Symmetries in Sorting

Computer Science > Logic in Computer Science [Submitted on 8 Dec 2025 (v1), last revised 1 Jun 2026 (this version, v2)] Title:Symmetries in Sorting View PDF HTML (experimental)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. Submission history From: Wind Wong [view email][v1] Mon, 8 Dec 2025 09:41:47 UTC (250 KB) [v2] Mon, 1 Jun 2026 09:56:26 UTC (278 KB) Current browse context: cs.LO References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.