A Standard for Aligning Mathematical Concepts

Mathematical knowledge is publicly available in dozens of different formats and languages, ranging from informal (e.g. Wikipedia) to formal corpora (e.g., Mizar). Despite an enormous amount of overlap between these corpora, few machine-actionable connections exist. We speak of alignment if the same concept occurs in different libraries, possibly with slightly different names, notations, or formal definitions. Leveraging these alignments would create a huge potential for knowledge sharing and transfer, e.g., integrating theorem provers or reusing services across systems. Formally describing and verifying alignments is extremely expensive, as it typically requires not only aligning two concepts but whole libraries together with their foundations, or may even be impossible in case of alignments between semi-formal or informal concepts. Therefore, we introduce a lightweight approach that focuses on identifying the alignments while abstracting from formal definitions. Notably, this is already sufficient for many practically valuable applications. We present a classification of alignments and design a simple format for describing alignments as well as an infrastructure for sharing them. We propose these as a centralized standard for the community to collect and curate alignments from the different kinds of mathematical corpora, including proof assistant libraries, computer algebra and programming language algorithms, and semi-formal libraries.