Abstract
Diagrammatic reasoning has the potential to be important in numerous application areas. This paper focuses on the simple, but widely used, Euler diagrams that form the basis of many more expressive logics. We have implemented a diagrammatic theorem prover, called Edith, which has access to four sound and complete sets of reasoning rules for Euler diagrams. Furthermore, for each rule set we develop a sophisticated heuristic to guide the search for a proof. This paper is about understanding how the choice of reasoning rule set affects the time taken to find proofs. Such an understanding will influence reasoning rule design in other logics. Moreover, this work specific to Euler diagrams directly benefits the many logics based on Euler diagrams. We investigate how the time taken to find a proof depends not only on the proof task but also on the reasoning system used. Our evaluation allows us to predict the best choice of reasoning system, given a proof task, in terms of time taken, and we extract a guide for defining reasoning rules for other logics in order to minimize time requirements.
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References
Barwise, J., Etchemendy, J.: Hyperproof. CSLI Press, Stanford (1994)
Benoy, F., Rodgers, P.: Evaluating the comprehension of Euler diagrams. In: 2nd International Workshop on Euler Diagrams, Paris, pp. 29–33 (2005)
Chow, S., Ruskey, F.: Drawing area-proportional Venn and Euler diagrams. In: Proceedings of Graph Drawing 2003. LNCS, vol. 2912, pp. 466–477. Springer, Berlin Heidelberg New York (2003)
Clark, R.: Failure mode modular de-composition using spider diagrams. In: 1st International Workshop on Euler Diagrams, Brighton 2004. ENTCS, vol. 134, pp. 19–31 (2005)
DeChiara, R., Erra, U., Scarano, V.: VennFS: A Venn diagram file manager. In: 7th International Conference on Information Visualisation, London, pp. 120-126. IEEE Computer Society Press, Los Alamitos, CA (2003)
Dechter, R., Pearl, J.: Generalized best-first search strategies and the optimality of A*. J. Assoc. Comput. Mach. 32(3), 505–536 (1985)
Dunn-Davies, H., Cunningham, R.: Propostional statecharts for agent interaction protocols. In: 1st International Workshop on Euler Diagrams, Brighton 2004. ENTCS, vol. 134, pp. 55–75. Elsevier, Amsterdam (2005)
Edelkamp, S.: Memory limitations in artificial intelligence. In: Meyer, U. et al. (eds.),: Algorithms for Memory Hierarchies. LNCS, vol. 2625, pp. 233-250. Springer, Berlin Heidelberg New York (2003)
Fish, A., Flower, J.: Investigating reasoning with constraint diagrams. In: Visual Languages and Formal Methods, Rome 2004. ENTCS, vol. 127, pp. 53–69. Elsevier, Amsterdam (2005)
Fish, A., Flower, J., Howse, J.: The semantics of augmented constraint diagrams. J. Visual Lang. Comput. 16, 541–573 (2005)
Fish, A., Stapleton, G.: Formal issues in languages based on closed curves. In: 11th International Conference on Distributed Multimedia Systems, International Workshop on Visual Languages and Computing, Knowledge Systems Institute, Grand Canyon, pp. 161–167 (2006)
Flower, J., Howse, J.: Generating Euler diagrams. In: 2nd International Conference on the Theory and Application of Diagrams, Callaway Gardens, GA, LNAI 2317, pp. 61–75. Springer, Berlin Heidelberg New York (2002)
Flower, J., Masthoff, J., Stapleton, G.: Generating proofs with spider diagrams using heuristics. In: 10th International Conference on Distributed Multimedia Systems, International Conference on Visual Languages and Computing, Knowledge Systems Institute, Banff, pp. 279–285, (2004)
Flower, J., Masthoff, J., Stapleton, G.: Generating readable proofs: a heuristic approach to theorem proving with spider diagrams. In: 3rd International Conference on the Theory and Application of Diagrams, Cambridge. LNAI, vol. 2980, pp. 166–181. Springer, Berlin Heidelberg New York (2004)
Flower, J., Rodgers, P., Mutton, P.: Layout metrics for Euler diagrams. In: 7th International Conference on Information Visualisation, London, pp. 272–280. IEEE Computer Society Press, Los Alamitos, CA (2003)
Flower, J., Stapleton, G.: Automated theorem proving with spider diagrams. In: Computing: The Australasian Theory Symposium, Dunedin. ENTCS, vol. 91, pp. 116–132. Elsevier, Amsterdam (2004)
Gil, J., Howse, J., Kent, S.: Formalising spider diagrams. In: IEEE Symposium on Visual Languages, pp. 130-137. IEEE Computer Society Press, Los Alamitos, CA (1999)
Gelernter, H.: Realization of a geometry theorem proving machine. In: Computers and Thought, pp. 134–152. McGraw-Hill, New York (1963)
Hammer, E.: Logic and Visual Information. CSLI Publications, Stanford (1995)
Hart, P., Nilsson, N., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. 4(2), 100–107 (1968)
Hayes, P., Eskridge, T., Saavedra, R., Reichherzer, T., Mehrotra, M., Bobrovnikoff, D.: Collaborative knowledge capture in ontologies. In: 3rd International Conference on Knowledge Capture, pp. 99-106. ACM Press, New York (2005)
Howse, J., Schuman, S.: Precise visual modelling. J. Softw. Syst. Model. 4(3), 310–325 (2005)
Howse, J., Molina, F., Shin, S.-J., Taylor, J.: On diagram tokens and types. In: 2nd International Conference on the Theory and Application of Diagrams, Georgia. LNAI, vol. 2317, pp. 76–90. Spinger, Berlin Heidelberg New York (2002)
Howse, J., Stapleton, G., Flower, J., Taylor, J.: Corresponding regions in Euler diagrams. In: 2nd International Conference on the Theory and Application of Diagrams, Callaway Gardens, GA. LNAI, vol. 2317, pp. 146–160. Springer, Berlin Heidelberg New York (2002)
Howse, J., Stapleton, G., Taylor, J.: Spider diagrams. LMS J. Comput. Math. 8, 145–194 (2005)
Jamnik, M.: Mathematical Reasoning with Diagrams. CSLI Publications, Stanford (2001)
Jamnik, M., Bundy, A., Green, I.: Automation of diagrammatic reasoning. In: 15th International Joint Conference on Artificial Intelligence, vol. 1, pp. 528–533. Morgan Kaufmann, San Mateo, CA (1997)
John, C.: Reasoning with projected contours. In: 3rd International Conference on the Theory and Application of Diagrams, Cambridge. LNAI, vol. 2980, pp. 147–150. Springer, Berlin Heidelberg New York (2004)
Kent, S.: Constraint diagrams: visualizing invariants in object oriented modelling. In: OOPSLA, 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. Bioinformatics 21(8), 1592-1595 (2005)
Kim, S.-K., Carrington, D.: Visualization of formal specifications. 6th Asia Pacific Software Engineering Conference, pp. 102–109. IEEE Computer Society Press, Los Alamitos, CA (1999)
Larkin, J., Simon, H.: Why a diagram is (sometimes) worth ten thousand words. Cogn. Sci. 11, 5–99 (1987)
Lemon, O., Pratt, I.: On spatial logic and the complexity of diagrammatic reasoning. Mach. Graph. Vis. 6(1), 89–108 (1997)
Lovdahl, J.: Towards a visual editing environment for the languages of the semantic web. Ph.D. thesis, Linkoping University (2002)
Luger, G.: Artificial Intelligence: Structures and Strategies for Complex Problem Solving. Addison-Wesley, Reading, MA (2002)
MacKenzie, D.: Computers and the sociology of mathematical proof. In: 3rd Northern Formal Methods Workshop, http://ewic.bcs.org/conferences/1998/3rdfacs/papers/paper13.pdf (1998)
McLaughlin, J., Gallinger, S.: Cancer epidemiology. In: The Basic Science of Oncology, pp. 4–24. McGraw-Hill, New York (2004)
Misue, K., Eades, P., Lai, W., Sugiyama, K.: Layout adjustment and the mental map. Technical Report IIAS-RR-94-1E, Fujitsu Laboratories (1994)
Molina, F.: Reasoning with Extended Venn-Peirce Diagrammatic Systems. Ph.D. thesis, University of Brighton (2001)
Oberlander, J., Stenning, K., Cox, R.: Hyperproof: the multimodal moral. In: 2nd Conference on Information-theoretic Approaches to Logic, Language, and Computation. Regent’s College, London (1996)
Patrascoiu, O., Thompson, S., Rodgers, P.: Tableaux for diagrammatic reasoning. In: 11th International Conference on Distributed Multimedia Systems, International Workshop on Visual Languages and Computing, Knowledge Systems Institute, Banff, pp. 279–286 (2005)
Peirce, C.: Collected Papers, vol. 4. Harvard University Press, Cambridge, MA (1933)
Puigsegur, J., Agusti, J.: Visual logic programming by means of diagram transformations. In: Joint Conference on Declarative Programming, La Coruna, pp. 311–328 (1998)
Rector, A.: Specifying values in OWL: value partitions and value Sets. W3C Editors Draft 02 (2005)
Rodgers, P., Mutton, P., Flower, J.: Dynamic Euler diagram drawing. In: IEEE Symposium on Visual Languages and Human Centric Computing, Rome, pp. 147–156. IEEE Computer Society Press, Los Alamitos, CA (2004)
Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach. Prentice-Hall, Englewood Cliffs, NJ (2003)
Sawamura, H., Kiyozuka, K.: JVenn: a visual reasoning system with diagrams and sentences. In: 1st International Conference on the Theory and Application of Diagrams, Edinburgh. LNAI, vol. 1889, pp. 271–285. Springer, Berlin Heidelberg New York (2000)
Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)
Stapleton, G.: A survey of reasoning systems based on Euler diagrams. In: 1st International Workshop on Euler Diagrams, Brighton 2004. ENTCS, vol. 134, pp. 127-151. Elsevier, Amsterdam (2005)
Stapleton, G., Howse, J., Taylor, J.: A constraint diagram reasoning system. In: 9th International Conference on Distributed Multimedia Systems, International Conference on Visual Languages and Computing, Knowledge Systems Institute, Miami, pp. 263–270 (2003)
Stapleton, G., Howse, J., Taylor, J.: A decidable constraint diagram reasoning system. J. Log. Comput. 15(6), 541–573 (2005)
Stapleton, G., Masthoff, J., Flower, J., Fish, A., Southern, J.: Appendices for automated theorem proving in Euler diagram systems. Technical Report VMG.06.2, University of Brighton, available from www.cmis.brighton.ac.uk/research/vmg/publications.htm (2006)
Stapleton, G., Thompson, S., Howse, J., Taylor, J.: The expressiveness of spider diagrams. J. Log. Comput. 14(6), 857–880 (2004)
Swoboda, N.: Implementing Euler/Venn reasoning systems. In: Anderson, M., Meyer, B., Oliver, P., (eds.), Diagrammatic Representation and Reasoning, pp. 371–386. Springer, Berlin Heidelberg New York (2001)
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)
Verroust, A., Viaud, M.-L.: Ensuring the drawability of Euler diagrams for up to eight sets. In: 3rd International Conference on the Theory and Application of Diagrams, Cambridge. LNAI, vol. 2980, pp. 128–141. Springer, Berlin Heidelberg New York (2004)
Winterstein, D., Bundy, A., Gurr, C.: Dr Doodle: a diagrammatic theorem prover. In: International Joint Conference on Automated Reasoning. LNCS, vol. 3097, pp. 331–335. Springer, Berlin Heidelberg New York (2004)
Winterstein, D., Bundy, A., Jamnik, M.: On differences between the real and physical plane. In: 3rd International Conference on the Theory and Application of Diagrams, Cambridge. LNAI, vol. 2980, pp. 29–31. Springer, Berlin Heidelberg New York (2004)
Zhao, Y., Lövdahl, J.: A reuse based method of developing the ontology for e-procurement. In: Nordic Conference on Web Services, pp. 101–112 (2003)
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Stapleton, G., Masthoff, J., Flower, J. et al. Automated Theorem Proving in Euler Diagram Systems. J Autom Reasoning 39, 431–470 (2007). https://doi.org/10.1007/s10817-007-9069-y
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DOI: https://doi.org/10.1007/s10817-007-9069-y