Measuring complexity in multidimensional systems with high degrees of freedom and a variety of types of information, remains an important challenge. The complexity of a system is related to the number and variety of components, the number and type of interactions among them, the degree of redundancy, and the degrees of freedom of the system. Examples show that different disciplines of science converge in complexity measures for low and high dimensional problems. For low dimensional systems, such as coded strings of symbols (text, computer code, DNA, RNA, proteins, music), Shannon’s Information Entropy (expected amount of information in an event drawn from a given distribution) and Kolmogorov‘s Algorithmic Complexity (the length of the shortest algorithm that produces the object as output), are used for quantitative measurements of complexity. For systems with more dimensions (ecosystems, brains, social groupings), network science provides better tools for that purpose. For highly complex multidimensional systems, none of the former methods are useful. Here, information related to complexity can be used in systems, ranging from the subatomic to the ecological, social, mental and to AI. Useful Information Φ (Information that produces thermodynamic free energy) can be quantified by measuring the thermodynamic Free Energy and/or useful Work it produces. Complexity can be measured as Total Information I of the system, that includes Φ, useless information or Noise N, and Redundant Information R. Measuring one or more of these variables allows quantifying and classifying complexity. Complexity and Information are two windows overlooking the same fundamental phenomenon, broadening out tools to explore the deep structural dynamics of nature at all levels of complexity, including natural and artificial intelligence.
