2 research outputs found
Heuristic algorithms for obtaining Polynomial Threshold Functions with low densities
In this paper we present several heuristic algorithms, including a Genetic
Algorithm (GA), for obtaining polynomial threshold function (PTF)
representations of Boolean functions (BFs) with small number of monomials. We
compare these among each other and against the algorithm of Oztop via
computational experiments. The results indicate that our heuristic algorithms
find more parsimonious representations compared to the those of non-heuristic
and GA-based algorithms.Comment: This paper will appear in the 13th Cologne-Twente Workshop on Graphs
& Combinatorial Optimizatio
Representing Boolean Functions Using Polynomials: More Can Offer Less
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 of the 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 monomials based on
the binary inputs by introducing extra variables, and at the same
time keeping the degree upper bound at . 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.Comment: A shorter version of this article appeared in LNCS 6677, 201