Extremely Complex Problems



In this chapter the problem of rectangle-free assignments of four colors to a grid will be explored that also can be considered as a problem of Ramsey numbers. It will also be shown that it is not sufficient to use logic equations; many other mathematical concepts also have to be used in order to solve the problem. We started this research because we wanted to know how far the power of logic equations and of ternary vectors will reach to solve a problem of this extreme complexity. The successful solution of such a problem can be taken as the borderline for the solution of similar problems.


  1. 2.
    Barrow, J.D.: The Constants of Nature: The Numbers That Encode the Deepest Secrets of the Universe. Knopf Doubleday Publishing Group, New York (2009). ISBN: 978-0-3754-2221-8zbMATHGoogle Scholar
  2. 3.
    Biere, A.: Lingeling, Plingeling, PicoSAT and PrecoSAT at SAT Race 2010. Technical report 1. Linz, Aug. 2010Google Scholar
  3. 4.
    Biere, A., et al. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009). ISBN: 978-1-58603-929-5Google Scholar
  4. 15.
    Gebser, M., et al.: clasp: a conflict-driven answer set solver. In: Baral, C., Brewka, G., Schlipf, J. (eds.) International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2007). Lecture Notes in Computer Science, vol. 4483, pp. 260–265. Springer, Berlin (2007). ISBN: 978-3-540-72199-4. CrossRefGoogle Scholar
  5. 25.
    Planck Collaboration. Planck 2013 results. I. Overview of products and scientific results. Astron. Astrophys. 571(11), 1–44 (2014).
  6. 56.
    Steinbach, B., Posthoff, C.: Utilization of permutation classes for solving extremely complex 4-colorable rectangle-free grids. In: Proceedings of the IEEE 2012 International Conference on Systems and Informatics ICSAI, Yantai, May 2012, pp. 2361–2370. ISBN: 978-1-4673-0197-8.
  7. 60.
    Steinbach, B., Posthoff, C.: Highly complex 4-colored rectangle-free grids – solution unsolved multiple-valued problems. J. Multiple-Valued Logic and Soft Comput. 24(1–4), 369–404 (2014). ISSN: 1542-3980Google Scholar
  8. 63.
    Steinbach, B., Posthoff, C.: The last unsolved four-colored rectangle- free grid: the solution of extremely complex multiple-valued problems. J. Multiple-Valued Logic Soft Comput. 25(4–5), 461–490 (2015). ISSN: 1542-3980Google Scholar
  9. 74.
    Steinbach, B., Wessely, W., Posthoff, C.: Several approaches to parallel computing in the Boolean domain. In: Chaudhuri, P., et al. (eds.) First International Conference on Parallel, Distributed and Grid Computing, PDGC 1, Solan, H.P., Oct. 2010, pp. 6–11Google Scholar
  10. 75.
    Steinbach, B., Posthoff, C., Wessely, W.: Approaches to shift the complexity limitations of Boolean problems. In: Computer-Aided Design of Discrete Devices CAD DD 2010, Proceedings of the Seventh International Conference. CAD DD 7, Minsk, Belarus, Nov. 2010, pp. 84–91. ISBN: 978-985-6744-63-4Google Scholar

Copyright information

© Springer International Publishing AG 2019

Authors and Affiliations

  1. 1.Computing and Information TechnologyUniversity of the West Indies (retired)ChemnitzGermany
  2. 2.Computer ScienceTU Bergakademie Freiberg (retired)ChemnitzGermany

Personalised recommendations