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Symmetric Transrationals: The Data Type and the Algorithmic Degree of its Equational Theory
Lecture Notes in Computer Science, Volume: Lecture Notes in Computer Science 13560, Pages: 63 - 80
Swansea University Author:
John Tucker
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DOI (Published version): 10.1007/978-3-031-15629-8_4
Abstract
We introduce and investigate an arithmetical data typedesigned for computation with rational numbers. Called the symmetrictransrationals, this data type comes about as a more algebraicallysymmetric modification of the arithmetical data type of transrationalnumbers [9], which was inspired by the tran...
| Published in: | Lecture Notes in Computer Science |
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| ISBN: | 9783031156281 9783031156298 |
| ISSN: | 0302-9743 1611-3349 |
| Published: |
Cham
Springer Nature Switzerland
2022
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa61535 |
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| Abstract: |
We introduce and investigate an arithmetical data typedesigned for computation with rational numbers. Called the symmetrictransrationals, this data type comes about as a more algebraicallysymmetric modification of the arithmetical data type of transrationalnumbers [9], which was inspired by the transreals of Anderson et.al. [1].We also define a bounded version of the symmetric transrationals therebymodelling some further key semantic properties of floating point arithmetic.We prove that the bounded symmetric transrationals constitutea data type. Next, we consider the equational theory and prove thatdeciding the validity of equations over the symmetric transrationals is1-1 algorithmically equivalent with deciding unsolvability of Diophantineequations over the rational numbers, which is a longstanding openproblem. The algorithmic degree of the bounded case remains open. |
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| Keywords: |
rational numbers, data types , computer arithmetic, common meadows, transrationals, diophantine problem, floating-point |
| College: |
Faculty of Science and Engineering |
| Start Page: |
63 |
| End Page: |
80 |