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I was just reading a chapter about LOOP-computable functions and I have the following question: Is it possible to numerate every LOOP program with an algorithm?

Formally: Is it possible to have a LOOP program M, s.t M(n,m)=P_n(m) (input is n,m. output is P_n(m), where P_n is the numeration of the loop programs).

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No. Assume M is a loop-program. Define program D(x) = M(x, x). Then D is trivially a loop-program. Let i be the index of D in the enumeration, i.e. P_i = D; then M(i, i) doesn't halt because it's an infinite loop (D(i) calls M(i,i) calls D(i) ...). Therefore M can't be a loop-program.

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  • Thank you, could you explain me the second part a little bit more in detail? If i is the index you mean D_i(x)=? Where is the difference between M(i,i) and M(x,x)?
    – Voyage
    Commented Mar 10, 2013 at 11:48
  • @Voyage "D(x) = M(x,x)" defines D for any argument x (In other words: D = lambda x. M(x, x)). "M(i,i)" and "D(i)" are calls of the functions with whatever the value of "i" might be (we don't know what it is, but its existence is guaranteed by the fact that D is a loop-program)
    – ebsddd
    Commented Mar 10, 2013 at 18:47

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