*All of the results in this post, and most of the informal observations/interpretations, are due to Sam Eisenstat. I think the Troll Bridge story, as a way to make the decision problem understandable, is due to Tsvi; but I'm not sure.*

## Pure Logic Version

Troll Bridge is a decision problem which has been floating around for a while, but which has lacked a good introductory post. The original post gives the essential example, but it lacks the "troll bridge" story, which (1) makes it hard to understand, since it is just stated in mathematical abstraction, and (2) makes it difficult to find if you search for "troll bridge".

The basic idea is that you want to cross a bridge. However, there is a troll who will blow up the bridge with you on it, * if* (and only if) you cross it "for a dumb reason" — for example, due to unsound logic. You can get to where you want to go by a worse path (through the stream). This path is better than being blown up, though.

We apply a Löbian proof to show not only that you choose not to cross, but furthermore, that your counterfactual reasoning is confident that the bridge would have blown up if you had crossed. This is supposed to be a counterexample to various proposed notions of counterfactual, and for various proposed decision theories.

The pseudocode for the environment (more specifically, the utility gained from the environment) is as follows:

IE, if the agent crosses the bridge and is inconsistent, then U=-10. (□⊥ means "PA proves an inconsistency".) Otherwise, if the agent crosses the bridge, U=+10. If neither of these (IE, the agent does not cross the bridge), U=0.

The pseudocode for the agent could be as follows:

This is a little more complicated, but the idea is supposed to be that you search for every "action implies utility" pair, and take the action for which you can prove the highest utility (with some tie-breaking procedure). Importantly, this is the kind of proof-based decision theory which eliminates spurious counterfactuals in 5-and-10 type problems. It isn't that easy to trip up with Löbian proofs. *(Historical/terminological note: This decision theory was initially called MUDT, and is still sometimes referred to in that way. However, I now often call it proof-based decision theory, because it isn't centrally a UDT. "Modal DT" (MDT) would be reasonable, but the modal operator involved is the "provability" operator, so "proof-based DT" seems more direct.)*

Now, the proof:

- Reasoning within PA (ie, the logic of the agent):
- Suppose the agent crosses.
- Further suppose that the agent proves that crossing implies U=-10.
- Examining the source code of the agent, because we're assuming the agent crosses, either PA proved that crossing implies U=+10, or it proved that crossing implies U=0.
- So, either way, PA is inconsistent -- by way of 0=-10 or +10=-10.
- So the troll actually blows up the bridge, and really, U=-10.

- Therefore (popping out of the second assumption), if the agent proves that crossing implies U=-10, then in fact crossing implies U=-10.
- By Löb's theorem, crossing really implies U=-10.
- So (since we're still under the assumption that the agent crosses), U=-10.

- Further suppose that the agent proves that crossing implies U=-10.
- So (popping out of the assumption that the agent crosses), the agent crossing implies U=-10.

- Suppose the agent crosses.
- Since we proved all of this in PA, the agent proves it, and proves no better utility in addition (unless PA is truly inconsistent). On the other hand, it will prove that not crossing gives it a safe U=0. So it will in fact not cross.

The paradoxical aspect of this example is not that the agent doesn't cross -- it makes sense that a proof-based agent can't cross a bridge whose safety is dependent on the agent's own logic being consistent, since proof-based agents can't know whether their logic is consistent. Rather, the point is that the agent's "counterfactual" reasoning looks crazy. *(However, keep reading for a version of the argument where it does make the agent take the wrong action.)* Arguably, the agent should be uncertain of what happens if it crosses the bridge, rather than certain that the bridge would blow up. Furthermore, the agent is reasoning as if it can control whether PA is consistent, which is arguably wrong.

In a comment, Stuart points out that this reasoning seems highly dependent on the code of the agent; the "else" clause could be different, and the argument falls apart. I think the argument keeps its force:

- On the one hand, it's still very concerning if the sensibility of the agent depends greatly on which action it performs in the "else" case.
- On the other hand, we can modify the troll's behavior to match the modified agent. The general rule is that the troll blows up the bridge if the agent would cross for a "dumb reason" -- the agent then concludes that the bridge would be blown up if it crossed. I can no longer complain that the agent reasons as if it were controlling the consistency of PA, but I can still complain that the agent thinks an action is bad because that action indicates its own insanity, due to a troublingly circular argument.

## Analogy to Smoking Lesion

One interpretation of this thought-experiment is that it shows proof-based decision theory to be essentially a version of EDT, in that it has EDT-like behavior for Smoking Lesion. The analogy to Smoking Lesion is relatively strong:

- An agent is at risk of having a significant internal issue. (In Smoking Lesion, it’s a medical issue. In Troll Bridge, it is logical inconsistency.)
- The internal issue would bias the agent toward a particular action. (In Smoking Lesion, the agent smokes. In Troll Bridge, an inconsistent agent crosses the bridge.)
- The internal issue also causes some imagined practical problem for the agent. (In Smoking Lesion, the lesion makes one more likely to get cancer. In Troll Bridge, the inconsistency would make the troll blow up the bridge.)
- There is a chain of reasoning which combines these facts to stop the agent from taking the action. (In smoking lesion, EDT refuses to smoke due to the correlation with cancer. In Troll Bridge, the proof-based agent refuses to cross the bridge because of a Löbian proof that crossing the bridge leads to disaster.)
- We intuitively find the conclusion nonsensical. (It seems the EDT agent should smoke; it seems the proof-based agent should not expect the bridge to explode.)

Indeed, the analogy to smoking lesion seems to strengthen the final point -- that the counterfactual reasoning is wrong.

I've come to think of Troll Bridge as "the real smoking lesion", since I'm generally not satisfied with how smoking lesion is set up.

But is proof-based decision theory *really* a version of EDT? I think there's more to say about the analogy, but a simple explanation is this: both EDT and proof-based decision theory evaluate actions by adding them to the knowledge base and seeing what the world looks like under that additional assumption. Or, to put it differently, proof-based DT reasons about actions as if they're observations. That's the fundamental idea of evidential decision theory.

Still, as good as the analogy might be, you still probably shouldn't be convinced until you've seen the Troll Bridge defeat *real* EDT.

## Probabilistic Version

For the purely logical version, I said that we can't fault the conclusion (because the agent can't prove that it is safe to cross the bridge) – only the *reasoning* is being critiqued. However, the probabilistic version better demonstrates the severity of the reasoning error; it illustrates an EDT agent making what seems to be an outright mistake.

It's reasonable to avoid crossing the bridge if you're only thinking about provable results. But we might expect an agent who balances risks and rewards would decide to cross, so long as it believes the probability of PA being inconsistent is low. Thus turns out not to be the case.

Here's the setup.

Suppose that the agent uses a probability distribution which "respects logic" in the sense that it assigns probability zero to anything which is logically refutable. (This assumes logical omniscience. I'll drop that assumption later.)

The agent is as follows: *if P(cross)=0, cross; if P(¬cross)=0, don't cross; else, take the action with highest expected utility, breaking ties by not crossing.* The idea is that we are trying to take the highest-expected-utility action; however, we must handle cases where this isn't well-defined. Choosing to take an action when the probability of the action is zero guarantees that the agent can never prove its own action; it is a version of the chicken rule. Breaking ties by not crossing is arbitrary, but gives the simplest version of the example, because this choice reduces the number of cases we need to check.

The troll is as follows: *if the agent crosses the bridge due to the P(cross)=0 clause, blow up the bridge.*

Reasoning in the logic of the agent:

- Suppose
- Suppose
- Examining the agent, either crossing had higher expected utility, or P(cross)=0. But we assumed , so it must be the latter. So the bridge gets blown up.

- Therefore, .

- Suppose
- Therefore,

Thus, by Löb's theorem, we see that .

Therefore, since the agent is certain of things which are provable, the agent assigns expected value -10 to crossing. Since the expected value of the other action will always be higher than this, and since we know the agent doesn't ever really enter into the P=0 conditions, the agent will choose not to cross.

Notice that this reasoning did not depend much on the values 10, 0, and -10. The utility of the bridge getting blown up could be -0.000001, and the agent still won't cross. *It isn't weighing the risk; it's decided that the worst outcome is inevitable.* In the case of proof-based agents, I said that the overall decision not to cross was understandable, since proof-based agents are unable to weigh the risks. A probabilistic agent, however, should intuitively be able to say "I don't know whether the bridge will get blown up, because it involves reasoning about properties of myself which I'm fundamentally uncertain about; but, the odds look decent." But that's not what happens: instead, it is sure that crossing is unfavorable, *no matter what overall probability it assigns to P(A=cross)=0*.

So, in this case we conclude that the Troll Bridge example results in a chicken-rule-based agent taking the wrong action overall. The agent shouldn't be sure that it would cross "for the right reason" (it should assign some probability to P(A=cross)=0, since it can't know that its own logic is consistent). However, intuitively, it should be able to assign some probability to this, and balance the risks. If the downside risk is U=-0.000001, and the probability it assigns to its logic being consistent is not similarly small, it should cross -- and in doing so, it would get +10.

As mentioned for the proof-based agent, the agent's code is a bit arbitrary, and it is worth asking how important the details were. In particular, the default in the case of a tie was to not cross. What if the default in case of a tie were to cross?

We then modify the troll's algorithm to blow up the bridge if and only if P(A=cross)=0 **or** there is a tie. The proof then goes through in the same way.

Perhaps you think that the problem with the above version is that I assumed logical omniscience. It is unrealistic to suppose that agents have beliefs which perfectly respect logic. (Un)Fortunately, the argument doesn't really depend on this; it only requires that the agent respects proofs which it can see, and eventually sees the Löbian proof referenced.

## Random Exploration

The frustrating thing about Troll Bridge is that it seems like the agent could just cross the bridge, and things would be fine. The proof that things *wouldn't* be fine *relies on the fact that the agent accepts that very proof as sufficient reason*; so can't we just ignore that kind of proof somehow?

One thing you might try is to consider a learning agent, and force random exploration so the agent just crosses the bridge sometimes. If the agent crosses the bridge, it should be able to see that it's safe, right?

However, we have to ask: what's the appropriate version of Troll Bridge for the exploring agent? Remember I said that the basic idea of Troll Bridge is that the troll blows up the bridge if the agent crosses "for a dumb reason" -- ie, for a reason other than "crossing is a good idea". Random exploration falls into this category.

Furthermore, random exploration is really closely analogous to the chicken rule. Consider:

- (A) The first chicken rule I mentioned: if you prove that you don't take an action, you immediately take it.
- (B) The second chicken rule mentioned: if P(A)=0, then do A.
- (C) Epsilon-chicken: if P(A)<epsilon, then do A.
- (D) Epsilon-exploration: randomly do A with probability epsilon.

(A) is very similar to (B) if the probability distribution respects proofs in the way I mentioned previously; if the probability distribution has the further property of assigning positive probability to anything not disproven, then they're exactly the same. (C) is a natural generalization of (B). (B) tells us what to do when the EDT decision rule isn't well-defined (we can't condition on probability zero events). (C) generalizes this by treating small-probability events in the same way (we might expect our conditional beliefs to be bad for small-probability events, since we presumably don't have much experience with them; so we treat it as if it's the undefined case). (B) ensures there are no zero-probability actions, which means we don't actually ever have to deal with the EDT decision rule being non-well-defined; similarly, (C) ensures no action is almost surely never taken, which helps ensure that our conditional expectations for all actions are empirically grounded.

(C) and (D) are very similar; (C) just supplies pseudorandomness vie the agent's own probabilities. Whenever the agent is too confident about what it does, it does something else. This makes the randomization unpredictable *too the agent*. (D) supplies randomness from an external source instead.

So we have a chain connecting the chicken rule to random exploration.

(Another way of thinking about the similarity: The chicken rule ensures that there are models of the agent's beliefs in which each action is taken, so that the agent can have sensible logical reasoning about each alternative. Epsilon-exploration ensures that the empirical experiences of the agent include all of the actions, so that the agent can have reasonable empirical estimates of consequences of each action.)

OK. So Troll Bridge for an exploring agent should blow up the bridge on exploration, *or* if PA is inconsistent. An agent could look like this:

- With small probability, explore.
- If exploring, flip a coin to cross or not cross.

- If P(cross)=0, cross.
- If P(¬cross)=0, don't cross.
- Otherwise, take the max-expected-value action, breaking ties randomly.

I'm still assuming that the agent's probability distribution respects proofs, as before. I'm also assuming this agent is playing the game repeatedly, and learning. I also must assume that the agent has found Now, the agent reasons:

- Suppose (cross u=-10) for a particular round.
- Further suppose I crossed on that round.
- By the first supposition, I knew the payout of crossing to be low; and I must also have known that the payout of not crossing is higher, since I can prove that. Since I can prove what both payouts are, the expected values must equal those, unless PA is inconsistent (in which case P(cross)=0 anyway, since my beliefs respect proofs). So I can only be crossing the bridge for two reasons -- either this is an exploration round, or P(cross)=0.
- In either case, crossing the bridge yields payout u=-10.

- Therefore, cross u=-10 in fact.

- Further suppose I crossed on that round.
- So (cross u=-10) (cross u=-10).

Since the agent proves that a proof of crossing being bad implies crossing is actually bad, the agent further must prove that crossing is bad in fact, by Löb.

I did this for the logically omniscient case again, but as before, I claim that you can translate the above proof to work in the case that the agent's beliefs respect proofs it can find. That's maybe a bit weird, though, because it involves a Bayesian agent updating on logical proofs; we know this isn't a particularly good way of handling logical uncertainty.

We can use logical induction instead, using an epsilon-exploring version of LIDT. We consider LIDT on a sequence of troll-bridge problems, and show that it eventually notices the Löbian proof and starts refusing to cross. This is even more frustrating than the previous examples, because LIDT might successfully cross for a long time, apparently learning that crossing is safe, and reliably gets +10 payoff. Then, one day, it finds the Löbian proof and stops crossing the bridge!

That case is a little more complicated to work out than the Bayesian probability case, and I omit the proof here.

## Non-examples: RL

On the other hand, consider an agent which uses random exploration but doesn't do any logical reasoning, like a typical RL agent. Such an agent doesn't need any chicken rule, since it doesn't care about proofs of what it'll do. It still needs to explore, though. So the troll can blow up the bridge whenever the RL agent crosses due to exploration.

This obviously messes with the RL agent's ability to learn to cross the bridge. The RL agent might never learn to cross, since every time it tries it, it looks bad. So this is sort of similar to Troll Bridge.

However, I think this isn't really the point of Troll Bridge. The key difference is this: the RL agent can get past the bridge if its prior expectation that crossing is a good idea is high enough. It just starts out crossing, and happily crosses all the time.

Troll Bridge is about the inevitable confidence that crossing the bridge is bad. We would be fine if an agent decided not to cross because it assigned high probability to PA being inconsistent. The RL example seems similar in that it depends on the agent's prior.

We could try to alter the example to get that kind of inevitability. Maybe we argue it's still "dumb" to cross only because you start with a high prior probability of it being good. Have the troll punish crossing *unless the crossing is justified by an empirical history of crossing being good*. Then RL agents do poorly no matter what -- no one can get the good outcome in order to build up the history, since getting the good outcome *requires* the history.

But this still doesn't seem so interesting. You're just messing with these agents. It isn't illustrating the degree of pathological reasoning which the Löbian proof illustrates -- **of course** you don't put your hand in the fire if you get burned every single time you try it. There's nothing wrong with the way the RL agent is reacting!

So, Troll Bridge seems to be more exclusively about agents who do reason logically.

## Conclusions

All of the examples have depended on a version of the chicken rule. This leaves us with a fascinating catch-22:

- We need the chicken rule to avoid spurious proofs. As a reminder: spurious proofs are cases where an agent would reject an action if it could prove that it would not take that action. These actions can then be rejected by an application of Löb's theorem. The chicken rule avoids this problem by ensuring that agents cannot know their own actions, since if they did then they'd take a different action from the one they know they'll take (and they know this, conditional on their logic being consistent).
- However, Troll Bridge shows that the chicken rule can lead to another kind of problematic Löbian proof.

So, we might take Troll Bridge to show that the chicken rule does not achieve its goal, and therefore reject the chicken rule. However, this conclusion is very severe. We cannot simply drop the chicken rule and open the gates to the (much more common!) spurious proofs. We would need an altogether different way of rejecting the spurious proofs; perhaps a full account of logical counterfactuals.

Furthermore, it is possible to come up with variants of Troll Bridge which counter some such proposals. In particular, Troll Bridge was originally invented to counter proof-length counterfactuals, which essentially generalize chicken rules, and therefore lead to the same Troll Bridge problems).

Another possible conclusion could be that Troll Bridge is simply too hard, and we need to accept that agents will be vulnerable to this kind of reasoning.

Written slightly differently, the reasoning seems sane: Suppose I cross. I must have proven it's a good idea. Aka I proved that I'm consistent. Aka I'm inconsistent. Aka the bridge blows up. Better not cross.

I agree with your English characterization, and I also agree that it isn't really obvious that the reasoning is pathological. However, I don't think it is so obviously sane, either.

onlybecause it looks insane (and it looks insane only because you avoid it, so you think you've gone mad if you take it). This principle seems potentially strong enough that you'd want to apply it to the Troll Bridge case as well, even though in Troll Bridge it won't actually help us make the right decision (it just suggests that expecting the bridge to blow up isn't a legit counterfactual).Troll Bridge is a rare case where agents that require proof to take action can prove they would be insane to take some action before they've thought through its consequences. Can you show how they could unwisely do this in chess, or some sort of Troll Chess?

I don't see how this agent seems to control his sanity. Does the agent who jumps off a roof iff he can (falsely) prove it wise choose whether he's insane by choosing whether he jumps?

The agent in Troll Bridge thinks that it can make itself insane by crossing the bridge. (Maybe this doesn't answer your question?)

I make no claim that this sort of case is common. Scenarios where it comes up and is relevant to X-risk might involve alien superintelligences trolling human-made AGI. But it isn't exactly high on my list of concerns. The question is more about whether particular theories of counterfactual are right. Troll Bridge might be "too hard" in some sense -- we may just have to give up on it. But, generally, these weird philosophical counterexamples are more about pointing out problems. Complex real-life situations are difficult to deal with (in terms of reasoning about what a particular theory of counterfactuals will actually do), so we check simple examples, even if they're outlandish, to get a better idea of what the counterfactuals are doing in general.

Correct. I am trying to pin down exactly what you mean by an agent controlling a logical statement. To that end, I ask whether an agent that takes an action iff a statement is true controls the statement through choosing whether to take the action. ("The Killing Curse doesn't crack your soul. It just takes a cracked soul to cast.")

Perhaps we could equip logic with a "causation" preorder such that all tautologies are equivalent, causation implies implication, and whenever we define an agent, we equip its control circuits with causation. Then we could say that A doesn't cross the bridge because it's not insane. (I perhaps contentiously assume that insanity and proving sanity are causally equivalent.)

If we really wanted to, we could investigate the agent that only accepts utility proofs that don't go causally backwards. (Or rather, it requires that its action provably causes the utility.)

You claimed this reasoning is unwise in chess. Can you give a simple example illustrating this?

The point here is that the agent described is acting like EDT is supposed to -- it is checking whether its action implies X. If yes, it is acting as if it controls X

in the sense thatit is deciding which action to take using those implications. I'm not arguing at all thatweshould think "implies X" is causal, nor even that the agent has opinions on the matter; only thatthe agent seems to be doing something wrong, and one way of analyzing what it is doing wrong is to take a CDT stance and say "the agent is behaving as if it controls X" -- in the same way that CDT says to EDT "you are behaving as if correlation implies causation" even though EDT would not assent to this interpretation of its decision.I think you have me the wrong way around; I was suggesting that certain causally-backwards reasoning would be unwise in chess, not the reverse. In particular, I was suggesting that we should not judge a move poor

becausewe think the move is something only a poor player would do, but always the other way around. For example, suppose we have a prior on moves which suggests that moving a queen into danger is something only a poor player would do. Further suppose we are in a position to move our queen into danger in a way which forces checkmate in 4 moves. I'm saying that if we reason "I could move my queen into danger to open up a path to checkmate in 4. However, only poor players move their queen into danger. Poor players would not successfully navigate a checkmate-in-4. Therefore, if I move my queen into danger, I expect to make a mistake costing me the checkmate in 4. Therefore, I will not move my queen into danger." That's an example of the mistake I was pointing at.Note: I do not personally endorse this as an argument for CDT! I am expressing these arguments because it is part of the significance of Troll Bridge. I think these arguments are the kinds of things one should grapple with if one is grappling with Troll Bridge. I have defended EDT from these kinds of critiques extensively elsewhere. My defenses do not work against Troll Bridge, but they do work against the chess example. But I'm not going into those defenses here because it would distract from the points relevant to Troll Bridge.

If I'm a poor enough player that I merely have evidence, not proof, that the queen move mates in four, then the heuristic that queen sacrifices usually don't work out is fine and I might use it in real life. If I can prove that queen sacrifices don't work out, the reasoning is fine even for a proof-requiring agent. Can you give a chesslike game where some proof-requiring agent can prove from the rules and perhaps the player source codes that queen sacrifices don't work out, and therefore scores worse than some other agent would have? (Perhaps through mechanisms as in Troll bridge.)

The heuristic can override mere evidence, agreed. The problem I'm pointing at isn't that the heuristic is fundamentally bad and shouldn't be used, but rather that it shouldn't circularly reinforce its own conclusion by counting a hypothesized move as

differentiallysuggesting you're a bad playerin the hypothetical where you make that move. Thinking that way seems contrary to the spirit of the hypothetical (whose purpose is to help evaluate the move). It's fine for the heuristic to suggest things are bad in that hypothetical (because you heuristically think the move is bad); it seems much more questionable to suppose thatyour subsequent moveswill be worse in that hypothetical, particularly if that inference is a lynchpin if your overall negative assessment of the move.What do you want out of the chess-like example? Is it enough for me to say the troll could be the other player, and the bridge could be a strategy which you want to employ? (The other player defeats the strategy if they think you did it for a dumb reason, and they let it work if they think you did it smartly, and they know you well, but you don't know whether they think you're dumb, but you do know that

ifyou were being dumbthenyou would use the strategy.) This is can be exactly troll bridge as stated in the post, but set in chess with player source code visible.I'm guessing that's not what you want, but I'm not sure what you want.

I started asking for a chess example because you implied that the reasoning in the top-level comment stops being sane in iterated games.

In a simple iteration of Troll bridge, whether we're dumb is clear after the first time we cross the bridge. In a simple variation, the troll requires smartness even given past observations. In either case, the best worst-case utility bound requires never to cross the bridge, and A knows crossing blows A up. You seemed to expect more.

Suppose my chess skill varies by day. If my last few moves were dumb, I shouldn't rely on my skill today. I don't see why I shouldn't deduce this ahead of time and, until I know I'm smart today, be extra careful around moves that to dumb players look extra good and are extra bad.

More concretely: Suppose that an unknown weighting of three subroutines approval-votes on my move: Timmy likes moving big pieces, Johnny likes playing good chess, and Spike tries to win in this meta. Suppose we start with move A, B or C available. A and B lead to a Johnny gambit that Timmy would ruin. Johnny thinks "If I play alone, A and B lead to 80% win probability and C to 75%. I approve exactly A and B.". Timmy gives 0, 0.2 and 1 of his maximum vote to A, B and C. Spike wants the gambit to happen iff Spike and Johnny can outvote Timmy. Spike wants to vote for A and against B. How hard Spike votes for C trades off between his test's false positive and false negative rates. If B wins, ruin is likely. Spike's reasoning seems to require those hypothetical skill updates you don't like.

Right, OK. I would say "sequential" rather than "iterated" -- my point was about making a weird assessment of your own future behavior, not what you can do if you face the same scenario repeatedly. IE: Troll Bridge might be seen as artificial in that the environment is explicitly designed to punish you if you're "dumb"; but, perhaps a sequential game can punish you more naturally by virtue of poor future choices.

Yep, I agree with this.

I concede the following points:

However, I still assert the following:

if the very reason why you think the action looks dumb is that the next actions are probably dumb if you take it.IE, you don't have a prior heuristic judgement that a move is one which you make when you're dumb; rather, you've circularly concluded that the move would be dumb -- because it's likely to lead to a bad outcome -- because if you take that move your subsequent moves are likely to be bad -- because it is a dumb move.

I don't have a natural setup which would lead to this, but the point is that it's a crazy way to reason rather than a natural one.

The question, then, is whether the troll-bridge reasoning is analogous to to this.

I think we should probably focus on the probabilistic case (recently added to the OP), rather than the proof-based agent. I could see myself deciding that the proof-based agent is more analogous to the sane case than the crazy one. But the probabilistic case seems completely wrong.

In the proof-based case, the question is: do we see the Löbian proof as "circular" in a bad way? It makes sense to conclude that you'd only cross the bridge when it is bad to do so, if you can see that proving it's a good idea is inconsistent. But does the proof that that's inconsistent "go through" that very inference? We know that the troll blows up the bridge if we're dumb, but that in itself doesn't constitute outside reason that crossing is dumb.

But I can see an argument that our "outside reason" is that we can't

knowthat crossing is safe, and since we're a proof-based agent, would never take the risk unless we're being dumb.However, this reasoning does not apply to the probabilistic agent. It can cross the bridge as a calculated risk. So its reasoning seems absolutely circular. There is no "prior reason" for it to think crossing is dumb; and, even if it did think it more likely dumb than not, it doesn't seem like it should be 100% certain of that. There should be some utilities for the three outcomes which preserve the preference ordering but which make the risk of crossing worthwhile.

I think the counterfactuals used by the agent are the correct counterfactuals for someone else to use while reasoning about the agent from the outside, but not the correct counterfactuals for the agent to use while deciding what to do. After all, knowing the agent's source code, if you see it start to cross the bridge, it is correct to infer that it's reasoning is inconsistent, and you should expect to see the troll blow up the bridge. But while deciding what to do, the agent should be able to reason about purely causal effects of its counterfactual behavior, screening out other logical implications.

Disagree that that's what's happening. The link between the consistency of the reasoning system and the behavior of the agent is because the consistency of the reasoning system controls the agent's behavior, rather than the other way around. Since the agent is selecting outcomes based on their consequences, it does make sense to speak of the agent choosing actions to some extent, but I think speaking of logical implications of the agent's actions on the consistency of formal systems as "controlling" the consistency of the formal system seems like an inappropriate attribution of agency to me.

I agree with everything you say here, but I read you as thinking you disagree with me.

Yeah, that's the problem I'm pointing at, right?

I think we just agree on that? As I responded to another comment here:

Suppose the bridge is safe iff there's a proof that the bridge is safe. Then you would forbid the reasoning "Suppose I cross. I must have proven it's safe. Then it's safe, and I get 10. Let's cross.", which seems sane enough in the face of Löb.

Heres what I imagine the agent saying in its defense:

Yes, of course I can control the consistency of PA, just like everything else can. For example, imagine that you're using PA and you see a ball rolling. And then in the next moment, you see the ball stopping and you also see the ball continuing to roll. Then obviously PA is inconsistent.

Now you might think this is dumb, because its impossible to see that. But why do you think its impossible? Only because its inconsistent. But if you're using PA, you must believe PA really might be inconsistent, so you can't believe its impossible. Remember also that in our agent design theres no clear boundary between perceiving and thinking that you could use to avoid that.

So whether PA is inconsistent depends on the ball: if it stopped and continued to roll, that would make PA inconsistent. Similarly, if my source code implies that I won't do something, and then I do it, I make PA inconsistent. You just don't really believe that your logic might be inconsistent. If you want me to do something else, give me a new theory of logic.

I'm not sure I believe the agent about this, but its not obviously worse than the interpretation given here.

This part, at least, I disagree with. If I'm using PA, I can prove that ¬(A&¬A). So I don't need to believe PA is consistent to believe that the ball won't stop rolling and also continue rolling.

On the other hand, I have no direct objection to believing you can control the consistency of PA by doing something else than PA says you will do. It's not

a prioriabsurd to me. I have two objections to the line of thinking, but both are indirect.definitelycollapse. It seems particularly absurd that, in some sense, the reason you think that is justbecause you think that.Sure, thats always true. But sometimes its also true that A&¬A. So unless you believe PA is consistent, you need to hold open the possibility that the ball will both (stop and continue) and (do at most one of those). But of course you can also prove that it will do at most one of

those. And so on. I'm not very confident whats right, ordinary imagination is probably just misleading here.The facts about what you think are theorems of PA. Judging from the outside: clearly if an agent with this source code crosses the bridge, then PA is inconsistent. So, I think the agent is reasoning correctly about the kind of agent it is. I agree that the outcome looks bad - but its not clear if the agent is "doing something wrong". For comparison, if we built an agent that would only act if it could be sure its logic is consistent, it wouldn't do anything - but its not doing anything wrong. Its looking for logical certainty, and there isn't any, but thats not its fault.

I think you're still just confusing levels here. If you're reasoning using PA, you'll hold open the possibility that PA is inconsistent, but you

won'thold open the possibility that A&¬A. You believe theworldis consistent. You're just not so sure about PA.I'm wondering what you mean by "hold open the possibility".

Do you? This sounds like PA is not actually the logic you're using. Which is realistic for a human. But if PA is indeed inconsistent, and you don't have some further-out system to think in, then what is the difference to you between "PA is inconsistent" and "the world is inconsistent"? In both cases you just believe everything and its negation. This also goes with what I said about thought and perception not being separated in this model, which stand in for "logic" and "the world". So I suppose that is where you would look when trying to fix this.

You do fully believe in PA. But it might be that you also believe its negation. Obviously this doesn't go well with probabilistic approaches.

Maybe this is the confusion.

I'mnot using PA. I'm assuming (well, provisionally assuming) PA is consistent.If PA is consistent, then an agent using PA believes the world is consistent -- in the sense of assigning probability 1 to tautologies, and also assigning probability 0 to contradictions.

(At least, 1 to tautologies it can recognize, and 0 to contradictions it can recognize.)

Hence, I (standing outside of PA) assert that (since I think PA is probably consistent) agents who use PA don't know whether PA is consistent, but, believe the world is consistent.

If PA were inconsistent, then we need more assumptions to tell us how probabilities are assigned. EG, maybe the agent "respects logic" in the sense of assigning 0 to refutable things. Then It assigns 0 to everything. Maybe it "respects logic" in the sense of assigning 1 to provable things. Then it assigns 1 to everything. (But we can't have both. The two notions of "respect logic" are equivalent if the underlying logic is consistent, but not otherwise.)

But such an agent doesn't have much to say for itself anyway, so it's more interesting to focus on what the consistent agent has to say for itself.

And I think the consistent agent

very much does not"hold open the possibility" that the world is inconsistent. It actively denies this.Seems to me that if an agent with a reasonable heuristic for logical uncertainty came upon this problem, and was confident but not certain of its consistency, it would simply cross because expected utility would be above zero, which is a reason that doesn't betray an inconsistency. (Besides, if it survived it would have good 3rd party validation of its own consistency, which would probably be pretty useful.)

I agree that "it seems that it should". I'll try and eventually edit the post to show why this is (at least) more difficult to achieve than it appears. The short version is that a proof is still a proof for a logically uncertain agent; so, if the Löbian proof did still work, then the agent would update to 100% believing it, eliminating its uncertainty; therefore, the proof still works (via its Löbian nature).

The proof doesn't work on a logically uncertain agent. The logic fails here:

A logically uncertain agent does not need a proof of either of those things in order to cross, it simply needs a positive expectation of utility, for example a heuristic which says that there's a 99% chance crossing implies U=+10.

Though you did say there's a version which still works for logical induction. Do you have a link to where I can see that version of the argument?

Edit: Now I still see the logic. On the assumption that the agent crosses but also proves that U=-10, the agent must have a contradiction somewhere, because that, and the logical uncertainty agents I'm aware of have a contradiction upon proving U=-10 because they prove that they will not cross, and then immediately cross in a maneuver meant to prevent exactly this kind of problem.

Wait but proving crossing implies U=-10 does not mean that prove they will not cross, exactly because they might still cross, if they have a contradiction.

God this stuff is confusing. I still don't think the logic holds though.

I've now edited the post to give the version which I claim works in the empirically uncertain case, and give more hints for how it still goes through in the fully logically uncertain case.

Sorry to necro this here, but I find this topic extremely interesting and I keep coming back to this page to stare at it and tie my brain in knots. Thanks for your notes on how it works in the logically uncertain case. I found a different objection based on the assumption of logical omniscience:

Regarding this you say:

However, this assumes that the Löbian proof exists. We show that the Löbian proof of A=cross→U=−10 exists by showing that the agent can prove □(A=cross→U=−10)→(A=cross→U=−10), and the agent's proof seems to assume logical omniscience:

If

□here means "provable in PA", the logic does not follow through if the agent is not logically omniscient: the agent might find crossing to have a higher expected utility regardless, because it may not have seen the proof. If□here instead means "discoverable by the agent's proof search" or something to that effect, then the logic here seems to follow through (making the reasonable assumption that if the agent can discover a proof for A=cross->U=-10, then it will set its expected value for crossing to -10). However, that would mean we are talking about provability in a system which can only prove finitely many things, which in particular cannot contain PA and so Löb's theorem does not apply.I am still trying to wrap my head around exactly what this means, since your logic seems unassailable in the logically omniscient case. It is counterintuitive to me that the logically omniscient agent would be susceptible to trolling but the more limited one would not. Perhaps there is a clever way for the troll to get around this issue? I dunno. I certainly have no proof that such an agent cannot be trolled in such a way.

Right, this is what you have to do.

Hmm. So, a bounded theorem prover using PA can still prove Löb about itself. I think everything is more complicated and you need to make some assumptions (because there's no guarantee a bounded proof search will find the right Löbian proof to apply to itself, in general), but you can make it go through.

I believe the technical details you're looking for will be in Critch's paper on bounded Löb.

Viewed from the outside, in the logical counterfactual where the agent crosses, PA can prove its own consistency, and so is inconsistent. There is a model of PA in which "PA proves False". Having counterfactualed away all the other models, these are the ones left. Logical counterfactualing on any statement that can't be proved or disproved by a theory should produce the same result as adding it as an axiom. Ie logical counterfactualing ZF on choice should produce ZFC.

The only unreasonableness here comes from the agent's worst case optimizing behaviour. This agent is excessively cautious. A logical induction agent, with PA as a deductive process will assign some prob P strictly between 0 and 1 to "PA is consistent". Depending on which version of logical induction you run, and how much you want to cross the bridge, crossing might be worth it. (the troll is still blowing up the bridge iff PA proves False)

A logical counterfactual where you don't cross the bridge is basically a counterfactual world where your design of logical induction assigns lower prob to "PA is consistent". In this world it doesn't cross and gets zero.

The alternative is a logical factual where it expects +ve util.

So if we make logical induction like crossing enough, and not mind getting blown up much, it crosses the bridge. Lets reverse this. An agent really doesn't want blown up.

In the counterfactual world where it crosses, logical induction assigns more prob to "PA is consistant". The expected utility procedure has to use its real probability distribution, not ask the counterfactual agent for its expected util.

I am not sure what happens after this, I think you still need to think about what you do in impossible worlds. Still working it out.

I've now edited the post to address uncertainty more extensively.

I don't totally disagree, but see my reply to Gurkenglas as well as my reply to Andrew Sauer. Uncertainty doesn't really save us, and the behavior isn't really due to the worst-case-minimizing behavior. It can end up doing the same thing even if getting blown up is only slightly worse than not crossing! I'll try to edit the post to add the argument wherein logical induction fails eventually (maybe not for a week, though). I'm much more inclined to say "Troll Bridge is too hard; we can't demand so much of our counterfactuals" than I am to say "the counterfactual is actually perfectly reasonable" or "the problem won't occur if we have reasonable uncertainty".

"(It makes sense that) A proof-based agent can't cross a bridge whose safety is dependent on the agent's own logic being consistent, since proof-based agents can't know whether their logic is consistent."

If the agent crosses the bridge, then the agent knows itself to be consistent.

The agent cannot know whether they are consistent.

Therefore, crossing the bridge implies an inconsistency (they know themself to be consistent, even though that's impossible.)

The counterfactual reasoning seems quite reasonable to me.

See my reply to Gurkenglass.

Does this way of writing "if" mean the same thing as "iff", i.e. "if and only if"?

No, but I probably should have said "iff" or "if and only if". I'll edit.

Interesting.

I have two issues with the reasoning as presented; the second one is more important.

First of all, I'm unsure about "Rather, the point is that the agent's "counterfactual" reasoning looks crazy." I think we don't know the agent's counterfactual reasoning. We know, by Löb's theorem, that "there exists a proof that (proof of L implies L)" implies "there exists a proof of L". It doesn't tell us what structure this proof of L has to take, right? Who knows what counterfactuals are being considered to make that proof? (I may be misunderstanding this).

Second of all, it seems that if we change the last line of the agent to [else, "cross"], the argument fails. Same if we insert [else if A()="cross" ⊢ U=-10, then output "cross"; else if A()="not cross" ⊢ U=-10, then output "not cross"] above the last line. In both cases, this is because U=-10 is now possible, given crossing. I'm suspicious when the argument seems to depend so much on the structure of the agent.

To develop that a bit, it seems the agent's algorithm as written implies "If I cross the bridge, I am consistent" (because U=-10 is not an option). If we modify the algorithm as I just suggested, then that's no longer the case; it can consider counterfactuals where it crosses the bridge and is inconsistent (or, at least, of unknown consistency). So, given that, the agent's counterfactual reasoning no longer seems so crazy, even if it's as claimed. That's because the agent's reasoning needs to deduce something from "If I cross the bridge, I am consistent" that it can't deduce without that. Given that statement, then being Löbian or similar seems quite natural, as those are some of the few ways of dealing with statements of that type.

The agent as described is using provable consequences of actions to make decisions. So, it is using provable consequences

as counterfactuals. At least, that's the sense in which I mean it -- forgive my terminology if this doesn't make sense to you. I could have said "the agent's conditional reasoning" or "the agent's consequentialist reasoning" etc.I think the benefit of the doubt does not go to the agent, here. If the agent's reasoning is sensible only under certain settings of the default action clause, then one would want a story about how to set the default action clause ahead of time so that we can ensure that the agent's reasoning is always sensible. But this seems impossible.

However, I furthermore think the example can be modified to make the agent's reasoning look silly in any case.

If the last line of the agent reads "cross" rather than "not cross", I think we can recover the argument by changing what the troll is doing. The general pattern is supposed to be: the troll blows up the bridge if we cross "for a dumb reason" -- where "dumb" is targeted at agents who do anything analogous to epsilon exploration or the chicken rule.

So, we modify the troll to blow up the bridge if PA is inconsistent OR if the agent reaches its "else" clause.

The agent can no longer be accused of reasoning as if it controlled PA. However, it still sees the bridge blowing up as a definite consequence of its crossing the bridge. This still doesn't seem sensible, because its very reason for believing this involves a circular supposition that it's provable -- much like my chess example where an agent concludes that an action is poor due to a probabilistic belief that it would be a poor player if it took that sort of action.

[Note that I have not checked the proof for the proposed variant in detail, so, could be wrong.]

That was my first rewriting; the second is an example of a more general algorithm that would go something like this. If we assume that both probabilities and utilities are discrete, all of the form q/n for some q, and bounded above and below by N, then something like this (for EU the expected utility, and Actions the set of actions, and b some default action):

Then the Löbian proof fails. The agent will fail to prove any of those "if" implications, until it proves "A()="not cross" ⊢ EU=0". Then it outputs "not cross"; the default action b is not relevant. Also not relevant, here, is the order in which a is sampled from "Actions".

I don't see why the proof fails here; it seems to go essentially as usual.

Reasoning in PA:

Suppose a=cross->u=-10 were provable.

Further suppose a=cross.

Note that we can see there's a proof that not crossing gets 0, so it must be that a better (or equal) value was found for crossing, which must have been +10 unless PA is inconsistent, since crossing implies that u is either +10 or -10. Since we already assumed crossing gets -10, this leads to trouble in the usual way, and the proof proceeds from there.

(Actually, I guess everything is a little messier since you haven't stipulated the search order for actions, so we have to examine more cases. Carrying out some more detailed reasoning: So (under our assumptions) we know PA must have proved (a=cross -> u=10). But we already supposed that it proves (a=cross -> u=-10). So PA must prove not(a=cross). But then it must prove prove(not(a=cross)), since PA has self-knowledge of proofs. Which means PA can't prove that it'll cross, if PA is to be consistent. But it knows that proving it doesn't take an action makes that action very appealing; so it knows it would cross, unless not crossing was equally appealing. But it can prove that not crossing nets zero. So the only way for

notcrossing to be equally appealing is for PA toalsoprove not crossing nets 10. For this to be consistent, PA has to prove that the agent doesn't not-cross. But now we have PA proving that the agent doesn't cross and also that it doesn't not cross! So PA must be inconsistent under our assumptions. The rest of the proof continues as usual.)You are entirely correct; I don't know why I was confused.

However, looking at the proof again, it seems there might be a potential hole. You use Löb's theorem within an assumption sub-loop. This seems to assume that from "A→(□B→B)", we can deduce "A→B".

But this cannot be true in general! To see this, set A=□B→B. Then A→(□B→B), trivially; if, from that, we could deduce A→B, we would have (□B→B)→B for any B. But this statement, though it looks like Löb's theorem, is one that we cannot deduce in general (see Eliezer's "medium-hard problem" here).

Can this hole be patched?

(note that if A→(□AB→B), where □A is a PA proof that adds A as an extra axiom, then we can deduce A→B).

I'm finding some of the text in the comic slightly hard to read.

Yep, sorry. The illustrations were not actually originally meant for publication; they're from my personal notes. I did it this way (1) because the pictures are kind of nice, (2) because I was frustrated that no one had written a good summary post on Troll Bridge yet, (3) because I was in a hurry. Ideally I'll edit the images to be more suitable for the post, although adding the omitted content is a higher priority.

How can you (in general) conclude something by examining the source code of an agent, without potentially implicating the Halting Problem?

Nothing stops the Halting problem being solved in particular instances. I can prove that some agent halts, and so can it. See FairBot in Robust Cooperation in the Prisoner's Dilemma.

In this case, we have (by assumption) an output of the program, so we just look at the cases where the program gives that output.