Complexity measures of sign matrices

What is complexity, and how should it be studied mathematically? In the interpretation that we adopt, there are several underlying common themes to complexity theories. The basic ground rules are these: There is a family F of some mathematical objects under consideration. The elements of some subset S ⊆F are deemed simple. Also, there are certain composition rules that allow one to put together objects in order to generate other objects in F . The complexity of an object is determined by the length of the shortest chain of steps to generate it from simple objects. In full generality one would want to get good estimates for all or many objects in the family F . Specifically, a major challenge is to be able to point out specific concrete objects that have high complexity. That is, elements that cannot be generated from simple objects using only a small number of composition steps. Arguably the currently most developed mathematical theory of complexity is to be found in the field of computational complexity. Typically (but