On Mathematics of Universal Computation with Generic Dynamical Systems

Book chapter, 2019

This chapter presents some challenging ideas regarding the possible use of dynamical systems for general-purpose computation, addressing possible strategies of inferring information about the computing capacity of such machines. It discusses some possibilities for developing a unifying theory of reservoir computing, which bridges rigorous mathematical thinking, the state of the art numerical sampling techniques, and the theory of dynamical systems. The chapter addresses the question of what one can actually compute with a reservoir computer. It provides a map of desired mathematical theorems that would be extremely useful for addressing the broad issues, and it as terra incognita of practical theorems. The chapter also presents some numerical examples, as an illustration of how validity tests of the probabilistic conditions from the previous section might be carried out. It explores the usual way of thinking about reservoir computing. The chapter also discusses generic open problems, focuses on a few specific ones, and also provides some fictitious theorems.