The MU Puzzle

1
The MU Puzzle

• Gödel, Escher, Bach An Eternal Golden Braid
• By Douglas Hofstadter

2
Kurt Gödel
Gödel is one of the most famous logicians of all
time. He is best known for his "Incompleteness
Theorem," which proved that in any axiom system,
there are statements that can be neither proved
nor disproved. Much later in his life, he showed
that the axiom of choice and continuum hypothesis
are consistent with the axioms of set theory,
which helped lead the way for Cohens proof of
the independence of these propositions.
3
Maurits Cornelis Escher
M.C. Escher is a very well known artist. His
lithographs are extremely popular and thought
provoking. These mathematically inspired
woodcuts, lithographs and mezzotints, feature
impossible constructions, explorations of
infinity, architecture, and tessellations. Some
of the most famous of this lithographs are
Drawing Hands, Relativity, Waterfall,
Metamorphosis, Ascending and Descending.
4
Eschers Artworks
Ascending and Descending
Waterfall
5
Eschers Artworks
Hand with Reflecting Sphere
Relativity
6
Johann Sebastian Bach
Johann Sebastian Bach was a German composer,
organist, violist, and violinist whose
ecclesiastical and secular works for choir,
orchestra, and solo instruments drew together the
strands of the Baroque period and brought it to
its ultimate maturity. Bachs most important
contribution is his work on Baroque fugue. He is
now regarded as the supreme composer of the
Baroque, and as one of the greatest of all time.
7
The MU Puzzle
"Can you produce MU?"
We start with String MI. We have to generate
string MU. At any stage one or more rules rules
can be applied to obtain alternate strings and
the choice of the rule to be applied depend upon
the judgement of the solver. But the string can
be changed only within the rules. This is called
The Requirement of Formality.
8
Rules to be Followed

• A U can be added at the end of any string whose
  last letter is I. For Example MI to MIU
• xI xIU
• String after a M can be doubled. For Example
  MIU to MIUIU
• Mx Mxx
• A III appearing anywhere in the string can be
  replaced by U. For example MIUIII to
  MIUU
• xIIIY xUy
• A UU appearing anywhere inside a string can be
  dropped. For Example MIUU to MI
• xUUy xy

I
9
Decision Procedures
Imagine a genie (or a super computer) who has all
the time in the world, and who enjoys using it to
produce theorems of the MIU-system, in a rather
methodical way. Here, for instance, is a possible
way the genie might go about it
Step 1 Apply every applicable rule to the axiom
MI. This yields two new theorems MIU,
MII. Step 2 Apply every applicable rule to the
theorems produced in step 1. This yields
three new theorems MIIU, MIUIU, MIIII. Step 3
Apply every applicable rule to the theorems
produced in step 2. This yields five
new theorems MIIIIU, MIIUIIU, MIUIUIUIU,
MIIIIIIII, MUI. And so on .
This method produces every single theorem sooner
or later, because the rules are applied in every
conceivable order. All of the lengthening-shorteni
ng alternations which we mentioned above
eventually get carried out. However, it is not
clear how long to wait for a given string
10
A systematically constructed "tree" of all the
theorems of the MIU-system. The N th level down
contains those theorems whose derivations contain
exactly N steps. The encircled numbers tell which
rule was employed. Is MU anywhere in this tree?
11
Solution
The answer to the MU Puzzle is No. It is
impossible to change the string MI into MU by
repeatedly applying the given rules.
Explanation
The number of I contained in the strings produced
in this case is not divisible by 3. This is
because only 2nd and 3rd rule changes the number
of I. The 2nd one doubles it while the 3rd one
reduces it 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.
12
AI and the MU Puzzle
We can read M-I-U as
M machineI intelligenceU understanding
Can we produce MU? (Starting with MI) -- This
question essentially explores the difference
between Machine Intelligence and Understanding.
In other words, the MU puzzle is really a
metaphor for some basic questions about
artificial (that is, machine-based) intelligence.
Minds can understand things. Minds also exhibit
the properties of intelligence and can derive
understanding from intelligence. Computers, on
the other hand, do not currently understand
things, but can and do behave logically, and
therefore can be imbued with a certain degree of
intelligence.
13
Continued ..
When enough intelligence is applied, the computer
may eventually exhibit output that is practically
indistinguishable from otherwise mind-like (that
is, human) understanding. But would such a
display be actual or real understanding?
By using the decision procedures like the one
described earlier might take ever to answer such
which might otherwise be trivial for a human mind