Mastering the Ackermann Function

Yesterday I came in touch with a curious, astonishing mathematical function. It’s called Ackermann function.

Mathematically speaking, it is a well-defined total function. That is, it has defined values for every integer input (= total function), and this value is not ambiguous (every input has one and one only possible output value) (= well-defined).

Speaking about computer science, this function is computable, but it’s not a primitive recursive function. In other words, you can implement an algorithm to express the function using while-loops (= computable), but still you cannot write an equivalent algorithm using only do-loops (= not primitive recursive). I suggest you to try this statement.

The function has this form:

As you can see, it’s form is fairly simple. But, even if it seems simple, its values explode quickly. A(4, 2), for example, has more than 19.000 digits.

Ackermann Function on the complex plane

Representation of the Ackermann Function on the complex plane

What I want to talk about is the possible implementation of such a function by means of a computer algorithm. I will use Python for the next examples, but the description is language-independent.

Recursive approach

The quicker approach to solving such a problem is by implementing a recursive algorithm, since the definition of the function itself is recursive. Let’s try with something like this:

import sys

def A(x, y):
  if x == 0:
    return y+1
    if y == 0:
      return A(x-1, 1)
      return A(x-1, A(x, y-1))

x = int(sys.argv[1])
y = int(sys.argv[2])

print "Result of A(%d, %d) is %d" % (x, y, A(x, y))

You will want to try it calling from command line:

python 4 2

You will face a brutal reality: this algorithm will not end for the input (4, 2), since it quickly reaches the maximum recursion depth provided by Python. We will get better results starting from lower inputs.

You will find that the maximum computable values, for each possible x are:

  • A(1, 997) = 999
  • A(2, 497) = 997
  • A(3, 6) = 509
  • A(4, 0) = 13

For every input (x, y) with x greater than 5 you will get no result, for each y. As you can see, the function quickly diverges in complexity when x = 3, and much more when x = 4.

What we can try to do is jump as much iterations as possible with a simple trick: saving already computed values in order to not compute them anymore. This could simplify the navigation through the recursion graph.

Let’s try this implementation (with some debug information):

import sys

res = {}
jumps = 0
recursions = 0

#------ Start Method -------#
def A(x, y):
  global res, jumps, recursions

  recursions += 1
    jumps += 1
    return res[x][y]
  except Exception, e:
    jumps -= 1
    res[x] = {}

  if x == 0:
    res[x][y] = y+1
    if y == 0:
      res[x][y] = A(x-1, 1)
      res[x][y] = A(x-1, A(x, y-1))

  return res[x][y]
#------ End Method -------#

x = int(sys.argv[1])
y = int(sys.argv[2])

  print "Result of A(%d, %d) is %d" % (x, y, A(x, y))
  print "%d total recursions, %d operations avoided" % (recursions, jumps)
except Exception, e:
  print "Exception occurred after %d recursions" % (recursions)

As you can see, the res variable is loaded with each computed value (at lines 20, 23, 25), and at the beginning of the function we try to return the value of A(x, y) if already computed (at line 14). The jumps variable keeps track of the number of the iterations saved by this implementation, while the recursions variable keeps track of the total number of single calls the the A function.

You can now still try to call the function to see what the maximum computable values are. You will find this:

  • A(1, 995) = 997
  • A(2, 496) = 995
  • A(3, 6) = 509
  • A(4, 0) = 13

We got three results: the maximum computable inputs have decreased ((1, 995) instead of (1,997) for example), the number of jumps increases with increasing values of the inputs, thus the computation time has decreased. Sadly, our target was not to reduce computable time, but reduce recursion depth.

We should try something different, to truly master the Ackermann Function. This something different involves computing the analytic form of the function. This approach will show a completely different world.

Probably I’ll talk about that in a future article. Let me know what you think about it, and submit your different implementations (different languages, different approaches) but still focusing on the recursion optimization.

48.877 thoughts on “Mastering the Ackermann Function

  1. Excellent beat ! I would like to apprentice while you amend your site, how can i subscribe for a blog site? The account helped me a acceptable deal. I had been a little bit acquainted of this your broadcast offered bright clear idea|

  2. It’s appropriate time to make some plans for the future and it’s time
    to be happy. I’ve read this post and if I could I wish to suggest you some interesting things or advice.
    Maybe you can write next articles referring to this article.
    I wish to read more things about it!

  3. Your style is really unique in comparison to other people I have read stuff from.
    I appreciate you for posting when you’ve got the opportunity, Guess I will just bookmark this site.

  4. Have you ever considered about adding a little bit more than just your articles?

    I mean, what you say is fundamental and all. Nevertheless just imagine if you added some great
    graphics or videos to give your posts more, “pop”!
    Your content is excellent but with pics and video clips,
    this site could undeniably be one of the very best
    in its niche. Terrific blog!

  5. I simply wanted to send a quick remark so as to express gratitude to you for all of the pleasant information you are giving on this website. My particularly long internet lookup has finally been recognized with beneficial facts to write about with my relatives. I would point out that we website visitors are definitely fortunate to be in a very good place with so many wonderful professionals with valuable advice. I feel truly happy to have used your web pages and look forward to tons of more cool minutes reading here. Thank you again for a lot of things.

  6. Usually I do not learn article on blogs, however I would like to say that this write-up very forced me to try and do it! Your writing taste has been amazed me. Thank you, very nice article.

  7. That is really fascinating, You’re an overly skilled blogger.
    I have joined your rss feed and look forward to seeking more of your excellent post.
    Also, I have shared your site in my social networks

  8. Hello! I know this is somewhat off topic but I was wondering which blog platform are you using for this site?

    I’m getting tired of WordPress because I’ve had problems with hackers and I’m looking at options for
    another platform. I would be great if you could point me
    in the direction of a good platform.

Rispondi a Stacy Borich Annulla risposta

Il tuo indirizzo email non sarà pubblicato. I campi obbligatori sono contrassegnati *