HIERARCHIES OF GENERALIZED KOLMOGOROV COMPLEXITIES AND NONENUMERABLE UNIVERSAL MEASURES COMPUTABLE IN THE LIMIT
(some equations may be hard to read without knowledge of LATEX)

JÜRGEN SCHMIDHUBER
juergen@idsia.ch - http://www.idsia.ch/~ juergen
IDSIA, Galleria 2, 6928 Manno (Lugano), Switzerland

International Journal of Foundations of Computer Science 13(4):587-612, 2002.
Submitted 10 July 2001, accepted 21 August 2001. Edited by Ming Li.
(C) World Scientific Publishing Company

Original work: Dec 2000 (quant-ph/0011122)

Abstract:

The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomonoff-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the ``true'' information content of some (possibly infinite) bitstring $x$ is the size of the shortest nonhalting program that converges to $x$ and nothing but $x$ on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's ``number of wisdom'' Omega, that any approximable measure of $x$ is small for any $x$ lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of $x$ is small for any $x$ lacking a short enumerating program. We briefly mention consequences for universes sampled from such priors.

Keywords: computability in the limit, generalized algorithmic probability, generalized Kolmogorov complexity hierarchy, halting probability Omega, cumulatively enumerable measures, computable universes.