Doklady Mathematics

, Volume 98, Issue 3, pp 564–567 | Cite as

Dualization Problem over the Product of Chains: Asymptotic Estimates for the Number of Solutions

  • E. V. DjukovaEmail author
  • G. O. Maslyakov
  • P. A. Prokofjev


A key intractable problem in logical data analysis, namely, dualization over the product of partial orders, is considered. The important special case where each order is a chain is studied. If the cardinality of each chain is equal to two, then the considered problem is to construct a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form, which is equivalent to the enumeration of irreducible coverings of a Boolean matrix. The asymptotics of the typical number of irreducible coverings is known in the case where the number of rows in the Boolean matrix has a lower order of growth than the number of columns. In this paper, a similar result is obtained for dualization over the product of chains when the cardinality of each chain is higher than two. Deriving such asymptotic estimates is a technically complicated task, and they are required, in particular, for proving the existence of asymptotically optimal algorithms for the problem of monotone dualization and its generalizations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. S. Jonson, M. Yannakakis, and C. H. Papadimitriou, Inf. Process. Lett. 27, 119–123 (1988).CrossRefGoogle Scholar
  2. 2.
    M. L. Fredman and L. Khachiyan, J. Algorithms 21, 618–628 (1996).MathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Boros, K. Elbassioni, V. Gurvich, L. Khachiyan, and K. Makino, SIAM J. Comput. 31 (5), 1624–1643 (2002).MathSciNetCrossRefGoogle Scholar
  4. 4.
    E. V. Djukova, Dokl. Akad. Nauk SSSR 233 (4), 527–530 (1977).MathSciNetGoogle Scholar
  5. 5.
    E. V. Djukova and P. A. Prokofjev, Comput. Math. Math. Phys. 55 (5), 891–906 (2015).MathSciNetCrossRefGoogle Scholar
  6. 6.
    E. V. Djukova, G. O. Maslyakov, and P. A. Prokofjev, Mashin. Obuchenie Anal. Dannykh 3 (4), 239–249 (2017).Google Scholar
  7. 7.
    V. N. Noskov and V. A. Slepyan, Kibernetika, No. 1, 60–65 (1972).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • E. V. Djukova
    • 1
    Email author
  • G. O. Maslyakov
    • 2
  • P. A. Prokofjev
    • 3
  1. 1.Federal Research Center “Computer Science and Control,”Russian Academy of SciencesMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Mechanical Engineering Research InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations