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In order for neural networks to learn complex languages or grammars, they must have sufficient computational power or resources to recognize or generate such languages. Though many approaches have been discussed, one ob- vious approach to enhancing the processing power of a recurrent neural network is to couple it with an external stack memory - in effect creating a neural network pushdown automata (NNPDA). This paper discusses in detail this NNPDA - its construction, how it can be trained and how useful symbolic information can be extracted from the trained network. In order to couple the external stack to the neural network, an optimization method is developed which uses an error function that connects the learning of the state automaton of the neural network to the learning of the operation of the external stack. To minimize the error function using gradient descent learning, an analog stack is designed such that the action and storage of information in the stack are continuous. One interpretation of a continuous stack is the probabilistic storage of and action on data. After training on sample strings of an unknown source grammar, a quantization procedure extracts from the analog stack and neural network a discrete pushdown automata (PDA). Simulations show that in learning deterministic context-free grammars - the balanced parenthesis language, 1*n0*n, and the deterministic Palindrome - the extracted PDA is correct in the sense that it can correctly recognize unseen strings of arbitrary length. In addition, the extracted PDAs can be shown to be identical or equivalent to the PDAs of the source grammars which were used to generate the training strings.

By looking closely at the dynamics of learning, it was discovered that for different input the states of network tended to cluster around three values plus the initial state. These four states can be considered as possible states of an actual finite state machine and the movement between these states as a function of input can be interpreted as the state transition of a state machine. This four state machine constructed is a perfect state machine that recognize the dual parity grammar. It recognizes dual parity strings with arbitrary length. This rule extraction generalization power is qualitatively different from that of the 'data interpolation' paradigm which is usually true for a feedforward neural net.