Formalizing Sleeping Beauty

General
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I’ve had this idea for formalizing anthropics problems by making separate models for the real world, and for the world of experiences – and connecting these with an adjoint functor pair.

With Sophie Libkind, Owen Lynch, and Joe Moller, we were able to get a simple version of this working!

Next, I wanted to try formalizing a classic anthropics problem: Sleeping Beauty.

In this problem, there will be a fair coinflip. If it comes up heads, then Sleeping Beauty will be woken up once. If it comes up tails, she will be woken up twice, with a memory-loss drug in-between so she can’t tell whether it’s her first or second time waking up.

Every time she’s woken, she will be asked to give her probability that the coin landed Heads. The dilemma is whether she’ll say 1/2 (since she knows the coinflip was fair), or 1/3 (since knows she’ll be woken twice as often in the Tails case).

Motivation

The motivation for this whole adjoint functor approach comes from a simple idea.

A sentence like “Adele sees red.” may be True or False. Furthermore, everyone should (in principle) agree with the veracity of this sentence.

On the other hand, a sentence like “I see red.” may also be True or False, but it depends now on who is saying it. It could very well be the case that “I see red.” is simultaneously true for me and false for you. This sort of sentence is known as an indexical.

Because of this difference, I thought it might be interesting to think of indexical statements as existing in a parallel version of logic, where instead of “True” and “False”, we called the truth values “Here” and “Not Here”, and which are understood to be relative to a specific entity.

I called these the “0th Person World” and “1st Person World” respectively, but this seemed mostly to confuse people.

Anyway, the upshot of thinking of it as its own logic is that we could use it as a foundation to define more complex mathematical objects, such as probabilities (via e.g. Cox’s theorem) which would have indexical meanings parallel to the traditional probabilities.

The question then arises, how do these two parallel worlds relate to each other? My intuition was that it would be an adjoint pair of functors, which I called “SENSATIONS” and “REALIZATION”. The idea was that SENSATIONS should take a piece of the world to the experiences occurring within it, and REALIZATION should take a part of experience to the parts of the world capable of instantiating it.

Formalization attempt

For formalization purposes, I’ve tried to simplify this set-up as much as possible. In particular, the sleeping and being asked states are collapsed (perhaps we probe her brain while she sleeps), and she has no episodic memory (so no memory-loss drug is needed).

On the other hand, I’ve made it so that she sees the result of the coinflip upon waking. This is because I think her having a subjective experience of the coinflip is important, as I’ll elaborate below.

Let W be our set of world states. For Sleeping Beauty, there are 6 world states:
(Monday, Heads, Sleeping)
(Monday, Heads, Awake)
(Monday, Tails, Sleeping)
(Monday, Tails, Awake)
(Tuesday, Tails, Sleeping)
(Tuesday, Tails, Awake)

Now let E be our set of experience states. This time there are three:
[Seeing Nothing]
[Seeing Heads]
[Seeing Tails]

And then we have our function f:W\to E
(_, _, Sleeping) \mapsto Seeing Nothing
(_, Heads, Awake) \mapsto Seeing Heads
(_, Tails, Awake) \mapsto Seeing Tails

So there’s the day, the coin, and the state of Sleeping Beauty. Her “Sleeping” state is a light sleep, in that she still is having the experience of “Seeing Nothing”. Once she’s awake, she has the experience of seeing the result of the coin flip.

From this, we can get an adjunction:

For each subset of world states, f_* maps it to the set of experiences contained within – giving it the right semantic meaning to be SENSATIONS. And f^* maps each set of experiences to the set of worlds which could instantiate it, thus making it work as REALIZATION!

Finally, we have two special elements: \mathsf{WORLDS}\in 2^{2^W}, and \mathsf{PRESENT}\in E.
\mathsf{WORLDS} represents the set of “worlds” which Sleeping Beauty still believes are possible. Each “world” is a set of world states that are mutually compatible (such as all the states in a sequence of states as the world evolves). \mathsf{PRESENT} is the experiential state that Sleeping Beauty is actually experiencing.

In particular, \mathsf{WORLDS} contains the following two sets:
H := {(Monday, Heads, Sleeping), (Monday, Heads, Awake)}
and
T := {(Monday, Tails, Sleeping), (Monday, Tails, Awake),
(Tuesday, Tails, Sleeping), (Tuesday, Tails, Awake))}

And when we probe Sleeping Beauty for her probabilities, \mathsf{PRESENT} will be [Seeing Nothing].

World Beliefs

First, let’s consider how Sleeping Beauty’s beliefs about the world will be updated as she experiences the \mathsf{PRESENT}.

First, let’s get the set of world states compatible with her \mathsf{PRESENT} experience.

f^*(\{\mathsf{PRESENT}\})

Next, we get the set of world states which she has considered to be live possibilities. This is \mu(\mathsf{WORLDS}), where \mu is from the powerset monad structure, and is essentially the union.

So the intersection f^*(\{\mathsf{PRESENT}\})\cap \mu(\mathsf{WORLDS}) represents the set of world states she considers live, updating from her \mathsf{PRESENT}.

Now we take k[f^*(\{\mathsf{PRESENT}\})\cap \mu(\mathsf{WORLDS})]\cap \mu(\mathsf{WORLDS}) to get the new value for \mathsf{WORLDS}, where k takes a set of world states to the subset of the powerset of world states where each set contains a value from the input.

Now let’s see what Sleeping Beauty thinks before she’s seen the result, i.e. when \mathsf{PRESENT} = [Seeing Nothing].

f^*(\{\mathsf{PRESENT}\}) is the set of states where she is sleeping:
(Monday, Heads, Sleeping)
(Monday, Tails, Sleeping)
(Tuesday, Tails, Sleeping)

So the intersection with \mu(\mathsf{WORLDS}) will be the same set (since we didn’t include any superfluous world states in this example).

Then k[W] is the set of worlds in \mathsf{WORLDS} which have at least one of these valid states. This is just \mathsf{WORLDS}, and so Sleeping Beauty’s \mathsf{WORLDS} does not change on the basis of this experience. Using the uniform prior, this implicitly makes her a “halfer”.

Anticipations

But now let’s say that we want to see what Sleeping Beauty anticipates seeing upon waking. After all, anticipations of future experiences are very often what people actually have in mind when considering probabilities.

First, we’ll define a function \mathbf{next}:E\to 2^E which takes an experience to the set of experiences which can validly follow it. For Sleeping Beauty, this takes [Seeing Nothing] to {[Seeing Heads], [Seeing Tails]}, and everything else to the empty set.

Then f^*(\mathbf{next}(\mathsf{PRESENT})) is the set of world states which house a possible next experience for Sleeping Beauty.

We the intersect this with \mu(\mathsf{WORLDS}) to get the next experiences Sleeping Beauty thinks are actually possible.

A := f^*(\mathbf{next}(\mathsf{PRESENT}))\cap \mu(\mathsf{WORLDS})

Now, we consider all the possible ways we can assign HERE values to the experiences on these world states.

\mathbf{YouAreHere}_i : 1\to A \xrightarrow{f} E

In our case, A consists of:
(Monday, Heads, Awake)
(Monday, Tails, Awake)
(Tuesday, Tails, Awake)

and so we have three ways to get an anticipated experience:
[Seeing Heads]
[Seeing Tails] (via Monday)
[Seeing Tails] (via Tuesday)

Using the uniform prior again, this means that before she opens her eyes, Sleeping Beauty anticipates [Seeing Heads] with probability 1/3.

(I feel like this is pretty messy and hope that there’s a cleaner way to do this. I feel like there’s a way in which this doesn’t quite mesh with the logical interpretation I want the HERE values to have, but as usual I’m having a hard time justifying this intuiton.)

Conclusion

I’m very pleased that the adjoint functor works essentially how I envisioned!

I like that this approach gives us both answers once we disambiguate the question. The “objective” answer is 1/2 and the “subjective” answer is 1/3, which feels intuitive to me.

I don’t like how janky and ad-hoc everything after the adjoint part feels, and I hope there’s a more natural way to reframe or express things.

I’m also of course hoping that this framework can be extended to more complex and structured models of the world or experiences. Ultimately, I want to grow this into a mathematical theory of anthropics.

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I should have caught this when I read this before, but I think that when you say f^\ast(\{\mathsf{PRESENT}\}) is all of W, you actually just mean that Sleeping Beauty doesn’t know what day it is, or what the coin flip is, but she does know that she is sleeping! So f^\ast(\{\mathsf{PRESENT}\}) should be half of W, if I understand the rest of what is going on here.

I think there’s also something interesting going on here, where you have chosen \mathbf{next} to go from the current state to “Sleeping Beauty’s next perceived state”. But you could also choose \mathbf{next} to increment the day, in which case I think the probabilities might be different.

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Ah thanks, I’ve fixed that now.

The way I envision anticipation in the full framework, I think it will work by including memories as parts of experiences. Then generally, the non-memory part of an experience will get saved to the memory of the next experience as part of the physical operation of Sleeping Beauty (or whoever). Anticipations will then look for the set of experiences where the non-memory part of PRESENT is part of the memory for them. The reason for doing it this way is to avoid presupposing time. It also lets us deal in a more uniform way questions where weird things happen with memory, since the memory mechanism is explicit and we can modify it according to the given shenanigans.

But I don’t think we can have \mathbf{next} be incremented by time for this example, since Sleeping Beauty does not have a way to experience time – e.g. she doesn’t know when she’s sleeping that she’s sleeping on Monday. We could introduce an internal clock as part of her experience if we wanted to explore this, which I think is interesting… (and we could do weird things like setting her clock back surreptitiously that could be fun to think about)

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Sleeping Beauty could be uncertain of the world clock, though. I.e., when she is woken up with heads, she does know it’s a Monday!

I feel like this is really the relational/nondeterministic version of a hidden Markov model. Which is such a cool extension of hidden Markov models!

I wonder if we can talk about nondeterministic Bayesian updating, which seems to be what is happening here where Sleeping Beauty has some “prior” about the logical relations between world states that is being updated.

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Could you say more about how you’re thinking about indexicals existing in a separate/parallel version of logic? I can’t quite understand how it differs from the usual account of indexicals (there, truth values \Omega are not \{0, 1\} but rather the W-indexed set \coprod_{w \in W} \{0, 1\} \to W; so it’s the difference between being true and being true at a world).

I think the details of the assumptions in the formalization confused me a little bit. Why is it important that Sleeping Beauty sees the result of the coin flip? And why do we need to represent the sleeping states or seeing nothing experiences at all?

I would think a simpler way to go would be to just have three worlds w_1 = (\text{Monday, H}), w_2 = (\text{Monday, T}), and w_3 = (\text{Tuesday, T}) and two experiential states e_1 = \text{Tails}, e_2 = \text{Heads}.

I also think you can give a more explicit construction of E from W. In particular, E is a quotient of W. In your set up, the equivalence relation would be something like “being experientially identical after observing the result of the coin flip.” And the map f: W \to E is then just the projection \pi: W \to W/\sim. This way of going also gives you explicit descriptions of the maps f^* and f_*.

I think viewing things this way is really important! Imagine the alternative experiment in which instead of being told the result of the flip, Sleeping Beauty is told what day of the week it is. Or how many times she’s been woken up. These result in different quotientings, and so different judgments about whether Sleeping Beauty ought to be a halfer or thirder.

That is, I think that’s really the whole ballgame here. There are different ways of representing our evidentiary/experiential states (different ways of quotienting) and in general, there is no consistent way to assign uniform credence with respect to each of these representations. The Principle of Indifference is underdetermined, it turns out.

(So the only reason you got the result that Sleeping Beauty is an “implicit halfer” or that halfer is the “objective answer” is because of how you chose to represent experiential states. And that result is not invariant across representations)

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