Circuit complexity of a Boolean function is defined to be the minimum number of gates in circuits computing the function. In general, the circuit complexity is established by deriving two types of bounds on the complexity. On one hand, an upper bound is derived by showing a circuit, of the size given by the bound, to compute a function. On the other hand, a lower bound is established by proving that a function can not be computed by any circuit of the size. There has been much success in obtaining good upper bounds, while in spite of much efforts few progress has been made toward establishing strong lower bounds. In this paper, after surveying general results concerning circuit complexity for Boolean functions, we explain recent results about lower bounds, focusing on the method of approximation.

This paper presents a new connected associative memory neural network. In this network, a threshold function which has two dynamical parameters is introduced. After analyzing the dynamical behaviors and giving an upper bound of the memory capacity of the conventional connected associative memory neural network, it is demonstrated that these parameters play an important role in the recalling processes of the connected neural network. An approximate method of evaluationg their optimum values is given. Further, the optimum feedback stopping time of this network is discussed. Therefore, in our network, the recalling processes are ended at the optimum feedback stopping time whether a state energy has been local minimum or not. The simulations on computer show that the dynamical behaviors of our network are greatly improved. Even though the number of learned patterns is so large as the number of neurons, the statistical properties of the dynamical behaviors of our network are that the output series of recalling processes approach to the expected patterns on their initial inputs.

We review the role of optics in interconnects, analog processing, neural networks, and digital computing. The properties of low interference, massively parallel interconnections, and very high data rates promise extremely high performance for optical information processing systems.