Positive reducibilities, extreme numberings, and completeness

M. Kh. Faizrahmanov

Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 176-191.

Résumé

In the present article, we study universal, minimal, and complete numberings of families of arithmetic sets. We show that, for every $m\in\mathbb{N}$ and every nontrivial $\Sigma^0_{m+2}$-computable family $\mathcal{S}$, there exists a $\Sigma^0_{m+2}$-computable numbering that is not universal with respect to positive reducibilities and is complete with respect to each element $B\in\mathcal{S}$. For finite families of computably enumerable sets, we obtain necessary and sufficient conditions for existence of numberings that are complete, computable, and not universal with respect to positive reducibilities. For every infinite $\Sigma^0_{m+2}$-computable family $\mathcal{S}$ and every element $B\in\mathcal{S}$, we construct a $\Sigma^0_{m+2}$-computable numbering that is complete with respect to $B$ and minimal with respect to classical and positive reducibilities.

Bibliographie

[1] S. A. Badaev, “Minimalnye numeratsii”, Tr. In-ta matematiki SO RAN, 25, 1993, 3–34

[2] S. A. Badaev, S. S. Goncharov, “O polureshetkakh Rodzhersa semeistv arifmeticheskikh mnozhestv”, Algebra i logika, 40:5 (2001), 507–522

[3] S. A. Badaev, S. S. Goncharov, “Obobschenno vychislimye universalnye numeratsii”, Algebra i logika, 53:5 (2014), 555–569

[4] S. A. Badaev, S. S. Goncharov, A. Sorbi, “Neskolko zamechanii o popolneniyakh numeratsii”, Sib. matem. zhurn., 49:5 (2008), 986–991

[5] S. A. Badaev, S.Yu. Podzorov, “Minimalnye nakrytiya v polureshetkakh Rodzhersa”, Sib. matem. zhurn., 43:4 (2002), 769–778

[6] S. S. Goncharov, A. Sorbn, “Obobschenno-vychislimye numeratsii i netrivialnye polureshetki Rodzhersa”, Algebra i logika, 36:6 (1997), 621–641

[7] A. N. Degtev, “O svodimostyakh numeratsii”, Matem. sb., 112(154):2 (6) (1980), 207–219

[8] A. N. Degtev, “O $p$-svodimosti vychislimykh numeratsii”, Matem. zametki, 69:1 (2001), 31–35

[9] A. N. Degtev, M. L. Platonov, “O $e$-glasnykh numeratsiyakh”, Sib. matem. zhurn., 49:2 (2008), 299–307

[10] Yu. L. Ershov, Teoriya numeratsii, Nauka, M., 1977

[11] A. N. Maltsev, “Polno numerovannye mnozhestva”, Algebra i logika, 2:2 (1963), 4–29

[12] M. L. Platonov, “O verkhnikh polureshetkakh $e$-stepenei vychislimykh numeratsii”, Vestnik TyumGU, 5 (2006), 160–162

[13] S. Yu. Podzorov, “O predelnosti naibolshego elementa polureshetki Rodzhersa”, Matem. tr., 7:2 (2004), 98–108

[14] S. Yu. Podzorov, “O lokalnom stroenii polureshetok Rodzhersa $\Sigma_n^0$-vychislimykh numeratsii”, Algebra i logika, 44:2 (2005), 148–172

[15] X. Rodzhers, Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972

[16] M. Kh. Faizrakhmanov, “O $p$-universalnykh i $p$-minimalnykh numeratsiyakh”, Sib. matem. zhurn., 63:2 (2022), 440–449

[17] M. Kh. Faizrakhmanov, “Svodimost po perechislimosti i pozitivnaya svodimost numeratsii semeistv arifmeticheskikh mnozhestv”, Sib. matem. zhurn., 64:1 (2023), 204–212

[18] S. A. Badaev, S. S. Goncbarov, A. Sorbi, “Completeness and universality of arithmetical numberings”, Computability and Models, The University Series in Mathematics, Springer, Boston, 2003

[19] S. B. Cooper, “Enumeration reductibility, nondeterministic computations and relative computability of partial functions”, Recursion Theory Week, Lecture Notes in Mathematics, 1432, eds. Ambos-Spies K., Muller G.H., Sacks G.E., Springer, Berlin–Heidelberg, 1990, 57–110