Symmetries in reversible programming: from symmetric rig groupoids to reversible programming languages

The Pi family of reversible programming languages for boolean circuits is presented as a syntax combinators witnessing type isomorphisms algebraic data types. In this paper, we give denotational semantics language, using weak groupoids à la Homotopy Type Theory, and show how to derive an equational theory it, by 2-combinators equivalences isomorphisms. We establish correspondence between the syntactic groupoid language formally univalent subuniverse finite relates 1-combinators 1-paths, 2-paths in universe, which shown be sound complete both levels, forming equivalence groupoids. use Curry-Howard-Lambek Reversible Logic, Programming Languages, Symmetric Rig Groupoids, showing that free symmetric rig groupoid, given sets bijections. Using formalisation our results, perform normalisation-by-evaluation, verification synthesis logic gates, motivated examples from quantum computing. also reason about transfer theorems different representations circuits.