Inductive Inference Systems for Learning Classes of Algorithmically Generated Sets and Structures

Abstract

Computability theorists have extensively studied sets A the elements of which can be enumerated by Turing machines. These sets, also called computably enumerable sets, can be identified with their Godel codes. Although each Turing machine has a unique Godel code, different Turing machines can enumerate the same set. Thus, knowing a computably enumerable set means knowing one of its infinitely many Godel codes. In the approach to learning theory stemming from E.M. Gold’s seminal paper [9], an inductive inference learner for a computably enumerable set A is a system or a device, usually algorithmic, which when successively (one by one) fed data for A outputs a sequence of Godel codes (one by one) that at a certain point stabilize at codes correct for A. The convergence is called semantic or behaviorally correct, unless the same code for A is eventually output, in which case it is also called syntactic or explanatory. There are classes of sets that are semantically inferable, but not syntactically inferable.