Representation and Measure of Structural Information

We introduce a uniform representation of general objects that captures the regularities with respect to their structure. It allows a representation of general class of objects such as geometric patterns and images as well as computation in a sparse, modular, hierarchical, and recursive manner. The intuitive complexity, or the amount of information in the object, corresponds to the size of the description. In particular, intuitively finite object has a finite description, even if its raw signal description is infinite, as in the case of geometric objects. A set of rules that does not depend on the instance of the object dictates how the described objects are related to the raw signal. Using the representation, we also introduce a measure of information in general objects relative to structures that are specified by a set of maps. We prove that the measure is equivalent to the Kolmogorov complexity in the case of strings.