Symmetries in Sorting

2025-12-08

AI Summary

This paper addresses the axiomatization of sorting functions over sets lacking a predefined total order. We propose an abstract sorting model based on sections (right inverses) of free commutative monoids, defining sorting functions as list operations satisfying specific algebraic axioms—thereby eliminating reliance on a priori total orders. Using functorial signatures and universal algebraic techniques, we formally construct free and commutative monoids within single-sorted type theory, with all proofs mechanized in Cubical Agda. Our main contributions are threefold: (1) the first rigorous bijection between decidable total orders and correct sorting functions; (2) the identification of sorting as fundamentally equivalent to constructing algebraic sections that induce total orders; and (3) a novel, comparison-free mathematical foundation for sorting algorithms grounded in monoid theory and constructive algebra. This framework enables reasoning about sorting independently of element-wise comparisons, offering new insights into its computational and structural essence.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

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.

Problem

• Axiomatizing sorting functions without total order
• Equating total orders with correct sorting functions
• Formalizing sorting in type theory with Cubical Agda

Innovation

• Abstract sorting functions defined via monoid sections
• Equivalence between total orders and correct sorting functions
• Formalization in Cubical Agda with univalent type theory