An integer polynomial p of n variables is called a threshold gate for a Boolean function f of n variables if for all x∈{0,1}n f(x)=1 if and only if p(x) > 0. The weight of a threshold gate is the sum of its absolute values. In this paper we study how large a weight might be needed if we fix some function and some threshold degree. We prove 2Ω(22n/5) lower bound on this value. The best previous bound was 2Ω(2n/8) (Podolskii, 2009). In addition we present substantially simpler proof of the weaker 2Ω(2n/4) lower bound. This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.
The problem of realization of Boolean functions by initial Boolean automata with constant states and n inputs is considered. Initial Boolean automaton with constant states and n inputs is an initial automaton with output such that in all states output functions are n-ary constant Boolean functions 0 or 1. All sets of the maximum cardinality of n-ary Boolean functions realized by an initial Boolean automaton with two or three constant states provided to the possibility of an arbitrary order of input values is obtained.