Modular Neural Nets Research Articles

Integrating Fast Karnough Map and Modular Neural Networks for Simplification and Realization of Complex Boolean Functions

In this paper a new fast simplification method is presented. Such method realizes Karnough map with large number of variables. In order to accelerate the operation of the proposed method, a new approach for fast detection of group of ones is presented. Such approach implemented in the frequency domain. The search operation relies on performing cross correlation in the frequency domain rather than time one. It is proved mathematically and practically that the number of computation steps required for the presented method is less than that needed by conventional cross correlation. Simulation results using MATLAB confirm the theoretical computations. Furthermore, a powerful solution for realization of complex functions is given. The simplified functions are implemented by using a new desigen for neural networks. Neural networks are used because they are fault tolerance and as a result they can recognize signals even with noise or distortion. This is very useful for logic functions used in data and computer communications. Moreover, the implemented functions are realized with minimum amount of components. This is done by using modular neural nets (MNNs) that divide the input space into several homogenous regions. Such approach is applied to implement XOR function, 16 logic functions on one bit level, and 2-bit digital multiplier. Compared to previous nonmodular designs, a clear reduction in the order of computations and hardware requirements is achieved. Keywords—Boolean Functions, Simplification, Karnough Map, Implementation of Logic Functions, Modular Neural Networks.

Complexity results on learning by neural nets

We consider the computational complexity of learning by neural nets. We are interested in how hard it is to design appropriate neural net architectures and to train neural nets for general and specialized learning tasks. Our main result shows that the training problem for 2-cascade neural nets (which have only two non-input nodes, one of which is hidden) is N℘-complete, which implies that finding an optimal net (in terms of the number of non-input units) that is consistent with a set of examples is also N℘-complete. This result also demonstrates a surprising gap between the computational complexities of one-node (perceptron) and two-node neural net training problems, since the perceptron training problem can be solved in polynomial time by linear programming techniques. We conjecture that training a k-cascade neural net, which is a classical threshold network training problem, is also N℘-complete, for each fixed k≥2. We also show that the problem of finding an optimal perceptron (in terms of the number of non-zero weights) consistent with a set of training examples is N℘-hard. Our neural net learning model encapsulates the idea of modular neural nets, which is a popular approach to overcoming the scaling problem in training neural nets. We investigate how much easier the training problem becomes if the class of concepts to be learned is known a priori and the net architecture is allowed to be sufficiently non-optimal. Finally, we classify several neural net optimization problems within the polynomial-time hierarchy.

Complexity Results on Learning by Neural Nets

We consider the computational complexity of learning by neural nets. We are interested in how hard it is to design appropriate neural net architectures and to train neural nets for general and specialized learning tasks. Our main result shows that the training problem for 2-cascade neural nets (which have only two non-input nodes, one of which is hidden) is \(\Re P\)-complete, which implies that finding an optimal net (in terms of the number of non-input units) that is consistent with a set of examples is also \(\Re P\)-complete. This result also demonstrates a surprising gap between the computational complexities of one-node (perceptron) and two-node neural net training problems, since the perceptron training problem can be solved in polynomial time by linear programming techniques. We conjecture that training a k-cascade neural net, which is a classical threshold network training problem, is also \(\Re P\)-complete, for each fixed k ≥ 2. We also show that the problem of finding an optimal perceptron (in terms of the number of non-zero weights) consistent with a set of training examples is \(\Re P\)-hard.Our neural net learning model encapsulates the idea of modular neural nets, which is a popular approach to overcoming the scaling problem in training neural nets. We investigate how much easier the training problem becomes if the class of concepts to be learned is known a priori and the net architecture is allowed to be sufficiently non-optimal. Finally, we classify several neural net optimization problems within the polynomial-time hierarchy.