Suppose that an agent $A_1$ reasons in a sound theory $T_1$, and an agent $A_2$ reasons in a theory $T_2$, such that $T_1$ proves that $T_2$ is sound. Now suppose $A_1$ is trying to reason in a way that is legible to $A_2$, in the sense that $A_2$ can rely on $A_1$ to reach correct conclusions. One way of doing this is for $A_1$ to restrict itself to some weaker theory $T_3$, which $T_2$ proves is sound, for the purposes of any reasoning that it wants to be legible to $A_2$. Of course, in order for this to work, not only would $A_1$ have to restrict itself to using $T_3$, but $A_2$ would to trust that $A_1$ had done so. A plausible way for that to happen is for $A_1$ to reach the decision quickly enough that $A_2$ can simulate $A_1$ making the decision to restrict itself to using $T_3$.

Example application #1: An agent communicating with its past and future selves. In this scenario, consider an agent that designs successors it can trust, and wants to be legible to its successors.

Let $T$ be some recursively axiomatizable first-order theory of arithmetic which is powerful enough to decide the important questions the agent wants to know the answer to that don't involve reasoning about agents (Peano Arithmetic, for instance). For a computable ordinal $\alpha$, let $T+\alpha$ refer to $T$ extended with a soundness schema for $T+\beta$ for every $\beta <\alpha$. So $T+0=T$, $T+(\alpha +1)=T+\text{Sound}(T+\alpha )$, and $T+\gamma ={\bigcup }_{\alpha <\gamma }T+\alpha$ for computable limit ordinals $\gamma$. Consider an agent initially (at time $0$) reasoning with $T+{\alpha }_0$ for some large computable ordinal ${\alpha }_0$. At each time step, the agent must design a successor that it can prove uses sound reasoning. So at each time-step $t$ (until ${\alpha }_t=0$), the agent can pick ${\alpha }_{t+1}<{\alpha }_t$ and have its successor reason with $T+{\alpha }_{t+1}$. Of course, this cannot go on forever, but there are astronomically long decreasing sequences of ordinals starting at ordinals that are straightforward to describe, and which require a very minimal amount of computation to get from one ordinal in the sequence to the next (for instance, if you're familiar with the proof that the Hydra game is impossible to lose, the sequence of ordinals below ${\epsilon }_0$ corresponding to the sequence of states of some Hydra if Hercules is using some simple strategy), so the lack of such infinite sequences does not seem like a serious obstacle. The sequence can be made arbitrarily long, and the agent can procrastinate arbitrarily long on giving an upper bound for how long the sequence needs to be, and can procrastinate arbitrarily long on giving an upper bound for when it will have to decide how long the sequence needs to be, and so on.

If this agent has proved something in the past and stored the result, how can the agent now take this as a demonstration that the result is true, since all it knows is that it proved it in the past, and it does not trust its own reasoning system (much less its past self's stronger reasoning system)? To deal with this, whenever the agent (on time-step $n$) stores a note to its future self in memory, it could tag the note with an ordinal $\beta \le {\alpha }_n$ such that it verified the note's accuracy using $T+\beta$. Then on step $m>n$, if $\beta <{\alpha }_m$, the agent can read the note from its past self and conclude that the note's contents are accurate without proving them again from scratch, because it knows that its past self must have already checked this in $T+\beta$, which the agent trusts. If the contents of the note do not involve reasoning about agents, then by the assumption I made about $T$, the agent should be able to tag them with $0$, and these will be legible to its future self until the last step. If $n>m>k$, then the agent at step $n$ must tag the fact that it trusts $T_m$ with some ordinal $\beta >{\alpha }_m$, and thus it cannot make this fact legible to itself at step $m$. However, if the agent at step $n$ tries to reason about itself at step $k$, it should be able to do so in $T+({\alpha }_k+1)$, so it should be able to share its conclusions with itself at step $m$ under the very weak assumption that ${\alpha }_m\ge {\alpha }_k+2$.

Example application #2: The Agent Simulates Predictor problem (from the Predictor's perspective). This is Newcomb's problem, in which the Predictor knows that the Agent's reasoning system is sound, and the Agent has more computing power than the Predictor does. We want to find a way to replicate the standard solution to Newcomb's problem, in which the Predictor simulates the Agent to find out what the Agent does, and the Agent recognizes that the Predictor will predict it correctly. The fact that the Agent has more computing power than the Predictor does is not a serious obstacle to this, since the Agent does not have to use all of the computing power available to it. If the Agent is able to reason probabilistically, it might assign high probability to the Predictor's reasoning being sound, and one-box on that basis. However, there is still the case in which the Predictor's reasoning is sound, but the Agent does not believe this with sufficiently high probability. In this case, the problem becomes a somewhat unfair one for the Agent; it is being asked to get good results in a situation in which its beliefs are false.

The Predictor's motivations (if it has any) are usually left unspecified, but let's assume here that the Predictor wants the Agent to one-box, and is attempting to incentivize the Agent to do so. In this case, the Predictor could commit to using a weaker reasoning system that the Agent knows is sound, and then simulate the Agent, and predict that the Agent will act as it did in the simulation. Running a program and predicting that the program will output the same thing next time it is run does not require any particularly heavy feats of logic, so commiting itself to using a weak reasoning system while it does this shouldn't be a problem. The Agent can then simulate the Predictor up to the point that it restricts itself to a reasoning system that the Agent trusts, at which point the Agent can conclude that the Predictor will predict correctly, and one-box.