Automated conjecturing III

Property-relations conjectures
  • C. E. LarsonEmail author
  • N. Van Cleemput


Discovery in mathematics is a prototypical intelligent behavior, and an early and continuing goal of artificial intelligence research. We present a heuristic for producing mathematical conjectures of a certain typical form and demonstrate its utility. Our program conjectures relations that hold between properties of objects (property-relation conjectures). These objects can be of a wide variety of types. The statements are true for all objects known to the program, and are the simplest statements which are true of all these objects. The examples here include new conjectures for the hamiltonicity of a graph, a well-studied property of graphs. While our motivation and experiments have been to produce mathematical conjectures—and to contribute to mathematical research—other kinds of interesting property-relation conjectures can be imagined, and this research may be more generally applicable to the development of intelligent machinery.


Automated conjecturing Automated mathematical discovery Property-relations conjectures 

Mathematics Subject Classification (2010)

05C45 05C69 05-04 


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The authors are grateful to the referees, whose comments helped us clarify and improve this paper.


  1. 1.
    Aouchiche, M., Caporossi, G., Hansen, P., Laffay, M.: Autographix: a survey. Electron. Notes Discret. Math. 22, 515–520 (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bagai, R., Shanbhogue, V., Żytkow, J.M., Chou, S.-C.: Automatic theorem generation in plane geometry. In: Methodologies for Intelligent Systems, pp 415–424. Springer (1993)Google Scholar
  3. 3.
    Bagai, R., Shanbhogue, V., Zytkow, J.M., Chou, S.-C.: Discovery of geometry theorems: avoiding isomorphic situation descriptions. In: Fifth International Conference on Computing and Information, 1993. Proceedings ICCI’93, pp. 354–358. IEEE (1993)Google Scholar
  4. 4.
    Borwein, J.M., Bailey, D.H.: Mathematics by experiment: Plausible reasoning in the 21st century. AK Peters Natick, MA (2004)zbMATHGoogle Scholar
  5. 5.
    Brigham, R., Dutton, R.: INGRID: A software tool for extremal graph theory research. Congr. Numer. 39, 337–352 (1983)zbMATHGoogle Scholar
  6. 6.
    Caporossi, G., Hansen, P.: Variable neighborhood search for extremal graphs: 1 the autographix system. Discret. Math. 212(1), 29–44 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caporossi, G., Hansen, P.: Variable neighborhood search for extremal graphs. 5. three ways to automate finding conjectures. Discret. Math. 276(1), 81–94 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Christophe, J., Dewez, S., Doignon, J.-P., Fasbender, G., Grégoire, P., Huygens, D., Labbé, M., Elloumi, S., Mélot, H., Yaman, H.: Linear inequalities among graph invariants: using GraPHedron to uncover optimal relationships. Networks 52(4), 287–298 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chvátal, V., Erdös, P.: A note on hamiltonian circuits. Discret. Math. 2 (2), 111–113 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Colton, S.: Refactorable numbers—a machine invention. J. Integer Sequences 2 (99.1), 2 (1999)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Colton, S.: Automated theory formation in pure mathematics. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  12. 12.
    Colton, S.: Automated conjecture making in number theory using HR, Otter and Maple. J. Symb. Comput. 39(5), 593–615 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Colton, S., Bundy, A., Walsh, T.: Automatic concept formation in pure mathematics. In: Proceedings of the 16th International Joint Conference on Artificial Intelligence, IJCAI’99, vol. 2, pp. 786–791. Morgan Kaufmann Publishers (1999)Google Scholar
  14. 14.
    Conforti, M., Cornuéjols, G., Kapoor, A., Vušković, K.: Even-hole-free graphs part i: Decomposition theorem. J. Graph Theory 39(1), 6–49 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cvetković, D., Gutman, I.: The computer system GRAPH: A useful tool in chemical graph theory. J. Comput. Chem. 7(5), 640–644 (1986)CrossRefGoogle Scholar
  16. 16.
    Davis, R., Lenat, D.B.: Knowledge-based systems in Artificial Intelligence. McGraw-Hill International Book Co. (1982)Google Scholar
  17. 17.
    DeLaVina, E., Pepper, R., Waller, W.: Independence, radius and Hamiltonian paths. MATCH Commun. Math. Comput. Chem. 58(2), 481–510 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dutton, R.D., Brigham, R.C., Gomez, F.: INGRID: A graph invariant manipulator. J. Symb. Comput. 7(2), 163–177 (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    Epstein, S.L.: On the discovery of mathematical theorems. In: IJCAI, pp 194–197 (1987)Google Scholar
  20. 20.
    Epstein, S.L.: Learning and discovery One system’s search for mathematical knowledge. Comput. Intell. 4(1), 42–53 (1988)CrossRefGoogle Scholar
  21. 21.
    Fajtlowicz, S.: On conjectures of Graffiti. V. In: Graph Theory, Combinatorics, and Algorithms, Vol. 1, 2 (Kalamazoo, MI, 1992), Wiley-Interscience Publishing, pp 367—376. Wiley, New York (1995)Google Scholar
  22. 22.
    Favaron, O., Mahéo, M., Saclé, J.-F.: On the residue of a graph. J. Graph Theory 15(1), 39–64 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gale, D.: A curious Nim-type game. American Mathematical Monthly, pp. 876–879 (1974)Google Scholar
  24. 24.
    Gould, R.J.: Updating the Hamiltonian problem—a survey. J. Graph Theory 15(2), 121–157 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gould, R.J.: Advances on the Hamiltonian problem–a survey. Graphs Comb. 19(1), 7–52 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Griggs, J.R., Kleitman, D.J.: Independence and the Havel-Hakimi residue. Discrete Math. 127(1-3), 209–212 (1994). Graph theory and applications (Hakone, 1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Larson, C.E.: A survey of research in automated mathematical conjecture-making. In: Graphs and Discovery, volume 69 of DIMACS Series Discrete Mathematics of Theoretical Computer Science, pp 297–318. American Mathematics Society, Providence, RI (2005)Google Scholar
  28. 28.
    Larson, C.E., Van Cleemput, N.: Automated conjecturing I: Fajtlowicz’s Dalmatian heuristic revisited. Artif. Intell. 231, 17–38 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lenat, D.B.: The ubiquity of discovery. Artif. Intell. 9(3), 257–285 (1977)CrossRefGoogle Scholar
  30. 30.
    Lenat, D.B.: On automated scientific theory formation: a case study using the am program. Mach. Intell. 9, 251–286 (1979)Google Scholar
  31. 31.
    Lenat, D.B.: The nature of heuristics. Artif. Intell. 19(2), 189–249 (1982)CrossRefGoogle Scholar
  32. 32.
    McCune, W.: Solution of the Robbins problem. J. Autom. Reason. 19(3), 263–276 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mélot, H.: Facet defining inequalities among graph invariants: the system GraPHedron. Discret. Appl. Math. 156(10), 1875–1891 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Simon, H.A., Newell, A.: Heuristic problem solving: The next advance in operations research. Oper. Res. 6(1), 1–10 (1958)CrossRefGoogle Scholar
  35. 35.
    Turing, A.: Intelligent machinery. The Essential Turing, pp 395–432 (2004)Google Scholar
  36. 36.
    Van Den Heuvel, J.: Hamilton cycles and eigenvalues of graphs. Linear Algebra Appl. 226, 723–730 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, H.: Toward mechanical mathematics. IBM J. Res. Develop. 4, 2–22 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wilf, H.S., Zeilberger, D.: Towards computerized proofs of identities. Bullet. Amer. Math. Soc. 23(1), 77–83 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA
  2. 2.Department of Applied Mathematics, Computer Science and StatisticsGhent UniversityGhentBelgium

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