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A multilayer neural network based on multi-valued neurons (MLMVN) is considered in the paper. A multi-valued neuron (MVN) is based on the principles of multiple-valued threshold logic over the field of the complex numbers. The most important properties of MVN are: the complexvalued weights, inputs and output coded by the kth roots of unity and the activation function, which maps the complex plane into the unit circle. MVN learning is reduced to the movement along the unit circle, it is based on a simple linear error correction rule and it does not require a derivative. It is shown that using a traditional architecture of multilayer feedforward neural network (MLF) and the high functionality of the MVN, it is possible to obtain a new powerful neural network. Its training does not require a derivative of the activation function and its functionality is higher than the functionality of MLF containing the same number of layers and neurons. These advantages of MLMVN are confirmed by testing using parity n, two spirals and "sonar" benchmarks and the Mackey-Glass time series prediction.

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A method for optimization of Fixed Polarity ReedMuller expressions (FPRM) using the dual polarity property has been presented in [7]. In [2], this method has been extended to optimization of Kronecker expressions by introducing the notion of extended dual polarity property. In this paper, we propose a generalization of this method to optimization of Fixed polarity Galois field (GF) expressions for quaternary functions. The proposed method exploits a simple relationship between fixed polarity GF expressions for dual polarities.

Genetic algorithm behavior is determined by the explorationrexploitation balance kept throughout the run. When this balance is disproportionate, the premature convergence problem will probably appear, causing a drop in the genetic algorithm's efficacy. One approach presented for dealing with this problem is the distributed genetic algorithm model. Its basic idea is to keep, in parallel, several subpopulations that are processed by genetic algorithms, with each one being independent from the others. Furthermore, a migration operator produces a chromosome exchange between the subpopulations. Making distinctions between the subpopulations of a distributed genetic algorithm by applying genetic algorithms with different configurations, we obtain the so-called heterogeneous distributed genetic algorithms. In this paper, we present a hierarchical model of distributed genetic algorithms in which a higher level distributed genetic algorithm joins different simple distributed genetic algorithms. Furthermore, with the union of the hierarchical structure presented and the idea of the heterogeneous distributed genetic algorithms, we propose a type of heterogeneous hierarchical distributed genetic algorithms, the hierarchical gradual distributed genetic algorithms. Experimental results show that the proposals consistently outperform equivalent sequential genetic algorithms and simple distributed genetic algorithms.

In this paper, we introduce the interior-outer-set model for calculating a fuzzy risk represented by a possibility-probability distribution. The model involving combination calculus is very difficult to follow. In this paper, we transform it into a matrix algorithm. Although the algorithm is still difficult to follow, fortunately, it is easy to make a computer program for realizing. This algorithm consists of MOVING-subalgorithm and INDEX-subalgorithm. The former works out leaving and joining matrices. The latter is a combination algorithm to get index sets. An example is presented showing how a user can calculate a risk of strong earthquake with the algorithm.

In this paper, we use the information matrix technique to extract fuzzy if-then rules from data including noise. With a normal diffusion function, we change all crisp observations of a given sample into fuzzy sets to make an information matrix. We extract rules according to the centroids of the rows of an information matrix. These rules are integrated into an additive fuzzy system with the same rule weight. Such fuzzy systems can be used as adaptive function approximators. Simulations show that this method is very effective compared with the conventional least-squares method and neural network. The best advantage of the suggested method is that, it may be the simplest way to extract fuzzy if-then rules from data. r 2004 Elsevier Inc. All rights reserved.