Parity with two layer feedforward nets

Universal perceptron and DNA-like learning algorithm for binary neural networks: LSBF and PBF implementations

Universal perceptron (UP), a generalization of Rosenblatt's perceptron, is considered in this paper, which is capable of implementing all Boolean functions (BFs). In the classification of BFs, there are: 1) linearly separable Boolean function (LSBF) class, 2) parity Boolean function (PBF) class, and 3) non-LSBF and non-PBF class. To implement these functions, UP takes different kinds of simple topological structures in which each contains at most one hidden layer along with the smallest possible number of hidden neurons. Inspired by the concept of DNA sequences in biological systems, a novel learning algorithm named DNA-like learning is developed, which is able to quickly train a network with any prescribed BF. The focus is on performing LSBF and PBF by a single-layer perceptron (SLP) with the new algorithm. Two criteria for LSBF and PBF are proposed, respectively, and a new measure for a BF, named nonlinearly separable degree (NLSD), is introduced. In the sense of this measure, the PBF is the most complex one. The new algorithm has many advantages including, in particular, fast running speed, good robustness, and no need of considering the convergence property. For example, the number of iterations and computations in implementing the basic 2-bit logic operations such as AND, OR, and XOR by using the new algorithm is far smaller than the ones needed by using other existing algorithms such as error-correction (EC) and backpropagation (BP) algorithms. Moreover, the synaptic weights and threshold values derived from UP can be directly used in designing of the template of cellular neural networks (CNNs), which has been considered as a new spatial-temporal sensory computing paradigm.

Minimal Feedforward Parity Networks Using Threshold Gates

This article presents preliminary research on the general problem of reducing the number of neurons needed in a neural network so that the network can perform a specific recognition task. We consider a single-hidden-layer feedforward network in which only McCulloch-Pitts units are employed in the hidden layer. We show that if only interconnections between adjacent layers are allowed, the minimum size of the hidden layer required to solve the n-bit parity problem is n when n ≤ 4.

On the exact solution of the Parity-N problem using ordered neural networks

In this paper, an exact solution of the Parity-N function representation problem is provided using a fully connected feedforward (ordered) neural network. The solution requires neither training nor adaptation, and employs (N - 1) neurons. It is shown that any function sigma: R --> R with the property: sigma(- 1) = sigma(1) = 1 and sigma(0) = 0, can be used as the neural network activation function.

Solving the N-bit parity problem using neural networks

In this letter, a constructive solution to the N-bit parity problem is provided with a neural network that allows direct connections between the input layer and the output layer. The present approach requires no training and adaptation, and thus it warrants the use of the simple threshold activation function for the output and hidden layer neurons. It is previously shown that this choice of activation function and network structure leads to several solutions for the 3-bit parity problem obtained using linear programming. One of the solutions for the 3-bit parity problem is then generalized to obtain a solution for the N-bit parity problem using left floor N/2 right floor hidden layer neurons. It is shown that through the choice of a "staircase" type activation function, the left floor N/2 right floor hidden layer neurons can be further combined into a single hidden layer neuron.

Analysis of clinical data using neural nets

Clinical studies investigate the interdependence of dosing regimen, efficacy, and side effects. These relationships often involve complex dynamical functions. The study of phenomena intermediate between dosing and efficacy, e.g., pharmacokinetics (PK), helps one identify and understand these interdependences. However, efficacy still tends to be a complicated function of PK parameters, and indeed these parameters are becoming more complex as a function of dosing regimen, viz., studies involving immunosuppressants, biotechnology drugs, sophisticated delivery systems, and dosing strategies. Stationary and time-dependent neural nets can help one identify and model such unknown complex dynamical functions with few assumptions and limited data (1-7). Neural nets can relate dosing directly to efficacy, dosing to PK, PK to efficacy, or any component in the complex associations among treatments, pharmacodynamics, efficacy, and side effects. Neural nets can also assist one in the design of clinical trials involving complex and sophisticated procedures, e.g., randomized controlled clinical trials.