Ahh, there’s just a single day left until the Easter holidays start, and
I’m only having three hours tomorrow accourding to my schedule!

Not that I’ve been studying all that much the last couple of days… I sat
down yesterday and tried to make the Turing Machine that would take an
input tape with “`aaaaaa#aaaaaaaaa`

” and turn it into “`aaa`

“.

In other words: calculate the greatest common divisor of *m* and *n*
represented by the input `a^m#a^n`

. In the above case, the greatest
common divisor of 6 and 9 is 3 if I’m not mistaken.

It turned out that Turing Machines are extremely annoying to program! Even
the simplest operation gets enormously complicated. Somehow I managed to
sit up all night trying to get things to work — I just kept tweeking and
changing my program, hoping to catch that finaly special case that ruined
the result. Amazingly I didn’t get that tired, it’s as if you reach a
point and from then on you can carry on. But I still have a bad feeling
that I’m going to get really tired soon…

I used a rather nice program called gturing to test my program, which
you can find here. The comments are in Danish, sorry about that, but
I used the program in a Danish hand-in exercise and when I rewrote things
this morning I changed the English comments into Danish ones so that they
would be easier to use in the report. I believe that the program now
handles all correct input correctly. Correct input is input that matches
the RegularExpression “a^m#a^n”.

The program also handles cases like “`#`

“, “`aaa#`

“, and
“`#aaa`

“. With the first input the machine calculates gcd(0, 0) =
0, the two other inputs represent gcd(3, 0) = gcd(0, 3) = 3. The reason
why gcd(0, 0) = 0 is, that gcd(*m*, *n*) is defined as the unique
whole number *d* that satisfies

Div(*d*) = Div(*m*) intersect Div(*n*)

where Div(*n*) = { *d* in **N** | *d* divides *n* }. Here we have that
Div(0) = **N**, that is, all natural numbers divide 0 — take a natural
number and multiply it with 0 to see for your self. So we look for a
number *d* that satisfies that Div(*d*) = **N**. The only number that has
this property is 0 itself, so this means that gcd(0, 0) = 0.