On the computability of Solomonoff induction and AIXI

Abstract

How could we solve the machine learning and the artificial intelligence problem if we had infinite computation? Solomonoff induction and the reinforcement learning agent AIXI are proposed answers to this question. Both are known to be incomputable. We quantify this using the arithmetical hierarchy, and prove upper and in most cases corresponding lower bounds for incomputability. Moreover, we show that AIXI is not limit computable, thus it cannot be approximated using finite computation. However there are limit computable ε-optimal approximations to AIXI. We also derive computability bounds for knowledge-seeking agents, and give a limit computable weakly asymptotically optimal reinforcement learning agent.