We provide a general introduction to the AlgebraicJulia ecosystem and AlgebraicRewriting.jl, which allows for integrating general-purpose code with computation of many graph transformation constructions in a broad variety of categories. Practical applications of graph transformation depend on being able to apply sequences of rewrites in a controlled manner: we present work on a graphical language for the construction and composition of such programs, including computation of normal forms as well as scientific agent-based model simulations. This graphical language can be given semantics in many different contexts (e.g. deterministic, nondeterministic, probabilistic) and can be functorially migrated, which yields graph rewriting programs that operate in other categories.

Software interoperability is a recurring issue in nearly every bigger software project where two or more (legacy) software systems are involved. One aspect of interoperability, that is considered especially tricky is *semantic interoperability*, i.e., aligning the concepts, entities, data structures from multiple systems with each other. The model-driven software engineering community has investigated this issue under different names: model management, model synchronisation, inter-model consistency. From a theoretical side, there are two noteworthy common approaches: Goguen’s *Colimit-approach* (for general systems' theory) and *Triple Graph Grammars (TGGs)*. The former describes that idea that one may always consider a “global integrated system” as the colimit of a diagram of interacting systems, while the latter is the foundation for a powerful framework of binary model synchronizers derived from a declarative description (grammar). In our investigation, it, however, turned out that both approaches each have significant drawback when considering them in practical applications: The colimit turns out to be a “forgetful” operation and TGGs are limited to a binary setting (it is a well-known fact from logic and databases that there are multi-ary relations that cannot be factored into a system of binary relations). Thus, we invented a novel formalism, called *comprehensive systems*, first introduced in 1, and theoretically flashed out in 2 and 3. Comprehensive systems provide an alternative to the colimit approach, which can be thought of as instead of “merging” a diagram into a singular object, they “flatten” the whole diagram. Moreover, they are designed for a general n-ary ($n \geq 2$) settings and thus can be considered as a generalization of triple graphs. In a series of papers, we have shown that comprehensive systems admit SPO and DPO rewriting in the setting of a weak adhesive HLR category. From a practical perspective, comprehensive systems serve as the theoretical foundation of a prototypical software interoperability tool called CorrLang.

In this talk, I will provide a brief historical overview over interoperability, model management, and model synchronisation, provide the motivation for comprehensive systems, sketch their theoretical properties (with an emphasis on*partial morphisms*), and, if time allows, demonstrate how comprehensive systems are reified in a concrete tool (CorrLang).

In this talk, I will provide a brief historical overview over interoperability, model management, and model synchronisation, provide the motivation for comprehensive systems, sketch their theoretical properties (with an emphasis on