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Concept
Computation
Summary
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computations are mathematical equations and computer algorithms.
Mechanical or electronic devices (or, historically, people) that perform computations are known as computers. The study of computation is the field of computability, itself a sub-field of computer science.
The notion that mathematical statements should be ‘well-defined’ had been argued by mathematicians since at least the 1600s, but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing Machine. Other (mathematically equivalent) definitions include Alonzo Church's lambda-definability, Herbrand-Gödel-Kleene's general recursiveness and Emil Post's 1-definability.
Today, any formal statement or calculation that exhibits this quality of well-definedness is termed computable, while the statement or calculation itself is referred to as a computation.
Turing’s definition apportioned “well-definedness” to a very large class of mathematical statements, including all well-formed algebraic statements, and all statements written in modern computer programming languages.
Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes the halting problem and the busy beaver game. It remains an open question as to whether there exists a more powerful definition of ‘well-defined’ that is able to capture both computable and 'non-computable' statements.
Some examples of mathematical statements that are computable include:
All statements characterised in modern programming languages, including C++, Python, and Java.
All calculations carried by an electronic computer, calculator or abacus.
Official source
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