Abstract
Euler diagrams are a common visual representation of set-theoretic statements, and they have been used for visualising the results of database search queries or as the basis of diagrammatic logical constraint languages for use in software specification. Such applications rely upon the ability to automatically generate diagrams from an abstract description. However, this problem is difficult and is known to be NP-complete under certain wellformedness conditions. Therefore methods to identify when and how one can decompose abstract Euler diagrams into simpler components provide a vital step in improving the efficiency of tools which implement a generation process. One such decomposition, called diagram nesting, has previously been identified and exploited. In this paper, we make substantial progress, defining the notion of a disconnecting contour and identifying the conditions on an abstract Euler diagram that allow us to identify disconnecting contours. If a diagram has a disconnecting contour, we can draw it more easily, by combining the results of drawing smaller diagrams. The drawing problem is just one context which benefits from such diagram decomposition - we can also use the disconnecting contour to provide a more natural semantic interpretation of the Euler diagram.
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References
Chow, S., Ruskey, F.: Drawing area-proportional Venn and Euler diagrams. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 466–477. Springer, Heidelberg (2004)
Chow, S.C.: Generating and Drawing Area-Proportional Euler and Venn Diagrams. Ph.D thesis, University of Victoria (2007)
DeChiara, R., Erra, U., Scarano, V.: VennFS: A Venn diagram file manager. In: Proceedings of Information Visualisation, pp. 120–126. IEEE Computer Society, Los Alamitos (2003)
DeChiara, R., Erra, U., Scarano, V.: A system for virtual directories using Euler diagrams. In: Proceedings of Euler Diagrams 2004. Electronic Notes in Theoretical Computer Science, vol. 134, pp. 33–53 (2005)
Euler, L.: Letters a une princesse dallemagne sur divers sujets de physique et de philosophie. Letters 2, 102–108 (1775)
Fish, A., Flower, J.: Abstractions of Euler diagrams. In: Proceedings of Euler Diagrams 2004, Brighton, UK. ENTCS, vol. 134, pp. 77–101 (2005)
Fish, A., Flower, J.: Investigating reasoning with constraint diagrams. In: Visual Language and Formal Methods 2004, Rome, Italy. ENTCS, vol. 127, pp. 53–69. Elsevier, Amsterdam (2005)
Flower, J., Fish, A., Howse, J.: Euler diagram generation. Journal of Visual Languages and Computing (2008), http://dx.doi.org/10.1016/j.jvlc.2008.01.004
Flower, J., Howse, J.: Generating Euler diagrams. In: Proceedings of 2nd International Conference on the Theory and Application of Diagrams, Georgia, USA, April 2002, pp. 61–75. Springer, Heidelberg (2002)
Flower, J., Howse, J., Taylor, J.: Nesting in Euler diagrams. In: International Workshop on Graph Transformation and Visual Modeling Techniques, pp. 99–108 (2002)
Flower, J., Howse, J., Taylor, J.: Nesting in Euler diagrams: syntax, semantics and construction. Software and Systems Modelling 3, 55–67 (2004)
Hammer, E., Shin, S.J.: Euler’s visual logic. History and Philosophy of Logic, 1–29 (1998)
Harel, D.: On visual formalisms. In: Glasgow, J., Narayan, N.H., Chandrasekaran, B. (eds.) Diagrammatic Reasoning, pp. 235–271. MIT Press, Cambridge (1998)
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 Press, New York (2001)
Howse, J., Stapleton, G., Flower, J., Taylor, J.: Corresponding regions in Euler diagrams. In: Proceedings of 2nd International Conference on the Theory and Application of Diagrams, Georgia, USA, April 2002, pp. 76–90. Springer, Heidelberg (2002)
Howse, J., Stapleton, G., Taylor, J.: Spider diagrams. LMS Journal of Computation and Mathematics 8, 145–194 (2005)
Kent, S.: Constraint diagrams: Visualizing invariants in object oriented modelling. In: Proceedings of OOPSLA 1997, October 1997, pp. 327–341. ACM Press, New York (1997)
Kestler, H., Muller, A., Gress, T., Buchholz, M.: Generalized Venn diagrams: A new method for visualizing complex genetic set relations. Journal of Bioinformatics 21(8), 1592–1595 (2005)
Ruskey, F.: A survey of Venn diagrams. Electronic Journal of Combinatorics (1997), www.combinatorics.org/Surveys/ds5/VennEJC.html
Shimojima, A.: Inferential and expressive capacities of graphical representations: Survey and some generalizations. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 18–21. Springer, Heidelberg (2004)
Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)
Stapleton, G., Thompson, S., Howse, J., Taylor, J.: The expressiveness of spider diagrams. Journal of Logic and Computation 14(6), 857–880 (2004)
Swoboda, N., Allwein, G.: Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference. Journal on Software and System Modeling 3(2), 136–149 (2004)
Venn, J.: On the diagrammatic and mechanical representation of propositions and reasonings. Phil. Mag. (1880)
Verroust, A., Viaud, M.-L.: Ensuring the drawability of Euler diagrams for up to eight sets. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 128–141. Springer, Heidelberg (2004)
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Fish, A., Flower, J. (2008). Euler Diagram Decomposition. In: Stapleton, G., Howse, J., Lee, J. (eds) Diagrammatic Representation and Inference. Diagrams 2008. Lecture Notes in Computer Science(), vol 5223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87730-1_7
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DOI: https://doi.org/10.1007/978-3-540-87730-1_7
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