Symmetries in Reversible Programming: From Symmetric Rig Groupoids to Reversible Programming Languages

The $\mathit{\Pi}$ family of reversible programming languages for boolean circuits is presented as a syntax of combinators witnessing type isomorphisms of algebraic datatypes. In this paper, we give a denotational semantics for this language, using the language of weak groupoids \`a la Homotopy Type Theory, and show how to derive an equational theory for it, presented by 2-combinators witnessing equivalences of reversible circuits. We establish a correspondence between the syntactic groupoid of the language and a formally presented univalent subuniverse of finite types. The correspondence relates 1-combinators to 1-paths, and 2-combinators to 2-paths in the universe, which is shown to be sound and complete for both levels, establishing full abstraction and adequacy. We extend the already established Curry-Howard correspondence for $\mathit{\Pi}$ to a Curry-Howard-Lambek correspondence between Reversible Logic, Reversible Programming Languages, and Symmetric Rig Groupoids, by showing that the syntax of $\mathit{\Pi}$ is presented by the free symmetric rig groupoid, given by finite sets and permutations. Our proof uses techniques from the theory of group presentations and rewriting systems to solve the word problem for symmetric groups. Using the formalisation of our results, we show how to perform normalisation-by-evaluation, verification, and synthesis of reversible logic gates, motivated by examples from quantum computing.