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By Daniel E. Cohen

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14 Show that, for any r, the set {n; 11 takes at least r values} is listable, and so is {n; 11 is defined for at least r numbers}. What happens if 'at least' is replaced by 'exactly'? What happens if we consider the set of those pairs (r, n) satisfying the various conditions? 15 LetA be listable. Show that {n;A meets the range of 11 } and {n;A meets the domain of 11 } are listable. What happens if we consider {n;A contains the range of 11 } and {n;A contains the domain of 11 }?

As a result they claim that minimisation of partial functions does not preserve computability, and they have to look at minimisation only for total functions. 5 RICE'S THEOREM The next lemma requires a more detailed knowledge of what a program is than we need elsewhere in this chapter. Consequently, only a sketch proof is given. The ideas of this proof apply for many programming languages. 10. 11. Sec. 6 Let F:N 2 ~ N be a computable function. for every n, the function with index gn sends x to F(n,x)for all x.

Then, by definition, fm would also be defined and equal to ( m, m) + 1. • It is customary to denote the function sendingx to (n,x) by m and to refer ton as an index of this function. Thus any computable function from N to N has at least one index; we shall see later (and it is easy to see directly, by adding irrelevant steps to a program) that it has infinitely many indexes. Corollary 2 Let A be a set such that if nEA then n is total and such that every total computable function from N to N has at least one index in A.

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