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

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 eveloped conversion tool (referred to as LaCASt) to translate formulae from the NIST Digital Library of Mathematical Functions to the computer algebra systems Maple and Mathematica. 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 Maple and Mathematica of the formulae in the NIST Digital Library of Mathematical Functions

Simple algorithm for judging equivalence of differential-algebraic equation systems

Mathematical formulas play a prominent role in science, technology, engineering, and mathematics (STEM) documents; understanding STEM documents usually requires knowing the difference between equation groups containing multiple equations. When two equation groups can be transformed into the same form, we call the equation groups equivalent. Existing tools cannot judge the equivalence of two equation groups; thus, we develop an algorithm to judge such an equivalence using a computer algebra system. The proposed algorithm first eliminates variables appearing only in either equation group. It then checks the equivalence of the equations one by one: the equations with identical algebraic solutions for the same variable are judged equivalent. If each equation in one equation group is equivalent to an equation in the other, the equation groups are judged equivalent; otherwise, non-equivalent. We generated 50 pairs of equation groups for evaluation. The proposed method accurately judged the equivalence of all pairs. This method is expected to facilitate comprehension of a large amount of mathematical information in STEM documents. Furthermore, this is a necessary step for machines to understand equations, including process models

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

Formalizing graphical notations

The thesis describes research into graphical notations for software engineering, with a principal interest in ways of formalizing them. The research seeks to provide a theoretical basis that will help in designing both notations and the software tools that process them. The work starts from a survey of literature on notation, followed by a review of techniques for formal description and for computational handling of notations. The survey concentrates on collecting views of the benefits and the problems attending notation use in software development; the review covers picture description languages, grammars and tools such as generic editors and visual programming environments. The main problem of notation is found to be a lack of any coherent, rigorous description methods. The current approaches to this problem are analysed as lacking in consensus on syntax specification and also lacking a clear focus on a defined concept of notated expression. To address these deficiencies, the thesis embarks upon an exploration of serniotic, linguistic and logical theory; this culminates in a proposed formalization of serniosis in notations, using categorial model theory as a mathematical foundation. An argument about the structure of sign systems leads to an analysis of notation into a layered system of tractable theories, spanning the gap between expressive pictorial medium and subject domain. This notion of 'tectonic' theory aims to treat both diagrams and formulae together. The research gives details of how syntactic structure can be sketched in a mathematical sense, with examples applying to software development diagrams, offering a new solution to the problem of notation specification. Based on these methods, the thesis discusses directions for resolving the harder problems of supporting notation design, processing and computer-aided generic editing. A number of future research areas are thereby opened up. For practical trial of the ideas, the work proceeds to the development and partial implementation of a system to aid the design of notations and editors. Finally the thesis is evaluated as a contribution to theory in an area which has not attracted a standard approach

Institutional approaches to programming language specification

Formal specification has become increasingly important in software engineering, both as a design tool, and as a basis for verified software design. Formal methods have long been in use in the field of programming language design and implementation, and many formalisms, in both the syntactic and semantic domains, have evolved for this purpose. In this thesis we examine the possibilities of integrating specifications written in different formalisms used in the description of programming languages within a single framework. We suggest that the theory of institutions provides a suitable background for such integration, and we develop descriptions of several formalisms within this framework. While we do not merge the formalisms themselves, we see that it is possible to relate modules from specifications in each of them, and this is demonstrated in a small example

Emerging trends proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics: TPHOLs 2004

technical reportThis volume constitutes the proceedings of the Emerging Trends track of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004) held September 14-17, 2004 in Park City, Utah, USA. The TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research cate- gory, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the con- ference and publication in volume 3223 of Springer?s Lecture Notes in Computer Science series. In keeping with longstanding tradition, TPHOLs 2004 also offered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief introductory talk and then discuss their work at a poster session. The work-in-progress papers are held in this volume, which is published as a 2004 technical report of the School of Computing at the University of Utah

Canonical queries as a query answering device (Information Science)

Issued as Annual reports [nos. 1-2], and Final report, Project no. G-36-60