I am aware that Ackermann's function is total and complete, meaning it is decidable for every input, regardless of how long it takes to compute the result. I'm doing a project in which I decided to prove that the function is decidable through a Turing Machine that I designed.
I also don't know if I designed this TM correctly.
So Ackermann's Function is defined as $A=(m,n)$, where m,n are non-negative integers, such that $(m,n)$ are defined by the following:
\begin{cases} n+1 & m=0 \\ A(m-1,1) & m>0 ,n=0 \\ A(m-1,A(m,n-1) & m>0 , n>0 \end{cases}
So, following the instructions above I came up with a pseudocode for a TM that would tell me if the function is decidable, here's what I have:
read m
read n
if m = 0
write n+1 into A
halt
if m > 0 and if n = 0
write m-1 to m in another TM
write 1 to n in another TM
run other TM
copy A from other TM to A
halt
if m>0 and n>0
write m to m in another TM
write n-1 to n in another TM
run other TM
copy A from other TM to A
write m-1 to m in another TM
write A to n in another TM
run other TM
copy A from other TM to A
halt
can someone tell me if this looks about right? Also, how does one decide what the initial values for the 0's and 1's are on the tape? Or are they just randomly placed?