Achilles A. Beros

LEARNING THEORY IN THE ARITHMETIC HIERARCHY

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

A DNC function that computes no effectively bi-immune set

\Sigma_5^0

Normal Numbers and Limit Computable Cantor Series

-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

Effective Bi-immunity and Randomness

\Delta_2^0

From Eventually Different Functions to Pandemic Numberings.

enumeration witnessing failure.

Classifying the Arithmetical Complexity of Teaching Models

Jockusch and Lewis (J Symb Logic 78:977–988, 2013) proved that every DNC function computes a bi-immune set. They asked whether every DNC function computes an effectively bi-immune set. We construct a DNC function that computes no effectively bi-immune set, thereby answering their question in the negative.

Anomalous Vacillatory Learning

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 Delta^0_{n+1} basic sequence with respect to which no Delta^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.

Canonical immunity and genericity

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing \(\mathrm {DNR}\) functions relative to X that compute no Martin-Lof random set. We show that the same holds when Martin-Lof 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.

Index Sets of Universal Codes

A function is strongly non-recursive (SNR) if it is eventually different from each recursive function. We obtain hierarchy results for the mass problems associated with computing such functions with varying growth bounds. In particular, there is no least and no greatest Muchnik degree among those of the form \({{\mathrm{SNR}}}_f\) consisting of SNR functions bounded by varying recursive bounds f.

Teachers, Learners and Oracles

This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly r.e. families with finite teaching dimension, and (2) the class of uniformly r.e. families with finite positive recursive teaching dimension witnessed by a uniformly r.e. teaching sequence. We also derive the arithmetical complexity of several other decision problems in teaching, such as the problem of deciding, given an effective coding \(\{{\mathcal {L}}_0,{\mathcal {L}}_1,{\mathcal {L}}_2,\ldots \}\) of all uniformly r.e. families, any e such that \({\mathcal {L}}_e = \{L^e_0,L^e_1,\ldots ,\}\), any i and d, whether or not the teaching dimension of \(L^e_i\) with respect to \({\mathcal {L}}_e\) is upper bounded by d.