Symmetric Transrationals: The Data Type and the Algorithmic Degree of its Equational Theory

Bergstra, Jan A.; Tucker, John V.

doi:10.1007/978-3-031-15629-8_4

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…In practice, these semantic conventions can be found in theorem provers, common calculators, exact numerical computation and, of course, floating point computation. Indeed, we have developed a new model, the symmetric transrationals [21]. However, their fracterm calculi are not without complications.…”

Section: Division By Zeromentioning

confidence: 99%

Fracterm Calculus for Signed Common Meadows

Bergstra,

Tucker

2024

Transmath

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A common meadow is an enrichment of a field with a division operator and an error value to make division total. A signed common meadow enriches a common meadow with a sign function that can be equationally axiomatised; the sign function can simulate an ordering on the underlying field but is not limited to orderings. In particular, of mathematical interest are the weakly signed common meadows. The prime example of a weakly signed common meadow is an expansion of a common meadow of complex numbers with a weak sign function. We show that all common meadows may be enlarged to a weakly signed common meadow. A special case is the 4-signed common meadows, which are precisely the enlargements of ordered fields. To illustrate the equational calculus for signed common meadows, we use it as a foundation for building a probability calculus and derive some classical formulae.

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Section: Division By Zeromentioning

confidence: 99%

Fracterm Calculus for Signed Common Meadows

Bergstra,

Tucker

2024

Transmath

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“…Symmetric transrationals, where the error element ⊥, two external signed infinities +∞, −∞, two infinitesimals +ι, −ι are added so that division is totalised by setting 1/0 = ⊥, as with common meadows, and the other elements are used to manage overflows and underflows, [19]; specifically, totality is separated from over and under flows.…”

Section: Background To the Problem Of Division By Zeromentioning

confidence: 99%

On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness

Bergstra

Tucker

2022

The Computer Journal

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Common meadows are arithmetic structures with inverse or division, made total on $0$ by a flag $\bot $ for ease of calculation. We examine some axiomatizations of common meadows to clarify their relationship with commutative rings and serve different theoretical agendas. A common meadow fracterm calculus is a special form of the equational axiomatization of common meadows, originally based on the use of division on the rational numbers. We study axioms that allow the basic process of simplifying complex expressions involving division. A useful axiomatic extension of the common meadow fracterm calculus imposes the requirement that the characteristic of common meadows be zero (using a simple infinite scheme of closed equations). It is known that these axioms are complete for the full equational theory of common cancellation meadows of characteristic $0$. Here, we show that these axioms do not prove all conditional equations which hold in all common cancellation meadows of characteristic $0$.

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Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero

Bergstra,

Tucker

2024

J of Log Lang and Inf

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Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. Our dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion.

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Symmetric Transrationals: The Data Type and the Algorithmic Degree of its Equational Theory

Cited by 4 publications

References 17 publications

Fracterm Calculus for Signed Common Meadows

Fracterm Calculus for Signed Common Meadows

On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness

Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero

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