GENERALIZED HOPFIELD NEURAL-NETWORK FOR CONCURRENT TESTING

This paper deals with testing of digital circuits, spectral techniques , and neural networks: It shows how an optimization problem that appea rs in the design of concurrent testable circuits can be expressed by u sing digital spectral techniques in such a way that makes it possible to compute an acceptable solution by using neural networks. To build a concurrent testable circuit, it is necessary to use some specific che cking circuitry for analyzing the outputs of that circuit while it is in operation. The goal is to find the checking circuitry that, with le ss hardware complexity than a given bound, allows us to detect as many errors as possible at the outputs of the circuit under test. This pap er provides a measure, the aliasing probability, to evaluate the perfo rmances of the checking circuitry. It also shows how, by using spectra l techniques based on the Reed-Muller transform, the aliasing probabil ity can be expressed as a function of the Reed-Muller coefficients. In this way, to obtain the checking circuitry means to select a set of R eed-Muller spectral coefficients, with less elements than a given boun d, that minimizes the aliasing probability: It is an NP-complete probl em as in the majority of combinatorial optimization problems. It has b een suggested [1] that neural networks, in particular the Hopfield neu ral network, may be used to solve combinatorial optimization problems, as it always moves toward decreasing an energy function that is quadr atic in terms of the processor states. To apply the neural networks to design the checking circuitry for concurrent testing, the aliasing pr obability has been used as an energy function, and the Hopfield neural network has been modified to have an associated energy function with any type of polynomial dependence on the processor states. This paper analyzes the so-called generalized Hopfield neural network, considers its application to solve optimization problems with an objective funct ion taking any nonlinear form, and, as an example, uses the generalize d Hopfield network in the design of the extra modules of a concurrent testable circuit.