Reversible Programs Have Reversible Semantics

During the past decade, reversible programming languages have been formalized using various established semantic frameworks. However, these semantics fail to effectively specify the distinct properties of reversible languages at the metalevel, and even neglect the central question of whether the defined language is reversible. In this paper, we build on a metalanguage foundation for reversible languages based on the category of sets and partial injective functions. We exemplify our approach through step-by-step development of the full semantics of an r-Turing complete reversible while-language with recursive procedures. This yields a formalization of the semantics in which the reversibility of the language and its inverse semantics are immediate, as well as the inversion of programs written in the language. We further discuss applications and future research directions for reversible semantics.