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Computer Theorem Proving for Verifiable Solving of Geometric Construction Problems

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Automated Deduction in Geometry (ADG 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9201))

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Abstract

Over the last sixty years, a number of methods for automated theorem proving in geometry, especially Euclidean geometry, have been developed. Almost all of them focus on universally quantified theorems. On the other hand, there are only few studies about logical approaches to geometric constructions. Consequently, automated proving of \(\forall \exists \) theorems, that correspond to geometric construction problems, have seldom been studied. In this paper, we present a formal logical framework describing the traditional four phases process of geometric construction solving. It leads to automated production of constructions with corresponding human readable correctness proofs. To our knowledge, this is the first study in that direction. In this paper we also discuss algebraic approaches for solving ruler-and-compass construction problems. There are famous problems showing that it is often difficult to prove non-existence of constructible solutions for some tasks. We show how to put into practice well-known algebra-based methods and, in particular, field theory, to prove RC-constructibility in the case of problems from Wernick’s list.

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Notes

  1. 1.

    The English word ruler designates a tool with measurement in opposition to straightedge. In this paper, however, we will conform to the habits and use the terms ruler-and-compass constructibility or resolvability, in short RC-constructibility or RC-resolvability, for straightedge and compass constructibility or resolvability.

  2. 2.

    In later stages of the solution, the given condition \(\varPhi _a(X)\) may be extended to some condition \(\varPhi \) for which (3) holds.

  3. 3.

    This formula may involve disjunctions corresponding to different “cases” for X and Y. For instance, \((A \ne B \wedge midpoint(C,A,B)) \vee (A=B \wedge C=A)\).

  4. 4.

    Strictly speaking, functions \(RC_{i,k}\) may involve more than only ruler and compass. For instance, it may be the case that only one intersection point of two circles can be picked (e.g. “that is different from the point...”, “that is not on the same side...”, etc.). Also, some of \(RC_{i,k}\) may be non-deterministic, for instance “pick a point on the line ...”.

  5. 5.

    One can try a finite number of predicates over X.

  6. 6.

    All proofs can be found here: http://www.matf.bg.ac.rs/~vesnap/adg2014_wernick6.thy.

  7. 7.

    The ArgoTriCS tool, along with the list of lemmas used, is available on: http://argo.matf.bg.ac.rs/?content=downloads.

  8. 8.

    The technical report can be found here: http://www.mmrc.iss.ac.cn/pub/mm15.pdf/gao.pdf.

  9. 9.

    http://www.maplesoft.com/.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Belgrade, Serbia

    Vesna Marinković & Predrag Janičić

  2. ICube, UMR CNRS 7357, Université de Strasbourg, Strasbourg, France

    Pascal Schreck

Authors
  1. Vesna Marinković
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  2. Predrag Janičić
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  3. Pascal Schreck
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Correspondence to Vesna Marinković .

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Editors and Affiliations

  1. University of Vigo, Pontevedra, Spain

    Francisco Botana

  2. University of Coimbra, Coimbra, Portugal

    Pedro Quaresma

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Marinković, V., Janičić, P., Schreck, P. (2015). Computer Theorem Proving for Verifiable Solving of Geometric Construction Problems. In: Botana, F., Quaresma, P. (eds) Automated Deduction in Geometry. ADG 2014. Lecture Notes in Computer Science(), vol 9201. Springer, Cham. https://doi.org/10.1007/978-3-319-21362-0_5

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  • DOI: https://doi.org/10.1007/978-3-319-21362-0_5

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