Derived sets and inductive inference

Abstract

The paper deals with using topological concepts in studies of the Gold paradigm of inductive inference. They are — accumulation points, derived sets of order α (α — constructive ordinal) and compactness. Identifiability of a class U of total recursive functions with a bound α on the number of mindchanges implies $U^{(\alpha + 1)} = \not 0$. This allows to construct counter-examples — recursively enumerable classes of functions showing the proper inclusion between identification types: EX _α⊂EX _α+1.

The presence of an accumulation point in a class W determines whether or not all FIN strategies can be split into two families so that any finite team identifying W contains strategies from both families. A combinatorial idea, used to show the absence of such a splitting in the case when the derived set $W^d = \not 0$, leads to new identification types (FIN(2: *), etc.) which may be irreducible to the team identification types (e. g. FIN(k: m)).

Supported by Grant No. 93-599 from Latvian Councile of Science