---

author:

• 'Tsvi Benson-Tilsen' title: 'Open problem: thin logical priors' ...

• Background / Motivation

• Problem statement

  • Type signature
  • Desiderata

• Comments

In short, and at a high level, the problem of thin priors is to understand how an agent can learn logical facts and make use of them in its predictions, without setting up a reflective instability across time. Before the agent knows the fact, it is required by logical uncertainty to “care about” worlds where the fact does not hold; after it learns the fact, it might no longer care about those worlds; so the ignorant agent has different goals than the knowing agent. This problem points at a hole in our basic understanding, namely how to update on logical facts; logical induction solves much of logical uncertainty, but doesn’t clarify how to update on computations, since many logical facts are learned “behind the scenes” by traders.

*****

The ideas in this post seem to have been discussed for some time. Jessica brought them up in a crisper form in a conversation a while go with me, and also came up with the name; this post is largely based on ideas in that conversation and some subsequent ones with other people, possibly refined / reframed.

Background / Motivation

It would be nice to have a reflectively stable decision theory (i.e. a decision theory that largely endorses itself to continue making decisions over other potential targets of self-modification); this the most basic version of averting / containing instrumental goals, which is arguably necessary in some form to make a safe agent. Agents that choose policies using beliefs that have been updated on (logical) observations seem to be unstable, presenting an obstacle. More specifically, we have the following line of reasoning:

• Updating on empirical evidence leads to reflective instability. If $A_1$ is uncertain about the future even given all its observations; and its future instantiation $A_2$ would choose actions based on further data; then $A_1$ has an incentive to precommit / self-modify to not choose its future actions by updating its beliefs on future observations.

  For example, say that $A_1$ is looking forward to a counterfactual mugging with a quantum coin, and $A_2$ is going to model the world as having some particular unknown state that is then observed when the coin is revealed. Then $A_2$ would not pay up on tails, so $A_1$ wants to precommit to paying up. Doing so increases expected value from $A_1$’s perspective, since $A_1$ still has 1/2 probability on heads.

• We can view this reflective instability as stemming from $A$’s utility function changing. On the one hand, $A_1$ cares about the heads world; that is, it makes decisions that trade off utility in the tails world for utility in the heads world. On the other hand, once it has seen the coin and updated its world model, $A_2$ no longer thinks the heads worlds are real. Then $A_2$ doesn’t base its decisions on what would happen in the heads world, i.e. $A_2$ no longer cares about the heads worlds.

  Then it is not surprising that $A_1$ is incentivized to self-modify: $A_2$ has a different utility function, so its interests are not aligned with $A_1$’s.

• This can’t obviously be neatly factored into uncertainty about the world and a utility function and dealt with separately. That is, it isn’t (obviously) possible to coherently have a utility function that “only cares about real worlds”, while capturing all of the “free-parameter value judgements” that the agent has to make, and have the agent just be uncertain about which worlds are real.

  The issue, in the empirical realm, is that $A$’s observations are always going to be consistent with multiple possible worlds; that is, $A$ will be uncertain. In particular, $A$ will have to make tradeoffs between influencing different possible worlds. This usually comes in the form of a “simplicity prior”—a prior probability over worlds that is very non-dogmatic. Whether this is expressed as utility differences or probability differences, this “caring measure” on worlds changes in $A_2$. So the thing that $A$ cares about—the function on worlds that dictates how $A$ trades off between effects of actions—changes even if only $A$’s “uncertainty” changes.

• $A$ can be updateless with respect to empirical facts. That is, we can define $A$ to take actions following a policy selected according to judgments made by a fixed prior over worlds. The policy can take empirical observations as input and take different actions accordingly, but the policy itself is selected using a model that doesn’t depend on empirical observations.

• If $A$ is empirically updateless then it avoids some reflective instability. For example, in the counterfactual mugging with an empirical coin, $A_2$ will choose a policy using the prior held by $A_1$. That policy will say to pay up, so $A_2$ will pay up. Thus $A_1$ has no incentive (or at least doesn’t have the same incentive as above) to self-modify.

• The above line of reasoning can be repeated with logical evidence in place of empirical evidence… We have logical observations, i.e. the results of computations, in place of empirical observations; we have logical uncertainty (forced by computational boundedness) in place of empirical uncertainty (forced by limited observational data); therefore agents have a caring measure that incorporates logical uncertainty (i.e. that places positive caring on logically inconsistent worlds); so agents that update on logical facts have a changing caring measure and are reflectively unstable.

• …but it’s not clear how to be updateless with respect to logical facts. This is one construal of the open problem of thin logical priors: define a computable prior over logical facts or counterfactuals that has reasonable decision-theoretic counterfactual beliefs, but “only knows a fixed set of logical facts” in the sense relevant to logical updatelessness. More broadly, we might ask for some computable object that can be used as a general world model, but doesn’t imply (non-analyzable) conflict between differently informed instances of the same agent.

If we could write down a prior over logical statements that was thin enough to be computable, but rich enough to be useful for selecting policies (which may depend on or imply further computations), then we might be able write down a reflectively stable agent.

Problem statement

Desiderata

• Updateless. The prior should be “easy enough” to compute that it can be used as an updateless prior as described above. That is, in the course of being refined by thinking longer (but without explicitly conditioning on any logical facts), the prior should not incorporate any additional logical facts. A prior “incorporates a logical fact” (by being computed to more precision) when it starts penalizing worlds for not satisfying that logical fact. Incorporating logical facts is bad because it sets up a dynamic inconsistency across versions of the agent learning the fact.

  We could weaken this desideratum to allow the prior to be “updateless enough”, where enough is perhaps judged by reflective stability of the resulting agent.

• Knows consequences of policies. The prior is supposed to be useful as the beliefs that generate a system of action-counterfactuals. So the prior had better know, in some sense, what the consequences of different policies are.

• Can learn from computations. Since the world is complicated [citation needed], the agent will have to take advantage of more time to think by updating on results of computations (aka logical facts). Thus a thin prior should, at least implicitly, be able to take advantage of the logical information available given arbitrarily large amounts of computing time.

• Thin, not small. I think that Paul has suggested something like a “small” prior: a finite belief state that is computed once, and then used to decide what computations to run next (and those computations decide what to do after that, and so on). This is also roughly the idea of Son of X.

  A small-prior agent is probably reflectively stable in a somewhat trivial sense. In particular, this doesn’t immediately look useful in terms of analyzing the agent in a way that lets us say more specific things about its behavior, stably over time; all we can say is “the agent does whatever was considered optimal at that one point in time”. A thin prior would hopefully be more specific, so that a stably-comprehensible agent design could use the prior as its beliefs.

  On the other hand, a small prior that knows enough to be able to learn from future computations, and that we understand well enough for alignment purposes, should qualify.

Type signature

A natural type for a thin prior is $\Delta \left(2^{\omega }\right)$, a distribution on sequence space. We may want to restrict to distributions that assign probability 1 to propositionally consistent worlds (that is, we may want to fix an encoding of sentences). We may also want to restrict to distributions that are computable or efficiently computable—that is, the function $\lambda \overline{o}.P\left(\overline{o}\right)$ is computable using an amount of time that is some reasonable function of $|\overline{o}|$, where $\overline{o}$ is a finite dictionary of results of computations.

Another possible type is $\text{Obs}\to \Delta \left(2^{\omega }\right)$. That is, a thin “prior” is not a prior, but rather a possibly more general system of counterfactuals, where $P\left[\overline{o}\right]\left(\varphi \right)$ is intended to be interpreted as the agent’s “best guess at what is true in the counterfactual world in which computations behave as specified by $\overline{o}$”. Given the condition that $$P[\overline{o}\cup \{\psi \left\}\right]\left(\varphi \right)=\frac{P\left[\overline{o}\right](\varphi \wedge \psi )}{P\left[\overline{o}\right]\left(\psi \right)},$$ this is equivalent to just a fixed distribution in $\Delta \left(2^{\omega }\right)$. But since this condition can be violated, as in e.g. causal counterfactuals, this type signature is strictly more general. (We could go further and distinguish background known facts, facts to counterfact on, and unclamped facts.)

In place of $\Delta \left(2^{\omega }\right)$ we might instead put $\text{Act}\to \Delta \left(2^{\omega }\right)$, meaning that the prior is not just prior probabilities, but rather prior beliefs about counterfactual worlds given that the agent takes different possible actions.

Comments

• Although universal Garrabrant inductors don’t explicitly refer to logic in any way (and hence are perhaps more amenable to further analysis than logical inductors), UGIs do in fact update on logical facts, and they do so in an opaque / non-queryable way. (That is, we can’t get a reasonable answer from $P_n$ to the question “what would you have believed if computation $X$ had evaluated to 1?” if $X$ has finished by time $n$ and evaluated to 0.)

  To see that UGIs update on logical facts over time, consider conditioning a UGI on some initial segment ${PA}_k$ of $PA$, and then asking it about the ${10}^{100}$th binary digit of $\pi$. At best, $P_{10}\left(\pi \right({10}^{100})=0\mid {PA}_k)$ will be around $50\%$, since there has not been enough time to compute $\pi \left({10}^{100}\right)$, whereas (roughly speaking) $P_{{10}^{100}}\left(\pi \right({10}^{100})=0\mid {PA}_k)$ will be close to 1 or 0 according to the actual digit of $\pi$. The conditional beliefs of $\overline{P}$ have changed to reflect the result of the long-running computation $\pi \left({10}^{100}\right)$. We still have to condition on $PA$ statements in order to refer to the statement $\pi \left({10}^{100}\right)=0$ (so $k$ has to be 1000 or something, enough to define $\pi (-)$, exponentials, 10, and 100), but the fact of the matter has been learned by $\overline{P}$. In short: traders think longer to make more refined trades, and thereby learn logical facts and influence the market $\overline{P}$ based on those facts.

• Asking for a thin prior might not be carving decision theory at the joints. In particular, because counterfactuals may be partially subjective (in the same what that probability and utility is partially subjective), the notion of a good thin prior might be partially dependent on subjective human judgments, and so not amenable to math.

• This problem seems philosophically appealing; how can you meta-think without doing any actual thinking?

  In classical probability, if we have some space and some information about where we are in the space, we can ask: what belief state incorporates all the given information, but doesn’t add any additional information (which would be unjustified)? The answer is the maximum entropy prior. In the realm of logical uncertainty, we want to ask something like: what belief state incorporates all the given logical information (results of computations), but doesn’t add any “logical information”?

• It is ok for the thin prior to have some logical information “built in” at the outset. The agent won’t be counterfactually mugged using those logical facts, but that is fine. The problem is learning new facts, which creates a reflective instability.