Asymptotic logical uncertainty and the Benford test

Abstract

Almost all formal theories of intelligence suffer from the problem of logical omniscience, the assumption that an agent already knows all consequences of its beliefs. Logical uncertainty codifies uncertainty about the consequences of existing beliefs. This implies a departure from beliefs governed by standard probability theory. Here, we study the asymptotic properties of beliefs on quickly computable sequences of logical sentences. Motivated by an example we call the Benford test, we provide an approach which identifies when such subsequences are indistinguishable from random, and learns their probabilities.