The core problem remains computational complexity. 

Statements like "does this image look reasonable" or saying "you pay attention to regularities in the data", or "find the resolution by searching all possible resolutions" are all hiding high computational costs behind short English descriptions.

Let's consider the case of a 1280x720 pixel image. 
That's the same as 921600 pixels.

How many bytes is that?

It depends. How many bytes per pixel?^[1] In my post, I explained there could be 1-byte-per-pixel grayscale, or perhaps 3-bytes-per-pixel RGB using [0, 255] values for each color channel, or maybe 6-bytes-per-pixel with [0, 65535] values for each color channel, or maybe something like 4-bytes-per-pixel because we have 1-byte RGB channels and a 1-byte alpha channel.

Let's assume that a reasonable cutoff for how many bytes per pixel an encoding could be using is say 8 bytes per pixel, or a hypothetical 64-bit color depth.

How many ways can we divide this between channels?

If we assume 3 channels, it's 1953.
If we assume 4 channels, it's 39711.
Also if it turns out to be 5 channels, it's 595665.

This is a pretty fast growing function. The following is a plot.

Note that the red line is O(2^N) and the black line barely visible at the bottom is O(N^2). N^2 is a notorious runtime complexity because it's right on the threshold of what is generally unacceptable performance.^[2] 

Let's hope that this file isn't actually a frame buffer from a graphics card with 32 bits per channel or a 128 bit per pixel / 16 byte per pixel.

Unfortunately, we still need to repeat this calculation for all of the possibilities for how many bits per pixel this image could be. We need to add in the possibility that it is 63 bits per pixel, or 62 bits per pixel, or 61 bits per pixel.

In case anyone wants to claim this is unreasonable, it's not impossible to have image formats that have RGBA data, but only 1 bit associated with the alpha data for each pixel. ^[3]

And for each of these scenarios, we need to question how many channels of color data there are.

• 1? Grayscale.
• 2? Grayscale, with an alpha channel maybe?
• 3? RGB, probably, or something like HSV.
• 4? RGBA, or maybe it's the RGBG layout I described for a RAW encoding of a Bayer filter, or maybe it's CMYK for printing.
• 5? This is getting weird, but it's not impossible. We could be encoding additional metadata into each pixel, e.g. distance from the camera.
• 6? Actually, this question how how many channels there are is very important, given the fast growing function above.
• 7? This one question, if we don't know the right answer, is sufficient to make this algorithm pretty much impossible to run.
• 8? When we say we can try all of options, that's not actually possible.
• 9? What I think people mean is that we can use heuristics to pick the likely options first and try them, and then fall back to more esoteric options if the initial results don't make sense.
• 10? That's the difference between average run-time and worst case run-time.
• 11? The point that I am trying to make is that the worst case run-time for decoding an arbitrary binary file is pretty much unbounded, because there's a ridiculous amount of choice possible.
• 12? Some examples of "image" formats that have large numbers of channels per "pixel" are things like RADAR / LIDAR sensors, e.g. it's possible to have 5 channels per pixel for defining 3D coordinates (relative to the sensor), range, and intensity.

You actually ran into this problem yourself.

Similarly (though you'd likely do this first), you can tell the difference between RGB and RGBA. If you have (255, 0, 0, 255, 0, 0, 255, 0, 0, 255, 0, 0), this is probably 4 red pixels in RGB, and not a fully opaque red pixel, followed by a fully transparent green pixel, followed by a fully transparent blue pixel in RGBA. It could be 2 pixels that are mostly red and slightly green in 16 bit RGB, though. Not sure how you could piece that out.

Summing up all of the possibilities above is left as an exercise for the reader, and we'll call that sum K.

Without loss of generality, let's say our image was encoded as 3 bytes per pixel divided between 3 RGB color channels of 1 byte each.

Our 1280x720 image is actually 2764800 bytes as a binary file.

But since we're decoding it from the other side, and we don't know it's 1280x720, when we're staring at this pile of 2764800 bytes, we need to first assume how many bytes per pixel it is, so that we can divide the total bytes by the bytes per pixel to calculate the number of pixels.

Then, we need to test each possible resolutions as you've suggested.

The number of possible resolutions is the same as the number of divisors of the number of pixels. The equation for providing an upper bound is exp(log(N)/log(log(N)))^[4], but the average number of divisors is approximately log(N).

Oops, no it isn't!

Files have headers! How large is the header? For a bitmap, it's anywhere between 26 and 138 bytes. The JPEG header is at least 2 bytes. PNG uses 8 bytes. GIF uses at least 14 bytes.

Now we need to make the following choices:

1. Guess at how many bytes per pixel the data is.
2. Guess at the length of the header. (maybe it's 0, there is no header!)
3. Calculate the factorization of the remaining bytes N for the different possible resolutions.
4. Hope that there isn't a footer, checksum, or any type of other metadata hanging out in the sea of bytes. This is common too!

Once we've made our choices above, then we multiply that by log(N) for the number of resolutions to test, and then we'll apply the suggested metric. Remember that when considering the different pixel formats and ways the color channel data could be represented, the number was K, and that's what we're multiplying by log(N).

In most non-random images, pixels near to each other are similar. In an MxN image, the pixel below is a[i+M], whereas in an NxM image, it's a[i+N]. If, across the whole image, the difference between a[i+M] is less than the difference between a[i+N], it's more likely an MxN image. I expect you could find the resolution by searching all possible resolutions from 1x<length> to <length>x1, and finding which minimizes average distance of "adjacent" pixels.

What you're describing here is actually similar to a common metric used in algorithms for automatically focusing cameras by calculating the contrast of an image, except for focusing you want to maximize contrast instead of minimize it.

The interesting problem with this metric is that it's basically a one-way function. For a given image, you can compute this metric. However, minimizing this metric is not the same as knowing that you've decoded the image correctly. It says you've found a decoding, which did minimize the metric. It does not mean that is the correct decoding.

A trivial proof:

1. Consider an image and the reversal of that image along the horizontal axis. 
2. These have the same metric.
3. So the same metric can yield two different images.

A slightly less trivial proof:

1. For a given "image" of N bytes of image data, there are 2^(N*8) possible bit patterns.
2. Assuming the metric is calculated as an 8-byte IEEE 754 double, there are only 2^(8*8) possible bit patterns.
3. When N > 8, there are more bit patterns than values allowed in a double, so multiple images need to map to the same metric.

The difference between our 2^(2764800*8) image space and the 2^64 metric is, uhhh, 10^(10^6.8). Imagine 10^(10^6.8) pigeons. What a mess.^[5]

The metric cannot work as described. There will be various arbitrary interpretations of the data possible to minimize this metric, and almost all of those will result in images that are definitely not the image that was actually encoded, but did minimize the metric. There is no reliable way to do this because it isn't possible. When you have a pile of data, and you want to reverse meaning from it, there is not one "correct" message that you can divine from it.^[6] See also: numerology, for an example that doesn't involve binary file encodings. 

Even pretending that this metric did work, what's the time complexity of it? We have to check each pixel, so it's O(N). There's a constant factor for each pixel computation. How large is that constant? Let's pretend it's small and ignore it.

So now we've got K*O(N*log(N)) which is the time complexity of lots of useful algorithms, but we've got that awkward constant K in the front. Remember that the constant K reflects the number of choices for different bits per pixel, bits per channel, and the number of channels of data per pixel. Unfortunately, that constant is the one that was growing a rate best described as "absurd". That constant is the actual definition of what it means to have no priors. When I said "you can generate arbitrarily many hypotheses, but if you don't control what data you receive, and there's no interaction possible, then you can't rule out hypotheses", what I'm describing is this constant.

I think it would be very weird, if we were trying to train an AI, to send it compressed video, and much more likely that we do, in fact, send it raw RGB values frame by frame.

What I care about is the difference between:

1. Things that are computable.
2. Things that are computable efficiently.

These sets are not the same.
Capabilities of a superintelligent AGI lie only in the second set, not the first.

It is important to understand that a superintelligent AGI is not brute forcing this in the way that has been repeatedly described in this thread. Instead the superintelligent AGI is going to use a bunch of heuristics or knowledge about the provenance of the binary file, combined with access to the internet so that it can just lookup the various headers and features of common image formats, and it'll go through and check all of those, and then if it isn't any of the usual suspects, it'll throw up metaphorical hands, and concede defeat. Or, to quote the title of this thread, intelligence isn't magic.

1. ^{^}

   This is often phrased as bits per pixel, because a variety of color depth formats use less than 8 bits per channel, or other non-byte divisions.

2. ^{^}

   Refer to https://accidentallyquadratic.tumblr.com/ for examples. 

3. ^{^}

   A fun question to consider here becomes: where are the alpha bits stored? E.g. if we assume 3 bytes for RGB data, and then we have the 1 alpha bit, is each pixel taking up 9 bits, or are the pixels stored in runs of 8 pixels followed by a single "alpha" pixel with 8 bits describing the alpha channels of the previous 8 pixels?

4. ^{^}

   https://terrytao.wordpress.com/2008/09/23/the-divisor-bound/

5. ^{^}

   https://en.wikipedia.org/wiki/Pigeonhole_principle

6. ^{^}

   The way this works for real reverse engineering is that we already have expectations of what the data should look like, and we are tweaking inputs and outputs until we get the data we expected. An example would be figuring out a camera's RAW format by taking pictures of carefully chosen targets like an all white wall, or a checkerboard wall, or an all red wall, and using the knowledge of those targets to find patterns in the data that we can decode.

I was describing a file that would fit your criteria but not be useful. I was explaining in bullet points all of the reasons why that file can't be decoded without external knowledge.

I think that you understood the point though, with your example of data from the Hubble Space Telescope. One caveat: I want to be clear that the file does not have to be all zeroes. All zeroes would violate your criteria that the data cannot be compressed to less than 10% of it's uncompressed size, since all zeroes can be trivially run-length-encoded.

But let's look at this anyway.

You said the file is all zeroes, and it's 1209600 bytes. You also said it's pressure readings in kPa, taken once per second. You then said it's 2^11 x 3^3 x 5^2 x 7 zeroes -- I'm a little bit confused on where this number came from? That number is 9676800, which is larger than the file size in bytes. If I divide by 8, then I get the stated file size, so maybe you're referring to the binary sequence of bits being either 0 or 1, and then on this hardware a byte is 8-bits, and that's how those numbers connect.

In a trivial sense, yes, that is "deriving the structure but not the meaning".

What I really meant was that we would struggle to distinguish between:

• The file is 1209600 separate measurements, each 1-byte, taken by a single pressure sensor.
• The file is 604800 measurements of 1 byte each from 2 redundant pressure sensors.
• The file is 302400 measurements, each 4-bytes, taken by a single pressure sensor.
• The file is 241920 measurements, each a 4-byte timestamp field and a 1-byte pressure sensor value.

Or considering some number of values, with some N-byte width:

• The pressure sensor value is in kPa.
• The pressure sensor value is in psi.
• The pressure sensor value is in atmospheres.
• The pressure sensor value is in lbf.
• The pressure sensor value is in raw counts because it's a direct ADC measurement, so it needs to be converted to the actual pressure via a linear transform.
  • We don't know the ADC's reference voltage.
  • Or the data sheet for this pressure sensor.

Your questions are good.

1. Is it coherent to "understand the meaning but not the structure"?

Probably not? I guess there's like a weird "gotcha" answer to this question where I could describe what a format tells you, in words, but not show you the format itself, and maybe we could quibble that in such a scenario you'd understand "the meaning but not the structure".

EDIT: I think I've changed my mind on this answer since posting -- yes, there are scenarios where you would understand the meaning of something, but not necessarily the structure of it. A trivial example would be something like a video game save file. You know that some file represents your latest game, and that it allows you to continue and resume from where you left off. You know how that file was created (you pressed "Save Game"), and you know how to use it (press "Load Game", select save game name), but without some amount of reverse-engineering, you don't know the structure of it (assuming that the saved game is not stored as plain text). For non-video-game examples, something like a calibration file produced by some system where the system can both 1.) produce the calibration via some type of self-test or other procedure, and 2.) receive that calibration file. Or some type of system that can be configured by the user, and then you can save that configuration file to disc, so that you can upload it back to the system. Maybe you understand exactly what the configuration file will do, but you never bothered to learn the format of the file itself.

2. Would it be possible to go from "understands the structure but not the meaning" to "understands both" purely through receiving more data?

In general, no. The problem is that you can generate arbitrarily many hypotheses, but if you don't control what data you receive, and there's no interaction possible, then you can't rule out hypotheses. You'd have to just get exceedingly lucky and repeatedly be given more data that is basically designed to be interpreted correctly, i.e. the data, even though it is in a binary format, is self-describing. These formats do exist by the way. It's common for binary formats to include things like length prefixes telling you how many bytes follow some header. That's the type of thing you wouldn't notice with a single piece of data, but you would notice if you had a bunch of examples of data all sharing the same unknown schema.

3. If not , what about through interaction

Yes, this is how we actually reverse-engineer unknown binary formats. Almost always we have some proprietary software that can either produce the format, or read the format, and usually both. We don't have the schema, and for the sake of argument let's say we don't want decompile the software. An example: video game save files that are stored using some ad-hoc binary schema.

What we generally do is start poking known values into the software and seeing what the output file looks like -- like a specific date time, a naming something a certain field, or making sure some value or checkbox is entered in a specific way. Then we permute that entry and see how the file changes. The worse thing would be if almost the entire file changes, which tells us the file is either encrypted OR it's a direct dump of RAM to disk from some data structure with undefined ordering, like a hash map, and the structure is determined by keys which we haven't figured out yet. 

Likewise, we do the reverse. We change the format and we plug it back into the software (or an external API, if we're trying to understand how some website works). What we're hoping for is an error message that gives us additional information -- like we change some bytes and now it complains a length is invalid, that's probably related to length. Or if we change some bytes and it says the time cannot be negative, then that might be a time related. Or we change any byte and it rejects the data as being invalid, whoops, probably encrypted again.

The key here is that we have the same problem as question 2 -- we can generate arbitrarily many hypotheses -- but we have a solution now. We can design experiments and iteratively rule out hypotheses, over and over, until we figure out the actual meaning of the format -- not just the structure of it, but what values actually represent.

Again, there are limits. For instance, there are reverse-engineered binary formats where the best we know is that some byte needs to be some constant value for some reason. Maybe it's an internal software version? Who knows! We figured out the structure of that value -- the byte at location 0x806 shall be 203 -- but we don't know the meaning of it.

4. If not, what differences would you expect to observe between a world where I understood the structure but not the meaning of something and a world where I understood both?

Hopefully the above has answered this.

Replying to your other post here:

I don't think this algorithm could decode arbitrary data in a reasonable amount of time. I think it could decode some particularly structured types of data, and I think "fairly unprocessed sensor data from a very large number of nearly identical sensors" is one of those types of data.

My whole point is that "unprocessed sensor data" can be arbitrarily tied to hardware in ways that make it impossible to decode without knowledge of that particular hardware, e.g. ADC reference voltages, datasheets, or calibration tables.

RAW I'd expect is better than a bitmap, assuming no encryption step, on the hypothesis that more data is better.

The opposite. Bitmaps are much easier than RAW formats. A random RAW format, assuming no interaction with the camera hardware or software that can read/write that format, might as well be impossible. E.g. consider the description of a RAW format here. Would you have known that the way the camera hardware works involves pixels that are actually 4 separate color sensors arranged as (in this example) an RGBG grid (called a Bayer filter), and that to calculate the pixel colors for a bitmap you need to interpolate between 4 of those raw color sensors for each color channel on each pixel in the bitmap, and the resulting file size is going to be 3x larger? Or that the size of the output image is not the size of the sensor, so there's dead pixels that need to be ignored during the conversion? Again, this is just describing some specific RAW format -- other RAW formats work differently, because cameras don't all use the same type of color sensor. 

The point as I see it is more about whether it's possible with a halfway reasonable amount of compute or whether That Alien Message was completely off-base.

It was completely off-base.