Enumerable classes of total recursive functions: Complexity of inductive inference

Abstract

This paper includes some results on complexity of inductive inference for enumerable classes of total recursive functions, where enumeration is considered in more general meaning than usual recursive enumeration. The complexity is measured as the worst-case mindchange (error) number for the first n functions of the given class. Three generalizations are considered.

First: the numbering is computed in limit (with a fixed number of mind-changes). Then the complexity can be arbitrary fast growing recursive function. Second: a fixed number of functions are given by the enumbering function wrongly. In this case only universal strategies have large complexity function.

Third: every function given by the enumbering function can differ in a fixed number of points from the corresponding genuine function of the class. Two cases are considered: functions given by the enumbering function can be only partially defined or they must be total. In the first case there are unidentifiable classes. In the second case there are logarithmic algorithms for prediction and EX-identifying and linear algorithms for identifying of τ-indices.

This research was supported by Latvian Science Council Grant No. 9.599