On the Theoretical Foundation of Meta-Modelling in Graphically Extended BNF and First Order Logic

Meta-modeling plays an important role in model driven software development methodology. In our previous work, a graphic extension of BNF (GEBNF) was proposed to define the abstract syntax of graphic modeling languages. From a GEBNF syntax definition, a first order predicate logic language can be induced so that meta-modeling can be performed formally by specifying a predicate on the domain of syntactically valid models. In this paper, we investigate the theoretical foundation of this meta-modeling approach. We first formally define the semantics of GEBNF syntax definitions as algebras that contain no junk and satisfy constraints derived from GEBNF syntax rules. The semantics of the induced logic is then formally defined by regarding such algebras as models. We then formally prove that well-formed syntax definitions together with syntax morphisms form a category, where syntax morphisms represent the translations between modeling languages. The models (i.e. algebras) in a modeling language and the homomorphisms between them also form a category. Finally, we prove that the functors from GEBNF syntax definitions to the categories of models and to sentences in the induced first order logic form an institution. Therefore, GEBNF and its induced logics form a valid formal specification language for models.