On the Completeness of Spider Diagrams Augmented with Constants

  • Gem Stapleton
  • John Howse
  • Simon Thompson
  • John Taylor
  • Peter Chapman
Part of the Studies in Universal Logic book series (SUL)


Diagrammatic reasoning can be described formally by a number of diagrammatic logics; spider diagrams are one of these, and are used for expressing logical statements about set membership and containment. Here, existing work on spider diagrams is extended to include constant spiders that represent specific individuals. We give a formal syntax and semantics for the extended diagram language before introducing a collection of reasoning rules encapsulating logical equivalence and logical consequence. We prove that the resulting logic is sound, complete and decidable.


Spider diagrams Constants Soundness Completeness Monadic first-order logic Diagrammatic reasoning 

Mathematics Subject Classification (2010)

68R02 03B02 



This work is supported by the UK EPSRC grant “Defining Regular Languages with Diagrams” [EP/H012311/1].


  1. 1.
    Chow, S., Ruskey, F.: Drawing area-proportional Venn and Euler diagrams. In: Proceedings of Graph Drawing 2003, Perugia, Italy. LNCS, vol. 2912, pp. 466–477. Springer, Berlin (2003) Google Scholar
  2. 2.
    Clark, R.: Failure mode modular de-composition using spider diagrams. In: Proceedings of Euler Diagrams 2004. Electronic Notes in Theor. Comput. Sci., vol. 134, pp. 19–31 (2005) Google Scholar
  3. 3.
    De Chiara, R., Erra, U., Scarano, V.: VennFS: a Venn diagram file manager. In: Proceedings of Information Visualisation, pp. 120–126. IEEE Comput. Soc., Los Alamitos (2003) Google Scholar
  4. 4.
    Euler, L.: Lettres a une princesse d’Allemagne sur divers sujets de physique et de philosophie. Opera Omnia 2, 102–108 (1775) Google Scholar
  5. 5.
    Flower, J., Howse, J.: Generating Euler diagrams. In: Proceedings of 2nd International Conference on the Theory and Application of Diagrams, Georgia, USA, pp. 61–75. Springer, Callaway Gardens (2002) Google Scholar
  6. 6.
    Flower, J., Masthoff, J., Stapleton, G.: Generating proofs with spider diagrams using heuristics. In: Proceedings of Distributed Multimedia Systems, International Workshop on Visual Languages and Computing, pp. 279–285. Knowledge Systems Institute, San Francisco (2004) Google Scholar
  7. 7.
    Flower, J., Masthoff, J., Stapleton, G.: Generating readable proofs: a heuristic approach to theorem proving with spider diagrams. In: Proceedings of 3rd International Conference on the Theory and Application of Diagrams, Cambridge, UK. LNAI, vol. 2980, pp. 166–181. Springer, Berlin (2004) Google Scholar
  8. 8.
    Gurr, C.: Aligning syntax and semantics in formalisations of visual languages. In: Proceedings of IEEE Symposia on Human-Centric Computing Languages and Environments, pp. 60–61. IEEE Comput. Soc., Los Alamitos (2001) CrossRefGoogle Scholar
  9. 9.
    Hayes, P., Eskridge, T., Saavedra, R., Reichherzer, T., Mehrotra, M., Bobrovnikoff, D.: Collaborative knowledge capture in ontologies. In: Proceedings of the 3rd International Conference on Knowledge Capture, pp. 99–106 (2005) CrossRefGoogle Scholar
  10. 10.
    Howse, J., Molina, F., Shin, S.-J., Taylor, J.: Type-syntax and token-syntax in diagrammatic systems. In: Proceedings FOIS-2001: 2nd International Conference on Formal Ontology in Information Systems, Maine, USA, pp. 174–185. ACM, New York (2001) CrossRefGoogle Scholar
  11. 11.
    Howse, J., Molina, F., Taylor, J., Kent, S., Gil, J.: Spider diagrams: a diagrammatic reasoning system. J. Vis. Lang. Comput. 12(3), 299–324 (2001) CrossRefGoogle Scholar
  12. 12.
    Howse, J., Stapleton, G., Taylor, J.: Spider diagrams. LMS J. Comput. Math. 8, 145–194 (2005) MathSciNetMATHGoogle Scholar
  13. 13.
    Howse, J., Stapleton, G., Taylor, K., Chapman, P.: Visualizing ontologies: a case study. In: International Semantic Web Conference 2011. Springer, Bonn (2011) Google Scholar
  14. 14.
    Kent, S.: Constraint diagrams: visualizing invariants in object oriented modelling. In: Proceedings of OOPSLA97, pp. 327–341. ACM, New York (1997) Google Scholar
  15. 15.
    Kestler, H., Muller, A., Kraus, J., Buchholz, M., Gress, T., Kane, D., Zeeberg, B., Weinstein, J.: VennMaster: area-proportional Euler diagrams for functional go analysis of microarrays. BMC Bioinformatics 9(67) (2008) Google Scholar
  16. 16.
    Lovdahl, J.: Towards a visual editing environment for the languages of the semantic web. PhD thesis, Linkoping University (2002) Google Scholar
  17. 17.
    Mutton, P., Rodgers, P., Flower, J.: Drawing graphs in Euler diagrams. In: Proceedings of 3rd International Conference on the Theory and Application of Diagrams, Cambridge, UK. LNAI, vol. 2980, pp. 66–81. Springer, Berlin (2004) Google Scholar
  18. 18.
    Oliver, I., Howse, J., Stapleton, G., Nuutila, E., Törmä, S.: Visualising and specifying ontologies using diagrammatic logics. In: 5th Australasian Ontologies Workshop. Conf. Res. Pract. Inf. Technol., vol. 112. CRPIT, Melbourne (2009) Google Scholar
  19. 19.
    Rodgers, P., Zhang, L., Fish, A.: General Euler diagram generation. In: International Conference on Theory and Applications of Diagrams, pp. 13–27. Springer, Herrsching (2008) Google Scholar
  20. 20.
    Shimojima, A.: Inferential and expressive capacities of graphical representations: survey and some generalizations. In: Proceedings of 3rd International Conference on the Theory and Application of Diagrams, Cambridge, UK. LNAI, vol. 2980, pp. 18–21. Springer, Berlin (2004) Google Scholar
  21. 21.
    Stapleton, G.: Spider diagrams augmented with constants: a complete system. In: Visual Languages and Computing, pp. 292–299 (2008) Google Scholar
  22. 22.
    Stapleton, G., Delaney, A.: Evaluating and generalizing constraint diagrams. J. Vis. Lang. Comput. 19(4), 499–521 (2008) CrossRefGoogle Scholar
  23. 23.
    Stapleton, G., Thompson, S., Howse, J., Taylor, J.: The expressiveness of spider diagrams. J. Log. Comput. 14(6), 857–880 (2004) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Stapleton, G., Masthoff, J., Flower, J., Fish, A., Southern, J.: Automated theorem proving in Euler diagrams systems. J. Autom. Reason. 39, 431–470 (2007) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Stapleton, G., Taylor, J., Howse, J., Thompson, S.: The expressiveness of spider diagrams augmented with constants. J. Vis. Lang. Comput. 20(1), 30–49 (2009) CrossRefGoogle Scholar
  26. 26.
    Stapleton, G., Rodgers, P., Howse, J., Zhang, L.: Inductively generating Euler diagrams. IEEE Trans. Vis. Comput. Graph. 17(1), 88–100 (2011) CrossRefGoogle Scholar
  27. 27.
    Swoboda, N., Allwein, G.: Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference. J. Softw. Syst. Model. 3(2), 136–149 (2004) CrossRefGoogle Scholar
  28. 28.
    Zhao, Y., Lövdahl, J.: A reuse based method of developing the ontology for e-procurement. In: Proceedings of the Nordic Conference on Web Services, pp. 101–112 (2003) Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Gem Stapleton
    • 1
  • John Howse
    • 1
  • Simon Thompson
    • 2
  • John Taylor
    • 1
  • Peter Chapman
    • 1
  1. 1.Visual Modelling GroupUniversity of BrightonBrightonUK
  2. 2.School of ComputingUniversity of KentCanterburyUK

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