Acta Applicandae Mathematica

, Volume 68, Issue 1–3, pp 211–226

Arrangements of Hyperplanes and the Number of Threshold Functions

• A. A. Irmatov
Article

Abstract

Two approaches on estimating the number of threshold functions which were recently developed by the author are discussed. Let P(K,n) denote the number of threshold functions in K-valued logic. The first approach establishes that
$$P(K,n + 1) \geqslant \frac{1}{2}\left( {\mathop {K^{n - 1} }\limits_{\left\lfloor {n - 4 - 2\frac{n}{{\log _K n}}} \right\rfloor } } \right)P\left( {K,\left\lfloor {{\text{2}}\frac{n}{{\log _K n}} + 3} \right\rfloor } \right).$$
The key argument of investigation is the generalization of the result of Odlyzko on subspaces spanned by random selections of ±1-vectors. Let $$E_K = \{ 0,1 \ldots ,K - 1\}$$ and let E denote the set of all vectors $$w_i ,i = 1, \ldots ,K^n$$, which have the form $$(1,a_1 , \ldots ,a_n ),a_i \in E_K$$. Denote by $$\Lambda _n (K)$$ the number of all collections of different vectors $$(w_{i_1 } , \ldots ,w_{i_n } ),2 \leqslant i_1 , \ldots ,i_n \leqslant \mathbb{K}^n$$, such that, for any k,$$1 \leqslant k \leqslant n$$, the vector $$w_{i_k }$$ is minimal among all vectors from the set $$E \cap {\text{span}}(w_{i_k } , \ldots ,w_{i_n } )$$. The second approach is based on topology-combinatorical techniques and allows to establish the following inequality $$P(K,n) \geqslant 2\Lambda _n (K)$$.
arrangement of hyperplanes threshold function Möbius function

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