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Properties of Boolean functions with the extremal number of prime implicants
I. P. Chukhrov Institute of Computer Aided Design RAS, 19/18, Vtoraya Brestskaya Street, 123056 Moscow, Russia
Abstract:
The known lower bound for the maximum number of prime implicants (of maximal faces) of a Boolean function differs from the upper bound by O(√n) times and is asymptotically attained on a symmetric belt function. To study the properties of extremal functions, subsets of functions are defined that have the minimum and maximum vertices of the maximum faces in the belts n/3±rn and 2n/3±rn, respectively. In this case, the fraction of the number of vertices in each layer is not less than εn and for any such vertex the fraction of the number of maximum faces of the maximum possible number is not less than εn. Conditions are obtained for εn and rn under which a Boolean function from such a subset has the number of prime implicants equal to the maximum value asymptotically or in order of growth of the maximum value. Bibliogr. 10.
Keywords:
Boolean function, prime implicant, maximal face, number of prime implicants, asymptotics, order of growth.
Received: 08.09.2021 Revised: 08.09.2021 Accepted: 17.11.2021
Citation:
I. P. Chukhrov, “Properties of Boolean functions with the extremal number of prime implicants”, Diskretn. Anal. Issled. Oper., 29:1 (2022), 74–93
Linking options:
https://www.mathnet.ru/eng/da1294 https://www.mathnet.ru/eng/da/v29/i1/p74
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Abstract page: | 159 | Full-text PDF : | 20 | References: | 59 | First page: | 7 |
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