# MU puzzle

The MU puzzle is a puzzle stated by Douglas Hofstadter and found in Gödel, Escher, Bach. As stated, it is an example of a Post canonical system and can be reformulated as a string rewriting system.

## The puzzle

Has the dog Buddha-nature? MU!

Suppose there are the symbols M, I, and U which can be combined to produce strings of symbols called "words". The MU puzzle asks one to start with the "axiomatic" word MI and transform it into the word MU using in each step one of the following transformation rules:

1. Add a U to the end of any string ending in I. For example: MI to MIU.
2. Double the string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
3. Replace any III with a U. For example: MUIIIU to MUUU.
4. Remove any UU. For example: MUUU to MU.

Using these four rules is it possible to change MI into MU in a finite number of steps?

The production rules can be written in a more schematic way. Suppose x and y behave as variables (standing for strings of symbols). Then the production rules can be written as:

1. xI → xIU
2. Mx → Mxx
3. xIIIy → xUy
4. xUUy → xy

Is it possible to obtain the word MU using these rules? [4][5]

## Solution

The puzzle's solution is no. It is impossible to change the string MI into MU by repeatedly applying the given rules.

In order to prove assertions like this, it is often beneficial to look for an invariant, that is some quantity or property that doesn't change while applying the rules.

In this case, one can look at the total number of I in a string. Only the second and third rules change this number. In particular, rule two will double it while rule three will reduce it by 3. Now, the invariant property is that the number of I is not divisible by 3:

• In the beginning, the number of Is is 1 which is not divisible by 3.
• Doubling a number that is not divisible by 3 does not make it divisible by 3.
• Subtracting 3 from a number that is not divisible by 3 does not make it divisible by 3 either.

Thus, the goal of MU with zero I cannot be achieved because 0 is divisible by 3.

In the language of modular arithmetic, the number $n$ of I obeys the congruence

$n \equiv 2^a \not\equiv 0 \pmod 3.\,$

where $a$ counts how often the second rule is applied.[6]

## Relationship to provability

The result that MU cannot be obtained with these rules demonstrates the notion of independence in mathematical logic. The MIU system can be viewed as a formal logic in which a string is considered provable if it can be derived by application of the rules starting from MI. In this interpretation, the question is phrased as "Is MU provable in the MIU logic?".

Finding an invariant of the inference rules is a common method for establishing independence results.