Mathematics in Computer Science

, Volume 9, Issue 1, pp 23–39 | Cite as

A Graph Library for Isabelle

Article

Abstract

In contrast to other areas of mathematics such as calculus, number theory or probability theory, there is currently no standard library for graph theory for the Isabelle/HOL proof assistant. We present a formalization of directed graphs and essential related concepts. The library supports general infinite directed graphs (digraphs) with labeled and parallel arcs, but care has been taken not to complicate reasoning on more restricted classes of digraphs. We use this library to formalize a characterization of Euler Digraphs and to verify a method of checking Kuratowski subgraphs used in the LEDA library.

Keywords

Graph theory Isabelle HOL Euler Kuratowski 

Mathematics Subject Classification (2010)

Primary 05C20 Secondary 05C45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alkassar, E., Böhme, S., Mehlhorn, K., Rizkallah, C.: A framework for the verification of certifying computations. JAR (2013). doi:10.1007/s10817-013-9289-2
  2. 2.
    Ballarin, C.: Locales: A module system for mathematical theories. JAR (2013). doi:10.1007/s10817-013-9284-7
  3. 3.
    Bang-Jensen J., Gutin G.Z.: Digraphs: Theory, Algorithms and Applications. 2nd edn. Springer, New York (2009)CrossRefGoogle Scholar
  4. 4.
    Butler, R.W., Sjogren, J.A.: A PVS graph theory library. Tech. Rep., NASA Langley (1998)Google Scholar
  5. 5.
    Chou, C.: A formal theory of undirected graphs in higher-order logic. In: Proceedings of TPHOLs ’94. pp. 144–157. Springer, New York (1994)Google Scholar
  6. 6.
    Diestel R.: Graph Theory, GTM, vol. 173. 4 edn. Springer, New York (2010)Google Scholar
  7. 7.
    Duprat, J.: A Coq toolkit for graph theory. Rapport de recherche 2001-15. LIP ENS, Lyon (2001)Google Scholar
  8. 8.
    Esparza, J., Lammich, P., Neumann, R., Nipkow, T., Schimpf, A., Smaus, J.G.: A fully verified executable LTL model checker. In: Proceedings of CAV 2013, pp. 463–478 (2013)Google Scholar
  9. 9.
    Gonthier, G.: computer-checked proof of the Four Colour Theorem (2005)Google Scholar
  10. 10.
    Harary, F., Read, R.: Is the null-graph a pointless concept? In: Graphs and Combinatorics, pp. 37–44. Springer, New York (1974)Google Scholar
  11. 11.
    Hunt, Warren A., J., Kaufmann, M., Krug, R.B., Moore, J.S., Smith, E.W.: Meta reasoning in ACL2. In: Proceedings of TPHOLs ’05, pp. 163–178. Springer, New York (2005)Google Scholar
  12. 12.
    Kuratowski C.: Sur le problème des courbes gauches en topologie. Fundam. Math. 15(1), 271–283 (1930)MATHMathSciNetGoogle Scholar
  13. 13.
    Mehlhorn K., Näher S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (2000)Google Scholar
  14. 14.
    Nakamura Y., Rudnicki P.: Euler circuits and paths. Formaliz. Math. 6(3), 417–425 (1997)Google Scholar
  15. 15.
    Nipkow, T., Bauer, G., Schultz, P.: Flyspeck I: Tame graphs. In: Proc. IJCAR ’06. pp. 21–35. Springer, New York (2006)Google Scholar
  16. 16.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Springer, New York (2002)Google Scholar
  17. 17.
    Nordhoff, B., Lammich, P.: Dijkstra’s shortest path algorithm. Arch. Formal Proofs (2012). http://afp.sf.net/entries/Dijkstra_Shortest_Path
  18. 18.
    Noschinski, L.: Graph theory. Arch. Formal Proofs (2013). http://afp.sf.net/devel-entries/Graph_Theory.shtml, Formal proof development
  19. 19.
    Noschinski, L., Rizkallah, C., Mehlhorn, K.: Verification of certifying computations through AutoCorres and Simpl. In: Proceedings of NFM ’14. doi:10.1007/978-3-319-06200-6_4
  20. 20.
    Rizkallah, C.: An axiomatic characterization of the single-source shortest path problem. Arch. Formal Proofs (2013). http://afp.sf.net/entries/ShortestPath.shtml, Formal proof development
  21. 21.
    Traytel, D., Berghofer, S., Nipkow, T.: Extending Hindley-Milner type inference with coercive structural subtyping. In: APLAS ’11. pp. 89–104. Springer, New York (2011)Google Scholar
  22. 22.
    Volkmann L.: Fundamente der Graphentheorie. Springer, New York (1996)CrossRefMATHGoogle Scholar
  23. 23.
    Wong, W.: A simple graph theory and its application in railway signaling. In: Proceedings TPHOLs ’91. pp. 395–409. IEEE (1991)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Lehrstuhl XXI, Institut für Informatik, Technische Universität MünchenGarchingGermany

Personalised recommendations