Directed Projection Graph of N-Dimensional Hypercube and Subhypercube Decomposition of Balanced Linearly Separable Boolean Functions

2021 ◽  
Vol 31 (09) ◽  
pp. 2150138
Author(s):  
Wei Jin ◽  
Fangyue Chen ◽  
Qinbin He
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Boolean Function ◽  
Planar Graph ◽  
Boolean Functions ◽  
Conformal Transformation ◽  
Two Dimensional ◽  
Induced Subgraph ◽  
Projection Graph ◽  
Enumeration Scheme

A directed projection graph of the [Formula: see text]-dimensional hypercube on the two-dimensional plane is successfully created. Any [Formula: see text]-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on [Formula: see text]-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of [Formula: see text]-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.

scholarly journals REALIZATION AND BIFURCATION OF BOOLEAN FUNCTIONS VIA CELLULAR NEURAL NETWORKS

2005 ◽  
Vol 15 (07) ◽  
pp. 2109-2129 ◽  
Author(s):  
FANGYUE CHEN ◽  
GUANRONG CHEN
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Boolean Function ◽  
Boolean Functions ◽  
Cellular Neural Network ◽  
Cellular Neural Networks ◽  
Gene Bank ◽  
Engineering Applications ◽  
Binary Input

In this work, we study the realization and bifurcation of Boolean functions of four variables via a Cellular Neural Network (CNN). We characterize the basic relations between the genes and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set. Based on the analysis, we have rigorously proved that there are exactly 1882 linearly separable Boolean functions of four variables, and found an effective method for realizing all linearly separable Boolean functions via an uncoupled CNN. Consequently, any kind of linearly separable Boolean function can be implemented by an uncoupled CNN, and all CNN genes that are associated with these Boolean functions, called the CNN gene bank of four variables, can be easily determined. Through this work, we will show that the standard CNN invented by Chua and Yang in 1988 indeed is very essential not only in terms of engineering applications but also in the sense of fundamental mathematics.

Prediction of 2-D Lift Motions Using Artificial Neural Networks: An Evolutionary Approach

2005 ◽  
Vol 49 (14) ◽  
pp. 1405-1409
Author(s):  
Miguel A. Perez ◽  
Maury A. Nussbaum
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Network ◽  
Artificial Neural Networks ◽  
Model Performance ◽  
Human Movement ◽  
Two Dimensional ◽  
Modeling Paradigm ◽  
Neural Network Structure ◽  
Artificial Neural

Movement prediction is an important aspect of human simulation, where more efficient and accurate models are needed. Artificial neural networks could potentially serve as a modeling option in this realm. This investigation evaluates the performance of a particular artificial neural network structure in modeling sagittally symmetric two-dimensional lifting and lowering movements. Model performance was evaluated using three training datasets, each consisting of distinct representation levels of the overall dataset. Results are discussed in terms of their practical meaning, and suggestions for future improvements in the modeling scheme are provided. Overall, artificial neural networks show promise as a modeling paradigm for the prediction of human movement.

Sign in / Sign up

Export Citation Format

Share Document