We consider an agent interacting with an unknown environment. The environment is a function which maps natural numbers to natural numbers; the agent’s set of hypotheses about the environment contains all such functions which are computable and compatible with a finite set of known input-output pairs, and the agent assigns a positive probability to each such hypothesis. We do not require that this probability distribution be computable, but it must be bounded below by a positive computable function. The agent has a utility function on outputs from the environment. We show that if this utility function is bounded below in absolute value by an unbounded computable function, then the expected utility of any input is undefined. This implies that a computable utility function will have convergent expected utilities iff that function is bounded.
“Convergence of Expected Utility for Universal AI”, (2009-07-31):
We consider a sequence of repeated interactions between an agent and an environment. Uncertainty about the environment is captured by a probability distribution over a space of hypotheses, which includes all computable functions. Given a utility function, we can evaluate the expected utility of any computational policy for interaction with the environment. After making some plausible assumptions (and maybe one not-so-plausible assumption), we show that if the utility function is unbounded, then the expected utility of any policy is undefined.