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.

Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sigma 2-hard). We also study the complexity of going from Ato SUBSEQ(A) for several representations of A and SUBSEQ(A).

Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3,.... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.

this paper) and independently of the matter of considering Church's and Turing's thesis as de nitions (in any philosophical sense), or as empirical or mathematics thesis, we think that Church's thesis and Turing's thesis should have associated a true-value. This true-value is not known, and although is not possible a positive test, we thought that if should be possible a negative one