Explicit Hopcroft's Trick in Categorical Partition Refinement

TL;DR

This paper introduces a clear, generalized formulation of Hopcroft's trick using weighted trees and develops a functor-generic partition refinement algorithm applicable to various systems, leveraging fibrations for categorical abstraction.

Contribution

It presents a novel, explicit formulation of Hopcroft's trick via weighted trees and develops a functor-generic partition refinement algorithm using fibrations, broadening applicability.

Findings

The formulation of Hopcroft's inequality simplifies understanding of the trick.

The fibrational framework enables categorical analysis of partition refinement.

The algorithm applies to automata, Markov chains, and other systems.

Abstract

Algorithms for partition refinement are actively studied for a variety of systems, often with the optimisation called Hopcroft's trick. However, the low-level description of those algorithms in the literature often obscures the essence of Hopcroft's trick. Our contribution is twofold. Firstly, we present a novel formulation of Hopcroft's trick in terms of general trees with weights. This clean and explicit formulation -- we call it Hopcroft's inequality -- is crucially used in our second contribution, namely a general partition refinement algorithm that is functor-generic (i.e. it works for a variety of systems such as (non-)deterministic automata and Markov chains). Here we build on recent works on coalgebraic partition refinement but depart from them with the use of fibrations. In particular, our fibrational notion of $R$-partitioning exposes a concrete tree structure to which Hopcroft's inequality readily applies. It is notable that our fibrational framework accommodates such algorithmic analysis on the categorical level of abstraction.