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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
Mathematics
  • 4 Downloads

Abstract

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.

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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

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