Bounded Reducibility for Computable Numberings

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Abstract

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.

Original languageEnglish
Title of host publicationComputing with Foresight and Industry - 15th Conference on Computability in Europe, CiE 2019, Proceedings
EditorsFlorin Manea, Barnaby Martin, Daniël Paulusma, Giuseppe Primiero
PublisherSpringer-Verlag GmbH and Co. KG
Pages96-107
Number of pages12
ISBN (Print)9783030229955
DOIs
Publication statusPublished - 1 Jan 2019
Event15th Conference on Computability in Europe, CiE 2019 - Durham, United Kingdom
Duration: 15 Jul 201919 Jul 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11558 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference15th Conference on Computability in Europe, CiE 2019
CountryUnited Kingdom
CityDurham
Period15.07.201919.07.2019

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