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We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows:• ΩU[X] is random whenever X is Σn0-complete or Πn0-complete for some n ≥ 2.• However, for n ≥ 2, ΩU[X] is not n-random when X is Σn0 or Πn0. Nevertheless, there exists Δn+10 sets such that ΩU[X] is n-random.• There are Δ20 sets X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is Δn+10 and Σn0-hard such that ΩU[X] is not random.We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2≤ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2≤ω recursive in ∅′ ⊕ r, such that ΩU[X] = r.The same questions are also considered in the context of infinite computations, and lead to similar results.

We give an algorithm to compute an absolutely normal number so that the first i digits in its binary expansion are obtained in time polynomial in i; in fact, just above quadratic. The algorithm uses combinatorial tools to control divergence from normality. Speed of computation is achieved at the sacrifice of speed of convergence to normality.

The ÿrst example of an absolutely normal number was given by Sierpinski in 1916, twenty years before the concept of computability was formalized. In this note we give a recursive reformulation of Sierpinski's construction which produces a computable absolutely normal number.

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We prove that finite-state transducers with injective behavior, deterministic or not, realtime or not, with no extra memory or a single counter, cannot compress any normal word. We exhaust all combinations of determinism, real-time, and additional memory in the form of counters or stacks, identifying which models can compress normal words. The case of deterministic push-down transducers is the only one still open. We also present results on the preservation of normality by selection with finite automata. Complementing Agafonov's theorem for prefix selection, we show that suffix selection preserves normality. However, there are simple two-sided selection rules that do not.

The source code, results and visualization of some statistics are accessible from http://kapow.dc.uba.ar/patterns/.

In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result.