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Synthetic Fracterm Calculus

Jan Bergstra Orcid Logo, John Tucker Orcid Logo

JUCS - Journal of Universal Computer Science, Volume: 30, Issue: 3, Pages: 289 - 307

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.3897/jucs.107082

Abstract

Previously, in [Bergstra and Tucker 2023], we provided a systematic description of elementaryarithmetic concerning addition, multiplication, subtraction and division as it is practiced.Called the naive fracterm calculus, it captured a consensus on what ideas and options were widelyaccepted, rejected...

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Published in: JUCS - Journal of Universal Computer Science
ISSN: 0948-695X 0948-6968
Published: Pensoft Publishers 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa65233
first_indexed 2023-12-07T00:00:55Z
last_indexed 2024-11-25T14:15:38Z
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spelling 2024-04-10T10:09:53.3036017 v2 65233 2023-12-06 Synthetic Fracterm Calculus 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2023-12-06 MACS Previously, in [Bergstra and Tucker 2023], we provided a systematic description of elementaryarithmetic concerning addition, multiplication, subtraction and division as it is practiced.Called the naive fracterm calculus, it captured a consensus on what ideas and options were widelyaccepted, rejected or varied according to taste. We contrasted this state of the practical art witha plurality of its formal algebraic and logical axiomatisations, some of which were motivated bycomputer arithmetic. We identified a significant gap between the wide embrace of the naive fractermcalculus and the narrow precisely defined formalisations. In this paper, we introduce a newintermediate and informal axiomatisation of elementary arithmetic to bridge that gap; it is calledthe synthetic fracterm calculus. Compared with naive fracterm calculus, the synthetic fractermcalculus is more systematic, resolves several ambiguities and prepares for reasoning underpinnedby logic; indeed, it admits direct formalisations, which the naive fracterm calculus does not. Themethods of these papers may have wider application, wherever formalisations are needed to analyseand standardise practices. Journal Article JUCS - Journal of Universal Computer Science 30 3 289 307 Pensoft Publishers 0948-695X 0948-6968 fracterm calculus, partial meadow, common meadow, abstract data type 28 3 2024 2024-03-28 10.3897/jucs.107082 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Not Required 2024-04-10T10:09:53.3036017 2023-12-06T23:54:17.2628474 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan Bergstra 0000-0003-2492-506x 1 John Tucker 0000-0003-4689-8760 2 65233__29968__16f6f36e2104483ab67fe81d3a2b30ed.pdf 65233.VOR.pdf 2024-04-10T10:08:35.8090862 Output 266435 application/pdf Version of Record true This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY-ND 4.0). true eng https://creativecommons.org/licenses/by-nd/4.0/
title Synthetic Fracterm Calculus
spellingShingle Synthetic Fracterm Calculus
John Tucker
title_short Synthetic Fracterm Calculus
title_full Synthetic Fracterm Calculus
title_fullStr Synthetic Fracterm Calculus
title_full_unstemmed Synthetic Fracterm Calculus
title_sort Synthetic Fracterm Calculus
author_id_str_mv 431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv 431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author John Tucker
author2 Jan Bergstra
John Tucker
format Journal article
container_title JUCS - Journal of Universal Computer Science
container_volume 30
container_issue 3
container_start_page 289
publishDate 2024
institution Swansea University
issn 0948-695X
0948-6968
doi_str_mv 10.3897/jucs.107082
publisher Pensoft Publishers
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
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
document_store_str 1
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description Previously, in [Bergstra and Tucker 2023], we provided a systematic description of elementaryarithmetic concerning addition, multiplication, subtraction and division as it is practiced.Called the naive fracterm calculus, it captured a consensus on what ideas and options were widelyaccepted, rejected or varied according to taste. We contrasted this state of the practical art witha plurality of its formal algebraic and logical axiomatisations, some of which were motivated bycomputer arithmetic. We identified a significant gap between the wide embrace of the naive fractermcalculus and the narrow precisely defined formalisations. In this paper, we introduce a newintermediate and informal axiomatisation of elementary arithmetic to bridge that gap; it is calledthe synthetic fracterm calculus. Compared with naive fracterm calculus, the synthetic fractermcalculus is more systematic, resolves several ambiguities and prepares for reasoning underpinnedby logic; indeed, it admits direct formalisations, which the naive fracterm calculus does not. Themethods of these papers may have wider application, wherever formalisations are needed to analyseand standardise practices.
published_date 2024-03-28T05:31:22Z
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score 11.29607

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