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I have seen and understood the most definitions but i just could not understand how to show if a function is mu-partial recursive or not. I used search engines, but all I find are just more lectures with definitions...

Lets say I have a function like

f(x) = x^2

or

f(x)=root(x+5) + 5

Can you proof if those functions are mu-partial recursive? Explain why this function is mu-partial recurisve.

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    show that x is partial recursive and the product of two partial recursive is partial recursive. Then show the sum of two partial recursive is partial recursive, and show root is partial recursive.2012-12-09
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    Thx for comment!! but this is right now to difficult for me. I see all the definitions with variables, but i need one real example with real numbers :/. Something like a cookbook example2012-12-09
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    Recursive functions are, officially, only defined on natural numbers. You'll have to be more precise about what you mean by a recursive real-valued function.2012-12-09
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    hmm ok, can you show if f(x)=root(x+5) + 5 is mu-partial recursive? step by step2012-12-09
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    Exactly what definition you want to verify?2012-12-09
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    its simple: Just explain if the function f(x)=root(x+5)+5 is mu-partial recurisve. Argument why it is or why it is not mu-partial recursive. I do not need another definition. I need all those definitions used on a real function, that i have provided here.2012-12-09
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    @Gero: You have not addressed the question of how you are defining recursiveness of real-valued functions. (If you don't know the difference between a real number and a natural number, please go look that up first.) Of course, under any reasonable definition, $x \mapsto \sqrt{x + 5} + 5$ will be recursive, but I must stress _reasonable_.2012-12-09
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    Sorry, we had a misunderstanding. I did not mean real numbers as x element of R, but i meant to use natural numbers instead of variables. can you show if f(x)=root(x+5) + 5 is mu-partial recursive? step by step. Lets assume it is a question in a math test.2012-12-09
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    @Gero: You have just [asked this question on cstheory](http://cstheory.stackexchange.com/q/14676). Please don't cross-post. If you don't get adequate answers on one cite, request migration to another.2012-12-09
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    Ok. I was just stunned that nobody could answer such an easy question/function if f(x)=root(x+5)+5 is mu-partial recursive and why it is mu-partial recursive.2012-12-09

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I'm not sure what type of answer you're supposed to give, but partial recursive functions correspond to all computable functions. Thus, if you can write a computer program to represent the function, it's partial recursive. Either that or you can start from the definitions and build up to the functions you are given. In this case, both of your examples are clearly partial-recursive.

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    your answer is just once again another definition. I have a specific case and i am interested in an specific answer to the function f(x)=root(x+5)+52012-12-09
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    I know it's a definition, but using the definition and some thinking will allow you to come up with an answer. Plus, I told you the answer.2012-12-11