On the Information Carried by Programs About the Objects they Compute

Abstract

In computability theory and computable analysis, finite programs can compute infinite objects. Such objects can then be represented by finite programs. Can one characterize the additional useful information contained in a program computing an object, as compared to having the object itself? Having a program immediately gives an upper bound on the Kolmogorov complexity of the object, by simply measuring the length of the program, and such an information cannot usually be derived from an infinite representation of the object. We prove that bounding the Kolmogorov complexity of the object is the only additional useful information. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets. This article is an extended version of Hoyrup and Rojas (2015), including complete proofs and a new result (Theorem 9).