Showing papers by "Jan A. Bergstra published in 2020"

Arithmetical Datatypes, Fracterms, and the Fraction Definition Problem

Abstract: Datatypes and abstract datatypes are positioned as results of importing aspects of universal algebra into computer science and software engineering. It is suggested that 50 years later, by way of a transfer in the opposite direction, outcomes of research on datatypes can be made available via elementary arithmetic. This idea leads to the notions of an arithmetical signature, an arithmetical datatype and an arithmetical abstract datatype and to algebraic specifications for such entities. The area of fractions in elementary arithmetic is chosen as an application area and while taking a common meadow of rational numbers as the basis, an arithmetical datatype equipped with an absorptive element. The use of datatypes and signatures makes syntax available for giving precise definitions in cases where lack of precision is common place. Fracterm is coined as the name for a fraction when primarily understood as a syntactic entity. The main contribution of the paper is to provide a detailed terminology of fracterms. Subsequently the fraction definition problem is stated, a distinction between explicit definitions of fractions and implicit definitions of fractions is made, and an outline of a survey of both forms of definitions of the notion of a fraction is given.

The Transrational Numbers as an Abstract Data Type

Abstract: In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.

Arithmetical datatypes with true fractions

Abstract: We consider several novel congruences on the signature of meadows with the aim to survey different notions of fractions. In particular we suggest a notion of “true fraction”.

The Wheel of Rational Numbers as an Abstract Data Type

Abstract: In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element such as infinity \(\infty \) or error element \(\perp \). A wheel is an algebra in which division is totalised by setting \(1/ 0 = \infty \) but which also contains an error element \(\perp \) to help control its use. We construct the wheel of rational numbers as an abstract data type \(\mathbb {Q}_w\) and give it an equational specification without auxiliary operators under initial algebra semantics.

Fractions in Transrational Arithmetic

Abstract: In transrational arithmetic each closed fraction may be written as a simple fraction. It is shown that unlike in involutive meadows, in transrational arithmetic open fractions cannot be written as a sum of simple fractions (i.e. fractions the numerator and denominator of which are polynomials). It is also not the case that each open fraction with a single variable can be written as a mixed fraction.

Candidate Software Process Flaws for the Boeing 737 Max MCAS Algorithm and Risks for a Proposed Upgrade

Abstract: By reasoning about the claims and speculations promised as part of the public discourse, we analyze the hypothesis that flaws in software engineering played a critical role in the Boeing 737 MCAS incidents. We use promise-based reasoning to discuss how, from an outsider's perspective, one may assemble clues about what went wrong. Rather than looking for a Rational Alternative Design (RAD), as suggested by Wendel, we look for candidate flaws in the software process. We describe four such potential flaws. Recently, Boeing has circulated information on its envisaged MCAS algorithm upgrade. We cast this as a promise to resolve the flaws, i.e. to provide a RAD for the B737 Max. We offer an assessment of B-Max-New based on the public discourse.

Sumterms, Summands, Sumtuples, and Sums and the Meta-arithmetic of Summation

Abstract: Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.

Most general algebraic specifications for an abstract datatype of rational numbers

Abstract: The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced.Three examples of such specifications are given. A preference is formulated for a specification by means of infinitely many equations which can be presented via a finite number of so-called schematic equations phrased in terms of an infinite signature. On the basis of the latter specification three topics are discussed: (i) fracterm decomposition operators and the numerator paradox, (ii) foundational specifications of arithmetical datatypes, and (iii) poly-infix operations.

Promise Theory as a Tool for Informaticians

Abstract: Promise theory was designed and developed from 2005 onwards by Mark Burgess and his coworkers. It totalises the notion of a promise so that it applies to both animate and inanimate promisers. The focus of promise theory is on applications in informatics and systems design. This paper extends the account of promises by providing more detailed requirements on promises. In particular, the requirement of determinacy, the requirement of decomposition of aggregate agents, and the feature of a promise bias are introduced. The paper further includes an account of threats, as well as of risks, both viewed as an extension of promise theory. It is finally indicated by means of a series of informal examples how and where various kinds of promises and threats may occur in informatics.