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Random objects may be highly structured, in fact more so than non-random ones. There's no contradiction between Chaitin and Wolfram here. The whole field of "self-orgainzed complexity" is devoted to the emergence of complex structures from "random" causes. Highly structured means lots of information means high complexity. It takes lots of information to describe (for example) a random scattering of airplane parts, but the assembly manual (from Boeing?) describes an assembled plane easily.

We must be careful not to confuse complexity with usefulness. For example, the probabilty of getting a poker hand consisting of the Heart Ace, King, Queen, Jack , and Ten is equal to the probability of getting the Spade Two, Heart Jack, Diamond Six, Club Five, and Club Trey. The second hand is "worth" less in poker, not because of its probability, but because of the classes we assign hands to. We may treat all groups of scattered airplane parts as equivalent (not assembled into a plane) even though each group differes from the others by as much as they do from the working plane.

Thank you so much for the post and for all the information and leads!

Since my hypothesis is algorithm at inception is proof of intelligent design - I’m very much interested in self-organizing complexity. In that regard, Rocha’s work is especially engaging because he suggests how self organizing systems might evolve the symbols necessary for syntactic autonomy!

I've been off researching since reading your post, trying to figure this out myself - but it appears I will need your help:

Random objects may be highly structured, in fact more so than non-random ones. There's no contradiction between Chaitin and Wolfram here.

Mathematically speaking, “structure” doesn’t comport with what I understand to be the definition of algorithmic randomness Randomness and Complexity in Pure Mathematics – Chaitin - i.e. algorithmically irreducible information.

I also was under the impression that the consequence of initial random states was still at issue with Cellular Automata. Perhaps I misread. Twenty Problems in the Theory of Cellular Automata - i.e. problem 10, What is the correspondence between cellular automata and stochastic systems?

Cellular automata satisfy deterministic rules. But their initial states can have a random form. And the patterns they generate can have many of the properties of statistical randomness. As a consequence, the behaviour of cellular automata may have a close correspondence with the behaviour of systems usually described by basic rules that involve noise or probabilities. So for example domain walls in cellular automata execute essentially random walks, even though the evolution of the cellular automaton as a whole is entirely deterministic. Similarly, one can construct a cellular automaton that mimics say an Ising spin system with a fixed total energy (microcanonical ensemble) [32]. Apparently random behaviour occurs as a consequence of randomly-chosen initial conditions, just as in many systems governed by the deterministic laws of classical physics.

I would appreciate any help you can give me in understanding this.

For lurkers following our discussion: Toward A Mathematical Definition Of "Life" - Chaitin (ResearchIndex)