\section{Introduction}\label{sec:introduction}

To tackle the vast array of mathematical
publications, various ways of \emph{computerizing} mathematical
knowledge have been experimented with. As it is already difficult for
human mathematicians to keep even a subset of all mathematical
knowledge in their mind, a hope is that computerization will yield
great improvement to mathematical (and really any formal) research by
making the results of all collected publications readily available and
easy to search~\cite{onebrain}.

One research topic in this field is the idea of a \emph{tetrapodal
search} that combines four distinct areas of mathematical knowledge.
These four kinds being (1)~the actual formulae as \emph{symbolic
knowledge}, (2)~examples and concrete objects as \emph{concrete knowledge},
(3)~names and comments as \emph{narrative knowledge} and finally
(4)~identifiers, references and their relationships, referred to as
\emph{organizational knowledge}~\cite{tetra}.

Tetrapodal search aims to provide a unified search engine that indexes
each of the four different subsets of mathematical knowledge.  Because
all four kinds of knowledge are inherently different in their
structure, tetrapodal search proposes that each kind of mathematical
knowledge should be made available in a storage backend that fits
exactly with the kind of data it is providing. With all four areas
available for querying, tetrapodal search intends to then combine
the four indexes into a single query interface.

\subsection{Focus on Organizational Knowledge}

Currently, research is focused on providing schemas, storage backends
and indexes for the four different kinds of mathematical
knowledge. The focus of \emph{ulo-storage} is the area of
organizational knowledge.

A previously proposed way to structure such organizational data is the
\emph{upper level ontology} (ULO)~\cite{ulo}. ULO takes the form of an
OWL~ontology~\cite{uloonto} and as such all organization information
is stored as RDF~triplets with a unified schema of
ULO~predicates~\cite{owl}.  Some effort has been made to export
existing databases of formal mathematical knowledge to {ULO}. In
particular, there exist exports from Isabelle and Coq
libraries~\cite{uloisabelle, ulocoq}. The resulting data set is
already quite large, the Isabelle export alone containing more than
200~million triplets.

Existing exports from Isabelle and Coq result in single or multiple
RDF~files. This is a convenient format for exchange and easily
versioned using Git. However, considering the vast number of triplets,
it is impossible to query easily and efficiently in this state. This
is what \emph{ulo-storage} is focused on: Making ULO data sets
accessible for querying and analysis. We collected RDF files spread
over different Git repositories, imported them into a database and
then experimented with APIs for accessing that data set.

The main contribution of this project is twofold. First, (1) we built
up various infrastructure components that can make up building blocks
in a larger tetrapodal search system. Second, (2)~we ran sample
prototype applications and queries on top of this interface. While the
applications themselves are admittedly not very interesting, they can give
us insight about future development of~{ULO}.

\subsection{Components Implemented for \emph{ulo-storage}}\label{sec:components}

With RDF files exported and available for download as Git repositories
on MathHub, we have the goal of making the underlying data available
for use in applications.  It makes sense to first identify the various
components that might be involved in such a system. Figure~\ref{fig:components}
illustrates all components and their relationships.

\begin{figure}[]\begin{center}
    \includegraphics{figs/components}
    \caption{Components involved in the \emph{ulo-storage} system.}\label{fig:components}
\end{center}\end{figure}

\begin{description}
\item[Collecter] ULO triplets are present in various locations, be it Git
  repositories, available on web servers or on local disk.
  It is the job of a Collecter to assemble these {RDF}~files and
  forward them for further processing. This may involve cloning a Git
  repository or crawling the file system.

\item[Importer] With streams of ULO files assembled by the Collecter, this
  data then gets passed to an Importer. An Importer uploads
  received RDF~streams into some kind of permanent storage. For
  use in this project, the GraphDB~\cite{graphdb} triplet store was
  a natural fit.

  In this project, both Collecter and Importer ended up being one piece
  of software, but this does not have to be the case.

\item[Endpoint] Finally, with all triplets stored in a database, an
  Endpoint is where applications access the underlying
  knowledge base. This does not necessarily need to be any custom
  software, rather the programming interface of the underlying
  database itself could be understood as an endpoint of its
  own.

  Regardless, some thought can be put into designing an Endpoint as a
  layer that lives between application and database that is more
  convenient to use than the one provided by the database.
\end{description}

\subsection{An Additional Harvester Component}

These are the components realized for \emph{ulo-storage}. However,
additionally to these components, one could think of a
\emph{Harvester} component.  We assumed that the ULO triplets are
already available in RDF~format.  This is not necessarily true.  It
might be desirable to automate the export from third party formats to
ULO and we think this should be the job of a Harvester component.  It
fetches mathematical knowledge from some remote source and then
provides a volatile stream of ULO data to the Collecter, which then
passes it to the Importer and so on. The big advantage of such an
approach would be that exports from third party libraries can always
be up to date and do not have to be initiated manually.

We did not implement a Harvester for \emph{ulo-storage} but we suggest
that it is an idea to keep in mind. The components we did implement
(Collecter, Importer and Endpoint) provide us with an easy and
automated way of making RDF files ready for use with applications.  In
this introduction we only wanted to give the reader a general
understanding in the infrastructure that makes up \emph{ulo-storage};
the following sections will explain each component in more detail.
