Probabilistic inductive inference of indices in enumerable classes of total recursive functions

Abstract

We consider identification of indices of functions in recursively enumerable classes of total recursive functions. The number of changes of the hypotheses is used as the complexity measure of the inductive inference machine (IIM).

An efectively enumerable class U of total recursive functions is constructed such that for arbitrary E>0 there is a T-numbering of U such that:

• 1) every deterministic IIM needs a linear number of changes of hypotheses to identify T-indices for functions in U;

• 2) there is a probabilistic IIM which identifies T-indices for function in U with probability 1-E with a constant upper bound of changes of hypotheses.

For arbitrarily slowly growing total recursive function g(n) and arbitrary T-numbering of the class U there is a probabilistic IIM such that for arbitrary p<1 the IIM identifies U and with probability p the number changes of hypotheses does not exceed g(n).

There is a T-numbering of U such that no probabilistic IIM can identify T-indices for functions in U with a probability p (1/2<p<1) with a constant upper bound for the number of changes of hypotheses.