Reflective inductive inference of recursive functions

Abstract

In this paper, we investigate reflective inductive inference of recursive functions. A reflective IIM is a learning machine that is additionally able to assess its own competence.First, we formalize reflective learning from arbitrary, and from canonical, example sequences. Here, we arrive at four different types of reflection: reflection in the limit, optimistic, pessimistic and exact reflection.Then, we compare the learning power of reflective IIMs with each other as well as with the one of standard IIMs for learning in the limit, for consistent learning of three different types, and for finite learning.

On some open problems in reflective inductive inference

“…As Fin ⊆ RCons ⊆ Cons (this holds also for each of the reflection models) [WZ95,Gri08], we have the following corollary.…”

“…For Ex and Fin criteria this does not make a difference, though it does make a difference for consistent learning (see for example, [BB75,WZ95,Gri08]). Moreover, for optimistic, pessimistic or exact reflection (for any of the criteria of learning considered in this paper), considering only canonical sequences does make a difference (see [Gri08]).…”

“…For Ex and Fin criteria this does not make a difference, though it does make a difference for consistent learning (see for example, [BB75,WZ95,Gri08]). Moreover, for optimistic, pessimistic or exact reflection (for any of the criteria of learning considered in this paper), considering only canonical sequences does make a difference (see [Gri08]). We would not go into details of this, except mentioning that the results obtained hold in the strongest form: for the positive side, the learning can be done for arbitrary input sequences, whereas for the negative side, learning cannot be done even on canonical sequences.…”

“…Jantke [Jan95] and later Grieser [Gri08] considered learners that can reflect on their own capabilities. Such learners are equipped with a reflection function that indicates, in the limit, whether the input data represents a function in the class being learnt.…”

“…For each finite piece of input data σ, the learner is said to accept σ if R(σ) = 1; the learner is said to reject σ if R(σ) = 0. Grieser [Gri08] considered three different models of reflection, constraining the behaviour of R on finite pieces σ of input data.…”