Draft:Computational Method

In the field of computer science, a computational method is a formalized and generalized notion of repeatedly performing the same operations to a set of inputs, usually for the sake of getting an output.

Definition

A computational method is defined as a quadruple $(Q,I,\Omega ,f)$, where $Q$ is any set, $I,\Omega \subset Q$ and $f$ is a function $f:Q\to Q$. $I$ is called the set of inputs and $\Omega$ is called the set of outputs. Furthermore, every element $x\in I$ defines a computational sequence $x_{n}$, where $x_{0}=x$ and every other element of the sequence follows the recurrence relation $x_{n}=f(x_{n-1})$. The termination of a computational sequence $x_{n}$ is defined as the smallest number $k$ such that $x_{k}\in \Omega$. Intuitively, one can think of the computational sequence as concurrent steps performing the same operations on an input, and its termination as the amount of steps required to reach the output.

Relation to algorithms

An algorithm is a special case of a computational method which is finite. Formally speaking, a computational method is finite (and therefore an algorithm) if its termination is well-defined for the entire set $I$. This provides an elegant intuition for the formal notion of an algorithm; in this context, an algorithm is a procedure of steps that upon being repeated on any input will eventually result in an output.