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Evolution of Automated Deduction and Dynamic Constructions in Geometry

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Mathematics Education in the Age of Artificial Intelligence

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 17))

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Abstract

Given its formal, logical and spatial properties, geometry allows an integrated framework where model theory and proof theory approaches can be explored. The development of geometry automatic theorem proving systems and dynamic geometry systems began as separated enterprises but its merging in integrated systems is currently doing its way. In this text, the history of automated deduction in geometry is traced, from the early development of automated theorem provers for geometry and from the emergence of the dynamic geometry systems, to the current status where different application systems combine dynamic geometry and automated deduction. These tools enable their users to explore existing geometry knowledge in addition to creating new constructions and testing new conjectures.

This work is funded by national funds through the FCT—Foundation for Science and Technology, I.P., within the scope of the project CISUC—UID/CEC/00326/2020 and by European Social Fund, through the Regional Operational Program Centro 2020.

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Notes

  1. 1.

    Unfortunately most of the tiles were lost after the expelling of the Jesuits from Portugal by the Marquis of Pombal in 1759, the subsequent reform of the University of Coimbra and the construction of new buildings on the expense of the old ones.

  2. 2.

    Coherent logic is a fragment of (finitary) first-order logic which allows only the connectives and quantifiers \(\wedge \) (and), \(\vee \) (or), \(\top \) (true), \(\bot \) (false), \(\exists \) (existential quantifier).

  3. 3.

    Negative and positive orientation are only a syntactic convention to disambiguate between “different” geometric constructions built from the same set of points.

  4. 4.

    The Pythagoras difference is a generalization of the Pythagoras equality regarding the three sides of a right triangle, to an expression applicable to any triangle. For a triangle ABC with the right angle at B, it holds that \(\mathcal {P}_{ABC}=0\).

  5. 5.

    https://en.wikipedia.org/wiki/Automated_theorem_proving.

  6. 6.

    http://www.tptp.org/CASC/.

  7. 7.

    This is a recent contribution, by the author, to TPTP.

  8. 8.

    For the Euler line theorem, GCLC’s Wu Method took 0.012 s, Vampire took 0.062s.

  9. 9.

    https://en.wikipedia.org/wiki/Prolog.

  10. 10.

    http://maude.cs.illinois.edu/w/index.php/The_Maude_System.

  11. 11.

    The OpenGeometryProver github project: https://github.com/opengeometryprover/.

  12. 12.

    JGEX is currently not being supported/developed.

  13. 13.

    Portfolio problem solving is an approach in which for an individual instance of a specific problem, one particular, hopefully most appropriate, solving technique is automatically selected among several available ones and used. The selection usually employs machine learning methods.

  14. 14.

    http://polarprof-001-site1.htempurl.com/geometriagon/.

  15. 15.

    http://car.rene-grothmann.de/doc_en/index.html.

  16. 16.

    https://www.edukera.com/.

  17. 17.

    http://hilbert.mat.uc.pt/TGTP/index.php.

  18. 18.

    http://www.tptp.org/.

  19. 19.

    Pedro Quaresma and Pierluigi Graziani, Measuring the Readability of a Proof, submitted to publication.

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  1. CISUC, Mathematics Department, Universidade de Coimbra, Coimbra, Portugal

    Pedro Quaresma

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  1. Département de didactique, Université de Montréal, Montréal, QC, Canada

    Philippe R. Richard

  2. Departamento de Ingeniería Industrial, Universidad Antonio de Nebrija, Madrid, Spain

    M. Pilar Vélez

  3. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain

    Steven Van Vaerenbergh

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Quaresma, P. (2022). Evolution of Automated Deduction and Dynamic Constructions in Geometry. In: Richard, P.R., Vélez, M.P., Van Vaerenbergh, S. (eds) Mathematics Education in the Age of Artificial Intelligence. Mathematics Education in the Digital Era, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-86909-0_1

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