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

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Intelligent Computer Mathematics (CICM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11006))

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Abstract

We have developed an automated procedure for symbolic and numerical testing of formulae extracted from the National Institute of Standards and Technology (NIST) Digital Library of Mathematical Functions (DLMF). For the NIST Digital Repository of Mathematical Formulae, we have developed conversion tools from semantic 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 DLMF formulae extracted from the DLMF 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 is able to 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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Notes

  1. 1.

    The mention of specific products, trademarks, or brand names is for purposes of identification only. Such mention is not to be interpreted in any way as an endorsement or certification of such products or brands by the National Institute of Standards and Technology, nor does it imply that the products so identified are necessarily the best available for the purpose. All trademarks mentioned herein belong to their respective owners.

  2. 2.

    The macro is just one of many (currently unpublished) semantic DLMF semantic macros which have been developed by Bruce Miller for utilization in the DLMF. 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.

  3. 3.

    These semantic macros represent the complete and complementary elliptic integrals of the first kind [1, (19.2.8-9)].

  4. 4.

    These semantic macros represent the mathematical constants i, e and \(\pi \) [1, (1.9.1), (4.2.11), (3.12.1)].

  5. 5.

    A name in MAPLE is a sequence of one or more characters that uniquely identifies a command, file, variable, or other entity.

  6. 6.

    This dataset is available on request to the authors.

References

  1. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.) NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/. Release 1.0.19 of 2018-06-22

  2. Miller, B.R.: LaTeXML: A LaTeX to XML/HTML/MathML Converter, June 2018. http://dlmf.nist.gov/LaTeXML/

  3. Miller, B.R., Youssef, A.: Technical aspects of the Digital Library of Mathematical Functions. Ann. Math. Artif. Intell. 38(1–3), 121–136 (2003)

    Article  MathSciNet  Google Scholar 

  4. Schubotz, M., Greiner-Petter, A., Scharpf, P., Meuschke, N., Cohl, H.S., Gipp, B.: Improving the representation and conversion of mathematical formulae by considering their textual context. In: Joint Conference on Digital Libraries 2018, June 2018, Fort Worth, TX (2018)

    Google Scholar 

  5. Youssef, A.: Part-of-math tagging and applications. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds.) CICM 2017. LNCS (LNAI), vol. 10383, pp. 356–374. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62075-6_25

    Chapter  Google Scholar 

  6. Cohl, H.S., et al.: Semantic preserving bijective mappings of mathematical formulae between document preparation systems and computer algebra systems. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds.) CICM 2017. LNCS (LNAI), vol. 10383, pp. 115–131. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62075-6_9

    Chapter  Google Scholar 

  7. Schubotz, M., Meuschke, N., Hepp, T., Cohl, H.S., Gipp, B.: VMEXT: a visualization tool for mathematical expression trees. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds.) CICM 2017. LNCS (LNAI), vol. 10383, pp. 340–355. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62075-6_24

    Chapter  Google Scholar 

  8. Cohl, H.S., et al.: Growing the digital repository of mathematical formulae with generic LaTeX sources. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS (LNAI), vol. 9150, pp. 280–287. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20615-8_18

    Chapter  Google Scholar 

  9. Koepf, W.: On two conjectures of M. S. Robertson. Complex Variables. Theory Appl. Int. J. 16(2–3), 127–130 (1991)

    Article  MathSciNet  Google Scholar 

  10. Ballarin, C., Homann, K., Calmet, J.: Theorems and algorithms: an interface between Isabelle and Maple. In: Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, ISSAC 1995, pp. 150–157. ACM, New York (1995)

    Google Scholar 

  11. Harrison, J., Théry, L.: A skeptic’s approach to combining HOL and Maple. J. Autom. Reason. 21(3), 279–294 (1998)

    Article  MathSciNet  Google Scholar 

  12. Bailey, D.H., Borwein, J.M., Kaiser, A.D.: Automated simplification of large symbolic expressions. J. Symbolic Comput. 60, 120–136 (2014)

    Article  MathSciNet  Google Scholar 

  13. Bernardin, L., Chin, P., DeMarco, P., Geddes, K.O., Hare, D.E.G., Heal, K.M., Labahn, G., May, J.P., McCarron, J., Monagan, M.B., Ohashi, D., Vorkoetter, S.M.: Maple Programming Guide. Maplesoft, a Division of Waterloo Maple Inc., Waterloo (2016)

    Google Scholar 

  14. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC. With 1 CD-ROM (Windows, Macintosh and UNIX)

    Google Scholar 

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Acknowledgements

We are indebted to Bruce Miller and Abdou Youssef for valuable discussions and for the development of the custom macro set of semantic mathematical macros used in the DLMF, and for the development of the POM tagger, respectively. We also thank the DLMF editors for their assistance and support. We also greatly appreciate valuable discussions with JĂĽrgen Gerhard concerning MAPLE.

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Authors and Affiliations

  1. Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA, USA

    Howard S. Cohl

  2. Department of Computer and Information Science, University of Konstanz, Konstanz, Germany

    André Greiner-Petter & Moritz Schubotz

Authors
  1. Howard S. Cohl
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  2. André Greiner-Petter
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  3. Moritz Schubotz
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Corresponding author

Correspondence to Howard S. Cohl .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Bayern, Germany

    Florian Rabe

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    William M. Farmer

  3. University of Cambridge, Cambridge, United Kingdom

    Grant O. Passmore

  4. Department of Computer Science, The George Washington University, Washington, District of Columbia, USA

    Abdou Youssef

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Cohl, H.S., Greiner-Petter, A., Schubotz, M. (2018). Automated Symbolic and Numerical Testing of DLMF Formulae Using Computer Algebra Systems. In: Rabe, F., Farmer, W., Passmore, G., Youssef, A. (eds) Intelligent Computer Mathematics. CICM 2018. Lecture Notes in Computer Science(), vol 11006. Springer, Cham. https://doi.org/10.1007/978-3-319-96812-4_4

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  • DOI: https://doi.org/10.1007/978-3-319-96812-4_4

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  • Print ISBN: 978-3-319-96811-7

  • Online ISBN: 978-3-319-96812-4

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