Semantic Preserving Bijective Mappings of Mathematical Formulae Between Document Preparation Systems and Computer Algebra Systems

Cohl, Howard S.; Schubotz, Moritz; Youssef, Abdou; Greiner-Petter, André; Gerhard, Jürgen; Saunders, Bonita V.; McClain, Marjorie A.; Bang, Joon Seok; Chen, Kevin

doi:10.1007/978-3-319-62075-6_9

Cited by 10 publications

(17 citation statements)

References 7 publications

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“…For example, the translation (2) is direct, while a translation to the definition of the cosine function \cos@{z} M aple → (exp(I*z)+exp(-I*z))/2 is not a direct translation even though it is appropriate. Note that partial results of this paper have been published in (Cohl, Schubotz, Youssef, et al, 2017).…”

Section: For Example a Translation Such Asmentioning

confidence: 99%

Semantic preserving bijective mappings for expressions involving special functions between computer algebra systems and document preparation systems

Greiner-Petter

Schubotz

Cohl

et al. 2019

AJIM

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Purpose: Modern mathematicians and scientists of math-related disciplines often use Document Preparation Systems (DPS) to write and Computer Algebra Systems (CAS) to calculate mathematical expressions. Usually, they translate the expressions manually between DPS and CAS. This process is time-consuming and error-prone. Our goal is to automate this translation. This paper uses Maple and Mathematica as the CAS, and L A T E X as our DPS. Approach: Bruce Miller at the National Institute of Standards and Technology (NIST) developed a collection of special L A T E X macros that create links from mathematical symbols to their definitions in the NIST Digital Library of Mathematical Functions (DLMF). We are using these macros to perform rule-based translations between the formulae in the DLMF and CAS. Moreover, we develop software to ease the creation of new rules and to discover inconsistencies. Findings: We created 396 mappings and translated 58.8% of DLMF formulae (2,405 expressions) successfully between Maple and DLMF. For a significant percentage, the special function definitions in Maple and the DLMF were different. Therefore, an atomic symbol in one system maps to a composite expression in the other system. The translator was also successfully used for automatic verification of mathematical online compendia and CAS. Our evaluation techniques discovered two errors in the DLMF and one defect in Maple. Originality: This paper introduces the first translation tool for special functions between L A T E X and CAS. The approach improves error-prone manual translations and can be used to verify mathematical online compendia and CAS.

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Section: For Example a Translation Such Asmentioning

confidence: 99%

Semantic preserving bijective mappings for expressions involving special functions between computer algebra systems and document preparation systems

Greiner-Petter

Schubotz

Cohl

et al. 2019

AJIM

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“…Since there is no generally accepted definition of expression trees, we made several design decision to create semantic representations of the formulae in our dataset using MathML trees. In some cases, we created new macros to be able to create a MathML tree for our purposes using LaTeXML 5 . Table 1 lists the newly created macros.…”

Section: Gold Standardmentioning

confidence: 99%

“…To account for this circumstance, CAS typically restrict the set of permissible outputs by cutting the complex plane of permissible outputs. However, since the method of restricting the set of permissible outputs varies between systems, identical inputs can lead to drastically different results [5]. For example, multiple scientific publications address the problem of accounting for branch cuts when entering expressions in CAS, such as [8] for MAPLE.…”

Section: Background and Related Workmentioning

confidence: 99%

Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context

Schubotz

Greiner-Petter

Scharpf

et al. 2018

Proceedings of the 18th ACM/IEEE on Joint Conference on Digital Libraries

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Mathematical formulae represent complex semantic information in a concise form. Especially in Science, Technology, Engineering, and Mathematics, mathematical formulae are crucial to communicate information, e.g., in scientific papers, and to perform computations using computer algebra systems. Enabling computers to access the information encoded in mathematical formulae requires machine-readable formats that can represent both the presentation and content, i.e., the semantics, of formulae. Exchanging such information between systems additionally requires conversion methods for mathematical representation formats. We analyze how the semantic enrichment of formulae improves the format conversion process and show that considering the textual context of formulae reduces the error rate of such conversions. Our main contributions are: (1) providing an openly available benchmark dataset for the mathematical format conversion task consisting of a newly created test collection, an extensive, manually curated gold standard and task-specific evaluation metrics; (2) performing a quantitative evaluation of state-of-the-art tools for mathematical format conversions; (3) presenting a new approach that considers the textual context of formulae to reduce the error rate for mathematical format conversions. Our benchmark dataset facilitates future research on mathematical format conversions as well as research on many problems in mathematical information retrieval. Because we annotated and linked all components of formulae, e.g., identifiers, operators and other entities, to Wikidata entries, the gold standard can, for instance, be used to train methods for formula concept discovery and recognition. Such methods can then be applied to improve mathematical information retrieval systems, e.g., for semantic formula search, recommendation of mathematical content, or detection of mathematical plagiarism.

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“…In this paper, we describe the process of extracting mathematical formulae from the DLMF, and then take advantage of the powerful mathematical semantic coverage of semantic L A T E X used in the DLMF to convert to CAS representations whose sole purpose is to drive an automated scheme for verification of DLMF formulae. In order to improve the precision of our DLMF formulae testing reported on in [5], we did the following: (1) we updated our test formulae dataset; (2) fixed numerous translation bugs; (3) implemented a new method for handling constraints; and (4) added configuration files which make it easier to change the setup of our symbolic and numerical testing.…”

Section: Problem and Current Statementioning

confidence: 99%

“…In fact, the macro names used in this manuscript, have been more recently updated. However, the macro set we are utilizing, is for those in usage as of 9/16/2016 5. These semantic macros represent the complete and complementary elliptic integrals of the first kind [7, (19.2.8-9)] 6 These semantic macros represent the mathematical constants i, e and π [7, (1.9.1), (4.2.11), (3.12.1)].…”

mentioning

confidence: 99%

Automated Symbolic and Numerical Testing of DLMF Formulae Using Computer Algebra Systems

Cohl¹,

Greiner-Petter²,

Schubotz³

2018

Lecture Notes in Computer Science

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We have developed an automated procedure for symbolic and numerical testing of formulae extracted from the NIST Digital Library of Mathematical Functions (DLMF). For the NIST Digital Repository of Mathematical Formulae, we have developed conversion tools from semantic L A T E X to the Computer Algebra System (CAS) MAPLE which relies on Youssef's part-of-math tagger. We convert a test data subset of 4,078 semantic L A T E X DLMF formulae to the native CAS representation and then apply an automated scheme for symbolic and numerical testing and verification. Our framework is implemented using Java and MAPLE. We describe in detail the conversion process which is required so that the CAS can correctly interpret the mathematical representation of the formulae. We describe the improvement of the effectiveness of our automated scheme through incremental enhancement (making more precise) of the mathematical semantic markup for the formulae.

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Semantic Preserving Bijective Mappings of Mathematical Formulae Between Document Preparation Systems and Computer Algebra Systems

Cited by 10 publications

References 7 publications

Semantic preserving bijective mappings for expressions involving special functions between computer algebra systems and document preparation systems

Semantic preserving bijective mappings for expressions involving special functions between computer algebra systems and document preparation systems

Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context

Automated Symbolic and Numerical Testing of DLMF Formulae Using Computer Algebra Systems

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