On the complexity of the disjunctive normal form of threshold functions

O. V. Shabanin

Diskretnaya Matematika, Tome 12 (2000) no. 2, pp. 85-92

Résumé

We consider the problem on estimating the complexity of the disjunctive normal form (d.n.f.) of threshold functions in $n$ variables, where the complexity is the minimal number of simple implicants in the representation of the d.n.f. It is known that the complexity of the d.n.f. of almost all threshold functions is no less than $n^2/\log_2 n$. We prove inequalities, which connect the complexity $L \nu(f)$ of the d.n.f. of a threshold function $f$ with the Chow parameters. By using these inequalities we show that for almost all threshold functions, for sufficiently large $n$, $$ \log_2 L\nu(f)>n-2\sqrt{2n\log_2 n}(1+\delta(n)), $$ where $\delta(n)$ is an arbitrary function such that $\delta(n)\to 0$ and $n\delta(n)\to \infty$ as $n\to\infty$.