On the number of threshold functions

A.A. Irmatov


A Boolean function is called a threshold function if its truth domain is a part of the n-cube cut off by some hyperplane. The number of threshold functions of n variables P(2, n) was estimated in [1, 2, 3]. Obtaining the lower bounds is a problem of special difficulty. Using a result of the paper [4], Zuev in [3] showed that for sufficiently large n

P(2, n) > 2n²(1-10/ln n).

In the present paper a new proof which gives a more precise lower bound of P(2, n) is proposed, namely, it is proved that for sufficiently large n

P(2, n) > 2n²(1-7/ln n)P(2, [(7(n-1)ln2)/ln(n- 1)]).

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Discrete Mathematics and Applications provides the latest information on the development of discrete mathematics in Russia to a world-wide readership. The journal covers various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.