General Recursive Function in Automata Theory

Quiz

General recursive functions, also known as μ-recursive functions, are a fundamental class of functions that map natural numbers to natural numbers. They capture the concept of "computability" in a formal mathematical sense.

In this chapter, we will cover the basics of μ-recursive functions along with their connection to other models of computability, and see a detailed examples for a better understanding.

What are μ-Recursive Functions?

The concept of μ-recursive functions is simple. These are a class of partial functions from natural numbers to natural numbers. These functions are particularly important because they include all functions that can be computed by a Turing machine, which is an important concept in computability theory. This is related to the Church-Turing thesis, which states that any function that can be considered computable can be computed by a Turing machine or equivalently, is μ-recursive.

Relationship with Primitive Recursive Functions

Primitive recursive functions are a subset of μ-recursive functions. They are defined using basic functions and are closed under operations like composition and primitive recursion. However, not all μ-recursive functions are primitive recursive.

A classic example is the Ackermann function, which is total (meaning it is defined for all natural numbers) but not primitive recursive.

μ-Recursive Functions

Let us define the μ-recursive function formally. This is a function that can be constructed using a limited set of initial functions and a few basic operators. These include −

Initial Functions

Constant Functions − Functions that return a fixed natural number.
Successor Function − This function adds one to its input.
Projection Functions − Also known as identity functions, these return one of their arguments.

Operators

Composition − Combining functions such that the output of one function becomes the input of another.
Primitive Recursion − Defining a function based on its value at previous points.
Minimization (μ-operator) − This operator searches for the smallest input that causes the function to return zero. If no such input exists or if the function is undefined for an input, the resulting function will be undefined.

The μ-operator differentiates μ-recursive functions from primitive recursive functions. Where primitive recursion ensures that the function is total, the minimization operator introduces partiality by allowing functions to be undefined for some inputs.

Example of Ackermann Function

Let us see an example. The Ackermann function is such idea which is a μ-recursive function that is not primitive recursive.

It is A(m, n), and defined as follows −

$$\mathrm{n \:+\: 1 \:(if\: m \:=\: 0)}$$

$$\mathrm{A(m \:\: 1,\: 1)\: (if\: m \:\gt\: 0\: and\: n \:=\: 0)}$$

$$\mathrm{A(m \:\: 1,\: A(m,\: n \:\: 1))\: (if \:m \:\gt\: 0\: and\: n \:\gt\: 0)}$$

This function grows extremely fast and exceeds the bounds of primitive recursion. This shows the power of μ-recursive functions.

Basic Examples

Let us see some basic examples for simpler applications of μ-recursive functions −

Successor Function − The successor function S(x) = x + 1 is one of the simplest μ-recursive functions. It states the rule that each number is mapped to its successor, effectively adding one to the input.
Constant Function − A constant function C_k(x) = k returns the same natural number k regardless of the input. For instance, C₅(x) = 5, returns 5 for any input x.
Projection Function − The projection function $\mathrm{P_{i}^{k}\:(x_1,\: x_2,\: \dotso,\: x_k) \:=\: x_i}$. It simply returns the i-th input. For example, $\mathrm{P_2^3(7,\: 3,\: 8) \:=\: 3}$.

Construction Using Operators

We have seen the function definitions. Let us see the construction of more complex functions, using the composition and recursion operators −

Composition − Suppose we have functions like g(x) = x + 2 and h(x) = x². The composition f(x) = h(g(x)) would give us f(x) = (x + 2)², which is another μ-recursive function.
Primitive Recursion − Consider defining a function that calculates the sum of two numbers. Using primitive recursion, we define −
- Base step − f(0, y) = y
- Inductive step − f(x + 1, y) = f(x, y) + 1

This recursive definition shows the standard addition function.

Turing Machines and μ-Recursive Functions

As we discussed that any function which is computable by a Turing machine can also be expressed as a μ-recursive function. On the other hand, every μ-recursive function can be solved by a Turing machine. This equivalence supports the Church-Turing thesis as well.

Kleenes Normal Form Theorem

Another important theorem related to μ-recursive functions is Kleene's Normal Form Theorem. This theorem states that every μ-recursive function can be expressed in a normal form using a single application of the μ-operator to a primitive recursive function. This result further states the power of μ-recursive functions and their role in the theory of computation.

Conclusion

In this chapter, we have covered the concept of μ-recursive functions. We discussed their relationship with primitive recursive functions and discussed their formal definition.

Through examples like the Ackermann function, we demonstrated how μ-recursive functions can extend beyond the capabilities of primitive recursion. Finally, we explored other examples along with their equivalence with other models of computation, such as Turing machines, and Kleene's Normal Form Theorem.

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