Extracting finite-state representations from recurrent neural networks trained on chaotic symbolic sequences.

While much work has been done in neural-based modeling of real-valued chaotic time series, little effort has been devoted to address similar problems in the symbolic domain. We investigate the knowledge induction process associated with training recurrent neural networks (RNN's) on single long chaotic symbolic sequences. Even though training RNN's to predict the next symbol leaves the standard performance measures such as the mean square error on the network output virtually unchanged, the networks nevertheless do extract a lot of knowledge. We monitor the knowledge extraction process by considering the networks stochastic sources and letting them generate sequences which are then confronted with the training sequence via information theoretic entropy and cross-entropy measures. We also study the possibility of reformulating the knowledge gained by RNN's in a compact and easy-to-analyze form of finite-state stochastic machines. The experiments are performed on two sequences with different "complexities" measured by the size and state transition structure of the induced Crutchfield's epsilon-machines. We find that, with respect to the original RNN's, the extracted machines can achieve comparable or even better entropy and cross-entropy performance. Moreover, RNN's reflect the training sequence complexity in their dynamical state representations that can in turn be reformulated using finite-state means. Our findings are confirmed by a much more detailed analysis of model generated sequences through the statistical mechanical metaphor of entropy spectra. We also introduce a visual representation of allowed block structure in the studied sequences that, besides having nice theoretical properties, allows on the topological level for an illustrative insight into both RNN training and finite-state stochastic machine extraction processes.