Identifier Namespaces in Mathematical Notation

TL;DR

This paper proposes a novel method for automatically discovering identifier namespaces in mathematical notation by applying document clustering techniques to group identifiers, validated on source code and Wikipedia datasets.

Contribution

It introduces the first dataset and approach for automatic namespace discovery in mathematical notation, adapting document clustering methods to this new problem.

Findings

Partial recovery of namespaces from source code using identifiers

Effective extraction of namespaces from Wikipedia articles across languages

Hierarchical organization of namespaces using existing classification schemes

Abstract

In this thesis, we look at the problem of assigning each identifier of a document to a namespace. At the moment, there does not exist a special dataset where all identifiers are grouped to namespaces, and therefore we need to create such a dataset ourselves. To do that, we need to find groups of documents that use identifiers in the same way. This can be done with cluster analysis methods. We argue that documents can be represented by the identifiers they contain, and this approach is similar to representing textual information in the Vector Space Model. Because of this, we can apply traditional document clustering techniques for namespace discovery. Because the problem is new, there is no gold standard dataset, and it is hard to evaluate the performance of our method. To overcome it, we first use Java source code as a dataset for our experiments, since it contains the namespace information. We verify that our method can partially recover namespaces from source code using only information about identifiers. The algorithms are evaluated on the English Wikipedia, and the proposed method can extract namespaces on a variety of topics. After extraction, the namespaces are organized into a hierarchical structure by using existing classification schemes such as MSC, PACS and ACM. We also apply it to the Russian Wikipedia, and the results are consistent across the languages. To our knowledge, the problem of introducing namespaces to mathematics has not been studied before, and prior to our work there has been no dataset where identifiers are grouped into namespaces. Thus, our result is not only a good start, but also a good indicator that automatic namespace discovery is possible.