A Program-Size Complexity Measure for Mathematical Problems and Conjectures

  • Michael J. Dinneen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7160)


Cristian Calude et al. in [5] have recently introduced the idea of measuring the degree of difficulty of a mathematical problem (even those still given as conjectures) by the length of a program to verify or refute the statement. The method to evaluate and compare problems, in some objective way, will be discussed in this note. For the practitioner, wishing to apply this method using a standard universal register machine language, we provide (for the first time) some “small” core subroutines or library for dealing with array data structures. These can be used to ease the development of full programs to check mathematical problems that require more than a predetermined finite number of variables.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Calude, C.S.: Information and Randomness: An Algorithmic Perspective, 2nd edn. Revised and Extended. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  2. 2.
    Calude, C.S., Calude, E.: Evaluating the Complexity of Mathematical Problems. Part 1. Complex Systems 18, 267–285 (2009)MathSciNetMATHGoogle Scholar
  3. 3.
    Calude, C.S., Calude, E.: Evaluating the Complexity of Mathematical Problems. Part 2. Complex Systems 18, 387–401 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Calude, C.S., Calude, E.: The Complexity of the Four Colour Theorem. LMS J. Comput. Math. 13, 414–425 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Calude, C.S., Calude, E., Dinneen, M.J.: A new measure of the difficulty of problems. Journal of Multiple-Valued Logic and Soft Computing 12, 285–307 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Calude, C.S., Calude, E., Svozil, K.: The complexity of proving chaoticity and the Church-Turing Thesis. Chaos 20, 1–5 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Calude, C.S., Dinneen, M.J.: Exact approximations of Omega numbers. Intl. J. of Bifurcation and Chaos 17(6), 1937–1954 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Calude, C.S., Dinneen, M.J., Shu, C.-K.: Computing a glimpse of randomness. Experimental Mathematics 11(2), 369–378 (2002)MathSciNetMATHGoogle Scholar
  9. 9.
    Calude, C.S., Jürgensen, H., Legg, S.: Solving Finitely Refutable Mathematical Problems. In: Calude, C.S., Păun, G. (eds.) Finite Versus Infinite. Contributions to an Eternal Dilemma, pp. 39–52. Springer, London (2000)CrossRefGoogle Scholar
  10. 10.
    Calude, E.: The complexity of Riemann’s Hypothesis. Journal for Multiple-Valued Logic and Soft Computing (December 2010) (accepted)Google Scholar
  11. 11.
    Calude, E.: Fermat’s Last Theorem and chaoticity. Natural Computing (accepted, June 2011)Google Scholar
  12. 12.
    Calude, E.: The complexity of Goldbach’s Conjecture and Riemann’s Hypothesis, Report CDMTCS-370, Centre for Discrete Mathematics and Computer Science, p. 9 (2009)Google Scholar
  13. 13.
    Chaitin, G.J.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987) (third printing, 1990)CrossRefMATHGoogle Scholar
  14. 14.
    Dinneen, M.J., Gimel’farb, G., Wilson, M.C.: Introduction to Algorithms, Data Structures and Formal Languages, 2nd edn., p. 264. Pearson Education, New Zealand (2009)Google Scholar
  15. 15.
    Knuth, D.E.: The Art of Computer Programming, 1st edn. Combinatorial Algorithms, Part 1, vol. 4A, p. xv+883. Addison-Wesley, Reading (2011) ISBN 0-201-03804-8 Google Scholar
  16. 16.
    Hertel, J.: On the Difficulty of Goldbach and Dyson Conjectures, Report CDMTCS-367, Centre for Discrete Mathematics and Theoretical Computer Science, p. 15 (2009)Google Scholar
  17. 17.
    Hilbert, D.: Mathematical Problems. Bull. Amer. Math. Soc. 8, 437–479 (1901-1902)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Weisstein, E.W.: Chaitin’s Constant. From MathWorld–A Wolfram Web Resource (2011), http://mathworld.wolfram.com/ChaitinsConstant.html

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael J. Dinneen
    • 1
  1. 1.Department of Computer ScienceUniversity of AucklandNew Zealand

Personalised recommendations