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For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability

Jan A. Bergstra, John Tucker Orcid Logo

Theoretical Computer Science, Volume: 1034, Start page: 115124

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.1016/j.tcs.2025.115124

Abstract

Adding division to rings and fields leads to the question of how to deal with division by 0. From a plurality of options, we discuss in detail what we call Suppes-Ono division in which division by 0 produces 0. We explain the backstory of this semantic option and its associated notion of equality, a...

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Published in: Theoretical Computer Science
ISSN: 0304-3975 1879-2294
Published: Elsevier BV 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa68920
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spelling 2025-02-19T09:41:44.8737965 v2 68920 2025-02-19 For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2025-02-19 MACS Adding division to rings and fields leads to the question of how to deal with division by 0. From a plurality of options, we discuss in detail what we call Suppes-Ono division in which division by 0 produces 0. We explain the backstory of this semantic option and its associated notion of equality, and prove a result regarding the logical complexity of deciding equations over the rational numbers equipped with Suppes-Ono division. We prove that deciding the validity of the equations is computationally equivalent to the Diophantine Problem for the rational numbers, which is a longstanding open problem. Journal Article Theoretical Computer Science 1034 115124 Elsevier BV 0304-3975 1879-2294 Division by zero; Fracterm; Fracterm flattening; Diophantine equation; Decidability; 1-1 degrees 22 4 2025 2025-04-22 10.1016/j.tcs.2025.115124 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) Swansea University 2025-02-19T09:41:44.8737965 2025-02-19T09:33:53.2573601 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan A. Bergstra 1 John Tucker 0000-0003-4689-8760 2 68920__33635__11867fddf1534e9d837aab099b6ab1b1.pdf 68920.VOR.pdf 2025-02-19T09:38:32.8545551 Output 594279 application/pdf Version of Record true © 2025 The Authors. This is an open access article distributed under the terms of the Creative Commons CC-BY license. true eng http://creativecommons.org/licenses/by/4.0/
title For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
spellingShingle For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
John Tucker
title_short For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
title_full For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
title_fullStr For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
title_full_unstemmed For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
title_sort For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability
author_id_str_mv 431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv 431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author John Tucker
author2 Jan A. Bergstra
John Tucker
format Journal article
container_title Theoretical Computer Science
container_volume 1034
container_start_page 115124
publishDate 2025
institution Swansea University
issn 0304-3975
1879-2294
doi_str_mv 10.1016/j.tcs.2025.115124
publisher Elsevier BV
college_str Faculty of Science and Engineering
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hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
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description Adding division to rings and fields leads to the question of how to deal with division by 0. From a plurality of options, we discuss in detail what we call Suppes-Ono division in which division by 0 produces 0. We explain the backstory of this semantic option and its associated notion of equality, and prove a result regarding the logical complexity of deciding equations over the rational numbers equipped with Suppes-Ono division. We prove that deciding the validity of the equations is computationally equivalent to the Diophantine Problem for the rational numbers, which is a longstanding open problem.
published_date 2025-04-22T08:17:16Z
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score 11.055693

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