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Associative memory networks based on quaternionic Hopfield neural network are investigated in this paper. These networks are composed of quaternionic neurons, and input, output, threshold, and connection weights are represented in quaternions, which is a class of hypercomplex number systems. The energy function of the network and the Hebbian rule for embedding patterns are introduced. The stable states and their basins are explored for the networks with three neurons and four neurons. It is clarified that there exist at most 16 stable states, called multiplet components, as the degenerated stored patterns, and each of these states has its basin in the quaternionic networks.

This paper explores two types of multistate Hopfield neural networks, based on commutative quaternions that are similar to Hamilton's quaternions but with commutative multiplication. In one type of the networks, the state of a neuron is represented by two kinds of phases and one real number. The other type of the networks adopts the decomposed form of commutative quaternion, i.e., the state of a neuron consists of a combination of two complex values. We have investigated the stabilities of these networks, i.e., the energies monotonically decreases with respect to the changes of the network states.

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Asynchronous cellular arrays have gained attention as promising architectures for nanocomputers, because of their lack of a clock, which facilitates low power designs, and their regular structure, which potentially allows manufacturing techniques based on molecular self-organization. With the increase in integration density comes a decrease in the reliability of the components from which computers are built, and implementations based on cellular arrays are no exception to this. This paper advances asynchronous cellular arrays that are tolerant to transient errors in up to one third of the information stored by its cells. The cellular arrays require six rules to describe the interactions between the cells, implying less complexity of the cells as compared to a previously proposed (nonfault-tolerant) asynchronous cellular array that employs nine rules.