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!— Zen Koan[1]
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:
- Add a
U
to the end of any string ending inI
. For example:MI
toMIU
. - Double the string after the
M
(that is, changeMx
, toMxx
). For example:MIU
toMIUIU
. - Replace any
III
with aU
. For example:MUIIIU
toMUUU
. - Remove any
UU
. For example:MUUU
toMU
.
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:
-
xI → xIU
-
Mx → Mxx
-
xIIIy → xUy
-
xUUy → xy
Is it possible to obtain the word MU
using these rules?
[2]
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
I
s 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 of I
obeys the congruence
where counts how often the second rule is applied.[3]
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.
See also
References
- ↑ "The MU–Puzzle". Retrieved 29 July 2013.
- ↑ Justin Curry / Curran Kelleher (2007). Gödel, Escher, Bach: A Mental Space Odyssey. MIT OpenCourseWare.
- ↑ "Solution to MIU puzzle". Retrieved 29 July 2013.
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