Cellular automata and discrete neural networks

Abstract

Recent results about generalizations of the cellular automaton model to arbitrary digraphs and quasi-linear transition functions are surveyed. Computations by both types of devices can be abstractly regarded as dynamical systems on the Cantor set. This formulation is used as a basis for a systematic comparison of their computational power and their classification. Classifying cellular automata and neural networks on finite graphs by isomorphism of their dynamics diagraphs gives rise to difficult NP-hard and co NP-complete problems. On infinite locally finite graphs one obtains a proper generalization of Turing computability. Neural and automata networks on graphs of finite bandwidth are just as powerful as cellular automata. Computable self-maps of the Cantor set play here the role that partial recursive functions play in classical computability. A characterization of functions computable by neural networks (or equivalently, cellular automata) on a Cantor set in one time step is given. Finally, some open problems are briefly discussed.