In short, Turing answered the Entscheidungsproblem (Hilbert) with a negative. Then went on to develop the theoretical Turing Machine: a tape reader head that responds to inputted symbols and rewrites the answers back onto the tape. The Turing machine was able to answer mathematical computations in a true or false manner. He developed algorithms to solve the German Enigma during the second world war, when he worked for the British government at Bletchley Park. He put all his previous knowledge together, and understood that a single algorithm/machine could not be developed for each task, a general purpose computer was necessary to fulfill the requirements of the cryptography and mathematical branch. He developed the colossus computer, generally considered the first digital computer, along with Zusse and Von Neuman's work, the title is contested! Due to his contributions Turing's name was put to the award for computing excellence, basically the Nobel Prize for computing. The Turing Test is still used today, to show how computers and humans respond differently to logic problems and complex computations, proving that a machine cannot behave like a human.
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The Turing machine is the theoretical underpinning of all modern computing devices. The Turing machine is not a physically constructed device, but a way of conceptualizing computer algorithms. See link.
A Turing Machine is a theoretical computing machine in math to serve as an ideal model for mathematical calculation. A busy beaver is an n-state, 2 color Turing Machine which writes a maximum number of 1s before halting.
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The inventor of the electronic general purpose computer is widely acknowledged to be Alan Turing. Turing conceptualized the idea of a universal computing machine and played a crucial role in the development of the first electronic computer, the Colossus, during World War II. His theoretical work also laid the foundation for modern computer science.
Turing Decidable Languages are both Turing Rec and Turing Co-Recognizable. If a Language is Not Turing Decidable, either it, or it's complement, must be not Recognizable.