Serpentine Squiggles

A Closer Look At Time Travel and Probability

Abstract — I discuss several models for assigning probability to timelines under the assumption that time travel is possible, but paradoxes are absolutely impossible, as is the case in many fictional worlds. The models are mathematically precise, and illuminate issues that have previously confused many people about what sort of timelines are "most likely". I discuss an example due to /u/TimTravel  in a old post on /r/HPMOR, then analyse whether time travel can be used to solve the halting problem. I outline how timeline probability may interact with physical probabilities, often used to justify physics "conspiring" or contriving a certain outcome to prevent paradox.

Total length: ~5000 words, or about 15-20 minutes of reading.

Edit: commenters have pointed out similarities between this and the Ted Chiang story, What's Expected of Us. The similarity wasn't intentional, but it's pretty interesting.

Contents

Introduction

Let's say you're walking down the street one day when a wizard appears in a clap of thunder, and places a strange gray device of buttons and switches into your hands.

You're looking down at it, struggling to make heads or tails of it, and then you look up and the wizard is gone.

At the top of the device, there is a slider, already set to the leftmost extreme. Below it, two switches: a power switch already set to ON, and an stiff, unlabeled switch, the exact gray of the surface, rising so inconspicuously low off the surface you almost miss it. Below that, two LED buttons, both inactive.

Suddenly, the left LED glows blue. Confused, you press the button (it goes in with a satisfying click) and the light flashes off instantly.

Furrowing your brow, you decide to press the button again. The blue light quickly comes on while your finger's still moving, and it again winks out immediately as the button is depressed. You try pressing the button again and again, and each time the blue light turn on, seeming to predict or anticipate the button press.

Then, the other LED button glows red. You press it, and it turns off; several tries later, you conclude it behaves exactly the same.

You decide now to deliberately not press either button, even if the lights were to shine encouragingly. But nothing happens; neither light comes back on. You move your finger closer to a button, determined to arrest its motion at the last possible second. But the light doesn't come on, even when your skin is brushing the cool metal. You forget it and press the button. The light blinks bright blue milliseconds before you've even decided.

Now, you (you, dear reader, not the above character) have already read the title of this post. This is strange device sends information backward in time. Specifically, it sends a single bit back in time one second.

Or well, you fiddle with the slider, and notice it controls the interval; you can set it to one minute, an hour, or even a day.

All that established, it's time to test something. "Red is heads, and blue tails," you say. A coin from your pockets is flipping in the air until you catch it and slap it down on your wrist.

The device shines blue. You lift your hand. It's heads.

You push the blue button anyway, out of habit, the light flashing off. And then it hits you: you have to commit intently to pressing the right button even when (especially when) the device is wrong.

Another test: if the device shines red again, you'll press blue. But if it shines blue, you'll press still blue.

There's a noticeable delay before the device tentatively shines a light.

It's blue.

Call this act forcing. You can force the device to be red or blue.

You try the coin flip test just a few more times. Now, the device is always right, even if it seems to pause a random interval before shining a light.

The opposite of forcing would be splinting (for 'splinterpoint'). This is, pressing the button for whichever light comes on next, with no tricks and no conditionals.

Finally, the last thing you can do — for a broad notion of 'can' — is what we'll call crashing. This is: pressing the button of whichever light doesn't blink on. It's less that you can do this, and more that you can intend this, and reality responds to that.

You give it a try right now: you commit to crashing if your next coin toss doesn't come up heads.

You flip the coin, anxiously watching it's path through the air, catch it, slap it down on your wrist, spend a few seconds working up the nerve and then lifting your hand. It's tails.

You take a deep breath, and look expectantly at the device.

No light comes on. You're waiting for a few minutes.

And then it hits you; the device isn't binary, it's trinary. Sure, it can shine red or blue — but so too can it not shine at all! And if it either light leads to paradox, why would any light come on? The only winning move is not to play.

Is that it, then? Are your dreams of munchkinry doomed to fail? Was it just a coincidence that 'forcing' seemed to work earlier?

And then the red light comes on. You grin triumphantly, with not a little dread. You're about to destroy the universe! Before the implications catch up to, you're flinging your hand forward, jabbing it at the device. You don't want to lose your nerve.

You look down, and see that you missed, pressing the red button, rather than the blue like you planned.

Is this fate? Is the world itself conspiring to prevent paradox, just like in the stories? You want to give crashing another try, but the last thing you want is to wait those long minutes for the light to come on again. You glare down at the device, and then you notice the second switch. You'd almost forgotten about it.

You idly flick it, and immediately the blue light comes on.

It forces a prediction? Maybe your plans aren't doomed. You consider giving crashes another try, but maybe destroying the whole timeline is not worth the risk. You decide to spare the universe, and press the blue button.

You need to understand how this device works before you can really exploit it. And you have just the idea for another experiment. What if you splint, and if the splint comes out blue, you force blue again, but otherwise you just splint again. After two button presses, you turn off the device.

It's clear there are three possibilities: blue-blue, red-blue and red-red. But which are most likely?

You run this experiment a hundred times, and keep track of the results.

Call it the double blue experiment.

There are a few ways it could turn out:

Model A: Path Realism

It seems that consistent timelines are the only thing that matters. It's as if the universe has already set aside exactly the number of timelines there needs to be, and you're already in a certain timeline, you just don't know which one yet.

In the double blue experiment, there are three possibilities, and every one is equally likely. p(red,red) = p(red,blue) = p(blue,blue) = 1/3

You find it strange, as a follow-up experiment aptly demonstrates:

Splint once. If it comes out blue, force blue twenty-nine times. Otherwise, do nothing. Turn off the device.

On the face of it, it's crazy that you can even experience the second possibility. It's like winning the lottery half the time. Then again, maybe it's not so crazy? If you were to just force blue twenty-nine times, it's equally unlikely on the face of it; like flipping dozens of coins that all come up heads.

There's a weirder consequence, though. If you splint ten times, you can see any combination of reds and blues; red-blue-blue-red-red-red-blue-red-red-red and all the others, with uniform probability.

But if you splint ten times, and if and only if every splint came up blue, you splint ten more times, you'll find that the first set of splints come up all blue half the time!

This is easy to reconcile with path realism. There are 210 = 1024 through the ten splints. Each is as likely as the other.

But if you commit to doing ten more splints if and only if the first set comes up all blues, then there are 211 = ~2048 paths down the time-tree. If each is as likely as the other, then half of them are located under one branch!

Model B: Local Branch Realism

It seems that splints are basically coin tosses; it either comes up blue or it comes up red. The exception is if one of those options always leads to paradox. If you commit to causing paradox when the light shines blue, then it will always shine red. If you commit to splinting then crashing when the first splint comes out blue, then the splint will similarly always shine red.

The intermediate is more interesting: in the double blue experiment (you splint twice and crash if both splints are blue), then half the time the first splint will come out red, but if the first splint comes out blue, the next one always comes out red. In numbers, the possibilities are p(red,red) = p(red,blue) = 1/4, and p(blue,red) = 1/2.

It's like the universe is a savescumming as a gamer might: it effectively saves to a new slot to every time a time travel event is about to happen. If a paradox happens, it reloads from its newest saves one after another, finding the latest one that lets it avoid the paradox.

Model C: Reroll Realism (or, Bayesian Branch Realism)

You're not sure if paradoxes really don't happen. You've looked at the numbers. What it suggests is that, rather than avoiding paradoxes, paradoxes could simply cause the universe to restart.

The stats from the double blue experiment don't lie: p(red) = 2/3, p(blue,blue) = 1/3.

Imagine you were simulating the universe. 1/2 the time, red comes up and you're just fine. 1/2 the time, blue comes up. 1/2 the time after that (for a total of 1/4 the time), blue comes again, and you've got a paradox on your hands.

What if you just, restarted the universe (if only from the latest unbranched timeline), and hoped it didn't happen again?

Well, there's a 1/4 chance it will. Since you have a 1/4 chance of restarting in the first place, that's 1/16 of the time you'll restart twice. Luckily, it's getting exponentially less likely.

Looked at another way, the odds of it coming up red is the limit of the infinite sum: 1/2 + 1/4 * 1/2 + 1/16 * 1/2 + 1/64 * 1/2 + 1/256 * 1/2 ...

This series converges on 2/3.

But there's another interpretation, with seems less like the work of a lazy programmer and more like something a statistician would come up with.

Suppose, as we must, that the timeline is consistent. What is the posterior probability of that timeline being red, given that 100% of red timelines are consistent, and 50% of blue timelines are consistent?

Or, in symbols:

P(red | consistency) = (P(consistency | red) * P(red)) / P(consistency)
P(red | consistency) = (1 * .5 / .75) = 2/3

Even more intuitively: you have four balls (timelines) you paint half of the balls red and half blue (splinterpoint), and you take away one blue ball (paradox). 2/3 of the remainder is red.

You'll recognize this as Bayes' Theorem.

Model D: Weighted Branch Realism

The reality is more subtle than you thought. It seems that, while you've never seen a paradox, if a branch has a path through splinterpoints that ends in paradox, that fact subtracts probability from the branch and gives it to its counterfactual sibling. This happens in Local Branch Realism too, but not to this degree: the very possibility that a time-path has a paradox however many days or years down the line always shaves some degree of probability, if only just a sliver; but naturally, that sliver increases as the paradox gets closer.

Thus, the results of the experiment are: p(red, red) = p(red, blue) = 3/8, while p(blue, blue) = 1/4 = 2/8.

You can see it clearer with a more involved experiment. Take your device and a sheet of paper and:

Splint, call this splint A:

According to weighted branch realism, the probabilities look like: P(foo) = 20/32 = 5/8, P(bar) = 9/32, P(baz) = 3/32.

To understand this result, we have to define a notion of "static paradox fraction", or spf. If you intend to force blue, then the spf is 1/2. Why? To force blue you must (intend to) cause a paradox in the event that not-blue happens. Despite that fact that paradoxes never happen, static paradox fractions seems be a real quantity in Weighted Branch Realism. It is as if the device is looking at every possible and impossible timeline, and measuring which ones are paradoxical.

(Note that static paradox fractions are diminuted by splints. So if you splint and when the splint is blue you then force red, the spf of the first splint is 1/4, even if there is no second splint whenever the first is red. This distinguishes it from simply counting paradoxical timelines; 1/3 of the timelines are paradoxical, but a paradox behind a splinterpoint has lesser weight.)

Furthermore, let's have a notion of "intrinsic probability" or ip. The ip of both splint outcomes is 1/2, even if one of them is paradoxical.

Thus:

P(C = red) = 1/2 (ip) + 1 (sibling's spf) * 1/2 (sibling's ip) = 1/2
P(B = red) = 1/2 (ip) + 1/2 (sibling's spf) * 1/2 (sibling's ip) = 3/4
P(A = red) = 1/2 (ip) + 1/4 (sibling's spf) * 1/2 (sibling's ip) = 5/8

To reiterate:

p(foo) = p(A = red) = 5/8, and
p(bar) = p(A = blue)) * p(B = red) = 3/8 * 3/4 = 18/64 = 9/32, and
p(baz) = p(A = blue) * p(B = blue) * p(C = red) = 3/8 * 1/4 * 1 = 6 / 64 = 3/32

(Note for the pedants: normally, the ip is actually 1/3, and ditto for spf; we're ignoring that the device can not shine a light, because you can just flip a switch and force a light on. Even without the switching, committing to either turning the device off, or splinting endlessly once the the experiment is over means the probability of the device choosing to not shine drops exponentially while the alternatives remain constant.)

This model is somewhat unintuitive, because despite the name, it has more in common with Path Realism than the other two _ Branch Realisms. You can't emulate the probability distribution of WBR by running one timeline and restarting (either from the beginning (Bayesian), or from the nearest viable alternate splint (Local)). This is entirely the fault of a phenomena we can call "paradox by association"; in the foo-bar-baz experiment, in a certain sense, just as 1/8 of quasi-timelines are paradoxical because they end in crashing, 1/4 of the quasi-timelines ending in baz are paradoxical just because baz timelines are near to the paradox.

This accounts for the numbers: p(foo) is 5/8, 4/8 intrinsic + 1/8 from the paradox. p(bar) is 9/32: 8/32 intrinsic + 1/32 from baz's paradox by association. p(baz), lastly is 3/32 owing to loosing 1/32 from paradox by association.

(Why 1/4? Good question. There must be a reason, and it's clear this is the number that comes out of the equations. Alas, I'm not smart enough provide a reason in words and not symbols.)

Which Model is Best?

Path Realism and Local Branch Realism are both pretty wack. Path Realism discards all local information about plausibility, and allows munchkins to blow up the probability of their favorite timelines arbitrarily high. Local Branch Realism does the same thing from the opposite direction; wanton invocation of paradoxes intuitively should be penalized, but Branch Realism simply says I don't mind.

Between Weighted Branch Realism and Reroll Realism, I'm inclined to prefer the latter. WBR is the first I thought up, but RR is just more natural. It has two obvious interpretations, both things that anyone would come up with after thinking about it for a little while. WBR, in the other hand, is harder to conceptualize in terms of what mechanism would actually cause the probabilities to look like that (I've tried; the results are not pretty). "Paradox by association", while potential a fresh concept to use in a story, is a truly strange mechanism.

Now, how does the connect with TimTravel's ideas? Just as he proposed, it is, in some models the case that the most probable timelines are the ones in which time machines are never invented. In Local Branch Realism, this is not true (unless some bad actor arises in every single timeline and causes paradox. Time Beast, anyone?). In Path Realism, this is again never true without positing a Time Beast. However in WBR and RR, it's more or less true. In general, timelines with fewer instances of retrocausation are more likely, only because instances of retrocausation are a proxy for instances of paradox. Now, if paradoxes are rare, this argument would be weak. (But to be fair, most meaningful uses of time travel require copious paradox; it's the oil in the engine.)

That said, I believe it is admissible for a work to posit that the characters find themselves in the (slightly unlikely) timeline where retrocausation happens. After that, though, the principles constrain the probability space.

Example: The Time Thief Puzzle

In the somewhat flawed post which inspired this, /u/TimTravel  outlines a paradoxical puzzle:

Suppose Alice has a bag of money with a dollar on it. If anyone steals it, she'll go back in time and see who did it. Bob wants to steal it. He knows she has this policy. He decides he'll give himself the thumbs up just before he leaves the future if all goes well stealing it and she doesn't see him. If these policies are followed then it leads to a paradox, so something must prevent them both from simultaneously following their policies. Either Alice wins because Bob goes to the past without getting an honest thumbs up from himself or Bob wins because Bob sees the honest thumbs up and Alice doesn't go back and check who stole the money for some reason, or some third possibility prevents both.

There is no reason to think that either of them automatically wins in this situation. Timelines in which Alice wins should be about equally frequent as timelines in which Bob wins. Numerous characters have implicitly assumed that there is a reason to think one of them automatically wins in such situations.

We'll have to change this scenario a little bit to fit with the schema we've been using so far. (Besides, Tim's example is kind of unclear and it's not even obvious that paradox must occur in all permutations. If Bob doesn't get the thumbs up, wouldn't he not steal? Puzzle solved.)

Alice and Bob

Let's say that in the morning Alice has acquired the bag full of money with a dollar sign on it from sources unknown, and has come to an arrangement with a shadowy individual: leave a dufflebag full of money with a dollar sign on it at a dropoff location, and in exchange, the individual will leave a limited print run of all eleven books of Worth the Candle at the same location.

[Editor's note: at the time this was written, the series was not finished; hence the joke.]

Alice knows people want to steal that money, but part of the arrangement is that she can't be there guarding it when the shadowy individual arrives.

On Tuesday morning, the deal is still in its negotiation stage, and there are two places Alice can think of to arrange for dropoffs: atop the looming mountains outside of town, or deep into the mysterious catacombs below it. Both of these hiding places will take two hours to enter and two to leave. (Pretend the mountains have a rogue paramilitary that shoots down helicopters or something.)

Due to work obligations, Alice can only make the dropoff in the early morning, and return that evening to pick up the books.

Meanwhile, Bob, the thief, knows all this and certainly doesn't want to get caught. He can't go into either location until Alice has left, else he'll be seen. Lucky for him, that leaves a large window for him to do the deed.

Both of these characters have the same magical devices from the earlier section, and they'll naturally use them to ensure success; except, for obvious reasons, we'll call their predictions "catacombs" and "mountains".

Before she goes to hide the money at 5:00 AM, Alice consults her device for where to hide it.

Four hours after he has seen Alice leave, at 9:00, Bob consults his device to determine where she hid it. If the predict is wrong, he forces a paradox.

When Alice returns to get the money, at 17:00, if it's there, she confirms the location that the device advised. Otherwise, she presses the opposite button, forcing a crash via paradox.

What happens?

This requires introducing yet another notion.

Interlude: TIME FORCE

The TIME FORCE is any one in a billion freak accident that happens 100% of the time to prevent a paradox from occurring.

TIME FORCE is a quantum fluctuation that causes right neuron to misfire which butterflies into changing your whole decision. TIME FORCE is random air currents that causes a bird to fly by and drop a rock on the right button of the time-device. TIME FORCE is the lightning in the clear blue sky which spells out Do not mess with time in typographically perfect serifs.

There are a few things we can say about TIME FORCE.

Let's say that the general odds of TIME FORCE acting on a given person in a given second is extremely, astronomically unlikely. One in a billion, or one in a trillion sounds about right.

But from that, it follows that the odds of TIME FORCE acting over an interval of time is proportional to the length of that interval. (It's at least monotonic. Difficult/impossible to say how fast it grows: in one second, it might only be possible for single extremely unlikely fluctuation to intervene; whereas across one hour, several unlikely things that on their own wouldn't make difference could co-occur and together cause a significant fluctuation.)

It also follows that the odds of TIME FORCE acting is increased if an agent is acting in concert with it, and decreased if they are acting in opposition, proportional to the efficacy of that agent. I.e., an agent is defending against TIME FORCE, or attempting to utilize TIME FORCE.

(Consider: if Bob, after stealing, were to proceed to try to also steal Alice's device or persuade her to cancel her prediction herself (e.g., by faking a dire emergency which requires her foreknowledge to solve), then TIME FORCE would provide some boost to the probability of success.)

An obvious corollary to all this is that TIME FORCE is almost never relevant. If you had a bigger device that spat out 32 red/blue pairs at a time, you could predict the lottery without seriously worrying about TIME FORCE.

One common confusion which leads people to overstate the importance of TIME FORCE is the fact that parallel universes and timelines aren't necessarily the same thing.

Let's say you wanted to force a coin to come up heads. Turn on your device. Then, splint. If the result was blue, flip the coin. If the result was red, splint again. The idea is to have the device spawn as many timelines as possible. Pressing buttons (subtly) alters the configuration of your brain and muscles and the microcurrents of air in the room, and the hope is a certain combination of buttons at a certain rhythm is prod you into the right configuration to flip the coin heads. This is almost certainly true in this specific example, but if the coin is flipped before the device is turned on, time cannot help you. And if you don't have intimate control over the outcome, time cannot save you. E.g., if a meteor is flying towards your town, forcing a paradox if it hits true cannot avert its course. Of course, if you splint long enough, maybe the branches describe a powerful, quickly-createable, meteor-destroying technology in morse code. Or maybe it just spells out "You needed worthy opponents," and you give up and let the asteroid take you.

(There is one slight exception, and this is where the different formulations of Bayesian Branch Realism and Reroll Realism differ. In BBR, the universe is posited to either A) know before splintering the posterior probabilities of each branch or equivalently, B) have so many timelines that destroying paradoxical ones leaves the distribution looking as it should. However, in RR, paradoxes might cause the universe to restart from the beginning (or when the device was turned on). This means that in RR, simply flipping a coin and forcing a paradox if it's tails is all you need. That is, assuming quantum fluctuations making the coin heads is more likely than quantum making you decide not to crash, or failing to crash. Or dying instantly and having the wizard return to push the button.)

There's one last possibility, and that's if you posit that quantum randomness itself are biased by time travel, so each quantum measurement counts as a splinterpoint. I'm reluctant to do such, because the edict I've heard over and over again is that when worldbuilding, Do Not Mess With Physics.

I'm going to continue writing this article with the assumption that physical randomness is not biased by timelines. Extreme improbabilities are still extremely improbable, but, to mangle the quote, when you have eliminated the impossible, whatever remains, however improbable, must happen.

Back to Alice and Bob

So, with TIME FORCE in mind, what happens to Alice and Bob?

It's 4:50 in the morning. Alice is sitting beside her bag of money with a dollar sign on it, her device in front of her. If the device shows 'catacombs', she intends to, when she returns from work, press 'mountains' in case her bag was stolen and she doesn't have her book, or otherwise she will confirm 'catacombs' (and vice versa).

She waits. And the device doesn't say anything at all!

It's well known that sometimes there are random delays before the devices spit out answers. Some users interpret it as an omen, suggesting that whatever you're asking is so likely to lead to paradox, time itself has to work up the nerve to allow it to happen; the theorized mechanism is 'paradox aversion', where in some models, the odds turn against timelines long before the paradox is even nigh. (But as far as Alice knows, no one has never proved which model they live in.)

She decides to buck superstition and conjecture, and reaches out to flip the switch which forces an output.

Record scratch, freeze frame. What happens next?

A) TIME FORCE intervenes before Alice can flip the switch.

B) Alice flips the switch, but TIME FORCE subverts the resulting prophecy. (I.e., the bag is stolen, but events contrive to have the incorrect button on the device pressed anyway.)

C) Alice flips the switch, and TIME FORCE subverts Bob's prophecy instead, sending him to the wrong location. Her bag is not stolen, and she happily reads the ending of WtC.

D) Alice flips the switch, and TIME FORCE subverts both prophecies.

(Stop reading now if you want to try to work out an answer yourself.)

The correct answer is B, which is perhaps three times more likely than anything else, barring unspecified details.

A requires TIME FORCE to act in the acute interval before Alice presses the button, which is at best a few minutes long.

C requires TIME FORCE to act in the four hour interval of 9:00-13:00.

D is the conjunction of A and C, and less likely than both.

B is the winner, because it only requires the TIME FORCE to act on the long, twelve-hour interval of 5:00-17:00

I think this goes even if timelines nudge physical probabilities. Exercise for the reader, though.

(((Now, one may object that this formulation bears little resemblance to Tim's example. My only excuse is that Tim's model was too unclear for me to formalize specifically. When I tried, I got this scenario:

First, Alice gets a prediction from the device: stolen, or untouched. Iff it says stolen, she waits to see who the thief is, and gets them. Else, she goes about her day, secure knowing her money is safe.

Then, Bob consults his device as to whether his theft would be successful: if it says yes, then either 1) Alice is there, catches him, and he triggers a paradox, or 2) Alice isn't there, he gets away, and she triggers a paradox later. However, if it says no, then he just sighs, and fucks off, no paradox to worry about.

Even if I missed something/misinterpreted TimTravel and this situation is paradoxical all four ways, it still follows that Bob will probably win (if not so overwhelmingly so) because he spends less time in temporal limbo where TIME FORCE might fuck with him.)))

Example: Hypercomputers?

It's clear that if one were to disassemble the strange device and hook up a few wires to its circuit boards to a computer, you'd create a hybrid device capable of advanced feats of computation. What is the exact strength of this retrocausal computer?

As mathematicians are wont to do, we will dispense with practicalities like having to use at most as much space as actually exists, or needing our computations finish before the heat death of the universe. Given all this, if we have an idealized retrocausal computer, a la the idealized turing machine, what can we do?

Let's try the halting problem, a classic test of strength. Say we have a computer program, and we want to know if it's ever stops running. Well, either it does or doesn't.

Consider a slightly different device, instead of red/blue leds, it has magic screen which can display any integer. (For models where it matters, the intrinsic probability of an integer n is equal to 2-k, where k is smallest number with 2k > n and k > 0.) It also has a numpad now, which allows the input of any integer.

With this device, to determine when a program halts, given that it halts, is as simple and looking at what number comes up on its screen, and running the program for that many steps. If it halts before then, input when it halted (causing paradox). Otherwise, input the number it gave you. Otherwise otherwise, cause a paradox via your preferred means.

If the program might run forever, things are trickier. What you can do is interpret the number the screen outputs as the index of a proof of (not) halting. This isn't sufficient, however, as no computably-checkable proof system can prove that any turing machine (never) halts, essentially by definition. But we can use the fact that if a program runs forever it doesn't halt: simply try over and over again until 1) you learn the program does not, or 2) the odds of it halting given that you found no proof is as astronomically low as satisfies you.

By construction, the odds of the screen outputing the right halting time decreases exponentially as the halting time increases. If the halting time is in the millions, it takes a several hundred trials before you have even odds of the screen having already spat out the right answer. If the time is in the billions, it takes several hundred thousand.

(Model-specific tricks can alleviate this quite a bit. In Path Realism, you can use the path blowup technique to increase the probability of the correct halting time coming up. In Weighted and Reroll, you can inflate the static paradox fraction to arbitrary heights, reducing the odds of false negatives.)

From ordinary turing machines, this is a difference in degree (retrocausal machines are better at it), but not kind (retrocausal machines can never decide whether a machine halts or doesn't).

Long story short, retrocausation can increase the efficacy of your computers, but you're still stuck at 0.

Applications to More Permissive Time Travel Models

Our device is quite limited, in the world of retrocausation. There are at least two stronger types of models:

Bound Time Travel

It's clear how our models transfer the bound case; proper time travel is basically sending a whole bunch of information at once. There's another hurdle though: can you tell from when a time travel comes?

With our red/blue device, the slider at the top puts an upper bound on how long the device waits for stablization. If the system allows this, then great! It means there's a clean cutoff point after which we know the timeline is stable or not.

Otherwise, you probably want to make probability proportional to how far in the future the traveler comes from; if you're uniformly selecting a person that could exist between now and the heat death of the universe (without grandfathering themselves, granted), it's probably not going to be you from two weeks hence, of all people.

There's a more interesting question this is avoiding though. What can we say about what will probably step out of the time machine, aside from whence it came?

Well, it's helpful to assume that there's an organization controlling and regulating time travel. There's some failure modes that would be cripplingly common. For instance, doppelgangers.

Temporal doppelgangers are a variation of the bootstrap paradox (i.e., self-causation), where a mutant version of your steps out of the time machine, finds current you, and forcibly alters your mind to replicate its own (anthropically, it must know how to succeed at this).

This seems pretty inevitable from the premise, and it provides a nice, fresh justification for "you can't interact with your past self". Not out of fear that it might cause a paradox, but out of fear that it won't. If your mind is randomly altered repeatedly, even by slight amounts each time, the results are quickly going to not be pretty.

Other than that, this scheme of time travel seems somewhat tractable; while the odds of any given arrangement of matter is a specific person with a specific set of memories consistent with the past and future of the extant universe is very very very low, there is some wiggle room, especially depending on the specifics of the time machine.

The assumption baked into our models is that, in effect, the time travel mechanism is plucking a random configuration of matter from possibility space. Most arrangements of matter, even restricting to the stable ones, aren't neat blobs of protein and water. And the most of the ones that are, are random goop!

Now, requiring that the configurations which arise in the past-time machine are exactly 1-1 equivalent to what enters the future-time machine is very tight requirement. I doubt bodies will be too much worse for wear if a few atoms are a few picometers off. And you can relax the requirement even further, allow what appears in the present to be "close enough" to its future equivalent, and increase the possibilities further. Of course, this will have ramifications; cancer, prions, strange tastes in the mouth.

The organization controlling the time machines could require that everyone who walks out of a time machine undergo a medical examination, and make most crippling ailments thereby paradoxical. (And, likewise for the dead bodies which can't walk out anyway).

Free Time Travel

Free Time Travel is the trickiest of all, but it has a few felicities in addition to all the extra warts. There's not necessarily authoritative time travel device (or an immediately plausible time travel agency) that you can stick in to stealthily add in extra conditions and assert nice properties.

With FTT, a time traveler could pop up anywhere, and at any time. Unless you add in a time agency that can monitor for new arrivals, there's nothing you can do about doppelgangers, unless you bolt 'no interacty with the past self' into the rules of the system somehow.

You probably shouldn't have location be conserved; requiring that you come out exactly where you came tightens probabilities too tightly. Allowing leeway puffs them up a bit. The same goes for concerns about exact molecular matching.

All those caveats aside, it seems as tho you can otherwise treat BTT and FTT similary to our toy examples, where they line up, showing the benefits of the simplification.

Conclusion

Well, that turned out much longer than I'd expected (or wanted). It feels like it puttered out here at the end, but I've said everything I set out to say and then some.

I hope this served to sharpen your intuitions regard time travel, and make precise things which were previously vague.

I would like to thank the nice people on the /r/rational discord for inspiring this line of thinking and providing the impetus to refine it.

Thank you for coming to my TED talk.

P.S.: worth mentioning that Tim covered much of the same ground as me in their initial post. My post is less a refutation to theirs than me working out my own solution to the problems they pose, as I didn't understand or believe all of their arguments.