-oriented (OO) model has a static component, the set of allowable snapshots or system states, and a dynamic component, the set of filmstrips or sequences of snapshots. Diagrammatic notations, such as those in UML, each places constraints on the static and/or dynamic models. A formal semantics of OO modeling notations can be constructed by providing a formal description of (i) sets of snapshots and filmstrips, (ii) constraints on those sets, and (iii) the derivation of those constraints from diagrammatic notations. In addition, since constraints are contributed by many diagrams for the same model, a way of doing this compositionally is desirable. One approach to the semantics is to use first-order logic for (i) and (ii), and theory inclusion with renaming, as in Larch, to characterize composition. A common approach to (iii) is to bootstrap: provide a semantics for a kernel of the notation and then use the kernel to give a semantics to the other notations. This only works if a kernel which is sufficiently expressive can be identified, and this is not the case for UML. However, we have developed a diagrammatic notation, dubbed constraint diagrams, which seems capable of expressing most if not all static and dynamic constraints, and it is proposed that this be used to give a diagrammatic semantics to OO models.