Journal of Logic, Language and Information

, Volume 8, Issue 3, pp 297–321

# On Automating Diagrammatic Proofs of Arithmetic Arguments

• Mateja Jamnik
• Alan Bundy
• Ian Green
Article

## Abstract

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the “inference steps” of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive ω-rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here.

automated reasoning diagrammatic reasoning theorem proving

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### References

1. Baker, S., Ireland, A., and Smaill, A., 1992, “On the use of the constructive omega rule within automated deduction,” pp. 214–225 in International Conference on Logic Programming and Automated Reasoning – LPAR 92, St. Petersburg, A. Voronkov, ed., Lecture Notes in Artificial Intelligence, Vol. 624, Berlin: Springer-Verlag.Google Scholar
2. Barker-Plummer, D. and Bailin, S.C., 1997, “The role of diagrams in mathematical proofs,” Machine Graphics and Vision 6(1), 25–56.Google Scholar
3. Barwise, J. and Etchemendy, J., 1991, “Visual information and valid reasoning,” pp. 9–24 in Visualization in Teaching and Learning Mathematics, W. Zimmerman and S. Cunningham, eds., Washington, DC: The Mathematical Association of America.Google Scholar
4. Chandrasekaran, B., Glasgow, J., and Narayanan, N.H., eds., 1995, Diagrammatic Reasoning: Cognitive and Computational Perspectives, Washington, DC/Cambridge, MA: AAAI Press/The MIT Press.Google Scholar
5. Gelernter, H., 1963, “Realization of a geometry theorem-proving machine,” pp. 134–152 in Computers and Thought, E. Feigenbaum and J. Feldman, eds., New York: McGraw-Hill.Google Scholar
6. Hammer, E.M., 1995, Logic and Visual Information, Stanford, CA: Center for the Study of Language and Information.Google Scholar
7. Jamnik, M., 1999, “Automating diagrammatic proofs of arithmetic arguments,” Ph.D. Thesis, available from Edinburgh University.Google Scholar
8. Jamnik, M., Bundy, A., and Green, I., 1997, “Automation of diagrammatic reasoning,” pp. 528–533 in Proceedings of the 15th IJCAI, Vol. 1, International Joint Conference on Artificial Intelligence, M.E. Pollack, ed., San Mateo, CA: Morgan Kaufmann Publishers. Also published in the Proceedings of the 1997 AAAI Fall Symposium. Also available from Edinburgh University as DAI Research Paper No. 873.Google Scholar
9. Koedinger, K.R. and Anderson, J.R., 1990, “Abstract planning and perceptual chunks,” Cognitive Science 14, 511–550. Reprinted in J. Glasgow, N.H. Narayanan, and B. Chandrasekaran, eds., Diagrammatic Reasoning: Cognitive and Computational Perspectives, Washington, DC/Cambridge, MA: AAAI Press/The MIT Press, 1995, pp. 577-625.Google Scholar
10. Larkin, J.H. and Simon, H.A., 1987, “Why a diagram is (sometimes) worth ten thousand words,” Cognitive Science 11, 65–99. Reprinted in J. Glasgow, N.H. Narayanan, and B. Chandrasekaran, eds., Diagrammatic Reasoning: Cognitive and Computational Perspectives, Washington, DC/Cambridge, MA: AAAI Press/The MIT Press, 1995, pp. 69-109.Google Scholar
11. Nelsen, R.B., 1993, Proofs without Words: Exercises in Visual Thinking, Washington, DC: The Mathematical Association of America.Google Scholar
12. Pólya, G., 1945, How to Solve It, Princeton, NJ: Princeton University Press.Google Scholar
13. Pólya, G., 1965, Mathematical Discovery, two volumes, New York: John Wiley & SonsGoogle Scholar
14. Shin, S.J., 1995, The Logical Status of Diagrams, Cambridge: Cambridge University Press.Google Scholar
15. Simon, H.A., 1996, The Sciences of the Artificial, third edition, Cambridge, MA: The MIT Press.Google Scholar
17. Stenning, K. and Oberlander, J., 1995, “A cognitive theory of graphical and linguistic reasoning: Logic and implementation,” Cognitive Science 19, 97–140.Google Scholar