Neural Machine Translation for Mathematical Formulae

Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics Volume 1: Long Papers, pages 11534–11550 July 9-14, 2023 ©2023 Association for Computational Linguistics Neural Machine Translation for Mathematical Formulae Felix Petersen � � � , Moritz Schubotz � � � , André Greiner-Petter ��� , Bela Gipp ��� � Stanford University, � FIZ Karlsruhe, � University of Göttingen mail@felix-petersen.de, moritz.schubotz@fiz-karlsruhe.de greinerpetter@gipplab.com, gipp@uni-goettingen.de Abstract We tackle the problem of neural machine trans- lation of mathematical formulae between am- biguous presentation languages and unambigu- ous content languages. Compared to neural machine translation on natural language, math- ematical formulae have a much smaller vocab- ulary and much longer sequences of symbols, while their translation requires extreme preci- sion to satisfy mathematical information needs. In this work, we perform the tasks of translat- ing from LATEX to Mathematica as well as from LATEX to semantic LATEX. While recurrent, re- cursive, and transformer networks struggle with preserving all contained information, we find that convolutional sequence-to-sequence net- works achieve 95.1% and 90.7% exact matches, respectively. 1 Introduction Mathematical notations consist of symbolic rep- resentations of mathematical concepts. For the purpose of displaying them, most mathematical formulae are denoted in presentation languages (PL) (Schubotz et al., 2018) such as LATEX (Lam- port, 1994). However, for computer-interpretation of formulae, machine-readable and unambiguous content languages (CL) such as Mathematica or se- mantic LATEX are necessary. Thus, this work tackles the problem of neural machine translation between PLs and CLs as a crucial step toward machine- interpretation of mathematics found in academic and technical documents. In the following, we will illustrate the ambigui- ties of representational languages. Those ambigu- ities range from a symbol having different mean- ings over notational conventions that change over time to a meaning having multiple symbols. Con- sider the ambiguous mathematical expression (x)n. While Pochhammer (Pochhammer, 1870) himself used (x)n for the binomial coefficient �x n � , for mathematicians in the subject area of special func- tions, more precisely hypergeometric series, (x)n usually denotes the Pochhammer symbol, which is defined for natural numbers as (x)n := xn = n−1 � k=0 (x + k). (1) To further complicate matters, in statistics and com- binatorics, the same notation is defined as (x)n := xn = n−1 � k=0 (x − k). (2) This work uses LATEX as PL and Mathematica as well as semantic LATEX as CLs. Mathematica is one of the most popular Computer Algebra Systems (CASs); we use Mathematica’s standard notation (InputForm) as a CL (from now on, for simplicity, referred to as Mathematica.) Semantic LATEX is a set of LATEX macros that allow an unambiguous mathematical notation within LATEX (Miller and Youssef, 2003) and which has been developed at the National Institute for Standards and Technol- ogy (NIST) by the Digital Library of Mathematical Functions (DLMF) and the Digital Repository of Mathematical Formulae (DRMF). In LATEX, the Pochhammer symbol (x)n is sim- ply denoted as (x)_n. In semantic LATEX, it is de- noted as \Pochhammersym{x}{n} and compiled to LATEX …

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