Ackermann’s Function
May 25, 2012
Our function is a simple transliteration of the mathematical definition into code; note that in the second clause of the cond
it is already known that m is zero:
(define (a m n)
(cond ((zero? m) (+ n 1))
((zero? n) (a (- m 1) 1))
(else (a (- m 1) (a m (- n 1))))))
Here’s an example:
> (a 3 4)
125
You can run the program at http://programmingpraxis.codepad.org/lLpK9rFw.
Ackermann’s function was important in math, and is also important in computer science, being part of the computation of the time complexity of algorithms such as the disjoint-set data structure and others. It is also used as a test of recursion for compilers.
Given how heavily recursive this function is, I went with Haskell instead of Python:
[…] Programming News: Ackermann’s Function In the 1920s, Wilhelm Ackermann demonstrated a computable function that was not primitive-recursive, settling an important argument in the run-up to the theory of computation. Read full story => Programming Praxis […]
Feel free to laugh at me. I tried to use visual C# for this. Stack overflow or there is some arbitrary stack limit where visual C# in its infinite wisdom decided I didn’t know what I was doing. Fun times
Toecutter: If your language doesn’t provide a sufficient stack, you must create your own stack; that’s the point of the exercise. A(4,1)=65533 should be within range of your function.
Ha! Ackermann’s function. At least this can computer A(4,2) in python
def ackermann(m, n):
while m >= 4:
if n == 0:
n = 1
else:
n = ackermann(m, n - 1)
m = m - 1
if m == 3:
return (1 << n + 3) - 3
elif m == 2:
return (n << 1) + 3
elif m == 1:
return n + 2
else:
return n + 1
print ackermann(4,2)
http://codepad.org/QdneSl1Q
http://jsfiddle.net/vcKdF/3248/
How about some q
A:{[m; n]
if [m ~ 0; : n+1];
$ [n ~ 0;
if [0 < m;
: A[m-1; 1]];
if [0 < n;
: A[m-1; A[m; n-1]]]];
};
My solution written in Go lang: https://gist.github.com/2942073
Using the closed form, recursion only is necessary for Ackerman(m, n) with m >= 4.