Automata Minimization: a Functorial Approach

TL;DR

This paper presents a functorial framework for automata minimization, providing conditions for guaranteed minimization, and demonstrating how to lift category adjunctions to automata, with applications to classical algorithms.

Contribution

It introduces a novel functorial approach to automata minimization, unifying various algorithms and extending categorical techniques to automata theory.

Findings

Provided conditions for automata minimization guarantees

Lifted adjunctions between output categories to automata categories

Applied framework to classical minimization algorithms like Choffrut's and Brzozowski's

Abstract

In this paper we regard languages and their acceptors -- such as deterministic or weighted automata, transducers, or monoids -- as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: a) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. b) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. c) We show how this framework can be applied to several phenomena in automata theory, starting with determinization and minimization (previously studied from a coalgebraic and duality theoretic perspective). We apply in particular these techniques to Choffrut's minimization algorithm for subsequential transducers and revisit Brzozowski's minimization algorithm.