Making Presentation Math Computable

This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book

Making Presentation Math Computable

This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book

The Case of Wikidata

Since its launch in 2012, Wikidata has grown to become the largest open knowledge base (KB), containing more than 100 million data items and over 6 million registered users. Wikidata serves as the structured data backbone of Wikipedia, addressing data inconsistencies, and adhering to the motto of “serving anyone anywhere in the world,” a vision realized through the diversity of knowledge. Despite being a collaboratively contributed platform, the Wikidata community heavily relies on bots, automated accounts with batch, and speedy editing rights, for a majority of edits. As Wikidata approaches its first decade, the question arises: How close is Wikidata to achieving its vision of becoming a global KB and how diverse is it in serving the global population? This dissertation investigates the current status of Wikidata’s diversity, the role of bot interventions on diversity, and how bots can be leveraged to improve diversity within the context of Wikidata. The methodologies used in this study are mapping study and content analysis, which led to the development of three datasets: 1) Wikidata Research Articles Dataset, covering the literature on Wikidata from its first decade of existence sourced from online databases to inspect its current status; 2) Wikidata Requests-for-Permissions Dataset, based on the pages requesting bot rights on the Wikidata website to explore bots from a community perspective; and 3) Wikidata Revision History Dataset, compiled from the edit history of Wikidata to investigate bot editing behavior and its impact on diversity, all of which are freely available online. The insights gained from the mapping study reveal the growing popularity of Wikidata in the research community and its various application areas, indicative of its progress toward the ultimate goal of reaching the global community. However, there is currently no research addressing the topic of diversity in Wikidata, which could shed light on its capacity to serve a diverse global population. To address this gap, this dissertation proposes a diversity measurement concept that defines diversity in a KB context in terms of variety, balance, and disparity and is capable of assessing diversity in a KB from two main angles: user and data. The application of this concept on the domains and classes of the Wikidata Revision History Dataset exposes imbalanced content distribution across Wikidata domains, which indicates low data diversity in Wikidata domains. Further analysis discloses that bots have been active since the inception of Wikidata, and the community embraces their involvement in content editing tasks, often importing data from Wikipedia, which shows a low diversity of sources in bot edits. Bots and human users engage in similar editing tasks but exhibit distinct editing patterns. The findings of this thesis confirm that bots possess the potential to influence diversity within Wikidata by contributing substantial amounts of data to specific classes and domains, leading to an imbalance. However, this potential can also be harnessed to enhance coverage in classes with limited content and restore balance, thus improving diversity. Hence, this study proposes to enhance diversity through automation and demonstrate the practical implementation of the recommendations using a specific use case. In essence, this research enhances our understanding of diversity in relation to a KB, elucidates the influence of automation on data diversity, and sheds light on diversity improvement within a KB context through the usage of automation.Seit seiner Einführung im Jahr 2012 hat sich Wikidata zu der größten offenen Wissensdatenbank entwickelt, die mehr als 100 Millionen Datenelemente und über 6 Millionen registrierte Benutzer enthält. Wikidata dient als das strukturierte Rückgrat von Wikipedia, indem es Datenunstimmigkeiten angeht und sich dem Motto verschrieben hat, ’jedem überall auf der Welt zu dienen’, eine Vision, die durch die Diversität des Wissens verwirklicht wird. Trotz seiner kooperativen Natur ist die Wikidata-Community in hohem Maße auf Bots, automatisierte Konten mit Batch- Verarbeitung und schnelle Bearbeitungsrechte angewiesen, um die Mehrheit der Bearbeitungen durchzuführen. Da Wikidata seinem ersten Jahrzehnt entgegengeht, stellt sich die Frage: Wie nahe ist Wikidata daran, seine Vision, eine globale Wissensdatenbank zu werden, zu verwirklichen, und wie ausgeprägt ist seine Dienstleistung für die globale Bevölkerung? Diese Dissertation untersucht den aktuellen Status der Diversität von Wikidata, die Rolle von Bot-Eingriffen in Bezug auf Diversität und wie Bots im Kontext von Wikidata zur Verbesserung der Diversität genutzt werden können. Die in dieser Studie verwendeten Methoden sind Mapping-Studie und Inhaltsanalyse, die zur Entwicklung von drei Datensätzen geführt haben: 1) Wikidata Research Articles Dataset, die die Literatur zu Wikidata aus dem ersten Jahrzehnt aus Online-Datenbanken umfasst, um den aktuellen Stand zu untersuchen; 2) Requestfor- Permission Dataset, der auf den Seiten zur Beantragung von Bot-Rechten auf der Wikidata-Website basiert, um Bots aus der Perspektive der Gemeinschaft zu untersuchen; und 3)Wikidata Revision History Dataset, der aus der Bearbeitungshistorie von Wikidata zusammengestellt wurde, um das Bearbeitungsverhalten von Bots zu untersuchen und dessen Auswirkungen auf die Diversität, die alle online frei verfügbar sind. Die Erkenntnisse aus der Mapping-Studie zeigen die wachsende Beliebtheit von Wikidata in der Forschungsgemeinschaft und in verschiedenen Anwendungsbereichen, was auf seinen Fortschritt hin zur letztendlichen Zielsetzung hindeutet, die globale Gemeinschaft zu erreichen. Es gibt jedoch derzeit keine Forschung, die sich mit dem Thema der Diversität in Wikidata befasst und Licht auf seine Fähigkeit werfen könnte, eine vielfältige globale Bevölkerung zu bedienen. Um diese Lücke zu schließen, schlägt diese Dissertation ein Konzept zur Messung der Diversität vor, das die Diversität im Kontext einer Wissensbasis anhand von Vielfalt, Balance und Diskrepanz definiert und in der Lage ist, die Diversität aus zwei Hauptperspektiven zu bewerten: Benutzer und Daten. Die Anwendung dieses Konzepts auf die Bereiche und Klassen des Wikidata Revision History Dataset zeigt eine unausgewogene Verteilung des Inhalts über die Bereiche von Wikidata auf, was auf eine geringe Diversität der Daten in den Bereichen von Wikidata hinweist. Weitere Analysen zeigen, dass Bots seit der Gründung von Wikidata aktiv waren und von der Gemeinschaft inhaltliche Bearbeitungsaufgaben angenommen werden, oft mit Datenimporten aus Wikipedia, was auf eine geringe Diversität der Quellen bei Bot-Bearbeitungen hinweist. Bots und menschliche Benutzer führen ähnliche Bearbeitungsaufgaben aus, zeigen jedoch unterschiedliche Bearbeitungsmuster. Die Ergebnisse dieser Dissertation bestätigen, dass Bots das Potenzial haben, die Diversität in Wikidata zu beeinflussen, indem sie bedeutende Datenmengen zu bestimmten Klassen und Bereichen beitragen, was zu einer Ungleichgewichtung führt. Dieses Potenzial kann jedoch auch genutzt werden, um die Abdeckung in Klassen mit begrenztem Inhalt zu verbessern und das Gleichgewicht wiederherzustellen, um die Diversität zu verbessern. Daher schlägt diese Studie vor, die Diversität durch Automatisierung zu verbessern und die praktische Umsetzung der Empfehlungen anhand eines spezifischen Anwendungsfalls zu demonstrieren. Kurz gesagt trägt diese Forschung dazu bei, unser Verständnis der Diversität im Kontext einer Wissensbasis zu vertiefen, wirft Licht auf den Einfluss von Automatisierung auf die Diversität von Daten und zeigt die Verbesserung der Diversität im Kontext einer Wissensbasis durch die Verwendung von Automatisierung auf

Leveraging Formulae and Text for Improved Math Retrieval

Large collections containing millions of math formulas are available online. Retrieving math expressions from these collections is challenging. Users can use formula, formula+text, or math questions to express their math information needs. The structural complexity of formulas requires specialized processing. Despite the existence of math search systems and online community question-answering websites for math, little is known about mathematical information needs. This research first explores the characteristics of math searches using a general search engine. The findings show how math searches are different from general searches. Then, test collections for math-aware search are introduced. The ARQMath test collections have two main tasks: 1) finding answers for math questions and 2) contextual formula search. In each test collection (ARQMath-1 to -3) the same collection is used, Math Stack Exchange posts from 2010 to 2018, introducing different topics for each task. Compared to the previous test collections, ARQMath has a much larger number of diverse topics, and improved evaluation protocol. Another key role of this research is to leverage text and math information for improved math information retrieval. Three formula search models that only use the formula, with no context are introduced. The first model is an n-gram embedding model using both symbol layout tree and operator tree representations. The second model uses tree-edit distance to re-rank the results from the first model. Finally, a learning-to-rank model that leverages full-tree, sub-tree, and vector similarity scores is introduced. To use context, Math Abstract Meaning Representation (MathAMR) is introduced, which generalizes AMR trees to include math formula operations and arguments. This MathAMR is then used for contextualized formula search using a fine-tuned Sentence-BERT model. The experiments show tree-edit distance ranking achieves the current state-of-the-art results on contextual formula search task, and the MathAMR model can be beneficial for re-ranking. This research also addresses the answer retrieval task, introducing a two-step retrieval model in which similar questions are first found and then answers previously given to those similar questions are ranked. The proposed model, fine-tunes two Sentence-BERT models, one for finding similar questions and another one for ranking the answers. For Sentence-BERT model, raw text as well as MathAMR are used

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4