A Closer Look At Time Travel and Probability

; foreword : First posted many years ago elsewhere, now hosted here alongside my other essays. While my style has developed subtantially since this was first written, I think the contents remain worthwhile.

: related : - Time Travel and why everyone gets it "wrong"

• What's Expected of Us

; aside

: 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 analyze 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.

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.

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 a 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 refuse to 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) = ⅓

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. This results in one red or thirty blues in a row.

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 almost half the time!

This is easy to reconcile with path realism. There are 2^10^= 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 2^11^-1 = 2047 paths down the time-tree. If each is as likely as the other, then over 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) = ¼, and p(blue,red) = ½.

It's like the universe is savescumming, just 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 recent "saves" one after another, finding the newest 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 might simply cause the universe to effectively restart.

The stats from the double blue experiment don't lie: p(red) = ⅔; p(blue,blue) = ⅓.

Imag