Effectivity Questions for Kleene’s Recursion Theorem

Abstract

The present paper explores the interaction between two recursion-theoretic notions: program self-reference and learning partial recursive functions in the limit. Kleene’s Recursion Theorem formalises the notion of program self-reference: It says that given a partial-recursive function ψ _p there is an index e such that the e-th function ψ _e is equal to the e-th slice of ψ _p . The paper studies constructive forms of Kleene’s recursion theorem which are inspired by learning criteria from inductive inference and also relates these constructive forms to notions of learnability. For example, it is shown that a numbering can fail to satisfy Kleene’s Recursion Theorem, yet that numbering can still be used as a hypothesis space when learning explanatorily an arbitrary learnable class. The paper provides a detailed picture of numberings separating various versions of Kleene’s Recursion Theorem and learnability.