The MU Puzzle Is a First Contact Experiment
Hofstadter opens Gödel, Escher, Bach by presenting the reader with a formal system without first explaining what it is; he calls this an introduction to formal systems. In my opinion, it is also, entirely by chance, the most well-thought-out decoding experiment ever published. When we understand it this way, rather than as a lesson in logic, it shows several things about message design that the current literature on message design fails to do.
What the reader receives
The MIU system consists of three symbols and four rules. You begin with the single axiom, MI. When a string ends with an I, you can add a U to it. If you have Mx, then you can form Mxx. Three I’s in a row can be replaced by a U. Two U’s in a row can be deleted. The issue then is whether or not you can obtain MU.
Consider what is missing. No statement tells us what M, I, or U stand for, since they stand for nothing. No objective goes beyond the one set. Nothing to interpret, and a reader searching for meaning is justified in returning empty-handed, because Hofstadter has not included any such meaning.
Instead, the reader carries out the same actions as a recipient of an interstellar message: they apply the rules, produce various strings, observe the patterns that occur, and then formulate hypotheses about what might be reachable. Some discover the invariant and prove that MU cannot be reached; most do not find it, and that very moment of not finding the invariant constitutes the entire lesson.
We aim for the state of being stuck, not the moment when the problem is solved, and it helps to have been stuck yourself before you try to design anything.
The rules arrived in English
What matters is this, and it is easy to get through: the reader can play the game because Hofstadter has given them the rules in a language they already share with him.
We don’t have that; all the material which the MIU reader received for free is the whole problem as it stands on our side of the gap. The recipient has to reconstruct the rule set from the transmission, which comprises the same undifferentiated symbols the rules act upon.
The first design issue is how to transmit a rule set when it cannot be stated.
The solution appears on the following page of the book, where Hofstadter gives it without comment. Instead of showing only the end results, he displays the intermediate strings in sequence and states the rule at each stage: MI, then MII, then MIIII, then MIIIIU. If you read that sequence without the rule names, you can still recover the rules, since consecutive lines differ in just one position and that difference is due to the rule.
Here is the primary mechanism: he didn’t present a series of examples, but rather derivation traces. These traces show each step, with each step transforming the previous one. A receipient can see the trace with clear context. They can deduce a rule not from isolated evidence, but by seeing the rule applied, and the change shows exactly where it was applied.
If one wishes to teach a formal system, it should be constructed from traces; it is easy to say this but difficult to carry out, since traces require many more bits than statements, and this high bit cost has led the tradition to favor exhibition. I believe the tradition is mistaken on this point, and this constitutes a testable disagreement.
The trap in the good news
The MIU game has a trap, and a genuine message can’t afford it.
In a puzzle book, there’s no real harm if a reader gets stuck or makes a mistake; they just hit a dead end and try again. But with a real message, there’s no clear stopping point. If someone misunderstands the rules but stays consistent, they won’t realize anything is wrong, and everything will seem to work. The mistake keeps going and gets worse, and neither side ever notices.
A message needs to make sure only the intended meaning is possible, and doing this requires extra repetition. Show the same relationship in different ways. Add structures that stay balanced with the meaning you want, but fall apart with any other reading. Include a second rule that works with the first to rule out wrong options. Each step reduces the number of possible interpretations, and you can measure this narrowing, making it a real design goal.
In the end, write your whole message as if you can’t send the answer key along with it.