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The MIU system was created by Douglas Hoefstadter, author of GEB.

It is a simple, formal system. You have one MIU axiom, four MIU rules, and an MIU goal.

Incidentally, you can't solve the MIU system. See GEB for an interesting proof of this, as well as to see the hidden meanings in the MIU system.


There are four rules in the MIU system:

  1. Given a string of the form Mx, you may get Mxx.
  2. If a string ends in I, you may append U to it.
  3. Given III in an MIU system string, you may replace the I's with a U.
  4. Given UU in a string, you may remove them.


The MIU system has one goal:

Get from MI to MU.

This may be facilited by the MIU rules.

I encourage you to go through and try to get to the goal! It's fun. When you've given it a good try, go read the spoiler...

Spoiler Warning

Oh yeah. Incidentally, it's impossible. Harry explains this well.

I've reproduced Harry's node here, because somebody nuked the MIU Goal node. Ah, well.

Without further ado, Harry's node.

It's impossible.

  1. The goal contains 0 I's (a multiple of 3)
  2. We begin with 1 I (not a multiple of 3)
  3. Rules 2 and 4 do not change the number of Is.
  4. Rules 1 and 3 will only result in a multiple of 3 Is if it takes a&string with a multiple of 3 Is as in input.

It is impossible to go from a state without a multiple of 3 Is (start state) to a state with a multiple of 3 Is (goal state).

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