Learning with the knowledge of an upper bound on program size

Two learning situations are considered: machine identification of programs from graphs of recursive functions (modeling inductive hypothesis formation) and machine identification of grammars from texts of recursively enumerable languages (modeling first language acquisition). Both these learning models are extended to account for situations in which a learning machine is provided additional information in the form of knowledge about an upper-bound on the minimal size program (grammar) for the function (language) being identified. For a number of such extensions, it is shown that larger classes of functions (languages) can be algorithmically identified in the presence of upper-bound information. Numerous interesting relationships are shown between different models of learning, number of anomalies allowed in the inferred program (grammar), and number of anomalies allowed in the upper-bound information.