Binary strings of finite VC dimension

The complexity of a string can be measured by the richness of its substrings. For example in genetics a region of DNA is considered to be highly informative if many of the possible substrings of a certain length actually occur. Abstractly this kind of complexity is captured by the standard string complexity function. When dealing with binary strings, we have the additional feature that substrings can be viewed as subsets of an index set. This allows us to apply measures of subset complexity such as VC dimension. In this paper we define a notion of VC dimension for binary strings and investigate the structure of strings of finite VC dimension.