Finitary Reducibility on Equivalence Relations

We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π_(n+2)^0-complete under comp...

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Main Authors: Miller, Russell, Ng, Keng Meng
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2016
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Online Access:https://hdl.handle.net/10356/84705
http://hdl.handle.net/10220/41955
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-847052023-02-28T19:33:29Z Finitary Reducibility on Equivalence Relations Miller, Russell Ng, Keng Meng School of Physical and Mathematical Sciences Computable reducibility Computability We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π_(n+2)^0-complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is Π_(n+2)^0-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy. MOE (Min. of Education, S’pore) Published version 2016-12-28T07:46:18Z 2019-12-06T15:49:53Z 2016-12-28T07:46:18Z 2019-12-06T15:49:53Z 2016 Journal Article Miller, R., & Ng, K. M. (2016). Finitary Reducibility on Equivalence Relations. The Journal of Symbolic Logic, 81(4), 1225-1254. 0022-4812 https://hdl.handle.net/10356/84705 http://hdl.handle.net/10220/41955 10.1017/jsl.2016.23 en The Journal of Symbolic Logic © 2016 Association for Symbolic Logic. This paper was published in The Journal of Symbolic Logic and is made available as an electronic reprint (preprint) with permission of Association for Symbolic Logic. The published version is available at: [http://dx.doi.org/10.1017/jsl.2016.23]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. 30 p. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Computable reducibility
Computability
spellingShingle Computable reducibility
Computability
Miller, Russell
Ng, Keng Meng
Finitary Reducibility on Equivalence Relations
description We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π_(n+2)^0-complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is Π_(n+2)^0-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Miller, Russell
Ng, Keng Meng
format Article
author Miller, Russell
Ng, Keng Meng
author_sort Miller, Russell
title Finitary Reducibility on Equivalence Relations
title_short Finitary Reducibility on Equivalence Relations
title_full Finitary Reducibility on Equivalence Relations
title_fullStr Finitary Reducibility on Equivalence Relations
title_full_unstemmed Finitary Reducibility on Equivalence Relations
title_sort finitary reducibility on equivalence relations
publishDate 2016
url https://hdl.handle.net/10356/84705
http://hdl.handle.net/10220/41955
_version_ 1759855282485198848