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Discrete families of recursive functions and index sets. (English. Russian original) Zbl 0828.03014
Algebra Logic 33, No. 2, 85-94 (1994); translation from Algebra Logika 33, No. 2, 147-165 (1994).
Second order index sets of the discrete families of partial recursive functions (PRF) are studied.
Let $$\{\varphi_e\}_{e \in \omega}$$ be a Gödel numbering of the set of PRF. The $$e$$-th computable numbering is defined as follows: $$\nu^e : = \lambda i,x. \varphi_e (\langle i,x \rangle)$$, and let $$\nu^e$$ enumerate the family of PRF $$F^{(e)} : = \{ \lambda x. \nu^e (i,x)\}_{e \in \omega}$$. Denote by $$I(F)$$ the set $$\{e : F^{(e)} = F\}$$, and denote by $$R_1$$ the family of total recursive functions.
The authors show:
If $$A$$ is a non-recursive recursively enumerable set then $$\{\langle i,j \rangle : W_i \cap W_j = \emptyset\;\&\;W_i \cup W_j = A \cap W_i\;\&\;W_i$$ is nonrecursive} is $$\Pi_3$$-complete.
The family $$\{e : F^{(e)} \subseteq R_1$$ is effectively discrete} is $$\Sigma_3$$-complete.
The family $$\{e : F^{(e)} \subseteq R_1$$ is discrete}, as well as the family $$\{e : F^{(e)} \subseteq R_1$$ is discrete, but not effectively discrete} are both $$\Pi_3$$-complete.
The index set of any family $$F$$ of total recursive functions which has exactly one (i.e. $$|L \{F\} |= 1)$$ numbering up to numbering equivalence, is $$\Pi_4$$-complete.
The article contains a series of other results closely related to the above-mentioned ones. A number of open problem are listed.

##### MSC:
 03D20 Recursive functions and relations, subrecursive hierarchies 03D45 Theory of numerations, effectively presented structures
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