Abstract
The standard mathematical definition of signed integers, based on set theory, is not well-adapted to the needs of computer science. For this reason, many formal specification languages and theorem provers have designed alternative definitions of signed integers based on term algebras, by extending the Peano-style construction of unsigned naturals using “zero” and “succ” to the case of signed integers. We compare the various approaches used in CADP, CASL, Coq, Isabelle/HOL, KIV, Maude, mCRL2, PSF, SMT-LIB, TLA+, etc. according to objective criteria and suggest an “optimal” definition of signed integers.
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Notes
- 1.
In the present article, we distinguish between naturals (also: natural numbers), which are the elements of \({\mathbb {N}}\) (i.e., unsigned), and integers, which are the elements of \({\mathbb {Z}}\) (i.e., signed). We assume that \(0 \in {\mathbb {N}}\) and we write
. - 2.
We prefer using sort rather than type, which is often given a different meaning.
- 3.
And, hence, a bijection.
- 4.
In particular, are pairwise disjoint.
- 5.
- 6.
Notice that the IEEE 754 standard for floating-point numbers also defines two zeros, −0.0 and +0.0, which are equivalent in most cases but can still be distinguished, e.g., using the signbit primitive.
- 7.
See http://rewriting.loria.fr/systems.html to learn about such halted tools.
- 8.
- 9.
I.e., a datatype in the terminology of Coq.
- 10.
We name so this constructor, as a shorthand for NEGative One.
- 11.
This definition generalizes the Peano system (for \(k=1\)) and the approach presented in this section (for \(k=2\)).
- 12.
.
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Acknowledgements
The author is grateful to Radu Mateescu and Mihaela Sighireanu who, in 1997 at INRIA Grenoble, helped writing LOTOS algebraic equations to show that the approach 
was feasible. Acknowledgements are also due to Holger Hermanns and Gilles Nies (Saarland University), and to Jan Friso Groote (Technical University of Eindhoven) for their valuable remarks. Jan Bergstra and Alban Ponse (University of Amsterdam) provided the author with recent bibliographic references. This article benefited from the lively WADT’2016 discussions in Gregynog, especially with Alexander Knapp (Augsburg University), who mentioned KIV, with Narciso Marti-Oliet (Conplutence University of Madrid), who gave informed comments about Maude, and with Lutz Schröder (Friedrich-Alexander-Universität Erlangen-Nürnberg), who suggested the approach 
. Frédéric Lang, Wendelin Serwe, and Hugues Evrard (INRIA Grenoble) proof-read earlier versions of this article.
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Garavel, H. (2017). On the Most Suitable Axiomatization of Signed Integers. In: James, P., Roggenbach, M. (eds) Recent Trends in Algebraic Development Techniques. WADT 2016. Lecture Notes in Computer Science(), vol 10644. Springer, Cham. https://doi.org/10.1007/978-3-319-72044-9_9
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