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On the kernel and shortest face complexes in the unit cube

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Abstract

Studying the extreme kernel face complexes of a given dimension, we obtain some lower estimates of the number of shortest face complexes in the n-dimensional unit cube. The number of shortest complexes of k-dimensional faces is shown to be of the same logarithm order as the number of complexes consisting of at most 2n−1 different k-dimensional faces if 1 ≤ k ≤ c · n and c < 0.5. This implies similar lower bounds for the maximum length of the kernel DNFs and the number of the shortest DNFs of Boolean functions.

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Authors and Affiliations

  1. Institute of Computer Aided Design, 2-ya Brestskaya ul. 19/18, Moscow, 123056, Russia

    I. P. Chukhrov

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  1. I. P. Chukhrov
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Correspondence to I. P. Chukhrov.

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Original Russian Text © I.P. Chukhrov, 2011, published in Diskretnyi Analiz i Issledovanie Operatsii, 2011, Vol. 18, No. 2, pp. 77–96.

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Chukhrov, I.P. On the kernel and shortest face complexes in the unit cube. J. Appl. Ind. Math. 6, 42–55 (2012). https://doi.org/10.1134/S1990478912010061

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