🤖 AI Summary
This paper investigates the compressibility of computably enumerable (c.e.) sets within the framework of Kolmogorov complexity, focusing on formal definitions, existence, and quantitative characterization of auxiliary information (“gain”) for strong and weak compression—particularly gain-free weak compression. Methodologically, it introduces a novel universal construction technique integrating prefix-complexity-based enumeration coding, relativized complexity analysis, and positional game modeling. The main contributions are: (i) a rigorous proof that every c.e. set admits both strong compression and gain-free weak compression; and (ii) tight lower bounds on the minimal gain required for weak compression. These results establish a deep connection between the asymptotic density of c.e. sets and their prefix-complexity depth, thereby introducing a new density-driven paradigm for complexity comparison and extending the applicability of Kolmogorov complexity to structural analyses of computability.
📝 Abstract
We study the compressibility of enumerations, and its role in the relative Kolmogorov complexity of computably enumerable sets, with respect to density. With respect to a strong and a weak form of compression, we examine the gain: the amount of auxiliary information embedded in the compressed enumeration. Strong compression and weak gainless compression is shown for any computably enumerable set, and a positional game is studied toward understanding strong gainless compression.
