Also included is a little help (emphasis mine) for determining an information content for an infinite bit string (as was suggested by Doctor Stochastic.)
Ergodic theory can be described as the statistical and qualitative behavior of measurable group and semigroup actions on measure spaces. The group is most commonly N, R, R+, and Z…
A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold)...
The deterministically chaotic systems constitute a true subset of the set of untreatable problems. Moreover, unique connections between dynamical system properties--e.g., nonlinearity, non-integrability, and dynamical instability on the one hand, and algorithmic complexity and limited long-time computability on the other hand--cannot be established. There are effectively treatable (i.e., algorithmically computable) systems which do not admit of a closed-form solution as a function of time (e.g., transcendental equations); there are nonlinear systems which are integrable (e.g., solitons); there are linear systems which are uncomputable, or untreatable (Leiber, 1996b, pp. 38-40); in the framework of mathematical ergodic theory it has been shown that algorithmic Kolmogorov complexity is not synonymous with dynamical instability (or "deterministic randomness; Batterman, 1996); also, exponential instability is compatible with well-posedness, i.e., with the existence of a closed-form solution. In summary, for the instability hierarchy of dynamical systems there is no comprehensive characterization available in terms of computability concepts.
Juergen Schmidhuber's home page - Artificial Intelligence - New AI - Recurrent neural networks
Kolmogorov's (left) complexity K(x) of a bitstring x is the length of the shortest program that computes x and halts. Solomonoff's algorithmic probability of x is the probability of guessing a program for x. Chaitin's Omega is the halting probability of a Turing machine with random input (Omega is known as the "number of wisdom" because it compactly encodes all mathematical truth). All of this I generalized to non-halting but converging programs. This led to the shortest possible formal descriptions and to non-enumerable but limit-computable measures and Super Omegas, and even has consequences for computable universes and optimal inductive inference.
Generalized Algorithmic Information - Generalized Algorithmic Probability - Super Omegas
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 `Super Omegas' 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.