Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems

Abstract

Digital mathematical libraries assemble the knowledge of years of mathematical research. Numerous disciplines (e.g., physics, engineering, pure and applied mathematics) rely heavily on compendia gathered findings. Likewise, modern research applications rely more and more on computational solutions, which are often calculated and verified by computer algebra systems. Hence, the correctness, accuracy, and reliability of both digital mathematical libraries and computer algebra systems is a crucial attribute for modern research. In this paper, we present a novel approach to verify a digital mathematical library and two computer algebra systems with one another by converting mathematical expressions from one system to the other. We use our previously developed conversion tool (referred to as "Image missing" ) to translate formulae from the NIST Digital Library of Mathematical Functions to the computer algebra systems and . The contributions of our presented work are as follows: (1) we present the most comprehensive verification of computer algebra systems and digital mathematical libraries with one another; (2) we significantly enhance the performance of the underlying translator in terms of coverage and accuracy; and (3) we provide open access to translations for and of the formulae in the NIST Digital Library of Mathematical Functions.

Do the Math: Making Mathematics in Wikipedia Computable

“…The results t s (e, X) are in semantic L A T E X which is in L C . For the second step (the mapping), we rely on the original L A CAST implementation (from semantic L A T E X to CAS syntaxes) for t m (e) and presume that t m (e) is complete and appropriate [7], [8].…”

“…These steps are: (1) pre-processing Wikipedia articles to enable natural language processing on it, (2) constructing an annotated mathematical dependency graph, (3) generating semantic enhancing replacement patterns, and (4) performing CASspecific translations (see Figure 2). In addition, we perform automatic symbolic and numeric computations on the translated expressions to verify equations from Wikipedia articles [7], [8]. We show that the system is capable of detecting potential errors in mathematical equations in Wikipedia articles.…”

“…Our proposed pipeline tangents several well-known tasks from MathIR, namely descriptive entity recognition for mathematical expressions [15], [17]- [20], math tokenization [21], [22], math dependency recognition [23], [24], and automatic verification [7], [8]. Existing approaches to translate mathematical formulae from presentational languages, e.g., L A T E X or MathML, to content languages, e.g., content MathML or CAS syntax, do not analyze the context of a formula [24]- [26].…”

“…CAS, such as Maple [5] and Mathematica [6], are complex mathematical software tools that allow users to manipulate, simplify, plot, and evaluate mathematical expressions. Hence, translating mathematics in Wikipedia to CAS syntaxes enables automatic numeric and symbolic verification checks on complex mathematical equations [7], [8]. Integrating such verifications into the existing ORES system can significantly reduce the overload of moderating mathematical content and increasing credibility in the quality of Wikipedia articles at the same time [9].…”

Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems

“…Another feature we added to L A CAST is the support of packages in Maple. Some functions are only available in modules (packages) that must be preloaded, such as QPochhammer in the package QDifferenceEquations 16 . The general simplify method in Maple does not cover q-hypergeometric functions.…”