bench mark of compiler using Ackerman function

Ackermann Function Overview:

The Ackermann function is a classic example of a recursive function that grows extremely fast. It is one of the simplest examples of a function that is not primitive recursive, meaning it cannot be expressed using loops alone (iterative methods). Instead, it requires deep recursion to compute its results, making it an excellent stress test for recursion handling by both programming languages and compilers.

Definition:

The Ackermann function, denoted as A(m,n), is defined for non-negative integers m and n as follows:

Fig 1

Understanding the Ackermann Function:

Base Case: When m=0, the function simply returns n+1. This is the simplest scenario where the recursion terminates.
Recursive Cases:

If n=0 and m>0, the function reduces mmm by 1 and calls the function with n=1.
If both m>0 and n>0, the function recurses twice. First, it calls itself with n−1, and then uses the result of that call as the new value of n for another recursive call with m−1.

This double recursion for larger values of mmm causes the function to grow extremely fast, especially when m≥4.

Example: Expansion for A(1,2):

To illustrate the recursive nature, let's compute A(1,2) step by step:

A(1,2)=A(0,A(1,1))
A(1,1)=A(0,A(1,0))
A(1,0)=A(0,1)=2 (since A(0,n)=n+1)
A(1,1)=A(0,2)=3
A(1,2)=A(0,3)=4

Thus, A(1,2)=4.

Properties:

Termination: The function always terminates because the recursion eventually reduces the values of mmm and n to a base case where the function directly returns a value.
Extremely Fast Growth:

For small values of m, like m=1 or m=2, the function behaves similarly to simple arithmetic or polynomial growth.
For m ≥ 3, the function's growth is incredibly fast, quickly exceeding even exponential growth rates. For example, A(4,3) produces a value so large it cannot be computed within the physical constraints of most computing systems.

Deep Recursion: As the function is highly recursive, it tests the limits of recursion in programming languages. It can be used to evaluate how well a compiler or runtime environment optimizes recursion.

Advantages and Disadvantages of Recursion (as shown in Ackermann's Function):

Advantages:

Concise and Understandable: Recursive functions like the Ackermann function are easy to express and often more natural to understand compared to iterative solutions for complex tasks.

Disadvantages:

Performance Overhead: Each recursive call requires memory to store the current function state. As recursion deepens, the program may run slower due to overhead.
Stack Overflow Risk: If recursion is not carefully bounded (as in the Ackermann function), it can cause a "stack overflow" because each recursive call uses space on the call stack. The Ackermann function, for example, quickly causes stack overflows for even small values like A(3,11).
Memory Consumption: Recursive calls consume more memory since the state of each invocation must be preserved. The more nested the recursion, the more memory is required.

Applications of the Ackermann Function:

Benchmarking Compilers: Due to the deep recursion involved, the Ackermann function is often used to benchmark how well compilers handle recursive optimizations. It tests both the efficiency and limits of recursion.
Comparing Programming Languages: The Ackermann function is useful for comparing how different programming languages handle recursion and manage the call stack.

Why Ackermann Function is Infeasible for Large Inputs:

For small values like A(1,n) or A(2,n), the Ackermann function behaves similarly to simple arithmetic or polynomials. However, for higher values like A(3,n), the function grows exponentially, and for m=4, it grows even faster than exponential, leading to numbers that are impractically large to compute.

For example, A(4,3) produces an extremely large number, far exceeding the number of particles in the universe.

Java implementation

The provided code implements the Ackermann function using BigDecimal for handling large numbers in Java. However, it has a significant issue: the Ackermann function grows extremely fast, and recursion depth will quickly exceed the stack size for even moderately sized inputs (like Ack(3, 11)).

Code Breakdown:

Ackermann function definition:

A(m, n) is defined as:

Recursive

A(0, n) = n + 1

A(m, 0) = A(m - 1, 1) if m > 0

A(m, n) = A(m - 1, A(m, n - 1)) if m > 0 and n > 0

The code translates this into a recursive Java method using BigDecimal to handle large numbers.

Problem:

For m = 3 and n = 11, the Ackermann function will generate a massive recursion depth, likely causing a StackOverflowError.

Possible Improvements:

Recursion Optimization: Since the recursion grows deep, consider using memoization or converting the function into an iterative approach to prevent exceeding the stack size. However, both approaches will struggle with the extreme growth of the Ackermann function.
Stack Size: You can increase the JVM stack size using the -Xss JVM option, but that’s only a temporary workaround and won't help for larger inputs.

Example Implementation with Recursion Limits in Mind:

If you're testing with smaller numbers, this approach will work, but avoid inputs like Ack(3, 11) unless your system is capable of handling deep recursion.

java

import java.math.BigDecimal;

public class Ackerman {

private BigDecimal ack(BigDecimal m, BigDecimal n) {

BigDecimal one = BigDecimal.ONE;

BigDecimal zero = BigDecimal.ZERO;

if (m.equals(zero)) {

return n.add(one);

} else if (n.equals(zero)) {

return ack(m.subtract(one), one);

} else {

return ack(m.subtract(one), ack(m, n.subtract(one)));

}

public static void main(String[] args) {

Ackerman ackerman = new Ackerman();

// Test with smaller values due to the extremely fast growth of Ackermann function.

BigDecimal first = new BigDecimal(3);

BigDecimal second = new BigDecimal(3); // Use smaller second value for testing

BigDecimal result = ackerman.ack(first, second);

System.out.println("Ackermann(" + first + ", " + second + ") = " + result);

}

Output: 16381

Testing with Smaller Values:

Test the code with smaller inputs to avoid stack overflow:

Ack(2, 2) or Ack(2, 3) works fine.
Even Ack(3, 3) can already produce a large result but is still manageable.

For practical purposes, Java’s recursion and memory limitations will make it difficult to compute the Ackermann function for large inputs like Ack(3, 11)

Conclusion:

The Ackermann function is a classic example that highlights both the power and limitations of recursion in programming. It’s highly useful for theoretical exploration and benchmarking but impractical for large inputs due to its extremely rapid growth rate. Recursive functions, like the Ackermann function, must be carefully managed to prevent stack overflow and excessive memory usage.