The Fractal Dimension of SAT Formulas

  • Carlos Ansótegui
  • Maria Luisa Bonet
  • Jesús Giráldez-Cru
  • Jordi Levy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8562)


Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there is not a precise definition of the notion of structure.

Recently, there have been some attempts to analyze this structure in terms of complex networks, with the long-term aim of explaining the success of SAT solving techniques, and possibly improving them.

We study the fractal dimension of SAT instances with the aim of complementing the model that describes the structure of industrial instances. We show that many industrial families of formulas are self-similar, with a small fractal dimension. We also show how this dimension is affected by the addition of learnt clauses during the execution of SAT solvers.


Fractal Dimension Variable Node Random Instance Learn Clause Random Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Albert, R., Jeong, H., Barabási, A.-L.: The diameter of the WWW. Nature 401, 130–131 (1999)CrossRefGoogle Scholar
  2. 2.
    Ansótegui, C., Bonet, M.L., Levy, J.: On the structure of industrial SAT instances. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 127–141. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Ansótegui, C., Bonet, M.L., Levy, J.: Towards industrial-like random SAT instances. In: IJCAI 2009, pp. 387–392 (2009)Google Scholar
  4. 4.
    Ansótegui, C., Giráldez-Cru, J., Levy, J.: The community structure of SAT formulas. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 410–423. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Barabasi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Kadioglu, S., Malitsky, Y., Sabharwal, A., Samulowitz, H., Sellmann, M.: Algorithm selection and scheduling. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 454–469. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Katsirelos, G., Simon, L.: Eigenvector centrality in industrial SAT instances. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 348–356. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Mandelbrot, B.B.: The fractal geometry of nature. Macmillan (1983)Google Scholar
  9. 9.
    Papadopoulos, F., Kitsak, M., Serrano, M., Bogu, M., Krioukov, D.: Popularity versus similarity in growing networks. Nature 489, 537–540 (2012)CrossRefGoogle Scholar
  10. 10.
    Song, C., Gallos, L.K., Havlin, S., Makse, H.A.: How to calculate the fractal dimension of a complex network: the box covering algorithm. Journal of Statistical Mechanics: Theory and Experiment 2007(03), P03006 (2007)Google Scholar
  11. 11.
    Song, C., Havlin, S., Makse, H.A.: Self-similarity of complex networks. Nature 433, 392–395 (2005)CrossRefGoogle Scholar
  12. 12.
    Walsh, T.: Search in a small world. In: IJCAI 1999, pp. 1172–1177 (1999)Google Scholar
  13. 13.
    Williams, R., Gomes, C.P., Selman, B.: Backdoors to typical case complexity. In: IJCAI 2003, pp. 1173–1178 (2003)Google Scholar
  14. 14.
    Xu, L., Hutter, F., Hoos, H.H., Leyton-Brown, K.: SATzilla: portfolio-based algorithm selection for SAT. J. Artif. Int. Res. 32(1), 565–606 (2008)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Carlos Ansótegui
    • 1
  • Maria Luisa Bonet
    • 2
  • Jesús Giráldez-Cru
    • 3
  • Jordi Levy
    • 3
  1. 1.DIEIUniv. de LleidaSpain
  2. 2.LSIUPCSpain
  3. 3.IIIA-CSICSpain

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