1 Meaningful Information

Abstract — The information in an individual finite object (like a binary string) is commonly measured by its Kolmogorov complexity. One can divide that information into two parts: the information accounting for the useful regularity present in the object and the information accounting for the remaining accidental information. There can be several ways (model classes) in which the regularity is expressed. Kolmogorov has proposed the model class of finite sets, generalized later to computable probability mass functions. The resulting theory, known as Algorithmic Statistics, analyzes the algorithmic sufficient statistic when the statistic is restricted to the given model class. However, the most general way to proceed is perhaps to express the useful information as a recursive function. The resulting measure has been called the “sophistication ” of the object. We develop the theory of recursive functions statistic, the maximum and minimum value, the existence of absolutely nonstochastic objects (that have maximal sophistication— all the information in them is meaningful and there is no residual randomness), determine its relation with the more restricted model classes of finite sets, and computable probability distributions, in particular with respect to the algorithmic (Kolmogorov) minimal sufficient statistic, the relation to the halting problem and further algorithmic properties. Topics: Computational and structural complexity; Kolmogorov complexit

Sophistication revisited

Kolmogorov complexity measures the ammount of information in a string as the size of the shortest program that computes the string. The Kolmogorov structure function divides the smallest program producing a string in two parts: the useful information present in the string, called sophistication if based on total functions, and the remaining accidental information. We formalize a connection between sophistication (due to Koppel) and a variation of computational depth (intuitively the useful or nonrandom information in a string), prove the existence of strings with maximum sophistication and show that they are the deepest of all strings.