It is well known that the Hopfield Model (HM) for neural networks to solve the Traveling Salesman Problem (TSP) suffers from three major drawbacks. (1) It can converge on nonoptimal locally minimum solutions. (2) It can converge on infeasible solutions. (3) Results are very sensitive to the careful tuning of its parameters. A number of methods have been proposed to overcome (a) well. In contrast, work on (b) and (c) has not been sufficient; techniques have not been generalized to more general optimization problems. Thus this paper mathematically resolves (b) and (c) to such an extent that the resolution can be applied to solving with some general network continuous optimization problems including the Hopfield version of the TSP. It first constructs an Extended HM (E-HM) that overcomes both (b) and (c). Fundamental techniques of the E-HM lie in the addition of a synapse dynamical system cooperated with the current HM unit dynamical system. It is this synapse dynamical system that makes the TSP constraint hold at any final states for whatever choices of the IIM parameters and an initial state. The paper then generalizes the E-HM further to a network that can solve a class of continuous optimization problems with a constraint equation where both of the objective function and the constraint function are nonnegative and continuously differentiable.

The paper develops a mathematical framework for solving dynamic optimization problems with adaptive networks (AN's) based on Hopfield networks. The dynamic optimization problem (DOP) includes a dynamic traveling salesman problem (TSP), in which the distance between any pair of cities in the conventional TSP is extended into a time variable. Compared to previous deterministic networks, such as the Hopfield network, the adaptive network has the most distinguished feature: it can change its states, continually reacting to inputs from the outside environment. From the scientific viewpoint, our framework demonstrates mathematically rigorously that the adaptive network produces as final states locally minimum solutions to the DOP. From the engineering viewpoint, it provides a mathematical basis for developing engineering devices, such as very large scale integration (VLSI), that can solve real world DOP's efficiently.

A variety of real-world problems can be formulated as continuous optimization problems with variable constraint. It is well-known, however, that it is difficult to develop a unified method for obtaining their feasible solutions. We have recognized that the recent work of solving the traveling salesman problem (TSP) by the Hopfield model explores an innovative approach to them as well as combinatorial optimization problems. The Hopfield model is generalized into the Cohen-Grossberg model (CGM) to which a specific Lyapunov function has been found. This paper thus extends the Hopfield method onto the CGM in order to develop a unified solving-method of continuous optimization problems with variable-constraint. Specifically, we consider a certain class of continuous optimization problems with a constraint equation including the Hopfield version of the TSP as a particular member. Then we theoretically develop a method that, from any given problem of that class, derives a network of an extended CGM to provide feasible solutions to it. The main idea for constructing that extended CGM lies in adding to it a synapse dynamical system concurrently operating with its current unit dynamical system so that the constraint equation can be enforced to satisfaction at final states. This construction is also motivated by previous neuron models in biophysics and learning algorithms in neural networks.

We develop a neural network model that relieves time-consuming trial-and-error computer experiments usually performed in problem-solving with networks where problems, including the traveling salesman problem, pattern matching and pattern classification/learning, are formulated as optimization problems with constraint. First, we specify and uniquely distinguish the model as a set of constituent functions that should comply with restrictive conditions. Next, we demonstrate that it is unified, i.e., it yields most current networks. Finally, we verify that it is constructive, that is, we show a standard method that systematically constructs from a given optimization problem a particular network in that model to solve it.

This paper develops a theory for constructing 3-layered networks. The theory allows one to specify a finite discrete set of training data and a network structure (minimum intermediate units, synaptic weights and biases) that generalizes and approximates any given continuous mapping between sets of contours on a plane within any given permissible error.

Two fuzzy database query languages are proposed. They are used to query fuzzy databases that are enhanced from relational databases in such a way that fuzzy sets are allowed in both attribute values and truth values. A fuzzy calculus query language is constructed based on the relational calculus, and a fuzzy algebra query language is also constructed based on the relational algebra. In addition, a fuzzy relational completeness theorem such that the languages have equivalent expressive power is proved.>

A fuzzy query language (FQL) for relational databases is proposed. FQL, constructed as an enhancement of the relational domain calculus, has sufficient capability to represent all four types of the fuzzy statements distinguished by the work of L.A. Zadeh (1978). The idea for constructing FQL is in the formulation of the fuzzy matching degrees that assigns the appropriate values in the interval to any combination of fuzzy queries and tuples in relational databases. FQL helps to provide a human-oriented interface to relational databases that store a vast amount of information. Furthermore, fuzzy expert systems are expected to be provided with the facility to make use of fact data in relational databases through FQL. In addition, FQL is a theoretical basis for systematically developing a higher human-oriented interface with relational databases.>

This correspondence describes an algorithm for generating a set of test items and a test sequence which can be used at a minimal cost for testing protocol implementations for a layered protocol. "Set" and "weight" concepts are introduced and "test cost" is defined as the weighted sum of directed arcs.