Gödel, Turing, and K-Graph Machines

The notion of effective calculability is central for logic and the philosophy of mathematics, not to speak of computer science, artificial intelligence, and cognitive science. Turing gave in 1936 what is viewed as the most convincing analysis of this informal notion:1 he was led to the mathematical concept of computability by idealized machines and to the thesis that every effectively calculable number-theoretic function is computable by such a (Turing-) machine. In support of the thesis, Turing argued that all processes carried out by human computors — when effectively calculating the values of a number-theoretic function — can be performed by his machines. In 1964, Godel stated briefly and enigmatically: Turing’s work gives an analysis of the concept of “mechanical procedure” .... This concept is shown to be equivalent with that of a “Turing machine”.2

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