Open Access

Matthias Bröcheler

• The body of mathematical knowledge is rapidly increasing and constantly changing. Zentralblatt MATH, an abstracting and reviewing service in the field of mathematics, maintains a database of more than 1.6 million mathematical documents and reports an annual growth by 80,000 articles. Similarly, the open internet archive for for electronic preprints of scientific papers, arXiv.org, contains close to half a million documents. These figures suggest that neither a mathematician's memory nor the time she devotes to studying new publications can possibly suffice to cover a significant fraction of the accumulated wealth of mathematical knowledge. In addition to the exponential increase in the amount of information, one also observes an increase in the complexity of mathematical content with more interdependencies between different areas within and beyond mathematics. We propose a Mathematical Semantic Web to support mathematicians in efficiently managing and retrieving mathematical knowledge using computer systems and the internet. As with the World Wide Web, the Mathematical Semantic Web will allow authors of mathematics to publish their documents online which cumulate to a gigantic, decentralized and dynamic mathematical knowledge base. Authors semantically annotate their work in a special logical formalism, namely Description Logics, to allow computers to understand the actual knowledge contained therein. Based on these annotations, computer agents reason about the mathematical knowledge and provide novel services on the Mathematical Semantic Web, giving mathematicians efficient access to vast repositories of mathematics. This thesis first analyzes the utility of Description Logics for formalizing mathematical knowledge. This analysis concludes with the proposal of a Mathematical Semantic Web which is introduced in great detail subsequently. We elaborate on the architecture and individual building blocks before describing the authoring process for the Mathematical Semantic Web. To motivate the value of our proposal, some services operating on the Mathematical Semantic Web are specified. In an effort to improve knowledge retrieval even further, we introduce the combination of domain and structural semantics for reasoning processes, thereby effectively leveraging additional knowledge. Finally, we provide a list of Best Practices which aim at simplifying the knowledge modeling process and improving the quality of the resulting knowledge base. The Best Practices have been distilled from the experience gained during our case studies.