Introduction
- Recursive Function Theory is a branch of the theory of computation that studies computable functions, that is, functions that can be calculated using a finite procedure or algorithm.
- It provides a mathematical foundation for understanding what problems can be solved by a computer.
Definition
- Recursive Function Theory deals with functions that can be defined using basic functions and operations such as composition, recursion, and minimization.
Characteristics
- It explains which problems can be solved using algorithms and forms a key part of theoretical computer science.
- Recursive Function Theory provides a mathematical model to define computable functions.
- It helps in defining the concept of computability in mathematics.
Basic Functions
Recursive functions are built from three basic initial functions:-
- Zero Function
- This function returns 0 for any input.
- Example: Z(x) = 0
- Successor Function
- This function adds 1 to a number.
- Example: S(x) = x + 1
- Projection Function
- This function selects one of the input variables.
- Example: Pᵢⁿ(x₁, x₂, …, xₙ) = xᵢ
Operations to Build Recursive Functions
- Composition
- It combines simple functions to form complex functions.
- Primitive Recursion
- It defines a function in terms of itself using smaller values.
- Example: Factorial function.
- Minimization (µ-operator)
- This is used to define more complex functions by searching for the smallest value that satisfies a condition.
Types of Recursive Functions
- Primitive Recursive Functions
- It has been built using composition and primitive recursion only.
- General Recursive (µ-Recursive) Functions
- It includes minimization operations and is more powerful.
Importance
- It forms the foundation of computability theory.
- It is equivalent in power to Turing Machines.
- It helps classify decidable and undecidable problems.
0 Comments