pith. sign in

arxiv: 1906.11485 · v1 · pith:RHWU7HD4new · submitted 2019-06-27 · 💻 cs.DL · cs.SC

Semantic Preserving Bijective Mappings for Expressions involving Special Functions in Computer Algebra Systems and Document Preparation Systems

Andre Greiner-Petter , Moritz Schubotz , Howard S. Cohl , Bela Gipp This is my paper

Pith reviewed 2026-05-25 14:07 UTC · model grok-4.3

classification 💻 cs.DL cs.SC
keywords special functionsDLMFMapleLaTeXtranslationbijective mappingssemantic preservationcomputer algebra systems
0
0 comments X

The pith

396 mappings translate 2405 DLMF formulae to Maple while preserving meaning

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a rule-based translator that converts expressions with special functions between LaTeX documents and CAS such as Maple. It relies on existing DLMF LaTeX macros to define the mappings and handles cases where one system's atomic symbol corresponds to a composite expression in the other. The method was tested on the full DLMF collection and also applied to check consistency in mathematical databases and CAS implementations. This reduces the need for manual re-entry of formulae and surfaces discrepancies that would otherwise go unnoticed.

Core claim

The authors built 396 mappings and showed that these mappings successfully convert 2405 DLMF formulae (58.8 percent) into equivalent Maple expressions, with the same translator also exposing two errors in the DLMF and one defect in Maple.

What carries the argument

Rule-based translations driven by DLMF LaTeX macros that encode semantic links from symbols to their definitions

If this is right

  • The same mappings can be reused to verify consistency inside other mathematical online compendia.
  • The translator can flag defects inside CAS libraries such as Maple.
  • Manual translation steps between document systems and calculation systems become unnecessary for the covered expressions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be extended by adding mappings for the remaining 41.2 percent of DLMF formulae and for other CAS such as Mathematica.
  • A public repository of these mappings would allow document authors to generate verified CAS code directly from LaTeX sources.
  • The technique suggests a path toward machine-checkable semantic standards for special-function notation across publishing and computation platforms.

Load-bearing premise

The DLMF LaTeX macros already contain enough semantic detail to support meaning-preserving mappings even when the underlying definitions of special functions differ across systems.

What would settle it

A test set of additional DLMF formulae that the current mappings cannot translate without semantic loss, or a failure to detect any of the known DLMF errors, would show the coverage and verification claims do not hold.

Figures

Figures reproduced from arXiv: 1906.11485 by Andre Greiner-Petter, Bela Gipp, Howard S. Cohl, Moritz Schubotz.

Figure 1
Figure 1. Figure 1: Two plots of the real part for the arccotangent function with a branch cut at [ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Process diagram of a forward translation process. The PoM-tagger generates the PPT [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The PPT for the Jacobi polynomial example (1) using the DLMF/ [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The mathematical expres￾sion ‘(a+b)· x’ in infix, prefix, post￾fix and functional notation. does the opposite and places the operator to the right of/after its operands. The infix notation is commonly used in arithmetic and places the operator between its operands. This only makes sense if the operator is a binary operator. In mathematical expressions, notations are mostly mixed, depending on the case and … view at source ↗
Figure 5
Figure 5. Figure 5: The internal Maple DAG representation of [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A scheme of the backward translation process from Maple for the Jacobi polynomial [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

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 LaTeX as our DPS. Design/methodology/approach: Bruce Miller at the National Institute of Standards and Technology (NIST) developed a collection of special LaTeX 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 LaTeX and CAS. The approach improves error-prone manual translations and can be used to verify mathematical online compendia and CAS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript describes the development of 396 rule-based mappings using NIST DLMF LaTeX macros to enable bijective translations of special function expressions between the DLMF and CAS (Maple, Mathematica). It reports translating 2405 DLMF formulae (58.8% success rate), with many involving atomic-to-composite mappings due to differing definitions, and demonstrates the tool's use in verifying online compendia and CAS, uncovering two DLMF errors and one Maple defect.

Significance. If the mappings are semantically accurate, the work offers a novel automated solution for translating between DPS and CAS for special functions, improving on manual methods and enabling verification tasks. The reported success rate and error discoveries provide concrete evidence of utility, though the approach's reliance on macro-provided semantics for equivalence in differing-definition cases needs substantiation.

major comments (1)
  1. [Design/methodology] Design/methodology section: The central claim that the DLMF LaTeX macros supply sufficient semantics to guarantee bijective, meaning-preserving mappings (including atomic DLMF symbols to composite CAS expressions when definitions differ) lacks an explicit, independent verification procedure for semantic equivalence under conditions such as branch cuts, domains, and analytic continuation.
minor comments (1)
  1. [Abstract] Abstract: The purpose paragraph mentions both Maple and Mathematica as target CAS, but the findings paragraph reports results only for Maple and DLMF; clarify the scope for Mathematica translations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Design/methodology] Design/methodology section: The central claim that the DLMF LaTeX macros supply sufficient semantics to guarantee bijective, meaning-preserving mappings (including atomic DLMF symbols to composite CAS expressions when definitions differ) lacks an explicit, independent verification procedure for semantic equivalence under conditions such as branch cuts, domains, and analytic continuation.

    Authors: The DLMF LaTeX macros were developed precisely to link each symbol to its authoritative definition in the NIST handbook, which documents domains, branch cuts, and analytic continuation. Our 396 mappings were created by aligning these explicit DLMF definitions with CAS implementations, using composite expressions only when the documented definitions differ. Equivalence therefore follows from the source definitions rather than from post-hoc numerical checks. The practical validation comes from translating 2405 formulae and from the errors discovered in both DLMF and Maple. We will add a short subsection clarifying this construction process and the reliance on DLMF-documented properties. revision: partial

Circularity Check

0 steps flagged

No significant circularity; mappings constructed from external DLMF definitions

full rationale

The paper's central contribution is the manual creation of 396 rule-based mappings that leverage Bruce Miller's externally developed DLMF LaTeX macros to link symbols to NIST definitions. These mappings are then applied to translate 2405 DLMF expressions, with success measured against independent CAS implementations. Error discoveries in DLMF and Maple serve as external falsification rather than internal fits. No equations, parameters, or claims reduce by construction to the paper's own inputs; the derivation chain relies on external semantic sources and produces independently verifiable outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the semantic adequacy of DLMF macros and the feasibility of creating bijective mappings despite differing CAS definitions; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The NIST DLMF LaTeX macros accurately encode the semantics of special functions in a form usable for rule-based translation.
    Invoked in the design/methodology section as the foundation for all mappings.

pith-pipeline@v0.9.0 · 5837 in / 1286 out tokens · 28230 ms · 2026-05-25T14:07:04.435581+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

  1. [1]

    Do Open Source Developers Respond to Competition? The (La)TeX Case Study

    Alex, G. (June 2007). “Do Open Source Developers Respond to Competition? The (La)TeX Case Study”. In: Review of Network Economics 6.2. Beaumont, J. C., Bradford, R. J., Davenport, J. H., and Phisanbut, N. (2007). “Testing elementary function identities using CAD”. In: Appl. Algebra Eng. Commun. Comput. 18.6. Bernardin, L., Chin, P., DeMarco, P., Geddes, K...

    work page 2007

  2. [2]

    Growing the Digital Repository of Mathematical Formulae with Generic LATEX Sources

    Springer. Cohl, H. S., Schubotz, M., McClain, M. A., Saunders, B. V ., Zou, C. Y ., Mohammed, A. S., and Danoff, A. A. (2015). “Growing the Digital Repository of Mathematical Formulae with Generic LATEX Sources”. In: Proc. CICM. Ed. by M. Kerber, J. Carette, C. Kaliszyk, F. Rabe, and V . Sorge. V ol

    work page 2015

  3. [3]

    Semantic Preserving Bijective Mappings of Mathemat- ical Formulae Between Document Preparation Systems and Computer Algebra Systems

    Springer. Cohl, H. S., Schubotz, M., Youssef, A., Greiner-Petter, A., Gerhard, J., Saunders, B. V ., McClain, M. A., Bang, J., and Chen, K. (2017). “Semantic Preserving Bijective Mappings of Mathemat- ical Formulae Between Document Preparation Systems and Computer Algebra Systems”. In: Proc. CICM. Ed. by H. Geuvers, M. England, O. Hasan, F. Rabe, and O. T...

    work page 2017

  4. [4]

    The Challenges of Multivalued

    Ed. by J. Luca and E. R. Weippl. Vienna, Austria: Association for the Advancement of Computing in Education (AACE). Davenport, J. H. (2010). “The Challenges of Multivalued "Functions"”. In: Proc. CICM. Ed. by S. Autexier, J. Calmet, D. Delahaye, P. D. F. Ion, L. Rideau, R. Rioboo, and A. P. Sexton. V ol

    work page 2010

  5. [5]

    Branch cuts in Maple 17

    Springer. DLMF. NIST Digital Library of Mathematical Functions. Release 1.0.20 of 2018-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V . Saunders, editors.url: http://dlmf.nist.gov/. Drake, D. (June 2009). sagetex. https : / / ctan . org / tex - archive / macros / latex / contri...

    work page 2018

  6. [6]

    Semantic Preserv- ing Bijective Mappings for Expressions involving Special Functions in Computer Algebra Systems and Document Preparation Systems

    Springer. Greiner-Petter, A., Schubotz, M., Cohl, H. S., and Gipp, B. (July 2018). “Semantic Preserv- ing Bijective Mappings for Expressions involving Special Functions in Computer Algebra Systems and Document Preparation Systems”. In: Aslib Journal of Information Management 71.3. Hamblin, C. L. (Nov. 1962). “Translation to and from Polish Notation”. In: ...

    work page 2018

  7. [7]

    Miller, B

    Elsevier. Miller, B. R. (2004). LaTeXML: A LATEX to XML/HTML/MathML Converter. available at: http: //dlmf.nist.gov/LaTeXML/. Accessed June

    work page 2004

  8. [8]

    Technical Aspects of the Digital Library of Mathematical Functions

    Miller, B. R. and Youssef, A. (2003). “Technical Aspects of the Digital Library of Mathematical Functions”. In: Ann. Math. Artif. Intell. 38.1-3. Olver, F. W., Lozier, D. W., Boisvert, R. F., and Clark, C. W. (Apr. 30, 2010). NIST Handbook of Mathematical Functions. 1st. New York, NY , USA: Cambridge University Press. 968 pp. Schubotz, M., Greiner-Petter,...

    work page doi:10.1108/ajim-08-2018-0185 2003