Complex Systems and the Evolution of Computability

Abstract

The evolution of complexity in nature and society can be understood as the evolution of computational systems (Sect. 5.1). The present theory of computability enables us to distinguish complexity classes of problems, meaning the order of corresponding functions describing the computational time of their algorithms or computational programs (Sect. 5.2). Information dynamics in complex systems are analyzed by Shannon’s concept of information entropy and Kolmogorov-Sinai entropy. The degree of complexity of 1/f b noise can be linked to attractors in nonlinear dynamics (Sect. 5.3). In general, any stochastic process can be classified according to the degree of complexity of the probabilistic attractor. This offers deep insights into the power laws of complex systems, indicating the self-organization and emergence of order in nature and society (Sect. 5.4). Further on, we ask if more efficient information processing can be expected from quantum computers and quantum complexity theory (Sect. 5.5). Pattern formation in complex systems can be analyzed in the framework of cellular automata. Even chaos and randomness can be generated by simple rules of cellular automata. (Sect. 5.6).