Towards a Datatype Defining Rewrite System for Rational Arithmetic

In [3] the concept of datatype defining rewrite systems (DDRSs) is introduced. A DDRS is an equational specification of a datatype that yields a ground-complete term rewriting system (TRS) when its equations are interpreted from left to right as rewrite rules. In [3] a number of DDRSs are presented for terms in unary, binary, and decimal notation. The goal of the present work is to review whether it is possible to further extend these specifications so that they may model rational arithmetic. Some research has been done on this topic which we review in Section 2; in particular we compare and contrast the results proven in [2] and [1] regarding the existence of such specifications. In Section 3 we rephrase the approach for computing irreducible fractions proposed in [1] by using conditional rewrite rules, which results in the design for a conditional TRS (CTRS) that may give rise to the aforementioned specification.