The Quotient in Preorder Theories

TL;DR

This paper develops a unified theoretical framework for computing quotients in preorders with monotonic multiplication and involution, generalizing and subsuming many existing theories in computer science.

Contribution

It introduces the concept of admissibility and preordered heaps, providing conditions and formulas for the existence of quotients across various domains.

Findings

Many computer science theories are preordered heaps.

Admissibility guarantees the existence of quotients.

Sieved heaps extend the framework to multiple domains.

Abstract

Seeking the largest solution to an expression of the form A x <= B is a common task in several domains of engineering and computer science. This largest solution is commonly called quotient. Across domains, the meanings of the binary operation and the preorder are quite different, yet the syntax for computing the largest solution is remarkably similar. This paper is about finding a common framework to reason about quotients. We only assume we operate on a preorder endowed with an abstract monotonic multiplication and an involution. We provide a condition, called admissibility, which guarantees the existence of the quotient, and which yields its closed form. We call preordered heaps those structures satisfying the admissibility condition. We show that many existing theories in computer science are preordered heaps, and we are thus able to derive a quotient for them, subsuming existing solutions when available in the literature. We introduce the concept of sieved heaps to deal with structures which are given over multiple domains of definition. We show that sieved heaps also have well-defined quotients.