Here e investigate the relationship between Turing machines and computable functions. For convenience we will restrict ourselves to only look at numeric computations, this does not reflect any loss of generality since all computational problems can be encoded as number.

Kurt Goode used this fact in his famous incompleteness proof. We will show that, The functions computable by a Turing machine are exactly the u recursive functions. U recursive functions were developed by Goode and Stephen Kleenex. So between Turing, Church, Goode, and Kleenex we obtain the following equivalence relation Algorithms Turing machines Recursive function lulus In order to work towards a proof of this equivalence we start with primitive recursive function.

Summary It is not hard to believe that all such functions can be computed by some turning machine. What is a much deeper result is that every turning machine function corresponds to some recursive function Theorem. A function is to b computable “if” and only if is u recursive A primitive recursive function is built up from the base function zero, successor, and projection using the two operations composition and primitive recursion. There are computable functions that are not primitive recursive, such as Ackermann function.

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