Justifying the Anchients of MuMu

I have recently cracked into this book that I'm sure everyone was over 30 years ago, but have been deriving much pleasure and enjoyment from soon to be mentioned tome, namely, Godel, Escher, and Bach by Douglas Hofstadter. Among a great many other things Hofstadter posits a game to introduce the concept of formal systems he calls MU or MIU. The game is simple (in structure, perhaps you will find arriving at a solution simple as well, but that has not been my experience). The objective is to generate the string "MU" by applying several rules to the initial string (axiom) "MI".
The rules are such:

1. If you posses a string ending with "I", you may add "U" to the end.
2. If you have a string "Mx" where x = any combination of "I's" and "U's" you may make "Mxx".
3. If "III" occurs, you may replace it (trade if you rather) for "U".
4. If "UU" occurs, you may drop it.

Start: MI

I have at this point sunk more than a few days into what I initially thought was a simple problem. It has only gotten more dastardly the more time I have spent at it. In my naivete (the first time I thought I had solved it) I imagined that I could easily write a program that would endlessly generate solutions to the MU problem. Well with some (I thought) ingenuity, I was able to develop a quite effective program for applying the various rules, but I am presently not anywhere near smart enough to make a program anywhere near smart enough to solve this on its own. After a long evening of running numbers (on paper, not through the program mind you) I have started to grow afraid that it may be unsolvable (a possibility as suggested by the author) but I am many things, and stubborn is one of them, so I will persist for a while longer. The program I have written is called mumu.py. I couldn't help but include the reference to Wilson (since "mumu" is actually an impossible string to generate in this system) and one of my favorite books of all time. Good luck on your own quests should you endeavor to take them. Let me know if any intriguing solutions should arise out there, but be forewarned, double check yourself, I myself have "solved" this problem four times already, each turning out to be, in the end, based upon totally bogus assumptions and/ or math.