Information and Complexity, Or: Where Is the Information?

We demonstrate how a systematic theory of complexity emerges from information theoretical concepts. The complexity of a structure may refer to the difficulty of its description, the encoding of its regularities or the relations between its elements, components or parts. All such measures can be and usually are quantified with the help of information theoretical concepts. We first describe those concepts and then use them to analyze how complexity emerges from interactions between parts or conversely, can be decomposed into individual, joint, or complementary contributions of those parts. We can also use these theoretical concepts to understand the interaction between a system and its environment and the relations between different levels of aggregation in complex systems.