Definition:Computational Method
From ProofWiki
Definition
A computational method is an ordered quadruple $\left({Q, I, \Omega, f}\right)$ in which:
- $Q$ is a set representing the states of the computation;
- $I$ is a set representing the input to the computation;
- $\Omega$ is a set representing the output from the computation;
- $f: Q \to Q$ is a mapping representing the computational rule
subject to the following constraints:
- $I \subseteq Q$ and $\Omega \subseteq Q$;
- $\forall x \in \Omega: f \left({x}\right) = x$.
Each $x \in I$ defines a computational sequence $x_0, x_1, x_2, \ldots$ as follows:
- $x_0 = x$;
- $\forall k \ge 0: x_{k+1} = f \left({x_k}\right)$.
The computational sequence is said to terminate in $k$ steps if $k$ is the smallest integer for which $x_k \in \Omega$.
In this case, it produces the output $x_k$ from $x$.
Some computational sequences may never terminate.
An algorithm is a computational method which terminates in finitely many steps for all $x \in I$.
Historical Note
This definition is very nearly the same as that given by A.A. Markov, Jr. in his Theory of Algorithms (1954).
