Step by Step solution about How to resolve the algorithm Ackermann function step by step in the PHP programming language

The code you provided is an implementation of the Ackermann function in PHP. The Ackermann function is a mathematical function that is often used to test the efficiency of recursive algorithms.

The function takes two parameters, m and n, and computes the Ackermann value for those parameters. The Ackermann value is defined as follows:

• If m is equal to 0, then the Ackermann value is n plus 1.
• If n is equal to 0, then the Ackermann value is the Ackermann value for m minus 1 and 1.
• Otherwise, the Ackermann value is the Ackermann value for m minus 1 and the Ackermann value for m and n minus 1.

The code you provided implements the Ackermann function recursively. The recursive function calls itself to compute the Ackermann value for the given parameters. The base cases of the recursion are when m is equal to 0 or n is equal to 0. In these cases, the function returns the Ackermann value directly. Otherwise, the function calls itself to compute the Ackermann value for m minus 1 and n minus 1, and then returns the result of that call.

The following is a step-by-step trace of the code you provided:

1. The function is called with the parameters m equal to 3 and n equal to 4.
2. The function checks if m is equal to 0. It is not, so the function proceeds to the next step.
3. The function checks if n is equal to 0. It is not, so the function proceeds to the next step.
4. The function calls itself to compute the Ackermann value for m minus 1 and n minus 1. This results in a call to the function with the parameters m equal to 2 and n equal to 3.
5. The function checks if m is equal to 0. It is not, so the function proceeds to the next step.
6. The function checks if n is equal to 0. It is not, so the function proceeds to the next step.
7. The function calls itself to compute the Ackermann value for m minus 1 and n minus 1. This results in a call to the function with the parameters m equal to 1 and n equal to 2.
8. The function checks if m is equal to 0. It is not, so the function proceeds to the next step.
9. The function checks if n is equal to 0. It is not, so the function proceeds to the next step.
10. The function calls itself to compute the Ackermann value for m minus 1 and n minus 1. This results in a call to the function with the parameters m equal to 0 and n equal to 1.
11. The function checks if m is equal to 0. It is, so the function returns n plus 1, which is 2.
12. The function returns the result of the previous call, which is 2.
13. The function returns the result of the previous call, which is 2.
14. The function returns the result of the previous call, which is 125.

The final result of the function is 125.

function ackermann( $m , $n )
{
    if ( $m==0 )
    {
        return $n + 1;
    }
    elseif ( $n==0 )
    {
        return ackermann( $m-1 , 1 );
    }
    return ackermann( $m-1, ackermann( $m , $n-1 ) );
}

echo ackermann( 3, 4 );
// prints 125