Reflecting and Self-Confident Inductive Inference Machines

Abstract

Reflection denotes someones activity of thinking about oneself as well as about one's relation to the outside world. In particular, reflecting means pondering about ones capabilities and limitations. Reasoning about ones competence is a central issue of reflective behaviour. Reflection is a key issue of recent artificial intelligence. There is investigated the problem of automated reasoning about the competence of inductive inference machines. Reflective inductive inference machines are those which are able to identify whether or not some information presented exceeds its learning capabilities. An inductive inference machine is self-confident, if it usually trusts in its ability to solve the learning problem on hand. It is reflecting and self-confident, if it normally believes in its power, but recognizes problems exceeding its competence. The problem is formalized and studied within the setting of inductively learning total recursive functions. There is a crucial distinction of immediately reflecting inductive inference machines and those which need an a priori unknown amount of time for reasoning about its competence. The core result is a characterization of problem classes solvable by reflective inductive inference machines. Roughly speaking, for a given problem class $U \subseteq \mathcal{R}$ , one may develop a reflecting and self-confident inductive inference machine, if and only if the development of such a machine is not necessary at all, as the problem class can be reasonably extended such that reflection turns out to be unnecessary. A derived result exhibits that, in contrast to intuition, there is no difference in power between reflecting and immediately reflecting inductive inference machines. The ultimate goal of the present paper is to contribute to a better understanding of the reflection problem in artificial intelligence. The present paper is intended to be a launching pad for this endeavor.