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Diagrammatic Evaluation of Visual Mathematical Notations

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Diagrammatic Representation and Reasoning

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Abstract

This chapter discusses the specification of diagrams and diagram transformations with picture logic, a visual Horn clause language. Formalisation techniques for diagrammatic languages have previously mainly been investigated for the specification of static visual syntax. For reasoning about many types of diagrams, however, a formalisation of their dynamic aspects is indispensable. This is particularly true for many diagrammatic mathematical notations, because their evaluation rules or consequence relations correspond to visual or graphical transformations. The chapter presents constraint-based extensions of picture logic which render it suitable for the specification of such diagram notations and the required transformations.

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References

  1. Nelsen, R.B. (1993). Proofs without words. Washington, DC: Mathematical Association of America.

    MATH  Google Scholar 

  2. Allwein, G. and Barwise, J. (Eds) (1996). Logical reasoning with diagrams. New York: Oxford University Press.

    MATH  Google Scholar 

  3. Shin, S.-J. (1995). The logical status of diagrams. Cambridge, UK: Cambridge University Press.

    Book  Google Scholar 

  4. Brown, J.R. (1999). Philosophy of mathematics: An introduction to the world of pictures. London: Routledge.

    MATH  Google Scholar 

  5. Glasgow, J., Narayanan, N.H. and Chandrasekaran, B. (Eds) (1995). Diagrammatic reasoning. Menlo Par, CA: AAAI Press/Cambridge, MA: MIT Press.

    Google Scholar 

  6. Citrin, W., Hall, R. and Zorn, B. (1995). Programming with visual expressions. In IEEE workshop on visual languages. Darmstadt, Germany. Los Alamitos, CA: IEEE Computer Society Press, pp. 294–301.

    Google Scholar 

  7. Keenan, D. (1995). To dissect a mockingbird: A graphical notation for the lambda calculus with animated reduction. Technical report, Smalltalk Computing.

    Google Scholar 

  8. Bricken, W.M. (1988). An introduction to boundary logic with the losp deductive engine. Research report, University of Washington.

    Google Scholar 

  9. Harel, D. (1988). On visual formalisms. Communications of the ACM 31(5):514–530.

    Article  MathSciNet  Google Scholar 

  10. Kahn, K.M. and Saraswat, V.A. (1990). Complete visualizations of concurrent programs and their execution. In IEEE Workshop on visual languages, Los Alamitos, CA: Skokie, IL. IEEE Computer Society Press, pp. 7–15.

    Google Scholar 

  11. Hammer, E. (1996). Representing relations diagrammatically. In G. Allwein and J. Barwise (Eds), Logical reasoning with diagrams. New York: Oxford University Press.

    Google Scholar 

  12. Hammer, E. and Danner, N. (1996). Towards a model theory of diagrams. In G. Allwein and J. Barwise (Eds), Logical reasoning with diagrams. New York: Oxford University Press.

    Google Scholar 

  13. Tanaka, T. (1991). Definite clause set grammars: A formalism for problem solving. Journal of Logic Programming 10:1–17.

    Article  MathSciNet  Google Scholar 

  14. Marriott, K. (1994). Constraint multiset grammars. In IEEE symposium on visual languages, St Louis, MO. Los Alamitos, CA: IEEE Computer Society Press.

    Google Scholar 

  15. Ferrucci, F., Pacini, G., Tortora, G., Tucci, M. and Vitiello, G. (1991). Efficient parsing of multidimensional structures. In IEEE workshop on visual languages. Kobe, Japan. Los Alamitos, CA: IEEE Computer Society Press, pp. 105–110.

    Google Scholar 

  16. Bolognesi, T. and Latella, D. (1989). Techniques for the formal definition of the g-lotos syntax. In IEEE workshop on visual languages, Rome. Los Alamitos, CA: IEEE Computer Society Press, pp. 43–49.

    Google Scholar 

  17. Meyer, B. (1992). Pictures depicting pictures: On the specification of visual languages by visual grammars. In IEEE workshop on visual languages, Seattle, WA. Los Alamitos, CA: IEEE Computer Society Press, pp. 41–47.

    Google Scholar 

  18. Meyer, B. (1993). Logic and the structure of space: Towards a visual logic for spatial reasoning. In D. Miller (Ed.), International symposium on logic programming, Vancouver. Cambridge, MA: MIT Press.

    Google Scholar 

  19. Meyer, B. (1994). Visual logic languages for spatial information handling (in German). Doctoral thesis, FernUni Hagen.

    Google Scholar 

  20. Older, W. and Vellino, A. (1990). Extending prolog with constraint arithmetic on real intervals. In Canadian conference on electrical and computer engineering, Ottawa.

    Google Scholar 

  21. Borning, A., Maher, M., Martindale, A. and Wilson, M. (1989). Constraint hierarchies and logic programming. In G. Levi and M. Martelli (Eds), International conference on logic programming. Cambridge, MA: MIT Press, pp. 149–164.

    Google Scholar 

  22. Wilson, M. and Borning, A. (1993). Hierarchical constraint logic programming. Journal of Logic Programming 16(3,4):277–318.

    Article  MathSciNet  MATH  Google Scholar 

  23. Jampel, M., Freuder, E. and Maher, M. (Eds) (1996). Over-constrained systems. Berlin: Springer.

    Book  Google Scholar 

  24. DiBattista, G., Eades, P., Tamassia, R. and Tollis, I.G. (1999). Graph drawing. Upper Saddle River, NJ: Prentice Hall.

    Google Scholar 

  25. Borning, A., Marriott, K., Stuckey, P. and Xiao, Y. (1997). Solving linear arithmetic constraints for user interface applications. In UIST’97: ACM symposium on user interface software and technology, Banff, Canada, October.

    Google Scholar 

  26. Haarslev, V. (1998). A fully formalized theory for describing visual notations. In K. Marriott and B. Meyer (Eds), Visual language theory. New York: Springer, pp. 261–292.

    Chapter  Google Scholar 

  27. Gooday, J.M. and Cohn, A.G. (1996). Using spatial logic to describe visual programming languages. Artificial Intelligence Review 10:171–186.

    Article  Google Scholar 

  28. Clarke, B.L. (1981). A calculus of individuals based on “conncection”. Notre Dame Journal of Formal Logic 23(3):204–218.

    Article  Google Scholar 

  29. Kamada, T. and Kawai, S. (1991). A general framework for visualizing abstract objects and relations. ACM Transactions on Graphics 10(1).

    Google Scholar 

  30. Takahashi, S., Miyashita, K., Matsuoka, S. and Yonezawa, A. (1994). A framework for constructing animations via declarative mapping rules. In IEEE symposium on visual languages. St Louis, MO, pp. 314–322.

    Google Scholar 

  31. Marriott, K., Meyer, B. and Wittenburg, K. (1998). A survey of visual language specification and recognition. In K. Marriott and B. Meyer (Eds), Visual language theory. New York: Springer, pp. 5–86.

    Chapter  Google Scholar 

  32. Meyer, B. (1999). Constraint diagram reasoning. In Principles and practice of constraint programming: CP’99. Alexandria, VA, October. New York: Springer.

    Google Scholar 

  33. Meyer, B. (2000). A constraint-based framework for diagrammatic reasoning. In Applied artificial intelligence 14(4):327–344.

    Article  Google Scholar 

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Authors
  1. Bernd Meyer
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Editor information

Editors and Affiliations

  1. Department of Computer and Information Sciences, Fordham University, 441 East Fordham Road, Bronx, NY, 10458, USA

    Michael Anderson MSc, PhD

  2. School of Computer Science and Software Engineering, Monash University, PO Box 26, Clayton Campus, Victoria, 3800, Australia

    Bernd Meyer Dr rer nat

  3. Department of Computer Science, University of York, Heslington, York, YO10 5DD, UK

    Patrick Olivier MA, MSc, PhD

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© 2002 Springer-Verlag London

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Meyer, B. (2002). Diagrammatic Evaluation of Visual Mathematical Notations. In: Anderson, M., Meyer, B., Olivier, P. (eds) Diagrammatic Representation and Reasoning. Springer, London. https://doi.org/10.1007/978-1-4471-0109-3_15

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  • DOI: https://doi.org/10.1007/978-1-4471-0109-3_15

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-242-6

  • Online ISBN: 978-1-4471-0109-3

  • eBook Packages: Springer Book Archive

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