We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning, learning in the limit, behaviorally correct learning and anomalous learning in the limit. In proving the Σ 0 5 -completeness result for behaviorally correct learning we prove a result of independent interest; if a uniformly computably enumerable family is not learnable, then for any computable learner there is a ∆ 0 2 enumeration witnessing failure.2010 Mathematics Subject Classification. Primary 03D80, 68Q32.

Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a ∆ 0 n+1 basic sequence with respect to which no ∆ 0 n real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

Abstract. We study the relationship between randomness and effective biimmunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no MartinLöf random set. We show that the same holds when Martin-Löf randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set.We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every Martin-Löf random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

We prove the existence of a canonical form for semi-deterministic transducers with sets of pairwise incomparable output strings. Based on this, we develop an algorithm which learns semi-deterministic transducers given access to translation queries. We also prove that there is no learning algorithm for semideterministic transducers that uses only domain knowledge.

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