Identifier Namespaces in Mathematical Notation

In Computer Science, namespaces help to structure source code and organize it into hierarchies. Initially, the concept of namespaces did not exist for programming languages, and programmers had to manage the source code themselves to ensure there were no name conflicts. However, nowadays, namespaces are adopted by the majority of modern programming languages. The concept of namespaces is beneficial for mathematics as well: In mathematics, short one-symbol identifiers are very common, and the meaning of these identifiers is hard to understand immediately. By introducing namespaces to mathematics, we will be able to organize mathematical identifiers. Also, mathematicians will also benefit from a hierarchical organization of knowledge in the same way programmers do. In addition, the structure will make it easier to understand the meaning of each identifier in a document. 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. Namespaces are hard to prepare manually: building them requires a lot of time and effort. However, it can be done automatically, and we propose a method for automatic namespace discovery from a collection of documents. 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. To evaluate the results, we use the category information, and look for pure “namespace-defining” clusters: clusters where all documents are from the same category. In the experiments, we look for algorithms that discover as many namespace-defining clusters as possible. 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