@@ -40,15 +40,16 @@ SageMath has become extremely popular among mathematicians, especially those act
Due to its size and integrative nature, it has become somewhat of a cross--programming language packaging and distribution platform, via which researchers can disseminate specialized Open Source computation libraries.
It integrates a number of important mathematical datasets such as copies of some LMFDB datasets or GAP's small groups library.
The SageMath integration layer described above is largely non-semantic in the sense that it relies on custom ``glue code'' in Python that is unverified and can be broken by any update of one of the integrated systems or datasets.
Moreover, it is specific to SageMath and cannot be reused in other systems.
To have a more semantic, maintainable, and flexible interoperability platform, the OpenDreamKit project developed the Math-in-the-Middle (MitM) Framework~\cite{ODK-D6.5}.
There, all systems (boxes $A$-$H$ in Figure~\ref{fig:mistargraph}) export an interface specification (circles $a$-$h$) as symbolic data, whose concepts are aligned with a knowledge graph representing abstract --- i.e. system-independent --- mathematical knowledge (the MitM ontology in the center of~\ref{fig:mistargraph}).
Systems are connected by a MitM mediator system, which reads the interface specifications, the MitM ontology, and the alignments and translates between the OpenMath-based system dialects.
This mediator is built on top of and inherits most of the functionality from the MMT API~\cite{mmt:repo:on}, an open knowledge management system for symbolic data developed at \site{FAU}.
...
...
@@ -56,14 +57,15 @@ This mediator is built on top of and inherits most of the functionality from the
Figure~\ref{fig:mitm} shows a workflow and mathematical use case that makes use of the flexible delegation of sub-problems to external systems, which is not possible in SageMath alone.
Additionally, this MitM workflows can be set up just as easily for systems, e.g., where a GAP user delegates to SageMath and PARI/GP~\cite{KohMuePfe:kbimss17}.
In addition to computation systems and as a major inspiration for the \TheProject proposal, MitM also allows integrating mathematical \emph{databases}.
Here the MitM mediator directly connects to the database-level API and automatically decodes the encoded concrete mathematical data into mathematical objects via semantic annotations to the database tables.
The latter uses the mathematical schema technology developed in OpenDreamKit to specify the mathematical structure and encodings of concrete data.
\caption{OpenDreamKit: A MitM-based Workflow}\label{fig:mitm}
\end{figure}
In addition to computation systems and as a major inspiration for the \TheProject proposal, MitM also allows integrating mathematical \emph{databases}.
Here the MitM mediator directly connects to the database-level API and automatically decodes the encoded concrete mathematical data into mathematical objects via semantic annotations to the database tables.
The latter uses the mathematical schema technology developed in OpenDreamKit to specify the mathematical structure and encodings of concrete data.
\paragraph{Symbolic Representation Languages}
Symbolic data requires very sophisticated formal languages, such as type theories, programming languages, and logics to represent.
