OpenMath Technology for Interactive Mathematical Documents

Abstract. New technologies such as xml, xsl and both MathML and Open-Math make it possible to bring mathematics to the Internet. Indeed, Open-Math, a markup language for mathematical content, and OmDoc, its extension to mathematical documents, open a way of communicating mathematics between computers, between software applications and over the Internet without losing information. In this paper we describe the latest applications of OpenMath related technologies for Interactive Mathematical Documents. As an example we describe the way we incorporate these new technologies in a new version of Algebra Interactive, an interactive course on first and second year university algebra.

Linking HOL Light to Mathematica using OpenMath

One of the most important benefits of using a theorem prover system is the absolute accuracy of the obtained result. However, solving mathematical problems often requires both deductive reasoning and algebraic computation. This issue is due to the fact that many real-life problems can be described with equations for which we cannot find easily symbolic (or closed-form) solutions and therefore we are not able to formalize them using the theorem prover. In other cases, some applications require well developed libraries and a deep knowledge of the theories to formalize simple expressions. A straightforward way to overcome these issues is the use of computer algebra systems or numerical approaches which are known to be the most efficient tools in symbolic computation. However, to preserve the soundness of the computation, the results of these systems should be formally verified. In this thesis, we present a general architecture to connect HOL Light, a higher-order logic theorem prover, to any mechanized mathematical system that supports the mathematical standard OpenMath. We implemented a prototype, called HolMatica, which links HOL Light to the computer algebra system Mathematica through OpenMath. We describe our implementation of a HOL Light translator which converts HOL Light statements into OpenMath object and vice-versa

Dagstuhl News January - December 2001

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Gradual computerisation and verification of mathematics : MathLang's path into Mizar

There are many proof checking tools that allow capturing mathematical knowledge into formal representation. Those proof systems allow further automatic verifica- tion of the logical correctness of the captured knowledge. However, the process of encoding common mathematical documents in a chosen proof system is still labour- intensive and requires comprehensive knowledge of such system. This makes the use of proof checking tools inaccessible for ordinary mathematicians. This thesis provides a solution for the computerisation of mathematical documents via a num- ber of gradual steps using the MathLang framework. We express the full process of formalisation into the Mizar proof checker. The first levels of such gradual computerisation path have been developing well before the course of this PhD started. The whole project, called MathLang, dates back to 2000 when F. Kamareddine and J.B. Wells started expressing their ideas of novel approach for computerising mathematical texts. They mainly aimed at developing a mathematical framework which is flexible enough to connect existing, in many cases different, approaches of computerisation mathematics, which allows various degrees of formalisation (e.g., partial, full formalisation of chosen parts, or full formalisation of the entire doc- ument), which is compatible with different mathematical foundations (e.g., type theory, set theory, category theory, etc.) and proof systems (e.g., Mizar, Isar, Coq, HOL, Vampire). The first two steps in the gradual formalisation were developed by F. Kamareddine, J.B. Wells and M. Maarek with a small contribution of R. Lamar to the second step. In this thesis we develop the third level of the gradual path, which aims at capturing the rhetorical structure of mathematical documents. We have also integrated further steps of the gradual formalisation, whose final goal is the Mizar system. We present in this thesis a full path of computerisation and formalisation of math- ematical documents into the Mizar proof checker using the MathLang framework. The development of this method was driven by the experience of computerising a number of mathematical documents (covering different authoring styles)

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