Consider the sequence of bits observed by a camera situated within the physical universe (which we can imagine as a CA for concreteness).  If we draw a program uniformly at random (i.e., fixing a universal prefix free encoding) and condition on agreement with this prefix, what does the posterior (over programs) look like?

If the CA is deterministic and regular, then the observations have a (relatively) short description, given by writing down the initial conditions and rules and specifying how to extract the observation sequence from the world history (e.g., read off values from this cell, or watch for a particular pattern which appears near this coordinate, perhaps tracking physical continuity, or etc.) If the sequence is sufficiently unsurprising (for example, nearly constant) then it will have a very short description; otherwise, it will be specified by describing the entire universe and locating the observer within it.

If the CA is randomized, there is no short description. Pointing to an observation within the universe is intractable, as it involves specifying an amount of randomness which grows with the age and extent of the universe. Similarly, having pointed to the observation, specifying the continuing evolution of the universe requires too much randomness to be useful. So what should we believe about the shortest description of observations in a random universe?

Averaging over all sequences of observations within the universe, the best predictor is the one who is a given the description of the universe and a specification of how to locate observations within that universe, and which uses its initial observations to condition the resulting distribution (where the distribution depends on the averaging used to determine optimality). Therefore there is a short description which specifies the correct probability distribution over next observations. The fact that much randomness is used in the actual description is relevant only insofar as we may suspect that the observations alone are not adequate to determine the universe–it is now more likely that a sequence won’t contain enough surprises to rule out simpler models.

Returning to our universe, consider a human’s sequence of observations. Even fixing the laws of physics, all you can conclude from your observations is that you belong to the reference class of observers who have shared your experiences (and your cognitive architecture etc.)

In general, a CA is defined by a collection of cells, connected by some notion of locality and evolving over time. The state of a cell at time T+1 is determined by the state of its neighbors at time T. We will normally assume that the cells are regular in a strong sense, for example arranged in an N-dimensional grid with translationally invariant update rules.

There are many possible CA and some of their properties are useful exemplars of more general phenomena.

• Deterministic: The state of a deterministic CA at any time T is completely determined by knowledge of its state at any earlier time together with the update rule.
• Reversible: An automata is reversible if the state at time T-1 is uniquely determined by the state at time T. For example, if we have an automata on a line and each cell becomes equal to its left-hand neighbor on each step, we can reverse the evolution by shifting to the right each step.
• Randomized: In a randomized automaton, the state of a cell and its neighbors at time T determine only a probability distribution over possible states for the cell at time T+1. Knowledge of the initial configuration of a randomized CA is not enough to determine its state at future times: it may be necessary to specify a very large number of random bits in order to pin down the future state.
• Local: A CA is local if there is a constant D (the dimension) such that changing the value at any cell can affect the value of at most T^D other cells within T steps.
• Quantum: Quantum mechanics changes the way probability works; we can talk about a randomized CA using quantum probability (where now the local transitions are required to be unitary rather than Markov), and we obtain a quantum CA. These exhibit nearly all of the “weirdness” of physical quantum systems.