Linear programs in a simple reversible language

Abstract

Very simple reversible programming languages can be useful for the study of reversible transformations. For this purpose we define simple reversible language (SRL), a very simple reversible language, and analyse its properties. The language SRL is similar to the “loop” languages that have been used by several authors to characterise the set of primitive recursive functions. There are, however, important differences: SRL has domain Z instead of N and only reversible programs can be written in SRL. The reversibility of linear homogeneous SRL programs is related to the fact that the corresponding set of matrices has the algebraic structure of a group. We show that such programs implement exactly the linear transformations corresponding to the group of integer positive modular matrices, while in ESRL, an extended version of SRL, the set of transformations that can be implemented by linear homogeneous programs corresponds exactly to the group of integer modular matrices.