Turing's proof

Alan Turing is widely regarded as one of the greatest minds of the 20th century. His work in computer science and mathematics revolutionised the field and has had a lasting impact on our society. One of his most notable achievements was his 1936 paper, “On Computable Numbers, with an Application to the Entscheidungsproblem”, in which he described the concept of a universal Turing machine and posed the famous “halting problem”. In this paper, Turing also proved that there is no algorithm or process that can solve all problems, a result is now known as Turing’s proof.

Turing’s proof is based on the concept of computability, or the ability for a problem to be solved by a machine. Turing’s proof states that there is no algorithm or process that can determine whether a given computation will ever halt or not. In other words, there is no algorithm which can predict the future behaviour of a computation.

To prove this, Turing used a method known as diagonalisation. In this method, Turing takes a list of all possible algorithms and then constructs a new algorithm which, when run, produces a result that is not on the list. This new algorithm is then tested against the original list, and if it produces a result that is not on the list, then it is assumed that the original list was incomplete and that there can be no algorithm that solves all problems.

Turing’s proof is significant for two reasons.

Firstly, it proves that some problems are unsolvable, meaning that there are limits to what computers can do.
Secondly, it demonstrates the power of mathematical reasoning, by showing that a seemingly impossible problem can be solved through careful analysis and deduction.

Turing’s proof is still relevant today, as computer scientists and mathematicians continue to use the concept of computability to solve problems. Turing’s proof also serves as a reminder that, despite advances in technology, some problems may still be beyond our reach.