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Abstract. Blum and Blum (1975) showed that a class B of suitable recursive approximations to the halting problem is reliably EX-learnable. These investigations are carried on by showing that B is neither in NUM nor robustly EX-learnable. Since the definition of the class B is quite natural and does not contain any self-referential coding, B serves as an example that the notion of robustness for learning is quite more restrictive than intended. Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes U(A) are still EX-inferable but may fail to be reliably EX-learnable, for example if A is non-high and hypersimple. Additionally, it is proved that U(A) is neither in NUM nor robustly EX-learnable provided A is part of a recursively inseparable pair, A is simple but not hypersimple or A is neither recursive nor high. These results provide more evidence that there is still some need to find an adequate notion for “naturally learnable function classes.” 1.

Intuitively, a class of functions is robustly learnable if not only the class itself, but also all of the transformations of the class under natural transformations (such as via general recursive operators) are learnable. Fulk [Ful90] showed the existence of a non-trivial class which is robustly learnable under the criterion Ex. However, several of the hierarchies (such as the anomaly hierarchies for Ex and Bc) do not stand robustly. Fulk left open the question about whether Bc and Ex can be robustly separated. In this paper we resolve this question positively. 1

Blum and Blum (1975) showed that a class B of suitable recursive approximations to the halting problem K is reliably EX-learnable but left it open whether or not B is in NUM . By showing B to be not in NUM we resolve this old problem. Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes U(A) are still EX-inferable but may fail to be reliably EX-learnable, for example if A is non-high and hypersimple. Blum and Blum (1975) considered only approximations to K defined by monotone complexity functions. We prove this condition to be necessary for making learnability independent of the underlying complexity measure. The class ~ B of all recursive approximations to K generated by all total complexity functions is shown to be not even behaviorally correct learnable for a class of natural complexity measures. On the other hand, there are complexity measures such that ~ B is EX -learnable. A similar result is obtained for all classes ~ U(A). For natural complexity measures, B is shown to be not robustly learnable, but again there are complexity measures such that B and, more generally, every class U(A) is robustly EX-learnable. This result extends the criticism of Jain, Smith and Wiehagen (1998), since the classes defined by artificial complexity measures turn out to be robustly learnable while those defined by natural complexity measures are not robustly learnable. 1 Supported by the Deutsche Forschungsgemeinschaft (DFG) under Heisenberg grant no. Ste 967/1--1. 2 Supported by the Grant-in-Aid for Scientific Research in Fundamental Areas from the Japanese Ministry of Education, Science, Sports, and Culture under grant no. 10558047. Part of th...

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Studying the learnability of classes of recursive functions has attracted considerable interest for at least four decades. Starting with Gold’s (1967) model of learning in the limit, many variations, modifications and extensions have been proposed. These models differ in some of the following: the mode of convergence, the requirements intermediate hypotheses have to fulfill, the set of allowed learning strategies, the source of information available to the learner during the learning process, the set of admissible hypothesis spaces, and the learning goals. A considerable amount of work done in this field has been devoted to the characterization of function classes that can be learned in a given model, the influence of natural, intuitive postulates on the resulting learning power, the incorporation of randomness into the learning process, the complexity of learning, among others. On the occasion of Rolf Wiehagen’s 60th birthday, the last four decades of research in that area are surveyed, with a special focus on Rolf Wiehagen’s work, which has made him one of the most influential scientists in the theory of learning recursive functions.