Ackermanns Function in Automata Theory

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The Ackermann's Function was discovered by the German mathematician Wilhelm Friedrich Ackermann. This function is an example with which we can understand the limitations of primitive recursive functions. Unlike primitive recursive functions, Ackermann's function grows very rapidly and shows that not all total computable functions are primitive recursive.

In this chapter, we will see the basics of Ackermann's function and go through several examples for a better understanding.

Ackermann's Function

The Ackermann's function is a popular example in theoretical computer science because it is one of the simplest and earliest-discovered functions that is not primitive recursive.

The Ackermann's function is simple in structure but grows faster than any primitive recursive function. This property makes it a fundamental example in the study of computability theory. And this shows the power of μ functions, which is although total and computable.

Definition of Ackermann's Function

Ackermann's function is defined for two non-negative integers x and y as follows −

It is A(x, y), and defined as follows −

• y + 1 (if x = 0)
• A(x - 1, 1) (if x > 0 and y = 0)
• A(x - 1, A(x, y - 1)) (if x > 0 and y > 0)

This recursive function has three cases −

• If x is 0, the function simply returns y + 1.
• If x is greater than 0 and y is 0, the function returns the result of A(x − 1, 1)
• If both x and y are greater than 0, the function applies recursion to both arguments, making it highly complex.

For every non-negative integer x and y, the function A(x, y) can be computed. So, it is a total function.

A(x, 0) = A((x - 1), 1) = A((x - 2), A((x - 1), (1 - 1))) = A((x - 2), A((x - 1), 0)) [this will continue recursively].

Here, we are not getting a zero function, which proves that it is not primitive recursive.

Properties of Ackermann's Function

The Ackermann's function is total, meaning it provides a result for any non-negative integers x and y. However, the function is not primitive recursive, which means it cannot be constructed using only basic operations like zero functions, successor functions, and composition.

The rapid growth of Ackermann's function exceeds any primitive recursive function, which makes it a powerful example in understanding the boundaries of computability.

Examples of Ackermann's Function

To better understand Ackermann's function, let us go through some examples that demonstrate how it works.

Example 1: Calculating A(1, 3)

Let's compute A(1, 3) step by step −

• Start with A(1, 3) = A(0, A(1, 2)) because x > 0 and y > 0.
• Next, we calculate A(1, 2) = A(0, A(1, 1)).
• Then, calculate A(1, 1) = A(0, A(1, 0)).
• Now, A(1, 0) = A(0, 1), which equals 2.
• Substituting back, A(1, 1) = A(0, 2) = 3.
• Substituting again, A(1, 2) = A(0, 3) = 4.
• Finally, A(1, 3) = A(0, 4) = 5.

So, A(1, 3) = 5.

Example 2: Calculating A(2, 2)

Now, let's compute A(2, 2) −

• Start with A(2, 2) = A(1, A(2, 1)).
• Compute A(2, 1) = A(1, A(2, 0)).
• Next, A(2, 0) = A(1, 1) = 3 (from the previous example).
• Therefore, A(2, 1) = A(1, 3) = 5 (as calculated earlier).
• Now, A(2, 2) = A(1, 5).

Let's break down A(1,5) −

• A(1, 5) = A(0, A(1, 4))
• A(1, 4) = A(0, 5) = 6.
• Therefore, A(1, 5) = 7.
• Finally, A(2, 2) = 7.

Example 3: Calculating A(3, 2)

Let's compute A(3, 2) −

• A(3, 2) = A(2, A(3, 1)).
• Calculate A(3, 1) = A(2, A(3, 0)).
• Since A(3, 0) = A(2, 1) and we know A(2, 1) = 5, we get A(3, 1) = A(2, 5).

To break it down −

• A(2, 5) = A(1, A(2, 4))
• A(2, 4) = A(1, A(2, 3))
• A(2, 3) = A(1, A(2, 2)) = A(1, 7)

We already know that A(1, 7) will follow the pattern −

$$\mathrm{A(1, 7) \:=\: A(0, 8) \:=\: 9}$$

Therefore,

$$\mathrm{A(2, 3) \:=\: 9,\: A(2,4) \:=\: 10,\: and\: A(2, 5) \:=\: 11}$$

Finally, A(3, 2) = A(2, 11), and the computation will become increasingly complex from here, showing the rapid growth of Ackermann's function.

The Growth Rate of Ackermann's Function

As we have seen in the examples above, it shows how quickly Ackermann's function grows, even for small inputs. This rapid growth rate is what makes the function non-primitive recursive.

Primitive recursive functions can be computed with loops and basic recursion. The Ackermann's function requires deep levels of recursion that cannot be simplified to a primitive recursive form.

Practical Implications

The Ackermann's function is more than a theoretical idea; it can be treated as benchmark in the field of computer science. Its growth rate is often used to test the efficiency of algorithms and computational processes. For instance, in analyzing the performance of certain data structures like union-find, Ackermann's function can be used to express the time complexity in a compact form.

Conclusion

In this chapter, we explained the Ackermann's function in detail including its definition and the basic structure of its recursive cases. We discussed how this function, though total and computable, is not primitive recursive due to its rapid growth rate.

Through several examples, we demonstrated how Ackermann's function operates and how quickly its values escalate even for relatively small inputs.