Computing Ackermann [4,1]

[Daniel Andres]

How much RAM do we need to compute this one? I watched a video on youtube from computerphile where a professor explained how he used a very old computer and while it took him a long time to get the answer there was no overflow problems. If RAM plays a very important role, which I believe it does since we are abusing the heck out of the stack, someone please tell me how much so that I am able to run it on my computer

[Junilu Lacar]

It would be nice if you could provide a link / links to background material that will shed light on what exactly you're talking about. You shouldn't presume that folks will go out and try to find that stuff for themselves.

[Daniel Andres]

I apologize. I wrongfully assumed everyone would know what Ackermann. I just got caught up in the moment that I did not even think of explaining what it is. I will post a link and more details about it after class.

Thanks

[Stephan van Hulst]

It completely depends on how you write your application. You can write a naive implementation that just defines the function similar to it's mathematical definition:

final class Ackermann {

  static int A(int m, int n) {
    return m == 0 ? n+1
         : n == 0 ? A(m-1, 1)
         :          A(m-1, A(m, n-1));
  }

  public static void main(String... args) {
    System.out.println( A(4,1) );
  }
}

This however quickly results in a StackOverflowError. Instead, let's do some analysis on paper first. This is what I got:A(0, 0) = 1 A(0, 1) = 2 A(0, 2) = 3 A(0, n) = 1+n A(1, 0) = 2 A(1, 1) = 3 A(1, 2) = 4 A(1, n) = 2+n A(2, 0) = 3 A(2, 1) = 5 A(2, 2) = 7 A(2, n) = 2*n+3 A(3, 0) =  5 A(3, 1) = 13 A(3, 2) = 29 A(3, n) = 2^(n+3)-3 A(4, 0) = 13 A(4, 1) = ?
Adding special cases for m <= 3 and memorizing results we've calculated before yields the following code:

import java.util.*;

final class Ackermann {
  
  private static final Map<Integer, Map<Integer, Integer>> memory = new TreeMap<>();
  
  private static int memorizeResult(int m, int n, int result) {
    memory.computeIfAbsent(m, absent -> new TreeMap<>()).put(n, result);
    return result;
  }
  
  private static Optional<Integer> recallResult(int m, int n) {
    return Optional.ofNullable(memory.getOrDefault(m, Collections.emptyMap()).get(n));
  }
  
  static int A(int m, int n) {
    return (m == 0) ? 1+n
         : (m == 1) ? 2+n
         : (m == 2) ? 2*(n+3)-3
      // : (m == 3) ? (int) Math.pow(2, n+3)-3
         :            recallResult(m,n).orElseGet(() -> memorizeResult(m,n,
                        (n == 0) ? A(m-1, 1)
                                 : A(m-1, A(m, n-1))));
  }

  public static void main(String... args) {
    try {
      A(4,1);
    }
    
    finally {
      memory.forEach((m, map) -> map.forEach((n, result) -> System.out.printf("A(%d,%3d) = %5d%n", m, n, result)));
    }
  }
}

This program finishes running almost instantly and if I comment out line 20 it prints the following results, which should give you an indication of the amount of memory it uses:
A(3,  0) =     5 A(3,  1) =    13 A(3,  2) =    29 A(3,  3) =    61 A(3,  4) =   125 A(3,  5) =   253 A(3,  6) =   509 A(3,  7) =  1021 A(3,  8) =  2045 A(3,  9) =  4093 A(3, 10) =  8189 A(3, 11) = 16381 A(3, 12) = 32765 A(3, 13) = 65533 A(4,  0) =    13 A(4,  1) = 65533

[Daniel Andres]

Wow, I can't really follow up. I am not familiar with Maps or many of those terms you have used. Now that I'm on spring break I'll read ahead on the chapter on Maps ADT. What do you mean by

which should give you an indication of the amount of memory it uses

When we get the final results for n, is that an indication of memory use? If you have already explained it throughout the code sorry for asking about it. We had to program Ackermann for my CSC class computing the value, doing tracing like you did and also doing table look-up that stores already found values in a 2D array that way before we go into the if statements first we see if the previous recursive call is in the table already. If it is, then we can return it thus optimizing the program. But I am not that advanced yet in Java which is why I am lost in the second code you posted.

Thanks for the amazing detailed explanation. I'll get back to it when I'm ready!

For those unfamiliar with Ackerman watch this video on YouTube; Professor Brailsford explains what it is. There were certain things in the computing world that were so big that just had to be done recursively. Some people argued that they can just be done with for loops instead. Ackermann, a research student of mathematician David Hilbert was apparently the first to realize this and wondered if there actually was a problem that could only be done recursively. Very insightful video.

[Stephan van Hulst]

That was a really cool video, have a cow!

Daniel Andres wrote:doing table look-up that stores already found values in a 2D array

That's exactly what my program does as well. An array is a data structure that takes an integer, and returns a value associated with that integer. A two-dimensional array takes two integers and returns a value associated with the Cartesian product of those two integers. A Map also takes one value, and returns a value associated with it. Map<Integer, Map<Integer, Integer>> is really the same as int[][], and calling memorizeResult(m, n, result), is the same as performing array[m][n] = result. I chose Map because the size of an array needs to be known ahead of time, while a Map can grow as needed.

But I am not that advanced yet in Java which is why I am lost in the second code you posted.

Here's the same code, except in less concise form:

static int A(int m, int n) {
  if (m == 0) return 1+n;
  if (m == 1) return 2+n;
  if (m == 2) return 2*(n+3)-3;
  if (m == 3) return (int)Math.pow(2, n+3) - 3;

  Optional<Integer> memorizedResult = recallResult(m,n);
  if (memorizedResult.isPresent())
    return memorizedResult.get();

  int result;
  if (n == 0)
    result = A(m-1, 1);
  else
    result = A(m-1, A(m, n-1));

  memorizeResult(m, n, result);

  return result;
}

When we get the final results for n, is that an indication of memory use?

The results that my program print are ALL the calculations that it stored in the table in order to compute A(4,1) fast. It didn't store anything for m <= 2, because those values are calculated directly from the special cases that I added on lines 18 and 19 of the original code. If I disable those lines, the memory and time used become MUCH greater. This is not an indication of peak memory usage though, because it doesn't count stack frames. Indeed, the stack will blow up much sooner than the Map...

It's interesting that the professors in the video need so much time to calculate A(4,1), because that would mean they're not using memoization, and they have a huge stack to be able to make the recursive calls.

If I replace all Integers in my program with BigIntegers and enable the special case for (m == 3), my application is able to immediately calculate A(4,2):
200352993040684646497907235156025575044782547556975141926501697371089405955631145308950613088093334810103823434290726318 182294938211881266886950636476154702916504187191635158796634721944293092798208430910485599057015931895963952486337236720 300291696959215610876494888925409080591145703767520850020667156370236612635974714480711177481588091413574272096719015183 628256061809145885269982614142503012339110827360384376787644904320596037912449090570756031403507616256247603186379312648 470374378295497561377098160461441330869211810248595915238019533103029216280016056867010565164675056803874152946384224484 529253736144253361437372908830379460127472495841486491593064725201515569392262818069165079638106413227530726714399815850 881129262890113423778270556742108007006528396332215507783121428855167555407334510721311242739956298271976915005488390522 380435704584819795639315785351001899200002414196370681355984046403947219401606951769015611972698233789001764151719005113 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You can imagine that calculating A(4,3) or A(5,1) is not going to happen...

[Stephan van Hulst]

To test this yourself, here is the full program:

import java.math.*;
import java.util.*;

import static java.math.BigInteger.*;

final class Ackermann {
  
  private static BigInteger TWO   = BigInteger.valueOf(2);
  private static BigInteger THREE = BigInteger.valueOf(3);
  private static BigInteger FOUR  = BigInteger.valueOf(4);
  
  private static final Map<BigInteger, Map<BigInteger, BigInteger>> MEMORY = new TreeMap<>();
  
  private static BigInteger memorizeResult(BigInteger m, BigInteger n, BigInteger result) {
    MEMORY.computeIfAbsent(m, absent -> new TreeMap<>()).put(n, result);
    return result;
  }
  
  private static Optional<BigInteger> recallResult(BigInteger m, BigInteger n) {
    return Optional.ofNullable(MEMORY.getOrDefault(m, Collections.emptyMap()).get(n));
  }
  
  static BigInteger A(BigInteger m, BigInteger n) {
    if (m.equals(ZERO))  return n.add(ONE);
    if (m.equals(ONE))   return n.add(TWO);
    if (m.equals(TWO))   return n.add(THREE).multiply(TWO).subtract(THREE);
    if (m.equals(THREE)) return TWO.pow(n.add(THREE).intValue()).subtract(THREE);
  
    Optional<BigInteger> memorizedResult = recallResult(m,n);
    if (memorizedResult.isPresent())
      return memorizedResult.get();
  
    BigInteger result;
    if (n.equals(ZERO))
      result = A(m.subtract(ONE), ONE);
    else
      result = A(m.subtract(ONE), A(m, n.subtract(ONE)));
  
    memorizeResult(m, n, result);
  
    return result;
  }

  public static void main(String... args) {
    try {
      A(FOUR, TWO);
    }
    
    finally {
      memory.forEach((m, map) -> map.forEach((n, result) -> System.out.printf("A(%d,%3d) = %5d%n", m, n, result)));
    }
  }
}

[Campbell Ritchie]

A cow ... and a move to another forum as too difficult for here.

[Daniel Andres]

Thank you for posting your code @Stephan van Hulst. I will test it myself.

@Campbell I meant to post this under general java.

[Campbell Ritchie]

But the discussion has become more complicated since then.

[Daniel Andres]

Agree. I meant to say that instead of posting in beginners java it was supposed to be under general.