Complex systems, complexity measures, grammars and model-inferring

Abstract

Quantifying complex behavior has been the subject of intense research during the last few years. In spite of these efforts there neither exists a universal measure characterizing the complexity of a given system, nor is it even clear whether such a measure can be constructed at all. We discuss some of the attempts that have been made to quantify the complexity of nonlinear chaotic systems. We show why these measures do not meet the needs for a general procedure applicable to arbitrary (stochastic) processes, and we discuss another measure of complexity which is well suited for the study of non-stationary systems. Guided by the properties of formal languages and grammars, we argue that it will be necessary first to distinguish different classes of complex behavior. Chomsky's classification of formal languages may serve as an example how such classes can be identified. The look at formal language theory additionally shows that one of the main features of complex systems is their ability of information processing, from simple storage to universal computing.