Certifying algorithms and relevant properties of Reversible Primitive Permutations with Lean

TL;DR

This paper formalizes Reversible Primitive Permutations (RPP) in Lean, proving they can encode all Primitive Recursive Functions, and introduces a new reversible iteration scheme along with useful algorithm patterns.

Contribution

It provides a simplified, bug-fixed proof that RPP can encode all Primitive Recursive Functions and introduces a new primitive reversible iteration scheme.

Findings

RPP can encode all Primitive Recursive Functions

A new reversible iteration scheme is proposed

Lean formalization enables certified reversible algorithms

Abstract

Reversible Primitive Permutations (RPP) are recursively defined functions designed to model Reversible Computation. We illustrate a proof, fully developed with the proof-assistant Lean, certifying that: "RPP can encode every Primitive Recursive Function". Our reworking of the original proof of that statement is conceptually simpler, fixes some bugs, suggests a new more primitive reversible iteration scheme for RPP, and, in order to keep formalization and semi-automatic proofs simple, led us to identify a single pattern that can generate some useful reversible algorithms in RPP: Cantor Pairing, Quotient/Reminder of integer division, truncated Square Root. Our Lean source code is available for experiments on Reversible Computation whose properties can be certified.