by Brian Tomasik
First written: 6 Sep. 2016; last update: 6 Sep. 2016

This commentary on the 2014 film Time Lapse contains spoilers.

Time Lapse explores the ideas of predicting the future and fatalism. This is an old theme in literature, including in ancient works like Oedipus the King, as well as in the 2009 movie Timer, which I've also reviewed.

Elsewhere I've suggested a non-confusing way to think about moving backward in time, such as what occurs in the Terminator movies: Imagine the ordinary world as one simulation that runs until the time travel happens. At the point of time travel, take the characters who enter the time machine from the first world and stitch them into a new simulation that's at an earlier point in history. The new simulation, running at an earlier date but with the time travelers now inserted, will diverge from the old simulated history because of the influence of the time travelers.

Situations where the future is predicted are a bit trickier, because rather than just running the simulation forward with new initial conditions, we require that when the simulation does run forward, its output matches what was predicted. In the case of Time Lapse, this means that when the characters see a photo of the next day, the future must unfold such that what's in that photo actually comes to pass. Suppose it's day 1. Designate the contents of the photo predicting day 2 as p. (For instance, p could be the binary representation of a JPEG scan of the photo that predicts what happens on day 2.) Let x be all other state variables for day 1 (e.g., where the characters are, what they know, what the weather is like outside, etc.). Let P be a function that, given p and x, uses the laws of physics to perfectly predict what day 2's photo will look like. This is the function that Mr. Bezzerides's machine supposedly computes in the movie. Then, in order for the prediction to be accurate, it must be that P(p,x) = p. In other words, p must be a fixed pointa of the function P( ,x).b In general, it's probably very hard to find a fixed point like this for a system as complex as humans interacting with their environment, including the knowledge of what the next day's photo should be. However, Mr. Bezzerides's machine manages to get all its predictions correct in the movie, which is quite an accomplishment.

One way in which getting predictions right can be made easier is via self-fulfilling prophecies. Except for the initial prediction of knocking over a coat rack, the characters in the film always tried to make sure the prediction came true (for fear of what would happen if it didn't). And, as we learn at the end, even the coat rack falling over was intentional, since Callie told herself to do it. (We never see how Callie first learned to trust the machine's abilities, though. Maybe seeing tomorrow's photo of herself giving herself instructions was enough? Or plausibly she discovered what the machine was designed to do in Mr. Bezzerides's scientific journal, which she read aloud to Finn and Jasper when they first found the machine.)

It's tempting to wonder whether the machine wasn't actually capable of prediction at all but merely invented scenarios that the characters felt obligated to make come true. Maybe Callie never tested out whether the machine actually got predictions right on its own before she began to feel compelled, based on Mr. Bezzerides's journal, to make the machine's predictions happen. Maybe Callie's instructions to her yesterday self were ideas that the machine invented and that Callie just went along with. However, the hypothesis that the machine is just exploiting the "power of suggestion" doesn't explain the machine's ability to predict tomorrow's winners in the dog races. So -- unless the dog-race results are also influenced by the machine's predictions via self-fulfilling prophecy, or unless the machine's predictions and the dog-race results are both controlled by some mastermind troll -- it seems the movie is committed to a somewhat supernatural interpretation of the machine's abilities (or at least to the machine having insane amounts of computing power with which to simulate the dog races in enough detail to predict them accurately).

The film never shows us what happens when one of the machine's predictions fails, since they never do fail. The characters fear that they'll be killed if they don't act out what's in the photo, but we don't know if that fear is accurate. Maybe nothing significant happens when the machine gets a prediction wrong, in the same way as nothing significant happens when Gmail misclassifies an email as spam. If so, the characters' behavior can be seen as superstitious: They believe they have to carry out the ritual or else something bad will happen, and they're too scared to test out whether this assumption is wrong. (Similarly, the ancient Maya preferred to ritually sacrifice prisoners of war rather than test out whether the gods actually wanted human blood.)

However, it seemed to me as though Callie may have believed that if the prediction is wrong, then the day gets reset (a la Groundhog Day), until a fixed point of the prediction function is found. Maybe this explains how the predictions are consistently right: The simulator of the characters' world runs through tons of photos p for them to observe, most of which fail to materialize on the next day, and then the simulator keeps resetting the simulation and trying a new photo, until a photo that actually comes true the next day is found. We can imagine two different algorithms to search for fixed points.c One possibility, which I'll call Algorithm 1, is that new pictures are tried at random until a fixed point is found. Let X(p,x) be the function that runs the simulation on the picture and state variables for one day and returns the state variables for the next day.

// assume we start with some initial picture candidate p0 and initial conditions x0
p = p0
x = x0
while True:
   q <- P(p,x) // run the simulation one day forward given p and x and see what picture q gets taken
   while(q != p): // the machine's prediction that the picture would be p was wrong
      p <- new_random_picture() //randomly pick a new picture to present to the characters
      q <- P(p,x)
   x <- X(p,x) // advance to the next day; the Groundhog Day loop is broken for this day
   p <- new_random_picture()

The other possibility, which I'll call Algorithm 2, is that if the machine's prediction is wrong, the machine tries the picture that was actually taken as the new fixed-point candidate. This is called fixed-point iteration.

p = p0
x = x0
while True:
   q <- P(p,x)
   while(q != p):
      p <- q
      q <- P(p,x)
   x <- X(p,x)
   p <- q

It's possible that the inner "while" loops in both of these algorithms will run forever if there are no fixed points, but Algorithm 2 is especially vulnerable to infinite loops because there might be a cycle like this: Given some initial picture a, P(a,x) is some picture b, and then P(b,x) is a. Algorithm 2 would cycle forever between a and b.

That said, Algorithm 2 has the nice property that you can pass your own custom message to the next iteration of yourself on the previous day, rather than being given a random message (or, most likely, no written message at all) with Algorithm 1. With Algorithm 2, you can make the machine's picture wrong (q != p) and also give your own message (in the picture q) to the next trial.

If Callie believed that either Algorithm 1 or 2 was in operation, it might explain why she thought it was fine to shoot Finn at the end: She thought that as long as the photo at 8 pm didn't match the prediction (an empty room with police tape), the day would reset to 8 pm of the previous night, which would give her a chance to prevent the carnage that happened earlier in the morning.d And if Algorithm 2 in particular was in operation, her message to herself saying "Don't get caught at window" would be available for the next iteration of herself to learn from. This interpretation of Callie's behavior works, but I'm doubtful whether it's what the film's writers intended, since it involves a lot of theoretical extrapolation on my part.

At the end of the film, Callie tries to send a message to herself on the previous morning, telling herself to avoid getting caught by Finn so that her relationship with him would work out. However, this wasn't the ordinary 8 am photo, because the camera already flashed at 8 am when Callie told herself to "Swap photo with charade night" and "Kill Japser to save Finn". So how did she intend to send herself another message to the previous day? At the end of the movie, the camera takes another picture, which we assume was right after Callie got caught. This is off of the normal 8 am / 8 pm schedule. Presumably this was the one-off photo that Mr. Bezzerides received over a week earlier, and the blood-spattered room at the end of the movie matches with the photo that Dr. Heidecker shows Jasper. (This photo was taken in addition to all regular 8 am / 8 pm photos because it was one-off, not recurring on schedule.) But did Callie know when this one-off photo would be taken? Or was she just trying to influence the 8 pm photo? But in the latter case, why did she put up the "Don't get caught at window" sign so early, rather than waiting until almost 8 pm to put it up?

Also, it seems like she was banking on the idea that contradictions set back the clock -- i.e., that because her "Don't get caught at window" sign was up but hadn't been predicted by any previous photos, the clock would be reset 24 hours. But she had no precedent for believing that this Groundhog Day effect would work. (Even if she had already experienced countless Groundhog Day resets before, she wouldn't remember any of them, since she should only remember the times when she didn't get reset -- unless she's like Bill Murray's character in being the one person who does remember all the resets.)

Anyway, even if I'm correct that Callie had flawed (or at least tenuous) reasoning about what would happen, it doesn't mean the film overall is flawed, since it's easy to imagine that Callie was just confusede in her own reasoning. (We might call this a "lapse" in her reasoning about time, with reference to the film's title.) If so, and if time actually doesn't reset when the machine's prediction is wrong, then it wouldn't have mattered whether the "Don't get caught at window" sign fell down at the end or not, since her deeds would have been done and couldn't have been undone.

Overall, Time Lapse is an excellent movie. Not only is it intellectually stimulating, but the cinematography is pleasantly artistic. For example, the opening shot features a swirling red liquid, which we assume is just an abstract image until a paint brush is stuck into the water.

  1. For more on fixed points in the context of physical causation, see, e.g., pp. 1-3 of this paper.  (back)
  2. Here I'm ignoring the fact that there are actually two photos taken within a day. Incorporating that into my formalism would be tricky.  (back)
  3. Fixed-point-finding algorithms provide an answer to the bootstrap paradox, which asks, for example, how Finn first got ideas for his paintings, given that he painted what the picture showed, and the picture showed what he painted. The answer is that the fixed-point search algorithm stumbled upon a picture (containing a painting) that made the future consistent with that picture.  (back)
  4. This commenter seems to have had the same idea: "They basically are in a time loop until they get the picture right, So theoretically, if the picture stays up on the window, then she would see that instead of the picture of the police tape and she would, not knowing why, changed her future by not getting caught and not killing her BF."  (back)
  5. This comment agrees with me: "In the end, she makes a mistake and it is a little unclear as to why (I think). She thinks she can go back in time, like physically back in time to before she shoots Finn. This hasn’t ever happened in the movie where a character actually goes into the future or past. Why would she think she could go back in time? [...] Maybe she was confused [...]."  (back)