Inductive Inference and Unsolvability

Abstract

It is shown that many different problems have the same degree of unsolvability. Among these problems are: THE INDUCTIVE INFERENCE PROBLEM. Infer in the limit an index for a recursive function $f$ presented as $f(0), f(1), f(2),\ldots$. THE RECURSIVE INDEX PROBLEM. Decide in the limit if $i$ is the index of a total recursive function. THE ZERO NONVARIANT PROBLEM. Decide in the limit if a recursive function $f$ presented as $f(0), f(1), f(2),\ldots$ has value unequal to zero for infinitely many arguments. Finally, it is shown that these unsolvable problems are strictly easier than the halting problem.