Datatype defining rewrite systems for naturals and integers

TL;DR

This paper develops and proves the ground-completeness of datatype defining rewrite systems (DDRSs) for natural numbers and integers across unary, binary, and decimal notations, including alternative tree-based systems.

Contribution

It introduces new DDRSs for integers and various number representations, proving their ground-completeness and providing a unified algebraic framework for natural and integer arithmetic.

Findings

Two DDRSs for integers with twelve rules each are ground-complete.

DDRSs for unary, binary, and decimal notations are defined and proven ground-complete.

Alternative tree constructor DDRSs are also ground-complete, offering more complex expressions.

Abstract

A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: 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. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a 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 yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven.