TY - GEN

T1 - Bounded Reducibility for Computable Numberings

AU - Bazhenov, Nikolay

AU - Mustafa, Manat

AU - Ospichev, Sergei

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.

UR - http://www.scopus.com/inward/record.url?scp=85069464896&partnerID=8YFLogxK

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 GmbH and Co. KG

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

Y2 - 15 July 2019 through 19 July 2019

ER -