Why Are Computable Functions Also Called Recursive Functions? What Is Recursion About Them?

The term recursive has its roots in the field of mathematics and computer science, where it is used to describe any function that can be computed by an effective method or algorithm. This concept is deeply intertwined with the broader understanding of computability, which is the ability to solve problems using a computer or an equivalent mathematical model.

In the context of recursion theory, recursion refers to a particular method of computation where a function calls itself repeatedly until a certain condition is met. This self-referencing process is a cornerstone of many computational methods and programming paradigms. The term recursiveness is also synonymous with computation in the broader field of computability theory.

Recursive Functions: A Historical Perspective

The concept of recursive functions emerged from the work of mathematicians and logicians in the early 20th century. Kurt G?del and Jacques Herbrand, among others, developed the notion of recursive functions in 1933, three years before Alonzo Church developed the lambda calculus and Alan Turing introduced the concept of Turing machines. These theoretical frameworks laid the groundwork for understanding computable functions and the nature of recursive computation.

Computability and Effective Computation

The Church-Turing thesis is a foundational hypothesis in computer science that posits that any function that can be effectively computed can be computed by a Turing machine. This thesis, and its supporting evidence, breathe life into the idea of recursive functions. One of the key ways to support this thesis is by demonstrating the equivalence between different models of computation, such as Turing machines and the lambda calculus, which is a formal system for expressing computation.

Recursive Functions in Programming

In programming, recursive functions are a crucial concept, especially in functional programming languages. These languages, like Lisp and its derivatives, enforce a strict separation between data and behavior. As a result, traditional control structures such as while and for loops are either absent or discouraged. Instead, recursion is used to express repetitive processes. A function calling itself is a fundamental principle that enables complex operations to be broken down into simpler, manageable parts.

The recursive function is constructed using primitive recursive functions and operators. These operators, such as composition, primitive recursion, and minimisation, allow for the definition of more complex functions from simpler ones. Primitive recursion is a key operator that gives recursive functions their name. It enables the definition of a function by specifying its value for zero and a rule for computing its value from one argument to the next.

Examples and Applications

One classic example of a recursive function is the factorial function. The factorial of a non-negative integer n is defined as:

factorial(n)  n * factorial(n-1), factorial(0)  1

This function calls itself to compute the factorial of a smaller number until it reaches the base case where n is zero. This is a simple yet powerful demonstration of the power of recursion.

Conclusion

The concept of recursive functions is not only a theoretical construct but also a practical tool for solving complex problems in programming and computation. Understanding recursive functions and their relationship to computability theory and Turing machines is essential for computer scientists and programmers alike. Whether you are working in a functional programming language or a traditional imperative language, the principles of recursion and computability are fundamental to modern computing.

References

1. Church, A. (A note on the Entscheidungsproblem), Journal of Symbolic Logic, 1936.

2. Turing, A. M. (On computable numbers, with an application to the Entscheidungsproblem), Proceedings of the London Mathematical Society, 1936.

3. G?del, K. , Monatshefte für Mathematik und Physik, 1931.