Recursive Functions in Automata Theory

Quiz

In this chapter and in its subsequent few chapters, we will cover several mathematical functions related to automata theory. Recursive functions are quite useful in automata theory. They help in understanding the different operations and behaviors that can be defined mathematically. In this chapter, we will explain the basic concepts of recursive functions, including the types of functions used, their definitions, and examples for a better understanding.

Initial Functions in Recursive Function Theory

To start the discussion on Recursive functions, we will see the initial functions. These are the foundational functions in recursive function theory, and there are three of them −

Zero function
Successor function
Projection function

Let us take a closer look at them one by one.

Zero Function

The first initial function is the zero function, which we represent with the letter Z. The zero function is simple: It always returns zero, regardless of the input value. If any variable is given, the result will always be zero.

For instance, if we represent the zero function as Z(x), then Z(5) = 0 and Z(70) = 0. This function sets the value of any variable to zero, making it a basic part of recursive function theory.

Successor Function

The next type of function is the successor function, this is represented by the letter S. This function gives the next value of any variable by adding 1 to it.

For example, if you want to find the successor of 6, you simply calculate S(6) = 6 + 1 = 7. Similarly, S(70) = 70 + 1 = 71. The successor function is straightforward but important in defining other functions.

Projection Function

The third initial function is the projection function. This function is used to select a particular value from a set of natural numbers. If you have a set of numbers and we want to get a specific one, we use the projection function.

The projection function is represented as $\mathrm{P_{i}^{n}}$ , where n is the total number of values, and i is the position of the value you want to select.

For example −

$\mathrm{P_{1}^{4}}$ selects the first number from the set {3, 7, 8, 9}, giving an output of 3.
$\mathrm{P_{2}^{3}}$ selects the second number from the set {7, 9, 12}, giving an output of 9.
$\mathrm{P_{4}^{6}}$ selects the fourth number from the set {1, 2, 3, 4, 5, 6}, giving an output of 4.

These initial functions, the zero, successor, and projection makes the base upon which more complex functions are built.

Composite Functions in Recursive Function Theory

After understanding the initial functions, the next concept is composite functions. In a composite function, we combine two or more functions by substituting one function into another. This allows us to create more complex functions from simpler ones.

A composite function is defined when one function is substituted into another. For example, let us consider two functions −

$$\mathrm{f(x) \:=\: 3x \:+\: 2}$$

$$\mathrm{g(x) \:=\: x \:+\: 5}$$

If we want to compose these functions as f(g(x)), we substitute g(x) into f(x). So,

$$\mathrm{f(g(x)) \:=\: 3(x \:+\: 5) \:+\: 2 \:=\: 3x \:+\: 17}$$

We can also reverse the composition and create g(f(x)), which means substituting f(x) into g(x). So,

$$\mathrm{g(f(x)) \:=\: (3x \:+\: 2) \:+\: 5 \:=\: 3x \:+\: 7}$$

In recursive function theory, composite functions allow us to build complex operations by combining simpler ones, extending the range of possible computations.

Recursive Functions

Recursive Function is a function that repeats or uses previous term to calculate subsequent terms. This forms a sequence of terms. Usually, we get this function based on the arithmetic-geometric sequence, which has terms with a common difference between them.

Recursive functions are useful because they allow us to define operations that can continue indefinitely or until a specific condition is met.

Primitive Recursive Functions

A primitive recursive function is a special type of recursive function. It is defined as a function that can be obtained from initial functions through a finite number of composition and recursive steps.

In other words, a primitive recursive function is created by applying composition and recursion to the initial functions. The zero, successor, and projection functions. This type of function is useful in computation theory because it is both predictable and well-defined within a finite number of steps.

Partial Recursive Functions

Then the next type of function is partial recursive functions. A function is said to be partial recursive if it is defined only for some of its arguments, not all.

For example, consider the subtraction of two positive numbers M and N −

$$\mathrm{f(M, N) \:=\: M \:-\: N}$$

This function is defined only when M ≥ N. If M is less than N, the result would be negative, and the function would not be defined. Therefore, this function is a partial recursive function because it does not cover all possible inputs.

Total Recursive Functions

The next type is total recursive functions. A total recursive function is defined for all possible arguments. For example, addition −

$$\mathrm{f(M,\: N) \:=\: M \:+\: N}$$

This function is defined for any positive numbers M and N, and the result is always positive. Therefore, it is a total recursive function because it covers all input possibilities.

Conclusion

In this chapter, we explained the basics of recursive functions in automata theory. With the initial functions like zero, successor, and projection. Then, we explored composite functions, which combine simpler functions to create more complex ones.

After that, we introduced the concept of recursive functions, primitive recursive functions, composition, and recursion. Finally, we covered partial and total recursive functions with basic examples for a clear understanding.

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