The threshold order of a Boolean function
Discrete Applied Mathematics
Surveys in combinatorics, 1993
Discrete neural computation: a theoretical foundation
Discrete neural computation: a theoretical foundation
Classification by polynomial surfaces
Discrete Applied Mathematics
On the degree of Boolean functions as real polynomials
Computational Complexity - Special issue on circuit complexity
Perceptrons, PP, and the polynomial hierarchy
Computational Complexity - Special issue on circuit complexity
Fundamentals of Artificial Neural Networks
Fundamentals of Artificial Neural Networks
Testing Basic Boolean Formulae
SIAM Journal on Discrete Mathematics
On computing Boolean functions by a spiking neuron
Annals of Mathematics and Artificial Intelligence
Algebraic Techniques for Constructing Minimal Weight Threshold Functions
SIAM Journal on Discrete Mathematics
Extremal properties of polynomial threshold functions
Journal of Computer and System Sciences
SIAM Journal on Computing
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Polynomial threshold gates are basic processing units of an artificial neural network. When the input vectors are binary vectors, these gates correspond to Boolean functions and can be analyzed via their polynomial representations. In practical applications, it is desirable to find a polynomial representation with the smallest number of terms possible, in order to use the least possible number of input lines to the unit under consideration. For this purpose, instead of an exact polynomial representation, usually the sign representation of a Boolean function is considered. The non-uniqueness of the sign representation allows the possibility for using a smaller number of monomials by solving a minimization problem. This minimization problem is combinatorial in nature, and so far the best known deterministic algorithm claims the use of at most 0.75 × 2n of the 2n total possible monomials. In this paper, the basic methods of representing a Boolean function by polynomials are examined, and an alternative approach to this problem is proposed. It is shown that it is possible to use at most 0.5 × 2n = 2n-1 monomials based on the {0, 1} binary inputs by introducing extra variables, and at the same time keeping the degree upper bound at n. An algorithm for further reduction of the number of terms that used in a polynomial representation is provided. Examples show that in certain applications, the improvement achieved by the proposed method over the existing methods is significant.