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Properties of Boolean functions without three-argument implicents

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Published/Copyright: November 25, 2014
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 24 Issue 6

Abstract

Let Fn be the class of all Boolean n-argument functions and let Mn⊂Fn be the class of functions that do not admit implicents of three or less essential variables. Next, let n(3)max (m) be the largest n for which the class Mn contains a function of weight m. It will be shown that n(3)max (11) = 5, n(3)max (12) = 11, n(3)max (13) = 11 (heretofore, the exact values were known only for m ≤ 10). The tensor product of matrices is used to define the binary operation ξ:Fn1 × Fn2 → Fn1-n2. Moreover, it is shown that if one of the arguments ξ belongs to Mn1 and the other is a Boolean function f (x1, ..., xn2) such that f(x1, ...,xn2)⋁f(x̄1, ...,x̄n2) Belongs to Mn2, then the result of the operation always belongs to Mn1-n2. With relaxed conditions, the result of ξ may fail to belong to Mn1-n2. The operation ξ is used in the algorithm for construction of a Boolean functions without three-argument implicents with arbitrary small ratio of the weight of the function to the number of its variables.

Keywords: Boolean function; implicent; combinatorially complete matrix; combinatorially sufficient matrix
Published Online: 2014-11-25
Published in Print: 2014-12-1

© 2014 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the asymptotic normality of the number of empty cells in a scheme of group allocation of particles
  3. Branching processes in a Markov random environment
  4. Properties of Boolean functions without three-argument implicents
  5. On the fraction of matrices with maximal additive complexity
  6. On the maximum size of a tree in the Galton–Watson forest with a bounded number of vertices
  7. Lower bounds of temporal and spatial complexity of the substring search problem
https://doi.org/10.1515/dma-2014-0030
Keywords for this article
Boolean function; implicent; combinatorially complete matrix; combinatorially sufficient matrix

Articles in the same Issue

  1. Frontmatter
  2. On the asymptotic normality of the number of empty cells in a scheme of group allocation of particles
  3. Branching processes in a Markov random environment
  4. Properties of Boolean functions without three-argument implicents
  5. On the fraction of matrices with maximal additive complexity
  6. On the maximum size of a tree in the Galton–Watson forest with a bounded number of vertices
  7. Lower bounds of temporal and spatial complexity of the substring search problem
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