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Positive numerations of families with one-valued numerations. (English. Russian original) Zbl 0572.03023
Algebra Logic 22, 345-350 (1983); translation from Algebra Logika 22, No. 5, 481-488 (1983).
A result in the theory of numerations of families of recursively enumerable sets is proved. To state the main result, some definitions are necessary.
A numeration \(\nu\) :\({\mathbb{N}}\to S\) of a family S of r.e. sets is computable if \(\{\) (x,n): \(x\in \nu (n)\}\) is r.e. - equivalently if there is a recursive function g:\({\mathbb{N}}\to {\mathbb{N}}\) for which \(\nu (n)=W_{g(n)}\) for all n. A numeration \(\nu\) is reducible to numeration \(\mu\), \(\nu\leq \mu\), if there is a recursive function f:\({\mathbb{N}}\to {\mathbb{N}}\) for which \(\nu =\mu f\); \(\nu\) is equivalent with \(\mu\) if \(\nu\leq \mu\) and \(\mu\leq \nu\). A computable numeration \(\nu\) is minimal if \(\mu\leq \nu\) implies that \(\nu\leq \mu\) for every computable \(\mu\) ; \(\nu\) is smallest if \(\nu\leq \mu\) for every computable numeration \(\mu\) of the same family S of r.e. sets; \(\nu\) is positive if \(\{\) (n,m): \(\nu (n)=\nu (m)\}\) is r.e.; \(\nu\) is one-valued if \(n\neq m\) implies \(\nu\) (n)\(\neq \nu (m).\)
The main theorem proved now reads: Each family S admitting a one-valued computable numeration has either a smallest one-valued numeration or countably many nonequivalent positive numerations. This result follows from a lemma proved by a priority argument together with a previous result due to the author. The author has a counterexample showing that the main result cannot be improved to read ”nonequivalent one-valued numerations” in place of ”nonequivalent positive numerations”.
Reviewer: P.Clote

03D45 Theory of numerations, effectively presented structures
03D25 Recursively (computably) enumerable sets and degrees
03D20 Recursive functions and relations, subrecursive hierarchies
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[1] S. A. Badaev, ”On positive numerations,” Sib. Mat. Zh.,18, No. 3, 483–496 (1977). · Zbl 0358.02047
[2] S. S. Goncharov, ”The problem of the number of nonautoequivalent constructivizations,” Dokl. Akad. Nauk SSSR,25, No. 2, 271–274 (1980). · Zbl 0476.03045
[3] S. S. Goncharov, ”Computable one-valued numerations,” Algebra Logika,19, No. 5, 507–551 (1980).
[4] S. S. Goncharov, ”Nonautoequivalent constructivizations,” Tr. IM SO AN SSSR, Novosibirsk,2, (1982). · Zbl 0407.03040
[5] Yu. L. Ershov, Theory of Numerations [in Russian], Nauka, Moscow (1977).
[6] I. A. Lavrov, ”Computable numberings,” in: Logic, Foundations of Mathematics, and Computability Theory, pp. 195–206. · Zbl 0386.03023
[7] A. I. Mal’tsev, Algorithms and Recursive Functions [in Russian], Nauka, Moscow (1965).
[8] A. I. Mal’tsev, ”Positive and negative numerations,” Dokl. Akad. Nauk SSSR,160, No. 2, 278–280 (1965).
[9] S. S. Marchenkov, ”On computable numerations of families of general recursive functions,” Algebra Logika,11, No. 5, 588–607 (1972).
[10] H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, (1967). · Zbl 0183.01401
[11] V. L. Selivanov, ”Two theorems on computable numerations,” Algebra Logika,15, No. 4, 470–484 (1976). · Zbl 0358.02050
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