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On the co-absence of input terms in higher order neuron representation of boolean functions

Title	On the co-absence of input terms in higher order neuron representation of boolean functions
Author	Yapar, O., Öztop, Erhan
Publication Date:	2017
Publication Place	- Springer International Publishin
Subject	Boolean function, Higher order neuron, Sigma-pi neuron model, Polynomial sign representation, Weight elimination, Minimum fan-in representation
Type	Document
Language	English
Digital	Yes
Manuscript	No
Library:	Özyeğin University
Library Asset ID	2-s2.0-85021720828
Record ID	b4ae00a8-6bae-4376-a36d-930bbc7c4137
Library Location	Computer Science
Date	2017
Notes	Due to copyright restrictions, the access to the full text of this article is only available via subscription.
Sample Text	Boolean functions (BFs) can be represented by using polynomial functions when −1 and +1 are used represent True and False respectively. The coefficients of the representing polynomial can be obtained by exact interpolation given the truth table of the BF. A more parsimonious representation can be obtained with so called polynomial sign representation, where the exact interpolation is relaxed to allow the sign of the polynomial function to represent the BF value of True or False. This corresponds exactly to the higher order neuron or sigma-pi unit model of biological neurons. It is of interest to know what is the minimal set of monomials or input lines that is sufficient to represent a BF. In this study, we approach the problem by investigating the (small) subsets of monomials that cannot be absent as a whole from the representation of a given BF. With numerical investigations, we study low dimensional BFs and introduce a graph representation to visually describe the behavior of the two-element monomial subsets as to whether they cannot be absent from any sign representation. Finally, we prove that for any n-variable BF, any three-element monomial set cannot be absent as a whole if and only if all the pairs from that set has the same property. The results and direction taken in the study may lead to more efficient algorithms for finding higher order neuron representations with close-to-minimal input terms for Boolean functions.
DOI	10.1007/978-3-319-59081-3_43
Cilt	10262

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On the co-absence of input terms in higher order neuron representation of boolean functions

Author Yapar, O., Öztop, Erhan

Publication Date 2017

Publication Place - Springer International Publishin

Subject Boolean function, Higher order neuron, Sigma-pi neuron model, Polynomial sign representation, Weight elimination, Minimum fan-in representation

Type Document

Language English

Digital Yes

Manuscript No

Library Özyeğin University

Library Asset ID 2-s2.0-85021720828

Record ID b4ae00a8-6bae-4376-a36d-930bbc7c4137

Library Location Computer Science

Date 2017

Notes Due to copyright restrictions, the access to the full text of this article is only available via subscription.

Sample Text Boolean functions (BFs) can be represented by using polynomial functions when −1 and +1 are used represent True and False respectively. The coefficients of the representing polynomial can be obtained by exact interpolation given the truth table of the BF. A more parsimonious representation can be obtained with so called polynomial sign representation, where the exact interpolation is relaxed to allow the sign of the polynomial function to represent the BF value of True or False. This corresponds exactly to the higher order neuron or sigma-pi unit model of biological neurons. It is of interest to know what is the minimal set of monomials or input lines that is sufficient to represent a BF. In this study, we approach the problem by investigating the (small) subsets of monomials that cannot be absent as a whole from the representation of a given BF. With numerical investigations, we study low dimensional BFs and introduce a graph representation to visually describe the behavior of the two-element monomial subsets as to whether they cannot be absent from any sign representation. Finally, we prove that for any n-variable BF, any three-element monomial set cannot be absent as a whole if and only if all the pairs from that set has the same property. The results and direction taken in the study may lead to more efficient algorithms for finding higher order neuron representations with close-to-minimal input terms for Boolean functions.

DOI 10.1007/978-3-319-59081-3_43

Cilt 10262