E.10.INT:5.1 - An informative controller query

A controller must choose an input that gives output 5. It has two equally plausible candidate relations, y=x and y=-x, and can first query input 0 or input 1. “Choose the more surprising query” leaves the criterion unclear.

At input 0 both candidates return 0. At input 1 they return 1 and -1 respectively. The second query distinguishes the candidates: after observing 1, command 5; after observing -1, command -5. The recovered instruction is “query input 1 to distinguish the relations, then choose the command.”

A separate fair coin produces an outcome with one bit of surprisal, but its outcome is independent of which relation holds. Querying input 1 yields one bit of information about that relation under the stated equal prior. Tossing the coin yields none. If the controller is already given the relation, it can select the command immediately.

The example constructs an information-seeking policy. Whether a particular controller can execute it is a further capability question.