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A Survey of Languages for Formalizing Mathematics

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Intelligent Computer Mathematics (CICM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12236))

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Abstract

In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce human-readable documents. These goals are challenging to combine and state-of-the-art tools typically have to make difficult compromises.

In this paper we discuss languages that have been created for this purpose, including logical languages of proof assistants and other formal systems, semi-formal languages, intermediate languages for exchanging mathematical knowledge, and language frameworks that allow building customized languages.

We evaluate their advantages based on our experience in designing and applying languages and tools for formalizing mathematics. We reach the conclusion that no existing language is truly good enough yet and derive ideas for possible future improvements.

C. Kaliszyk—Supported by ERC starting grant no. 714034 SMART.

F. Rabe—Supported by DFG grant RA-1872/3-1 OAF and EU grant Horizon 2020 ERI 676541 OpenDreamKit.

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Authors and Affiliations

  1. University of Innsbruck, Innsbruck, Austria

    Cezary Kaliszyk

  2. University Erlangen-Nürnberg, Erlangen, Germany

    Florian Rabe

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  1. Cezary Kaliszyk
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  1. Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  2. National Institute of Standards and Technology, Gaithersburg, MD, USA

    Bruce Miller

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Kaliszyk, C., Rabe, F. (2020). A Survey of Languages for Formalizing Mathematics. In: Benzmüller, C., Miller, B. (eds) Intelligent Computer Mathematics. CICM 2020. Lecture Notes in Computer Science(), vol 12236. Springer, Cham. https://doi.org/10.1007/978-3-030-53518-6_9

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