Three Datatype Defining Rewrite Systems for Datatypes of Integers each extending a Datatype of Naturals

• Published in: arXiv.org
• Main Authors: Bergstra, Jan A, Ponse, Alban
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 18.07.2016
• Subjects: Arithmetic; Completeness; Digits; Integers; Number theory
• ISSN: 2331-8422

Integer arithmetic is specified according to three views: unary, binary, and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. The unary view comprises a specification of integer arithmetic based on 0, successor function \(S\), and predecessor function, with negative normal forms \(-S^i(0)\). Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a ground-confluent and terminating datatype defining rewrite system (DDRS), and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yield comparable normal forms, which for that binary and decimal view admits expressions that are algorithmically more involved. These DDRSes are incorporated because they are closer to existing literature. For these DDRSes we also provide ground-completeness results. Finally, we define a DDRS for the ring of Integers (comprising fifteen rewrite rules) and prove its ground-completeness.