Category Theory and Model-Driven Engineering: From Formal Semantics to Design Patterns and Beyond

Abstract

There is a hidden intrigue in the title. CT is one of the most abstract mathematical disciplines, sometimes nicknamed "abstract nonsense". MDE is a recent trend in software development, industrially supported by standards, tools, and the status of a new "silver bullet". Surprisingly, categorical patterns turn out to be directly applicable to mathematical modeling of structures appearing in everyday MDE practice. Model merging, transformation, synchronization, and other important model management scenarios can be seen as executions of categorical specifications.Moreover, the paper aims to elucidate a claim that relationships between CT and MDE are more complex and richer than is normally assumed for "applied mathematics". CT provides a toolbox of design patterns and structural principles of real practical value for MDE. We will present examples of how an elementary categorical arrangement of a model management scenario reveals deficiencies in the architecture of modern tools automating the scenario.

Symmetric delta lenses and spans of asymmetric delta lenses.

“…The present work owes its origins to the articles [2], [3] and [4] of Diskin and various coauthors. Delta lenses based on categories in both asymmetric and symmetric versions were first considered in those articles.…”

“…In the forward direction, given objects X and Y of X and Y that are synchronized by a corr r and an arrow x in X with domain X, the propagation returns an arrow y with domain Y and a corr witnessing the synchronization of the codomains of x and y. This is made precise in the following definition and is based on definitions in [3] and [4]. We denote the domain and codomain of an arrow x by d 0 (x) and d 1 (x) respectively.…”

“…Recall that Diskin et al [4] defined symmetric delta lenses (our fb-lenses), but they did not consider composing them. Like Hofmann et al [8] they would find that they need to consider equivalence classes of their symmetric delta lenses in order for the appropriate composition to be associative.…”

Categorical approach to modelling and to coupling of models

“…Apart from applications of category theory in the field of algebraic topology, see for example , in recent years it found various number of applications in probability theory , modelling of dynamical systems etc. In , the application of category theory to model‐driven engineering has been proposed, where the most attention has been drawn to software engineering from computer science point of view. Thus, up to our knowledge, there were no attempts to apply category theory for a general description of the modelling process.…”

Open and Original Problems in Software Language Engineering 2015 Workshop Report

“…Patterns found in category theory in the form of commuting diagrams (for pushouts, pullbacks and colimits in general), are directly applicable to mathematical modelling of structures appearing in everyday MDE practice: model merging, transformation, synchronisation, well-formedness and consistency management can all be seen as executions of categorical specifications [10]. Then, if an operation can be defined diagrammatically, composition of them amounts to trivial tiling of commuting squares [9].…”