Reflective Variants of Solomonoff Induction and AIXI

Abstract

Solomonoff induction and AIXI model their environment as an arbitrary Turing machine, but are themselves uncomputable. This fails to capture an essential property of real-world agents, which cannot be more powerful than the environment they are embedded in; for example, AIXI cannot accurately model game-theoretic scenarios in which its opponent is another instance of AIXI.

In this paper, we define reflective variants of Solomonoff induction and AIXI, which are able to reason about environments containing other, equally powerful reasoners. To do so, we replace Turing machines by probabilistic oracle machines (stochastic Turing machines with access to an oracle). We then use reflective oracles, which answer questions of the form, “is the probability that oracle machine \(T\) outputs 1 greater than p, when run on this same oracle?” Diagonalization can be avoided by allowing the oracle to answer randomly if this probability is equal to p; given this provision, reflective oracles can be shown to exist. We show that reflective Solomonoff induction and AIXI can themselves be implemented as oracle machines with access to a reflective oracle, making it possible for them to model environments that contain reasoners as powerful as themselves.