On the information carried by programs about the objects they compute

TL;DR

This paper explores the additional information programs provide about computable objects, showing it corresponds to upper bounds on Kolmogorov complexity and linking Markov- and Type-2-computability.

Contribution

It characterizes the extra information in programs as Kolmogorov complexity bounds and establishes the relationship between Markov- and Type-2-computability.

Findings

Programs encode upper bounds on Kolmogorov complexity.

Relationship between Markov- and Type-2-computability is clarified.

Results on the structure of Markov-semidecidable sets are obtained.

Abstract

In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an infinite tape. What additional information do programs provide? We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object. 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.