this paper, # 0 , # 1 , # 2 , . . . denotes an acceptable programming system [17], also known as a Godel numbering of the partial recursive functions [15]. The function # e is said to be computed by the program e.

this paper M 0 ; M 1 ; : : : is a standard list of all Turing machines, M

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "not-so-nearly" minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.

This work introduces a new query inference model that can access data and communicate with a teacher by asking finitely many boolean queries in a language L. In this model the parameters of interest are the number of queries used and the expressive power of L. We study how the learning power varies with these parameters. Preliminary results suggest that this model can help studying query inference in an resource bounded environment.