TY - GEN

T1 - Bounded Reducibility for Computable Numberings

AU - Bazhenov, Nikolay

AU - Mustafa, Manat

AU - Ospichev, Sergei

N1 - Funding Information:
The work was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342. The first author was partially supported by the grant of the President of the Russian Federation (No. MK-1214.2019.1). The third author was partially supported by the program of fundamental scientific researches of the SB RAS No. I.1.1, project No. 0314-2019-0002.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The theory of numberings gives a fruitful approach to studying uniform computations for various families of mathematical objects. The algorithmic complexity of numberings is usually classified via the reducibility ≤ between numberings. This reducibility gives rise to an upper semilattice of degrees, which is often called the Rogers semilattice. For a computable family S of c.e. sets, its Rogers semilattice R(S) contains the ≤ -degrees of computable numberings of S. Khutoretskii proved that R(S) is always either one-element, or infinite. Selivanov proved that an infinite R(S) cannot be a lattice. We introduce a bounded version of reducibility between numberings, denoted by ≤ bm. We show that Rogers semilattices Rbm(S), induced by ≤ bm, exhibit a striking difference from the classical case. We prove that the results of Khutoretskii and Selivanov cannot be extended to our setting: For any natural number n≥ 2, there is a finite family S of c.e. sets such that its semilattice Rbm(S) has precisely 2 n- 1 elements. Furthermore, there is a computable family T of c.e. sets such that Rbm(T) is an infinite lattice.

AB - The theory of numberings gives a fruitful approach to studying uniform computations for various families of mathematical objects. The algorithmic complexity of numberings is usually classified via the reducibility ≤ between numberings. This reducibility gives rise to an upper semilattice of degrees, which is often called the Rogers semilattice. For a computable family S of c.e. sets, its Rogers semilattice R(S) contains the ≤ -degrees of computable numberings of S. Khutoretskii proved that R(S) is always either one-element, or infinite. Selivanov proved that an infinite R(S) cannot be a lattice. We introduce a bounded version of reducibility between numberings, denoted by ≤ bm. We show that Rogers semilattices Rbm(S), induced by ≤ bm, exhibit a striking difference from the classical case. We prove that the results of Khutoretskii and Selivanov cannot be extended to our setting: For any natural number n≥ 2, there is a finite family S of c.e. sets such that its semilattice Rbm(S) has precisely 2 n- 1 elements. Furthermore, there is a computable family T of c.e. sets such that Rbm(T) is an infinite lattice.

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U2 - 10.1007/978-3-030-22996-2_9

DO - 10.1007/978-3-030-22996-2_9

M3 - Conference contribution

AN - SCOPUS:85069464896

SN - 9783030229955

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 96

EP - 107

BT - Computing with Foresight and Industry - 15th Conference on Computability in Europe, CiE 2019, Proceedings

A2 - Manea, Florin

A2 - Martin, Barnaby

A2 - Paulusma, Daniël

A2 - Primiero, Giuseppe

PB - Springer Verlag

T2 - 15th Conference on Computability in Europe, CiE 2019

Y2 - 15 July 2019 through 19 July 2019

ER -