Formal Foundations of Metamodeling: Abstract Words, Abstract Languages, and Edge Algebra

Textual languages are not the only way to write down models. Especially in the field of model-based engineering, graphical languages are gaining more and more popularity. In this regard well-known specification techniques for textual languages like context-free grammars are often substituted by metamodels. Nevertheless do we have to give up the theory of formal languages including their definitions for word and language if using metamodels for language specification. In this paper we introduce a formal framework for the purpose of metamodeling by defining an abstract word as a special type of a partially ordered multi-graph (called M-graph). We consciously avoid a definition of a metamodeling language itself in order to show that this approach is independent of any metamodeling language. We will instead introduce an algebra on edges which allows to express complex consistency conditions on models as well as querying large models.