Abstract
The class \(F_{n,k}\) of Boolean functions consisting of all functions of \(n\) variables each of which outputs \(1\) at exactly \(k\) \(n\)-tuples of values of the variables is considered. For small \(k\), for example, for \(k<\ln n\), an asymptotics for the complexity of implementation of every function in \(F_{n,k}\) by a circuit of functional elements in an irredundant basis containing \(x\to y\) and \(\overline{x\to y}\) is found.
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Notes
Here and in what follows, “\(\log\)” stands for logarithm base \(2\).
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Funding
This work was financially supported by the Russian Foundation for Basic Research under grant 18.01.00337 “Problems of synthesis, complexity, and reliability in the theory of control systems.”
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Red’kin, N.P. Asymptotics for the Complexity of Boolean Functions with Small Number of Ones. Math Notes 109, 256–261 (2021). https://doi.org/10.1134/S0001434621010296
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DOI: https://doi.org/10.1134/S0001434621010296