Learning Recursive Functions From Approximations

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\u27s 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

Structural measures for games and process control in the branch learning model

Process control problems can be modeled as closed recursive games. Learning strategies for such games is equivalent to the concept of learning infinite recursive branches for recursive trees. We use this branch learning model to measure the difficulty of learning and synthesizing process controllers. We also measure the difference between several process learning criteria, and their difference to controller synthesis. As measure we use the information content (i.e. the Turing degree) of the oracle which a machine need to get the desired power. The investigated learning criteria are finite, EX-, BC-, Weak BC- and online learning. Finite, EX- and BC-style learning are well known from inductive inference, while weak BC- and online learning came up with the new notion of branch (i.e. process) learning. For all considered criteria - including synthesis - we also solve the questions of their trivial degrees, their omniscient degrees and with some restrictions their inference degrees. While most of the results about finite, EX- and BC-style branch learning can be derived from inductive inference, new techniques had to be developed for online learning, weak BC-style learning and synthesis, and for the comparisons of all process learning criteria with the power of controller synthesis

Extremes in the degrees of inferability

Annals of Pure and Applied Logic663231-276APAL