This article investigates algorithmic learning, in the limit, of correct programs for recursive functionsffrom both input/output examples offand several interesting varieties ofapproximateadditional (algorithmic) information aboutf. Specifically considered, as such approximate additional information aboutf, are Rose's frequency computations forfand several natural generalizations from the literature, each generalization involving programs for restricted trees of recursive functions which havefas a branch. Considered as the types of trees are those with bounded variation, bounded width, and bounded rank. For the case of learning final correct programs for recursive functions, EX-learning, where the additional information involves frequency computations, an insightful and interestingly complex combinatorial characterization of learning power is presented as a function of the frequency parameters. For EX-learning (as well as for BC-learning, where a finalsequenceof correct programs is learned), for the cases of providing the types of additional information considered in this paper, the maximal probability is determined such that the entire class of recursive functions is learnable with that probability.
Case, J. et al. "Learning Recursive Functions From Approximations." Journal of Computer and System Sciences 55.1 (1997): 183-196.