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Formal specification and theorem proving breakthroughs in geometric modeling

  • Refereed Papers
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Theorem Proving in Higher Order Logics (TPHOLs 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1479))

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Abstract

An innovative attempt to develop formal techniques in geometric modeling is reported through the axiomatization of the combinatorial maps in the Calculus of Inductive Constructions. A hierarchical specification of ordered sorts is presented and validated by inductive proofs of consistency and completeness in the Coq prover. Classical difficulties in theorem proving like cohabitation of objects with their generalization, smooth handling of subtyping, completion of partial relations or objects, observationality v. constructivism, and symmetry of relations, are addressed. Geometrical modeling issues are thus solved in a new and unquestionable fashion, giving a great insight on the domain and a deep understanding of the model, and so validating the methodology.

This work is supported by the GDR-PRC of Programmation, and the GDR-PRC of Algorithmique, Modèles et Infographie (MENRT, CNRS, France).

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References

  1. R. Aharoni, G. Herman, and M. Loebl. Jordan Graphs. Graphical Models and Image Processing, 58(4):345–359, July 1996.

    Article  Google Scholar 

  2. C. Berge. Graphes et Hypergraphes. Dunod, Paris (France), 1973.

    Google Scholar 

  3. G. Bernot, M. Bidoit, and T. Knapik. Behavioural Approaches to Algebraic Specifications: a Comparative Study. Acta Informatica 31(7):651–671, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  4. Y. Bertrand and J.-F. Dufourd. Algebraic Specification of a 3D-Modeler Based on Hypermaps. Graphical Models and Image Processing, 56(1):29–60, 1994.

    Article  Google Scholar 

  5. Y. Bertrand, J.-F. Dufourd, J. FranÇon, and P. Lienhardt. Algebraic Specification and Development in Geometric Modeling. In TAPSOFT, pages 74–87, Orsay (France), 1993. Springer-Verlag, LNCS 668.

    Google Scholar 

  6. T. Coquand and G. Huet. Constructions: a Higher Order Proof System for Mechanizing Mathematics. In EUROCAL, Linz, 1985. LNCS 203.

    Google Scholar 

  7. R. Cori. Un Code pour les Graphes Planaires et ses Applications. Société Math. de France, Paris, Astérisque 27, 1975.

    Google Scholar 

  8. C. Cornes et al. The Coq Proof Assistant Reference Manual V6.1. INRIA-Rocquencourt, CNRS-ENS Lyon (France), December 1996.

    Google Scholar 

  9. J.-F. Dufourd. Algebras and Formal Specifications in Geometric Modeling. In The Visual Computer, pages 131–154. Spinger-Verlag, 1997.

    Google Scholar 

  10. J.-F. Dufourd and F. Puitg. Boundary Representation Specification Revisited with a New Quasi-map Concept. Submitted to Graphical Models and Image Processing, 1998.

    Google Scholar 

  11. J. FranÇon. Topologie Combinatoire en Imagerie. Research report 95/12, Université L. Pasteur, LSIIT, Strasbourg (France), 1995.

    Google Scholar 

  12. H. Griffiths. Surfaces. Cambridge University Press, 1981.

    Google Scholar 

  13. A. Jacques. Sur le Genre d'une Paire de Substitutions. Notes des membres et correspondants, volume 267, pages 625–627, 1968.

    MATH  MathSciNet  Google Scholar 

  14. A. Jacques. Constellations et Graphes Topologiques. In Combinatorial Theory and Applications, pages 657–673. Budapest (Hungary), 1970.

    Google Scholar 

  15. P. Lienhardt. Topological Models for Boundary Representation: A Survey. In Computer Aided Design, volume 23(1), pages 59–81. 1991.

    Article  MATH  Google Scholar 

  16. C. Parent. Synthesizing Proofs from Programs in the Calculus of Inductive Constructions. In Mathematics of Program Construction. LNCS 947, 1995.

    Google Scholar 

  17. C. Paulin-Mohring. Inductive Definitions in the System Coq — Rules and Properties. In Typed Lambda-calculi and Applications. LNCS 664, 1993.

    Google Scholar 

  18. J.-P. Petit. Le Topologicon. Belin, 1985.

    Google Scholar 

  19. F. Puitg and J.-F. Dufourd. Spécifications en Modélisation Géométrique par la Théorie des Constructions. In Proc. Preuves et Spécifications Algébriques, Grenoble (France), 1995. Journées du GDR de Programmation du CNRS.

    Google Scholar 

  20. F. Puitg and J.-F. Dufourd. Combinatorial Maps and Planarity: Formal Specifications and Proofs in the Calculus of Inductive Constructions. Tech. Rep. 98/05 (100 pages), LSIIT, Strasbourg (France). etsehttp://dpt-info.u-strasbg.fr/~puitg/rr98/main.htmletse, 1998.

    Google Scholar 

  21. J. Rouyer. Développements d'Algorithmes dans le Calcul des Constructions. PhD thesis, Institut National Polytechnique de Lorraine, Nancy (France), March 1994.

    Google Scholar 

  22. W. Tutte. Combinatorial Oriented Maps. In Canadian Journal of Mathematics, volume XXXI(2), pages 986–1004. 1979.

    MathSciNet  Google Scholar 

  23. M. Wirsing. Algebraic Specification. In Formal Models and Semantics, chapter 13, pages 675–788. Elsevier, North-Holland, Amsterdam, 1990.

    Google Scholar 

  24. M. Yamamoto et al. Formalization of Planar Graphs. In HOL Theorem Proving and Its Applications, p 369, Aspen Grove (USA), 1995. Springer-Verlag, LNCS 971.

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection LSIIT, UPRES-A 7005, PÔle API, boulevard S. Brant, F-67400, Illkirch Cedex, France

    FranÇois Puitg & Jean -FranÇois Dufourd

Authors
  1. FranÇois Puitg
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  2. Jean -FranÇois Dufourd
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Jim Grundy Malcolm Newey

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© 1998 Springer-Verlag Berlin Heidelberg

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Puitg, F., Dufourd, J.F. (1998). Formal specification and theorem proving breakthroughs in geometric modeling. In: Grundy, J., Newey, M. (eds) Theorem Proving in Higher Order Logics. TPHOLs 1998. Lecture Notes in Computer Science, vol 1479. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055149

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  • DOI: https://doi.org/10.1007/BFb0055149

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64987-8

  • Online ISBN: 978-3-540-49801-8

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