Symmetries in Sorting

TL;DR

This paper conceptualizes sorting algorithms as abstract functions, establishing a connection between total orders and correct sorting functions through universal algebra and type theory, with formal verification in Cubical Agda.

Contribution

It introduces a novel axiomatization of sorting functions independent of total order assumptions, linking them to algebraic structures and formalizing in type theory.

Findings

Sorting functions form sections of a surjection from lists to multisets.

Decidable total orders correspond to correct sorting functions.

Formalization achieved in Cubical Agda.

Abstract

Sorting algorithms are fundamental to computer science, and their correctness criteria are well understood as rearranging elements of a list according to a specified total order on the underlying set of elements. As mathematical functions, they are functions on lists that perform combinatorial operations on the representation of the input list. In this paper, we study sorting algorithms conceptually as abstract sorting functions. There is a canonical surjection from the free monoid on a set (lists of elements) to the free commutative monoid on the same set (multisets of elements). We show that sorting functions determine a section (right inverse) to this surjection satisfying two axioms, that do not presuppose a total order on the underlying set. Then, we establish an equivalence between (decidable) total orders on the underlying set and correct sorting functions. The first part of the paper develops concepts from universal algebra from the point of view of functorial signatures, and gives constructions of free monoids and free commutative monoids in (univalent) type theory. Using these constructions, the second part of the paper develops the axiomatisation of sorting functions. The paper uses informal mathematical language, and comes with an accompanying formalisation in Cubical Agda.